Howard B (1) - Petroleum Engineers Handbook, Part 4 [PDF]

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Chapter 34



Wellbore Hydraulics A.F. M.J. Fred L.K.



Bertuzzi, Phillips Petroleum Co.* Fetkovich, Phillips Petroleum Co. H. Poettmann, Colorado School of Thomas, Philhps Petroleum Co.



Mines*



Introduction Wellbore hydraulics is defined here as the branch of production engineering that deals with the motion of fluids (oil, gas, and water) in tubing, casing, or the annulus between tubing and casing. Consideration is given to the relationship among fluid properties, fluid motion, and the well system. More specifically, the material presented is intended to describe methods for solving problems associated with the determination of the relationship among pressure drop, fluid rates, and pipe diameters and length. To maintain the scope of this section within prescribed limits, some material and data that are pertinent to the solving of wellbore problems. but which can be found conveniently elsewhere, are not presented. The material not covered includes (1) methods of measurement and (2) complete data on fluid properties (See Chaps. 13, 16-19, 24). The theoretical discussion that follows provides a basis for the development of correlations and calculation procedures in subsequent parts of the section.



Theoretical Basis Fluids in Motion Energy Relationships. The energy relationships for a fluid flowing through tubing, casing, or annulus may be obtained by an energy balance. Energy is carried with the flowing fluid and also is transferred from the fluid to the surroundings or from the surroundings to the fluid. Energy carried with the fluid includes (1) internal energy. U, (2) energy of motion or kinetic energy (mv’/2g,.), (3) energy of position (potential energy m,gZ/g,.), and (4) pressure energy, pV. Energy transferred between a fluid and ‘Authors authors



of the orlgmal chapter on !hls and J K Welchon (deceased)



fop~c I” the 1962



edmon



Included



these



its surroundings includes (1) heat absorbed or given up, Q, and (2) work done by the flowing fluid or on the flowing fluid, W. The conservation of mass, or the first law of thermodynamics, states that the change in internal energy plus kinetic energy plus potential energy plus pressure energy is equal to zero. The following energy balance between points 1 and 2 in Fig. 34.1 and the surroundings illustrates the relationship for the previously listed energy terms for unit mass of fluid. 2



2



U,+~t~z2+P2Vz=U,+1’1+~z,



%c



g,



+p,V,+Q-W,



Q,.



....



g,



.... ....



. . . . (1)



where U v g,. g Z p V Q



= = = = = = = =



internal energy, velocity, conversion factor of 32.174, acceleration of gravity, difference in elevation, pressure, specific volume, heat absorbed by system from surroundings, and W = work done by the fluid while in flow.



This energy-balance equation is based on a unit mass of fluid flowing and assumes no net accumulation of material or energy between points 1 and 2 in the system.



PETROLEUM



34-2



ENGINEERING



HANDBOOK



If flow is isothermal and the fluid is incompressible, 4 may be simplified to



2 ; Nv2) ; &&7=-E P %c gc



Fig. 34.1-Illustration



of energy-balance



Point



2



Point



1



relationship.



p,



.



Eq.



(5)



where p =density . The dimensions of the energy terms in Eq. -5 are energy per unit mass of fluid, such as foot-pounds per pound. Quite often the force term is canceled (incorrectly) with that of the mass term resulting in the dimensions of length as of a column of fluid. For this reason, these terms frequently are referred to as “head,” such as feet of the fluid. For most practical cases, the ratio g/g, is essentially unity. Although the terms in Eq. 5 are sometimes expressed as feet of fluid, no serious error is involved. In fact, one can derive a very similar expression where the terms are expressed in feet of “head.” Eqs. 4 and 5 are the energy relationships that provide the basis for the computational methods of the sections to follow. Irreversibility Losses. The use of Eqs. 4 and 5 requires a knowledge of Et, the term that accounts for irreversibilities (such as friction) in the system. The term E, can be expressed as follows ’:



Eq. 1 also can be put in the form



au+~+Lz+a(pv)=Q-w. c gc



fiftv2 Et=- 2g,d,



.....



since



where f commonly is referred to as a friction factor, L is length, and d is pipe diameter. The friction factor, f, usually is expressed in terms of the physical variables of the system by correlations of experimental data. For single-phase flow, the dimensionless friction factor, f, has been correlated in terms of the dimensionless Reynolds number dvp/p with p being viscosity. A relationship is also suggested by application of dimensional analysis to the variables involved. In either case the result is



VI



Sl



and s2



TdS=Q+Ef



s Sl



. . . . . . . . . . . . . . . . . . . . . . . . . . . . . (6)



where T = temperature, S = entropy, and EP = irreversible energy



VI



f=FIE, losses,



Pl



Eq. 2 can be put in the more familiar P2



s



Pl



form



2



v@+K+&=-W-E~.



%c



_. .



gc



(3)



Since, in the system shown in Fig. 34.1, there is no work done by or on the flowing fluid, W is equal to zero and the following equation results.



-Et.



. . . . . . . . . . . . . . . . . . . . . . . . . . . ...(7) CL



and



.. ..... .



.



where F1 is a function of Reynolds number. Eq. 7 has been the basis for correlation of considerable experimental data for single-phase flow over the past years. Eqs. 5, 6, and 7 have been adapted to multiphase flow. Consideration of the character of pipe surfaces as absolute roughness, E (that is, the distance from peaks to valleys in pipe-wall irregularities), which may be expressed as a dimensionless relative roughness factor, t/d, has led to improvements in correlations of single-phase flow experimental data



f=F2[(3



(3,



where F2 is a function roughness.



(8) of Reynolds



number and relative



WE lLLBORE



34-3



HYDRAULICS



0.1 009 aQ8 007



0.05 0.04 0.03



“,3 NJO.06 8



J E G F 6u5 E



‘005



0015



004 0.03 ^^^_l/llI



I llllli



UUL3



0.015



001 0009 0.008



&j&r 2 3456Bl14



2 3456B15



IO



REYNOLDS Fig. 34.2-Friction



NUMBER



2 345681, IO A,, IO Re = = P



factor as a function of Reynolds number with relative



%%s E o.aX% 5 cl0004 ; oooo2 0.ooo1 fTMnAK j”-‘“ti



2 345681



roughness



lo8



as a parameter.



since v2/2g, and El are equal to zero. Since g/g, sumed to be unity,



p2 dp s PI



-+Az=o.



is as-



. . . . . . . . . . . . . . . . . . . . . . . ...(n)



P



For the case of a static-liquid column, it is usually satisfactory to use an average density for the column of liquid. Eq. 11 then can be expressed in the more convenient and familiar form as Ap=pAz.



. . . . . . . I.. . . . . . .



.. .



(12)



The preceding equations will provide a basis for the calculation procedures of the following sections for staticfluid columns.



Producing Wells



Static Fluids Many wellbore problems are associated with static-fluid columns, either oil, water, or gas, or combinations thereof. In the case of static-fluid columns, Eq. 4 is applicable in general and reduces to P2



PI



=0.000,005



2 345681s



Fig. 34.2 shows the correlation for single-phase flow according to Eq. 8. * Similar plots are found in the literature in which other friction factors are plotted as a function of Reynolds number. Care must be taken to avoid confusion, as the same name and symbol are used for various multiples off as plotted in Fig. 34.2 The laminar-flow region, which extends up to a Reynolds number of 2,000, is represented by a straight-line relationship f=44/NR, on Fig. 34.2. Between 2,000 and 4,000, flow isunstable. Above 4,000, turbulence prevails and the influence bf the physical properties decreases as the Reynolds number increases. In fact, it is shown that at very high Reynolds numbers the friction factor depends solely on the relative roughness factor c/d. The preceding theoretical discussion concerning irreversibility losses is based on considerations involving singlephase flow. Nevertheless, the material presented will provide a basis for considerations involving both single- and multiphase flow that appear in the following seCtions.



vdp+Qz=o



g i? r-r i



QooO,Ol lb3



s



s



$382 ___0.004 0.002



002



;;;



0.01



. . . . . . . . . . . . . . . . . .



. . . . .



gc



or



p2 dp -+542=0,. . s PIP gc



.... . . . .



Gas Wells Calculation of Static Bottomhole Pressures (BHP’s). Static BHP’s are used to determine the deliverability of gas wells (backpressure curve) and to develop reservoir information for predicting reservoir performance and deliverability. Several methods for calculating static BHP’s have appeared in the literature.3-6 The methods differ primarily as a result of the assumptions made. All start with Eq. 9 assuming g/g, is unity for a static column:



PETROLEUM



34-4



ENGINEERING



HANDBOOK



For a particular gas, RIM, which is equal to 53.2411~~ where 7X is the gas gravity (air= 1.O), is a constant. Therefore, Eq. 16 can be simplified to 53.241



PI s



YR pz



GAS GRAVITY



(AIR=0



. . . . . . . . . (17)



well fluids



53.241? s YR



If the column is vertical, aZ=L, where L is the length of the pipe string, and Eq. 9 can be put in the form



PI z -dp=L.



P2



.



.



.



. (18)



p



The method using Eq. 18 was suggested by Fowler.’ Poettmann4 made the solution of Eq. 18 practical by presenting tables of the function



PI



l’dp=L.



. .



It is at this point where certain assumptions are made and calculation procedures differ. Assumptions are made in regard to z and T. For any calculation procedure, four “surface” properties must be known: well-effluent composition, well depth, wellhead presske, and well temperature. The gas composition is used to calculate the pseudocritical properties ppC and TPC of the gas, from which is estimated the value of the compressibility factor z used in the calculations. Quite often, gas composition is not available and gas gravity must be used to estimate the pseudocritical properties (Fig. 34.3).4 A recommended method assumes constant and average temperature T and allows z to vary with pressure. With temperature being constant, Eq. 17 becomes



Fig. 34.3-Pseudocritical properties of condensate and miscellaneous natural gases.



s P2



zT*=L. P



PPr z



. . . . . . . . . . . . . . . . . . . . . . . . . . . ..(13) s



0.2



If the column is not vertical, cal by an angle 8, U=L



but inclined with the verti-



in terms of ppr and Tpr. The tables are presented as Table 34.1. It can be shown that



c0se



and again usiq



z sp’fdl’=s (p,r),--dp,,



L, Eq. 9 becomes



Pi7 PI



Vdp=L



sins.



... ..



.



. .(14)



(P,,)?



MP



. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (1%



PPr



0.2



Ppr



-dppr.



s 0.2



Subsequently, only the vertical column will be considered and Eq. 13 will be used. Since



= fppr’’kdppr



(PPJ > z



-



s P2



v=E.



here



. ..



..



. . . . . (19)



PPr



An advantage of this method is that it is a direct method of calculating >BHP. No trial and error is involved. In terms of ppr and T,, Eq. 18 becomes



L=-



53.241? YR



(P,,), [s 0.2



z p,,dp,r



- I( ‘““’



&dppr]



0.2



where z = compressibility factor, R = gas constant, and M = molecular weight,



(20) By rearranging,



(p,,), 2 Eq. 13, upon pbstitution,



becomes



I



0.2



-dppr PPr



L-y,



+



= F



53.241T



(PP’)> s o,2



.. .... ..........



. . . . . . . . . . . . . . . . . . . . . . . . . (16) Eq. 21 permits



a direct solution



z _



dppr.



PPT



. . . . . . . (21)



for the static BHP.



WELLBORE



HYDRAULICS



34-5



TABLE 34.1-VALUES



Pseud+ reduced PWSSUrE PO,



i: i: i! 08 Yo’ 1: I 3 I: I 6 I 7 I! 20 :: :: 2s :; 5:



:: :: 35



Pseudoreduced



I 05 I I IO



I I5



I 20



0 IO 0 350 0 350 0 615 0 619 0 805 0 816



0 0 35C 0 623 0 826



0 955; I 078 I I75 I 256, I32711



0 I 1 I



971 IO0 207 3W 375’1



0 I I I



3801 433 4b3: 492’ 510



I I I I I



438 500, 545 590 620,



527 I 544: I 560, I 575 I I 590 1 I



649, 670’ hW/ 708, 725



lpr



I 35



0 0 350 0 626 0 834



0 n 0 350 0 3jU 0 625 0 63U 0 83'1 0 844



0 0 3X) 0 632 0 848



i: ::



985 I24 23Y 335 420



0 YQl3 I 145 I 264 I 365 I455



I 01 I I I62 I 285 I 3Rb I47Y



I I I I I



022 178 300 403 500



I 032’ I 190, I 3131 I 417, I415



0 0 0 0 IO



I I I I I



435 550 602 654 6W i



I I I I I



528 hO0 657 713 757



I I I I I



552 625 684 742 7YI



I I I I I



573 645 709 772 ~24



I I I I I



591 666 731 7Y5 848



I I I I I



7Zh 754 782 808 833,



I I I I I



800 834 867 KY6 924



I I I I I



819 876 VI> Y44 975



I I I I 2



RI5 ‘117 9>H 991 027



I I I 2 2



9C”l 443 ~ ‘It35 ULZ 05Y



I 1 i 604 I 743, I 854~ I 947 6171 I 761 I 876’ I 971 631 i 1779’1 RY7; IVY4 644 I 7971 I 919 2 018 658: ,815, I 9M 2 041



2 2 2 2 2



00) 031 059 087 II5



2 2 2 2 2



057 086 II6 I45 I75



2 2 2 2 2



UC12 I25 I57 IW, 223



6721 I 830 I 685 1 845’ I 699, I MO I 712ll875’2012,2i2I 726 I 690, 2



2 140



2 137 2 159 2 180 2202 2 224



2 I98 1221 2 245 12bH 1 LYI



2 2491 2 275 ~ 2 1021 23281 2 354’



2 157 2 175 2 l92;2 2210 2 227



2 243 2 261 280 2298 2 317



2 >II 2 >?I 2 350 1370 2 3’10



2 376 2 397; 2 419 2440: 2 462



95R 976 994 030



740 I ‘XI4 2 046 754 I 918 2062 767 ,Y3212O78 781!1946i2094 795 I 9tQ 2 I10



I 049 R62



2 061 2 081 2 IO1



I 25



1974’2121 I 988 2 14U 2 GO2 2 I55



2243’2333 1 ii9 2 34’) 2 275’2 365



24fl7 248Oi 1 424 2 4Y8 2 4411 2 5171



2 Olh 030



2 166 170



212’11306



2 457 474



2 5351 533



2 2 2 2



1 2 2 2 2



2 2 2 1



044 058 073 OR7 101



201 216 12 111 336’2 2311 2 351 245 2 )06 260 2 381



2 381 397



I 40



pm



0 350 ,fl0 033 851



1: I 3



I 50



I 60



Tpr



I 70



I 80



I 90



0 350 0 350 IO0 350 [O0 350 0 634 0 63j 0 636 0 862 637 854,O 856,O 860,O



0 350 100 3% 0 638 864 0 86b 639



1:



I 682 I 746



I 6% I 758



I 761 I 836l



I 810 867 ! I 825 884 943 WI 038 079 II912



I 737,l I 810



75) I 828



I 847 9-36



I 882, Y3U,



I 903 962 ~I 911 973 ’1 I 920 984



I 2 2 2



964 012 060 IW 140~2



I 2 2 2



913 2 043’2 093’ 2 178’22072 136



021 072 I23 165’2



:: 2 3 2 4 2 5



,22 2 2 2



I60 2 1RL2 12b12 I53 2 IY3 2 222 2 22712 256’2 260 2 2% 2



212 176 249 285 521



2 2 2 2



215’2 252 288 325 362,



2R8 248 2 329 2 3b9 2 410 :2



272lL 513’2 334 192 354 2 375 395 2 417 436 2 459



2 2 2 2 3



2 ,2 2 I2 12



206 2 316, 2 344, 2 372 j 2 Q33 I 2



350 379 407 436 465



2 2 2 2 2



392 ’ 2 442 ~2 423 / 2 474 2 413 2 506 2 484 : 2 538 2 514 2 570 ,2



069 ~2 492 502, 2 525 534 ~2 ii7 567 2 5W 600, 2 623



318 347 375 404 432



’2 2 (2 i2 ,2



3 6 3 7 38



: 2 535’ 2 568 ’ 2 603, 2 664 ,2 556 2 5138 2 624 2 686 ;2576:2 bOY 2644'2708



~2 ~2 2 2 2



2 2 2 2



46 47 48 4 9 50



1942 I 955 I Y6V / I982 I 995



2115 2 I28 2 142 2 I55 2 169’



2274 2 238 2 301 2 315 2 329



2195 2 009, 2 423 2 437 2 451



24YI 2 507 2 522 2 5% 2 553,



2570 2651 2 506 2 6b6 2 601 2 682 2 617 2 697 2 632 ~2 713



4 4 4 4 5



: :



I2 2 009’ 024



2 I83 197



2 342 355



2 465 479l



2 567 581



2 046 hbl



2 728 743



552 I



5 3 5 4 5 5



~2 038 2 053 2 067



2 210 2 224 2 238



2 369 2 382 2 395



2 4Y2’ 2 5% 2 506 2 609 2 520 2 623



2 675 2 bW 2 704



2 758 2 773 2 78A



F; :;



~2 07’ O’JI) 2 102 II4



2 251 LO4 2 277 210



2 408 421 2 435 440



2 533 547 2 560 574



1 hiU 1636 2 663 677



2 718 2 731 L2 74i 75H



22 MI RI5 2 R42 RZR



60



2 I2h



1303



2 461



2 587



2 O’K)



2 772



2 855



2 2 2 2



2 726 2 748 2771



’ 2 035l 2 089~ 2 142 223Il225U 187,



I 770 I 845



I 923 I 96Y 2 014 ‘2OY3 2 054



6 7 8 9 0



I I 2 2



I 710 I 77’)



, b I 7 I a :;



I 875 889’ I I 902 , I 916 I 919



4Y0 506 523 ij9 555



I 45



Temperature,



6 7 8 9



:: 4 3 44 45



2 2 1 2



413 429 444 460 476



Pseudoreduced



Pseudo reduced Plf%SUR



I 30



!IBo8 I 622 I 835 :i



Temperature.



OF S‘PP’Ldq,, 0.2 PPI



2 2 2 2



047 IO2 157 204



1 7hb 1791 181j



2 7Y2 2 RI7 2843



2 3 3 3 3



3 3 3 3 3



56’ 586’ 9 602 619 635 6 7 8 Y 0



719 735 752, 768 784



~22814799



2 2 2 2 2



754 2 793’2 770 ’ 2 810’ 786 ~1 I326 802 2 043 RI8 ~2 WI,



2 I2 2 2



863 2 933 881 , 2 952 899 2 970 917, 2 989 935 3 007



22850’ 834 2892~2 876 / 22968’ 952’ 3042 3 024



9W DO9 027 046 065



33OY9 082



022 041 061 080 IM)



3 136 I I8



PETROLEUM



34-6



TABLE 34.1 -VALUES PSWd3 reduced PlfJSSUre PO, ~~



Pseudoreduced



I 05 _~



61 62



I IO



I I5 ’ I 20 ’ I 25



I 35



2703’2 2716 2 729



785 2799 2 Ml1



2 869 2882 2 896



3 I31



3216



P~0SS”E p!x



Pseudoreduced



I I 40



I 45



T,,



Temperature,



I 50



I 60



’ I 70 I I 80 , I 90 -/-



2 474 2486 2 499



2600 2bl2 2 025



2585.2755I2908’3034



i i 9 9 IO 0



/\22 610 597’2 2 767 780 ~22 919a 931 133 045,3 057 i 3 I53 14283 228 239 12 622 12 702 2 942 3 068 ~3 164 133 251 2 634 2 804 2 954, 3 080 3 175 13 263



IO I IO 2 I03 IO 4 105



2 646 2 658 i2671 ; 2 683 2 695’



2 816’2 2 828 2840 2 852, 2 864



IO IO IO 109 II



0



2 876 2 888 2 900’3 2912 2 924



II II II II II



I 2 3 4 5



2 707 ! 2 719 ‘2 732 (2744 / 2 756 I 2 768 2780 2 793 2 805 2 817



II II II II I2



6 7 8 9 0



2 829 2 841 2 854 2 866 2878



b 7 8



Pseudo-, reduced



I 30



(continued)



PPI



0.2



---I



2 139 2316 2l52~2328 2 16512 341



96



Tpr



Temperalure,



p, 2 -dpp, OF s



ENGINEERING



966’3 2 97R 2989 3 001 3 013,



092 3 103 3115 3 I26 3 I38



084



2 996 3 008 3 020 3 032 3044,3I92



144 156 I68 It33



3 3 3 3



I08 129 132’



3302



i!



2 943 2 956 1 IWO 2 YHl



3 376



2 2 3 3



984 997 OII 024



3 3 3 3



029 043 056 070



3 424 ’ 3 475



3 3 3 3



II I I25 I40 I54



3 3 3 3



I87 LO2 218 233



3 585 ’ 3 644



3 3 3 3



250 266 281 297



3 713



3 314 326 3 33A 3 350,



9 8 9 9 IO0



1 3 39Y 1HR 3 435, 447, 3 467 495 , 3 41 I 3 458 3 510 ‘3423,3470,3521~3610



3 576, 508, 3 599



3 6% bb7 ) 3 724 736 3 679’ 3 747 3691 3758



3 I87 I 3 274 ~3 361 3 199 3 286’ 3 372, 3211;3297,3382~ 3 223 ’ 3 309 3 393 3 235 13 320 13 404



IO1 IO 2 I03 I; ;



13434 ’ 1 446 3 457 3 464 3480



3622 3 633 3 h45 3 656 3669



3702 3 714 3 725 3 737 3748



~3 025 3 I50 3 ! 3 037, 3 I61 3 048’3 l73l 3 3060~31R4~3281t3366 13 072, 3 1% 3



2 936 I3 294R:3096 2 960’ 3 2 972, 3 2 984l3



6 I bl 6 3



3 208 3 3220,3315 3 231 / 3 3 243 3 3 255 3



246 ’ 3 332 1 3 416 258 I 3 343, 3 428 269;3 355 3 440 3452 292 3 378, 3 464 304 327 338 350



3 267’ 3 361 3 279 3 373 3 290 3 384 3 302 3 396 3314,3407,3488



3 389 / 3 34UIl3486 3 412 3 3 424 3 3 435 3 3 3 3 3



446 456 467 477



IO IO IO IO II



6 7 8 9 0



475 497 508, 519



3 529 3 543 3 550 3 561 3571



II



5



II 6 II 7 II R II 9 I2 0



3 3 3 3 3



3 544 3 555 3 5b7 3578



i 3 541 3 552, 56213 3 573 3 584 I



3 588 3 598 60913 3 619 3 629



i 3 679 3 758 3 689 3 769 700 3 779 ; 3 710 3 790 3 721 13 BOO I I 3 551 3 595 3 639 3 732 ~3 81 I 3562’3605’3650’3743’3822 3 574 3 616 3 660. 3 753 3 832 3 585 3 626 3 671 3 764 3 843 3 5Y7 / 3 637 1 3.631 ’ 3 775 3 854 3 3 3 3 3



492 504 51513 527 539



3482:3532 3 494 3 506 3 518, 3530



607 617 h!9 h14 h48



3 3 3 3 3



648, 65A 660 b79 bW



3 3 3 3 3



692 702 713 723 734



3 3 3 3 3



756 797 808 819 830



3 3 3 3 3



865 R7h 886 R97 908



3769 3 780’3 3 790 3 801 3812 3 3 3 3 3



823 834 844 855 866



3 3 3 3



292 308 323 339



3 7M) 3 772 783 3 795 3806 3817 628 ‘3 840 [ 3 851 3862 3 3 3 3 3



073 883 894 904 915



3 877 3 926 3888 3937 3 899 3 947 3 910 3 958 3 Y2I / 3 969 3 932 3 943 3 95514 3 966, 3,977,



3 980 3 991 W3 4 014 4 025



HANDBOOK



WELLBORE



34-7



HYDRAULICS



TABLE 34.1-VALUES



OF ippLdp,, 0.2



PP __~. 02 0 3



rempmure.



PSBudOreduCed



Pseudo reduced Pressure I 2ccl



220



260



,240



0 0 150



0 0 J50



ii:



00867639



00868640



0 640 869



i; it



I 050 216 I 489 360



lI 051 2lR I 492 %J



I llil 219 I 494 5114



1.0



, 602



1 I 607



I 608



I:; 13



i I 691 780 I 851



/ I 699 790 1 I 868



I 702 795 I 875



I?



/ I, 915 997



~ 2I 945 010



I 2I 954 019



2 074 2 III ; y;



, 2 083 I 2 141 I : ;“4;



2 2;5



Jo0



PP



: 150 0 640 0 8b9



; Ji” 0 CT40 0 869



: J50 0 640 0 at9



I I I I I



052 I 052 220 I 220 Jf 4 ~ I J64 4Oj 1 I 495 WI9 I 611)



I Jh4 I 49) I 6,”



I I I I 2



706 ~ I JUY 802 I hU8 1)MJ 1 I 2490 964 I )7? 027 I 2 UJ6



I I I I 2



711 RI2 89b YHU 045



090 2 100 I48 i 2 1% 205 2 217 256 2 267 347 ~ 2 317



2 2 2 2 2



11” lb9 227 279 3M



:: 6J 64 65



I ii:



66 67 68 4: 7 I 72 :: 75



2 2 2 2 2



I 2 29%



2 2 2 2 2



2.1



2.307 2 349 2 391 2 433 2 475



2 2 2 2 2



326 / 2 J37 366 2 JR0 407 / 2 422 447 2 465 488 2 507



2 2 2 2 2



350 394 4JJ 481 524



’2 2 2 2 2



ibl 404 448 491 5Ji



2 2 2 2 2



375 42U 4b5 itu 555



’



2 2 2 2 2



508 541 575 608 641



2 2 2 2 2



523 / 2 544 559 2 MI 594 2 617 630 2 654 665 / 2 691



2 2 2 2 2



562 599 bJ7 674 712



2 2 2 2 2



574 012 051 6k9 728



2 2 2 2 2



593 CiO hbtl JU5 743



~



2 2 2 2 2



670 ~ 2 694 1 2 722 700 2 723 2 753 729 2 J52 2 783 759 2 78) 2 814 788 2 810 2 845



:: 2.4 25 :; :; 30 3.1 :: J4 35



I



2 813 ; .s%



2 744 2 775 2807 2 BJM 2 a70



2 2 2 3 3



3 3 3 3 3



002 025 049 072 095



3 081 JO92 1 IOJ 3 I14 J I25



3 1145 3 iUb9 J lN4 J118 J 142



3 064 3 OR8 3 112 1136 3 160



48 49 5.0



3 048 3.074 306a 3095 JO881 3 II5 3 108 i 3 136 3 128, 3 157



3 3 3 3 3



II? 119 161 18J 205



3 J 3 3 3



147 I68 190 211 23)



3164 3 IPI, 3 2W 3 231 3 253



3182 1 zn3 3 22i 3 246 3 268



5. I 52



3 146 1 lh4



3 3 J 3 3



225 244 264 283 303



3 J 3 I J 1 3



253 27) 294 JI4 3J4



3 274 3 295



3 2HX 3 JUH



:::I , 3 ii?



J3J2xJ4H 3 JbB



321 319 356 374 392



3 3 J 3 3



352 JJO 389 407 425



3 175 3 39, J 412 1411, 3 448



3 JR6 I 4115 J 42J 1442 3 440



:;



2 983 3.aJ5 3 028



i ili



:: 55 56



1 3 I?? 3 IOil 3 2J5 3 255



3 235



3 3 3 3 3



273 291 309 127 345



911 YJB 966 99J 021



929 957 984 012 040



~ 3 008 3 010 , 3 053



; ;; ~: ;;



2 2 2 2 i 3



:tE



2 2 2 J J



4.1 42 4.3 4.4 4.5



I



-



loo



2 20



2 40



3 321 3 JJJ 3 154 3370 3 387



3 362 J 379 J 395 3412 3 429



3 J 3 3 J



4U9 426 44J 460 477



J 1 3 J 3



442 4j9 4?6 49J 510



J J J , 3



466 483 501 518 536



J J J 3 J



4i7 494 511 526 54;



3 3 3 J 1



402 417 432 447 462



3 3 3 3 3



444 459 475 490 505



3 J J 3 3



493 508 524 539 555



3 3 3 3 3



526 542 557 573 5139



3 3 3 3 3



551 507 582 598 613



J J J 3 3



561 577 592 608 624



3 3 3 3 3



477 491 506 520 535



3 520 I 3 534 1 3 549 J 563 3 578



3 3 3 3 3



570 584 599 613 628



5 3 3 3 3



604 618 633 647 662



3 628 3 643 6659 3 674 3 689



3 6J9 3 654 3670 3 685 3 700



3 3 3 3 J



548 562 575 5R9 602



591 605 618 bJ2 645



3 1 3 J 3



642 3 676 656 1 3 690 670 704 684 ( 3 718 690 / 3 732



3 3 3 3 3



703 718 JJ2 747 761



3 714 1728 3 742 3 756 3 770



J 658 3671 3 684 3 fJQ7 3 710



3 3 3 J 3



711 723 736 748 761



~3 3 3 3 3



745 758 771 784 797



3 3 3 3 3



774 788 801 815 R18



J J J J J



JRJ 796 810 82J 836



3 3 3 3 3



3 615 3627 1 640 3 652 3 665



z: 83 84 RI 86 07 88 :z 9 I ;:



a72 899 925 952 979



2 914 ~ 2 940



910 950 990 OJO 070



2 775 2 806 2 8%



2 7% 2 JW 2821 2 H52 2 883



2 2 2 2 ! 2



:; :.G!22 915890 4.0



1 2 836 ; g;



T,



Temperature.



260



z&300



~~~



1.6 1.7 I.8 1.9 2.0



059 116 172 219 265



Pseudoreduced



’ PseudoI reduced Pressure



280



---I 0 ’ 0 350



rp



(continued)



PPI



2: 96 97 98 99 IO 0



J 3 3 3 3



3 3 J J ~3



677 690 702 715 727



3 3 J 3 ’ 1



722 714 746 758 770



J 3 3 3 3



773 786 798 RII 823



810 a23 835 848 Ml



3 3 3 3 3



840 853 865 878 890



3 3 J : J , J



a49 862 875 888 901



3 3 / 3 3 3



719 7% 762 77J 785



3 / 3 3 J I 3



782 794 X06 RIB 830



: 3 3 3



~ 3 873 “8:s 1 3 885 3 897 859 871 ~ 3 999 481 3 921



3 3 3 3 3



902 915 927 940 952



/ 3 3 / 3 3 J



91) 925 9J8 950 962



3 I J 3 7 i



797 R(r) 820 RJZ 844



J J J J 3



042 854 865 R77 689



3 3 3 3 i



R95 907 918 930 942



3 3 3 3 3



93J 94; 957 969 981



3 3 3 3 4



964 976 987 999 OII



J J J 4 4



974 980 999 01 I U2J



IO IO IO In In.5



I 2 J 4



3 3 3 J J



855 867 RJR A90 901



1 3 3 3 3



900 911 923 934 945



3 3 3 3 3



953 965 976 988 999



3 4 4 4 4



992 004 015 027 038



4 02J 4 035 4 046 4 ow 4070



4 OJS 4 046 4 058 4 069 40.31



ICI 10 In 10 II



6 7 a 9 0



3 J 3 3 3



912 92J 9J3 944 955



J J 3 3 4



956 9hJ 978 989 000



4 4 4 4 4



010 021 UJI 042 053



4 4 4 4 4



049 060 071 082 093



4 4 4 4 4



4 4 4 4 4



4 4 4 4 4



011 022 033 044 055



4 4 4 4 4



Ob4 075 1187 098 109



4 4 4 4 4



IO4 I!6 127 I39 I50



4118 4150 4 101 4 17) 4 184



4149 4IMI 4 172 4 IRJ 4 1’14



4 121 4 132



4 4 4 4 4



I61 172 IRJ 194 205



4 I95 4 206 * 217 4228 4 2J9



1 4 20, 4 2lh 4 227 42Jir 4 249



II



I



3 ‘,f,6



II II II



2J 4 5



31 977 9H” 3 9’)9 4 0,”



II b II 7 II a II 9 12.0



4 4 4 4 4



1022 OJ4 04; 057 069



4 0117



081 093 104 116 127



092 IO4 II5 127 1%



PETROLEUM



34-a



Example Problem 1.4 Calculate the static BHP of a gas well having a depth of 5,790 ft; the gas gravity is 0.60, and the pressure at the wellhead is 2,300 psia. The average temperature of the flow string is 117°F. From Fig. 34.3,



Since a=(T, LI(T, -T2)



HANDBOOK



-T7-)lL,



=-=



In T,lT,



L



53.241



TLM



s -fg



PI dp z--,



pz



.



.



(26)



. .



(27)



p



then



T,,+-+%lZ-dEl-dW=O. A,< SC,



.(29)



.



.



Assuming that the kinetic-energy term is small and can be taken as zero, and recognizing that dW, work done by or on the fluid. is zero, Eq. 29 reduces to



.



For vertical



gas flow, dz=dL.



V=F . . . . . WJ



.



.



(30)



Since



. . . . . . . . . . . . . . . . . . (15)



.



(32)



Velocity can be expressed in terms of volumetric flow rate and pipe diameter. Pressure can be expressed in terms of reduced pressure. Substituting these terms in Eq. 32, integrating the equation, and converting to common units results in



s



(PP~’: (zlp,,)dp,, 1 +B(z/p,,)2



(Ppr) ,



-O.O1877y, =



j”‘F



.



(33)



Li



where B=



667fq R2T2 4’ppc2



Y,q = L= T= T=



f= 48 = di



=



Ppc = Ppr =



’



gas gravity (air = 1 .O), length of flow string, ft, temperature, “R, average temperature, “R, friction factor, dimensionless, flow rate, lo6 cu ft/D referred to 14.65 psia and 60”F, inside diameter of pipe, in., pseudocritical pressure, psia, and pseudoreduced pressure pip,,.



At this point, it is further assumed that temperature is constant at some average value. This permits direct integration of the right side of Eq. 33, as



s(PP), (zbpr)dppr



Vdp+ %lZ+dEr=O. g,



(31)



(p,r) I



0.01877 =-ygL, 1+ B(zlp,,) 2 T



..



.



(34)



where the limits of the integral are inverted to change the sign. If the temperature is linear with depth, the use of log mean temperature as the average temperature provides a rigorous solution to the right side of Eq. 34. This use of log mean temperature confines the effect of the assumption of constant temperature to the left side of the equation, where, for practical purposes, it is extremely small. Thus, errors introduced by the assumption of constant temperature are negligible. (continued



on Page 34-23)



PETROLEUM



34-10



TABLE 34.2-EXTENDED



ENGINEERING



HANDBOOK



SUKKAR-CORNELL INTEGRAL FOR BHP CALCULATION



‘Pg., W,,)dp,, I 1 + WP,,?



02 Pseudoreduced



Pp,



1.1



temperature 12



for B=O 13



0



1.4



15



2.2



2.4



26



2.8



3.0



17



18



0.000



0.000



0000



0.0000



0.0000



0



0



0.0000



o.oooo



0 8897



0.8966



0.9017



0.9079



0.9082



0.9108



0.9147



0.9177



09194



0.9206



09218



15334



1.5514



15654



15781



15623



15889



1.5986



1.6059



16111



1.6148



1.6184



1.8565



1.8911



1.9192



1 9422



1 9609



1.9693



1.9798



19951



2.0063



2.0151



2.0211



2.0274



20842 2.2507



21331



2.1709



2.2023



2 2273



22397



2.2893



2.3013



2.3100



2.3184



23607



2.3996



24307



2.4469



22536 2.4641



2.2744



23138



2.4900



2.5081



2.5234



2.5347



2 5452



00000



00000



0.0000



0.0000



00000



0 50



08387



08582



0.8719



0.8824



1.00



13774



14440



14836



15129



1.50



1.6048



1 7373



1.8078



2.00



17149



2.50



17995



19116 2.0298



20157 2.1631



1.9



2.0



16 0.000



020



3.00



1.8750



21255



22778



2.3813



24570



2.5125



2.5583



2.5947



26148



26354



2.6654



2.6863



2.7050



2.7189



2.7314



3 50



1.9473



22101



2 3746



2.4898



2 5762



2 6390



2.6909



2.7325



2.7561



27798



2.8138



28382



2.6589



2.8752



28896



400



2.0178



2.2822



24603



2.5845



2 6793



2 7480



2.8052



2.8515



2.8784



2.9050



2.9426



2.9699



2.9928



3.0114



3.0274



4 50



20889



2.3622



2 5390



2.6698



27715



2 8449



2.9065



2.9569



2.9867



3.0158



3.0571



30871



31119



3.1322



31496



500



21547



2.4330



26128



2.7484



2 8558



29330



2.9982



3.0523



3.0645



3.1158



3.1605



3.1930



3.2195



3.2413



32597



550



22214



25013



26833



2.8222



29341



30146



3.0828



31400



3.1742



3.2074



3.2552



3.2899



33178



3.3408



33600



6.00



22872



2 5577



27512



28926



30079



30911



31616



32215



3.2575



3.2924



3.3428



33795



34085



34325



34524



6.50



23522



2.6329



28171



29603



30781



31635



32360



32980



33355



3.3720



3.4245



34629



34931



35176



35381



7.00



24165



26971



28814



30258



31452



32324



33065



33704



3.4092



3.4470



3.5012



35411



35722



35973



36181



750



2.4802



27602



2.9442



30893



32100



32985



3 3740



34393



3.4792



3.5180



3.5738



36148



35467



36723



3fi934



8.00



25432



28223



30058



31512



32727



33623



34387



35052



35460



35857



3.6486



36847



3.7173



37432



3.7646



850



2.6057



28836



30664



32118



3.3338



34239



35012



35685



36101



36504



3.7144



37512



37844



38108



3.8323



900



26676



29441



31260



3 2713



3.3934



3 4838



35617



36297



36718



3 7126



3.7775



3.8148



38484



3.8750



9.50



3.8969 3.9588



27289



30039



3.1847



33296



3.4516



3 5422



36204



36889



37315



3 7727



3.6382



3.8760



39099



3.9357



1000



27896



30630



32427



33870



3.5087



3 5993



3 6776



3 7465



3 7894



3 8308



3.8969



3 9350



3.9690



3.9961



40182



10.50



2 8499



31215



3.2999



34436



3 5647



3.6552



3 7336



3 8026



38456



36672



39538



39921



40262



4.0533



4 0755



11.00



2 9096



31794



3.3565



34993



3 6198



3.7100



3 7883



3 8573



3.9004



3 9421



4 0090



4 0473



40814



4.1086



41309



1150



29690



32369



34126



35543



36741



3 7640



3.8420



39108



3.9540



39958



40627



4.1010



4.1351



4 1622



41845



1200



30280



32940



3.4681



36086



3 7277



3.8171



3 8948



3 9634



40065



40432



41150



41532



41872



42143



4 2366



1250



30867



33506



35231



36623



37806



38694



39467



40150



4.0579



4.0994



41660



42041



4 2380



4 2650



4 2872



1300



31452



34068



3 5777



3.7154



3 8328



3 9211



3 9977



4 0557



4.1084



4.1495



4 2158



42537



42875



43144



4.3365



1350



32033



34627



36319



3.7680



3 8644



39721



40480



4 1155



4 1580



4.1989



4 2845



43021



43357



43625



4.3846



1400



32612



35183



36857



88200



39354



40224



40977



4 1547



4 2067



4 2472



4 3122



4 3494



4.3829



4 4095



4.4316



1450



33189



35735



3 7391



38716



39859



40722



4 1400



4 2131



4 2546



4 2947



4 3589



43957



4 4289



44555



4.4775



1500



33763



36285



37922



39228



4.0349



41215



4 1950



42609



43018



43414



4.4047



4 4410



4 4741



4 5005



4 5224



1550



34335



36832



38450



39736



4.0855



4 1702



42428



43080



43483



4 3874



4.4497



4 4855



4 5183



4.5446



4 5663



16.00



34906



37376



38974



40240



41346



42185



42900



43546



43942



44327



4.4939



4.5291



45617



45878



46094



16.50



35474



37919



39497



40740



41833



42663



43388



44007



44395



44773



4.5374



4.5720



46042



46302



46518



1700



36041



38459



40016



41237



42316



43138



43830



44462



44843



45213



4.5802



4.6141



46461



46719



46933



1750



3.6606



38996



40533



41731



42795



43608



44289



44913



45285



45648



46223



4.6555



4.5872



47129



47341



1800



3 7170



39532



41048



42221



43271



4.4075



44743



45359



45722



46077



4 6638



4.6963



4.7276



4.7532



4 7743



1850



37732



40066



41560



42709



43744



44538



45193



45801



46154



46501



4.7048



4.7365



4.7675



4.7928



48138 48527



1900



38293



40599



42071



43195



44214



44998



4.5640



46239



46582



46921



47451



4.7761



4.6067



46319



1950



3.8853



41129



42579



43678



4 4681



45455



4 6053



46574



47006



47335



4 7850



4.8151



4.8454



4.8704



48911



2000



3.9411



41658



43086



44158



45145



4.5909



46522



47104



47425



4 7746



4.8244



4.8536



4 8835



49083



49288 49661



20 50



3.9969



42186



43590



44636



45606



46360



4.6959



4.7531



4.7841



48152



48633



4 8916



4 9211



4 9457



2100



40525



42712



4.4094



45112



46065



4.6808



4.7392



4 7955



4.8253



4 8554



49017



4.9291



49582



4 9827



5 0029



21 50



4.1080



43237



44595



45586



46522



47254



4.7822



48376



4.8662



4 8953



4.9397



49662



49949



50192



50392



2200



41634



43760



45095



46058



46976



4 7697



4.8250



48794



4.9068



4 9348



4.9774



5 0029



5 0311



50552



50751



22 50



4.2187



44282



4 5594



4.6528



4.7428



48138



4.8675



4.9209



4.9470



4.9739



5.0146



50391



50670



50908



5.1105



2300



4 2739



44803



4 6091



46996



47879



48577



4.9098



49621



49869



50128



50514



50750



5 1024



5 1260



5 1455



2350



4.3291



45323



46587



4.7463



48327



4.9014



4.9518



50031



5.0265



5.0513



50879



5 1104



5 1374



5 1608



5.1802



24.00



4.3841



45842



47081



4.7928



48773



49449



4.9935



5.0438



5.0659



5.0895



5 1241



5 1455



5.1720



5 1953



5.2144



24.50



4.4391



4 6360



47575



48391



49217



49882



5.0351



50843



5.1050



5.1275



5 1599



5 1803



52063



5 2294



5.2483



25.00



4.4940



4.6877



48067



48853



49660



50312



5.0764



51245



5.1438



5.1651



5.1955



5 2147



5 2403



5.2631



5.2819



2550



4.5488



4.7392



48558



49314



5.0101



50741



51176



51646



5.1824



5.2025



5.2307



5 2488



5 2739



5.2965



5.3151



2600



4.6036



4.7907



49048



49772



5.0541



51169



51585



5.2044



5.2208



2.2397



5.2656



5.2826



5.3073



5 3296



5 3480



2650



46583



4.8421



49536



50230



5.0979



5 1594



51993



52440



5.2589



5 2766



5.3003



5.3162



5.3403



5.3624



5.3806



2700



47129



4.8934



5.0024



50686



5.1415



5 2019



52398



52834



5.2968



5.3132



5.3347



5.3494



5.3730



5.3950



54129



2750



47675



4.9447



5.0511



51142



5.1850



5.2441



5.2802



53227



5.3345



53497



5.3588



5.3823



5.4054



5.4272



5.4450



2800



48220



49958



5.0997



51595



5.2284



5.2862



5.3204



53817



5.3720



53859



5.4027



5.4150



5.4376



5.4591



54767



2850



4.8764



50469



5.1462



52048



5.2716



5.3282



5.3605



54006



5.4094



5.4219



5.4363



5.4475



5.4695



5.4908



55082



2900



49306



50979



5.1966



52500



53147



5.3700



54004



54393



54465



5.4577



5.4697



5.4796



5.5012



5.5223



5 5394



29 50



4.9851



51488



5.2450



52950



5.3577



5.4117



5.4401



5.4779



5.4834



5.4933



5.5029



5.5116



5.5326



5.5535



5.5704



3000



5.0394



51997



5.2932



5.3400



54005



5.4532



5.4797



5.5163



5.5202



5.5287



5.5359



5.5433



5.5638



5 5844



5.6011



WELLBORE



HYDRAULICS



TABLE



34-11



34.2-EXTENDED



SUKKAR-CORNELL



INTEGRAL



FOR BHP CALCULATION



(continued)



‘Prv Wp,r)dp,, I ; 2 1 +wP,,)” Pseudoreduced



Pp, 0.20



11 0.0000



temperature 12 00000



for 6= 13



0.0000



5 0 14



00000



15



16



00000



00000



17 00000



18 00000



19 00000



20 00000



22 00000



24



26



28



30



00000



00000



00000



00000



0.50



0.0226



00220



00216



00214



00212



00210



00209



00207



00207



00206



00205



00205



00204



00204



00204



1.00



0.1036



00983



00954



00934



00921



00909



00901



00894



00890



00886



00881



00877



00874



00871



00869



1.50



0.2121



02052



01995



01954



01924



01901



01882



01668



01859



01850



01838



01829



01822



01816



0 1811



2.00 250



0.3002 0.3741



03125 04046



0.3102 04126



0.3066 04133



03034 04124



03007 04107



02983 04090



02965 04076



02954 04066



02943 04056



0 2926 04041



02914 04030



02904 04020



02896 04012



0 2889 04005



3.00



0.4419



04854



0.5032



0.5105



05137



05144



05143



05140



05138



05134



05125



05118



05112



05108



05103



3.50



0.5074



05594



05847



05983



06065



06101



06123



06138



06147



06152



06154



06155



06155



06157



06156



4.00



0.5715



06291



06594



06785



06915



06982



07029



07064



07087



07104



07121



07133



07140



07149



07154



4.50



0.6346



06957



0.7294



0.7530



07702



07797



07868



07927



07964



0 7994



08027



0 8051



0 8068



08084



08094



5.00



0.6966



0.7601



07960



0.8229



08440



08560



08653



08734



08785



08827



08879



08916



08941



08965



08980



5.50



0.7579



08225



08601



0.8895



09138



09280



09393



09493



09558



09611



09682



09732



09765



09795



09815 10604



600



0.6185



08836



09222



0.9536



09803



09965



10095



10213



10289



10354



10441



10504



10544



10580



6.50



0.8784



09437



09829



1.0156



10442



10620



10764



10896



10984



1 1060



1 1162



1 1236



1 1284



1 1324



1 1351



700



09378



10030



10423



10758



11058



1 1249



1 1406



1 1552



1 1649



1 1734



1 1848



1 1932



1 1987



17031



17060 12737



750



0.9967



10614



11005



11346



1.1656



11857



12024



12182



12286



12379



12504



12597



12657



12704



BOO



10551



1 1191



1 1578



11921



12237



12447



12621



12788



1 2900



1 2999



13i67



13234



1 3299



1 3349



1 3383



850



11131



11761



12142



12486



12805



13020



13201



13374



13492



13596



13773



13845



13914



13967



1 4003



900



11706



12325



1 2698



13041



13361



13579



13764



13943



14066



14173



14357



14434



14506



14561



14599



950



12275



1.2083



13240



I 3587



13907



14125



14313



14497



14623



14733



14927



15008



15077



15135



15174



1000



12841



13435



13791



14126



14443



14661



14851



15037



15165



15278



15472



15555



1 5630



1 5689



1 5729



1050



13403



13983



14328



14658



14970



15187



15377



15564



15694



15808



16006



1 6090



16167



16226



16267



1100



13961



14526



14860



15162



‘1 5490



15705



15894



16081



16211



16326



16526



16611



16687



16747



16789



1150



14515



15065



15387



15701



16002



16214



16401



16587



16718



16833



17034



17118



1 7195



1 7254



1 7296



1200



15067



15601



15910



16214



16509



16717



16901



17085



17215



17330



17530



17613



1 7689



17749



1 7790



1250



15616



16133



1.6429



16721



17010



17213



17393



17575



17704



17817



18015



18097



18172



18231



18271



13.00



1.6163



16662



16944



1 7224



17505



1 7704



17879



18057



18184



18295



18489



18569



18644



18701



18742



1350



16708



17168



17456



17722



17995



18188



18358



18532



18656



18765



18954



19032



19105



19161



19201



14.00



1 7250



17711



17965



18216



18480



18667



18830



19001



19121



19227



19410



19485



19556



19612



19651



1450



17791



18232



18470



18706



18960



19142



19298



19463



19580



19681



19858



19920



19998



2 0053



2 0091



1500



18330



18750



18973



19192



19436



19612



19760



19920



20032



20128



2 0298



2 0364



2 0432



2 0485



2 0523



1550



18867



19266



19472



19675



19909



20077



20217



20372



20478



20570



2 0730



2 0792



20857



20910



2 0946



1600



19402



19780



19970



2 0154



2 0377



20538



20669



20818



20918



2 1005



2 1155



2 1212



2 1275



2 1326



2 1362



1650



19936



20292



20465



2 0631



2 0842



20996



21117



21260



2 1353



2 1434



2 1574



2 1626



2 1686



2 1736



2 1770



1700



2.0469



20958 21449



2 1104 2 1575



2 1303 2 1762



21450 21900



21561 2 2000



21697 2 2131



21783 2 2209



2 1858 22276



22032



2 2090



2 2138



2 2172



21000



20802 21311



2 1987



1750



22394



22433



2 2488



2 2535



2 2567



1800



21530



21817



21937



2 2043



22217



22347



22437



2 2560



22630



22690



22795



22828



2 2880



2 2925



2 2956



1850



22059



22323



22424



22509



22670



22791



22869



22985



23046



23100



23191



23217



23266



23309



23339



1900



22587



22826



22909



22973



23120



23233



23299



23407



23459



23505



23582



23600



23646



23688



23717



1950



23113



23329



23393



23434



23567



23671



23725



23825



23868



23906



23969



23979



24022



24062



24089



20.00



23639



23830



23875



23893



24012



24107



24148



24241



24273



24303



24350



24353



24392



24431



24J56



2050



24164



2.4329



24355



24350



24455



24541



24568



24653



24675



24696



24728



24723



24758



24795



24819



2100



24688



2.4828



24834



24306



24895



24972



24986



25062



25074



25086



25101



25088



25119



25155



25177



2150



25210



2.5325



25311



25259



25333



25400



25401



25468



25470



25472



25471



25449



25477



25510



25531



22.00



25733



2.5822



25788



25711



2 5770



25827



2 5814



25872



25862



25855



25837



25806



25830



25861



2 5881



2250



26254



26317



26263



26161



26204



26252



26224



26273



26252



26235



26199



26159



26179



26209



26226



2300



26774



26811



26736



26610



26637



26674



26632



26672



26639



26612



26558



26508



26524



26552



76566



2350



27294



27304



27209



27057



27068



2 7095



2 7038



'27068



2 7023



26986



26913



26854



25866



26892



26906 2 7241



2400



2.7813



27796



27680



27503



2 7497



2 7514



2 7441



2 7462



2 7405



2 7357



2 7266



2 7197



2 7204



2 7229



24.50



28332



2.8288



2.6151



27947



2 7924



2 7981



2 7043



2 7854



2 7784



2 7726



2 7615



2 7536



2 7540



2 7562



2 7573



25.00



28849



28778



28620



28390



28351



28346



28243



28244



28161



28092



27961



2 7872



27872



2 7892



2 7901



25.50



29367



29268



2.9088



28832



28775



28760



28640



28532



28536



28456



28305



28205



28200



28219



28226



26.00



29883



29757



29556



29272



29196



29172



29037



29018



28908



28818



28646



28536



28526



28543



28548



26.50



30399



30245



30022



29711



29620



29583



29431



29402



29279



29177



28985



28864



28850



28864



28867



2700



30915



30733



30488



30149



30040



29993



29824



29785



29648



29534



29320



29189



29170



29182



29184



27.50



31429



3.1220



30953



3.0586



3.0459



30400



30215



30165



30014



29889



29654



29512



29488



29498



29497



2800



31944



3.1706



31417



3.1022



30877



30807



20604



30544



30379



30242



29985



29832



29803



29811



29809



28.50



32458



3.2191



31880



31457



31294



31212



30992



30922



30742



30593



30314



30149



30116



30122



30117



29.00



32971



32676



32343



3.1891



3.1710



31616



31379



31297



31103



30942



30641



30465



30426



30430



30424



29.50



33484



33160



32804



32324



32124



32019



31764



31672



31463



31289



30966



30778



30735



30736



30728



30.00



3.3997



33644



3.3265



3.2756



3.2537



3.2421



32148



32045



3.1821



31635



31268



3 1089



31040



31040



31029



34-12



PETROLEUM



TABLE



Pseudoreduced



A?-



1.1



34.2-EXTENDED



temperature 1.2



for B= 1.3



SUKKAR-CORNELL



INTEGRAL



ENGINEERING



FOR BHP CALCULATION



HANDBOOK



(continued)



10 0 1.4



15



16



17



18



0.0000



0 0000



0 0000



0 0000



0.0108



00107



00107



00106



0.0494



0.0486



0 0479



00474



0.20



0.0000



o.oooaooooo



0.0000



0.50



0.0115



0.0112



0.0110



1.00



00561



00525



00507



2.2



24



26



28



30



0000000000



19



20



0 0000



0 0000



0 0000



0 0000



0 0000



00105



00105



00105



00104



00104



00104



00103



00103



00470



00468



00465



0.0462



00460



0 0458



0 0456



00455 0 0990



1.50



0.1292



01187



0.1132



0.1098



0.1074



01056



0 1041



0 1031



01024



01018



01009



01003



00997



0 0994



200



02028



0.1968



0 1891



0.1837



0.1797



01767



01743



01725



01713



01703



0.1687



0 1676



0 1667



0 1660



0 1653



2.50



0.2684



0.2723



02677



0.2624



0 2578



02543



02513



02490



02475



0 2461



02440



02426



02413



0 2403



0 2394



3.00



0.3300



0.3422



03427



0.3399



0 3364



0 3332



03302



03278



03263



0 3248



03225



03210



03195



03184



03174



3.50



0.3897



0.4080



0.4130



0.4135



04123



04102



0 4080



0 4061



04047



04035



04014



03999



0 3985



0 3974



0 3964



4.00



0.4485



0.4708



04793



0.4832



0 4846



0 4841



04830



04820



04812



04803



04787



04776



04764



04755



0 4746



4.50



0.5065



0.5315



05423



0.5492



05533



05545



05547



05549



05549



0 5546



0 5538



0.5532



05523



05517



05511



500



05638



05904



06029



06122



06189



06217



06233



06248



06256



06260



06262



06263



06258



06256



06252



550



0.6204



0.6480



0.6617



0.6729



0.6818



06861



06891



06919



06934



06946



06959



06967



06967



06966



06967



600



06765



07045



07190



0.7316



0 7424



0 7481



0 7522



0 7563



0 7586



07605



0 7629



0 7645



0 7650



0 7654



07655 0 8317



6.50



0.7321



07602



07752



0 7888



08010



08079



0 8131



0 8182



0 8214



08240



08273



0 8297



08307



08314



7.00



0.7873



08153



08304



0.8447



08580



08659



0 8720



0 8781



0 8619



08852



08895



0 8925



08940



08950



0 8955



7 50



08421



0.8697



08846



0 8994



09134



09221



0 9290



0 9360



0 9404



0 9443



0 9494



09531



0 9550



0 9562



0 9568



8.00



0.8965



09236



09381



09531



0.9676



0.9770



0 9845



0.9921



09971



10015



10092



10115



10138



10152



10160



8.50



0.9506



0 9769



0 9909



10059



10207



1.0305



10385



1.0467



10522



10569



10653



10681



10706



10723



10732



9.00



1.0043



1.0296



10431



10580



10729



1.0829



10912



10999



11057



11108



11197



1 1228



1 1256



1 1275



11286



950



10575



10819



1.0947



1 1094



1 1242



11342



11428



11518



1 1579



11633



11726



1 1760



11790



11810



1 1822



1 1104



11338



1 1458



1 1601



11747



11847



1 1935



12027



12090



12145



12242



12278



12309



12331



12343 12850



1000 1050



1 1630



11852



1.1964



12102



12245



12344



12432



12525



12589



12645



12746



12783



12814



12836



11.00



12153



12363



12466



12598



12736



12834



12920



13013



13078



13135



13238



13275



13307



13329



13343



11.50



12674



12871



12964



1.3089



13222



13317



13402



13494



13559



13616



13719



13756



13788



13810



13824



12.00



13192



1.3376



13458



1.3574



13702



13794



13876



13967



14032



14088



14190



14227



14258



14280



14294



12.50



13708



13877



13949



14056



1.4178



14266



14345



14433



14497



14552



14653



14688



14719



14740



14753



13.00



14222



1.4377



14437



14533



1.4649



14733



14807



14893



14955



15008



15106



15140



15169



15139



15202



13.50



14734



14873



14921



15006



1.5115



15194



15264



15346



15406



15457



15551



15582



15611



15630



15642



14.00



15244



15368



15403



1.5476



1.5577



15652



15716



15794



15851



15899



15988



16016



16043



16062



16074



1450



15753



15860



15883



15942



1.6035



16104



16163



16237



16290



16335



16417



16443



16468



16486



16497



1500



16261



16351



16360



16405



1.6490



16553



16605



16575



16723



16764



16840



16862



16885



16902



16912



15.50



16767



16839



1.6835



16865



16941



16999



17043



17108



17151



17811



1 7256



1 7274



1 7296



17311



17320



16.00



17271



17326



1 7308



17323



17389



17440



17477



17537



17575



17607



17666



17679



1 7699



17713



17722



16.50



17775



17811



17778



17778



17834



1 7878



1 7906



17961



17993



18020



18070



18078



18096



18109



18116



17.00



18277



18294



1.8247



18230



18275



18314



18333



18382



18407



18429



18469



18472



18487



18499



18505



17.50



18778



18777



18714



18680



18714



18746



18756



18799



18818



18833



18862



18859



18872



18883



18888



18.00



19278



19257



1.9179



19127



19151



19175



1.9175



19212



19224



19232



19251



19242



19252



19261



19265



18.50



19777



19737



1.9643



19573



19585



19602



1.9592



19622



19626



19628



19634



19619



19626



19634



19637



19.00



20276



20215



20105



20017



20016



20026



2.0005



20029



20025



20020



20013



19992



19996



2 0002



20004



1950



20773



20692



20566



20458



20446



20447



2.0416



20433



20420



20408



20388



20359



2 0360



2 0365



20366



20.00



2.1269



2 1167



21026



20898



20873



20867



20824



20833



20812



20792



20759



20723



20721



20724



20723



2050



21765



21642



21484



21336



21298



2 1284



2 1229



21232



21201



21173



21126



21082



21077



2 1079



21077



21.00



22260



22116



21941



21773



21722



21699



21632



21627



21587



21551



21489



21438



21429



21429



21425



21.50



22754



22588



2.2396



2.2207



22143



22112



22033



22020



21970



21926



21848



21789



21777



21775



2 1770



22.00



23248



23060



22851



22641



22563



22523



22432



22411



22350



22298



22204



22137



22121



22118



22111



2250



23741



23531



23304



23073



2.2981



22932



22828



22799



22728



22667



22557



22481



22462



22457



22449



2300



2.4233



24001



23757



23503



2.3397



23340



23222



23185



23103



23033



22906



22822



22799



22792



22783



23.50



24725



24470



24208



2.3932



23812



23745



23615



23569



23476



23397



23253



23160



23133



23124



23113



24.00



2.5216



24938



24659



24360



2.4226



24149



24005



2 3951



2 3847



23758



23597



23494



23463



23453



2 3440



24.50



2.5706



25406



25108



24787



2.4637



2 4552



24394



2 4331



24215



24117



23937



23826



2 3791



23779



2 3765



25.00



2.6196



2.5873



25557



25212



2.5048



2.4953



2.4761



2.4709



2.4581



24473



24275



24155



24115



24102



2 4086



25.50



2.6685



2.6339



26005



25637



2.5457



2.5353



25166



2.5085



2.4946



2.4827



24611



24481



24437



24422



24404



2600



2.7174



26805



26452



26060



2.5865



2.5751



2.5550



2.5459



2.5308



2.5179



24944



24804



24756



24739



24719



2650



2.7663



2.7269



2.6898



26482



26272



2.6148



2.5932



2.5832



2.5668



25529



2.5275



25124



25073



25053



25032



27 00



2.8151



2.7734



2.7343



26904



2.6677



2.6543



2.6312



2.6203



2.6027



2.5877



2.5603



2 5443



25386



25365



25342



2750



2.6638



2.8197



2.7788



2.7324



2 7082



2.6938



2.6691



26573



26384



2.6223



2.5929



25758



25698



25675



2 5650 25955



2800



2.9125



2.8660



2.8232



2.7743



2.7485



2.7331



2.7069



26941



26739



26567



26253



2.6072



2.6007



2 5982



28.50



2.9612



2.9123



2.8675



2.8162



27887



27723



27446



27307



27092



26909



26575



2.6383



26314



26286



26258



29.00



3.0098



2.9585



2.9118



2.8579



28288



20114



27821



27673



27444



27250



2 6895



2.6692



2.6618



26589



26558



2950



3.0584



3.0046



2.9560



2.8996



28689



28504



28194



28036



27794



27589



2 7212



26999



26920



26889



26857



30.00



3.1069



3.0507



3.0001



2.9412



29088



28892



28567



28399



28143



27926



27528



27304



2 7221



27187



27153



WELLBORE



HYDRAULICS



34-13



TABLE 34.2-EXTENDED



SUKKAR-CORNELL



INTEGRAL FOR BHP CALCULATION



(continued)



‘Pv (z/p,,,Wp,, 0I* 1 + W/P,,)’ Pseudoreduced



temperature



for



B= 15 0



pp’



1.1 ~__~~



1.2



1.3



1.4



1.5



1.6



17



18



19



20



22



0.20



00000



0.0000



0.0000



0.0000



0.0000



0.0000



0.0000



0.0000



0.000(3



0.0000



0.0000



2.4 0.0000



26 0



2.8 0.0000



30 o.oooo



0.50



00077



0.0075



0.0074



0 0073



0.0072



0.0071



0.0071



0.0071



0.0070



0.0070



0.0070



0.0070



0.0069



0.0069



00069



1.00



00385



0.0359



0.0345



0.0336



0.0330



0.0325



0.0322



0.0319



0.0317



0.0316



0.0313



0.0311



0.0310



0.0309



00308



150



00939



0.0838



0.0793



0.0765



0.0746



0.0732



0.0721



0.0713



0.0708



0.0703



0.0696



0.0692



0.0687



0.0685



0 0682



2.00



0.1571



0.1453



0.1371



0.1319



0.1282



0.1257



0.1236



0 1220



0.1211



0.1202



0.1189



0.1180



0 1172



01167



0.1161



250



02162



0.2093



0.2008



01943



0.1892



0.1857



01827



01804



01790



0.1777



0.1758



0.1745



01733



0.1724



01716



300



02725



0.2710



0.2648



0.2587



0.2533



0.2493



0.2458



0.2431



0.2413



0.2397



0.2374



0.2357



02342



02331



02320



350



0.3275



0.3302



0.3267



03222



0.3176



0.3138



0.3102



03074



0.3055



0.3038



0.3012



0.2994



02978



02964



02952



400



03818



0.3874



0.3862



0.3837



03805



0.3774



0.3743



0.3717



0.3699



0.3683



03657



03639



03622



03608



0.3596



450



04355



04430



0.4435



0.4431



04415



0.4393



0.4369



0.4349



0.4335



0.4320



04298



04281



04265



04252



0.4240



500



04887



0.4975



0.4992



0.5004



0.5006



0.4994



0.4978



0.4966



0.4956



0.4945



04928



04914



04900



0488%



04877



550



0.5413



0.5508



0.5535



0.5561



05579



0.5577



0 5570



0.5566



0.5561



0.5554



0 5543



05534



05522



0 5512



0 5503



600



0.5936



0.6034



06066



0.6103



0.6135



0.6143



06144



06149



0.6149



0.6147



0.6143



06138



06129



06121



06113



650



06454



0.6553



06590



0.6634



06676



06694



0.6703



06715



0.6720



0.6724



0.6726



06727



0.6721



0.6715



06708



7.00



0.6969



0.7068



0.7105



0.7155



0 7205



0 7230



0 7246



0 7265



0 7276



0.7284



0.7293



0.7299



0 7296



0.7291



0 7286 07848



750



0.7482



0.7577



0.7613



0.7666



0.7722



07754



0 7776



0 7802



0 7817



0 7829



0 7844



0.7854



0.7855



0 7852



8.00



0.7991



0.8082



08114



0.8170



0.8230



08266



08293



0 8324



0 8344



0 8360



0 8391



0.8395



0.8398



0 8397



0 8394



8 50



0.8497



08582



0.8611



08666



08729



0 8768



0 8799



0 8835



0.885%



0.8878



0 8914



0.8920



08926



08927



08925



9.00



0.9000



0 9078



09102



09157



0.9220



09261



0 9295



09440



09442



09441



0 9570



0.9588



09641



09704



0 9746



0 9782



0 9360 0.9382 0.9852 0.9876



0 9432



0 9500



0 9334 0 9824



09423



950



09920



0 9932



09941



09944



09944



10.00



0.9998



10059



1.0071



10121



1.0181



1.0223



1 0260



10304



10334



10359



10407



10420



10430



10434



10435



1050



1.0492



10544



1.0549



10595



1.0653



1.0694



10731



10776



10806



10833



10883



10897



10908



10913



10914



11.00



10985



1 1026



1.1024



11065



1.1119



1.1159



1.1195



1 1239



11271



1 1298



1 1349



11364



1 1375



1 1380



1 1381



11 50



1 1475



11506



1.1496



1 1530



1.1580



1.1618



1 1653



1 1696



1 1728



1 1755



1 1807



1 1822



11832



11837



11839



12.00



1 1963



1 1983



1.1964



1 1992



1.2037



1.2072



1 2105



12147



12178



12205



12256



12270



12281



12285



12287



1250



1.2449



12458



1.2430



12449



12490



1.2522



1.2551



1.2592



1.2622



12648



12698



12711



12720



12724



12725



13.00



12934



12931



1.2893



12903



12939



1.2967



12993



1.3031



1.3060



1.3084



1.3131



13143



13152



13155



13156



13.50



1.3417



13402



1.3354



13354



13384



1.3408



13430



1.3465



1.3492



1.3514



1355%



13567



13575



13578



1.3578



14.00



1.3899



1 3870



1.3812



13862



13825



1.3845



13862



1.3894



1.3918



1.3938



1.3977



13984



13991



13993



13992



14.50



14380



14337



14268



14247



14263



1.4278



14290



14319



14339



1.4356



14390



14395



14400



14401



14400



1500



1.4860



14803



14722



14689



14698



14708



14714



14739



14756



14769



1.4797



14798



14802



14802



14800



15.50



1.5338



1 5266



1.5174



1.5129



15130



15135



15134



15155



15168



15177



15198



15196



15197



15197



15194



16.00



1.5815



15728



15625



1.5566



15559



15558



15551



15567



15575



15580



15594



15587



15587



15585



15582



1650



1.6291



16189



16073



16001



15985



1 5979



15964



15976



15978



15979



15984



15973



15971



15968



15964



16409



16397



16374



16381



16378



16373



16370



16354



16350



16346



16341



16812



16781



16783



16773



16764



16750



16730



16723



16718



16712



1700



1.6766



1.6649



16520



1.6434



1750



17241



17107



16966



16865



16830



1800



1.7714



1.7564



17410



1.7293



17249



17225



17186



17181



17166



17150



17127



17100



17091



17085



17078



18.50



1.8187



18020



17853



17720



17666



17635



17587



17577



17554



17533



17499



17466



17455



17447



17439



1900



1.8659



18475



18294



1.8146



18081



18043



17986



17970



17940



17912



17866



17828



17814



17805



17796



1950



19130



18929



18734



18569



18493



18449



18382



18360



18322



1828%



18280



18186



18169



18158



18148



2000



19600



19382



19173



18991



1.8904



18853



18776



1 a747



18702



18661



18590



18540



18519



18508



18496



2050



20070



19834



19611



19412



1.9314



1.9255



19168



19132



19079



1.9031



18947



18889



18866



18853



18840



21 00



2.0539



20285



2004%



19831



1.9721



19655



1.9557



19515



19453



19397



19300



19235



19209



19195



19180



21.50



21007



20736



20484



20248



20127



2.0054



1.9944



19895



19824



19761



19650



19578



19549



19532



19517



22.00



2 1475



2 1185



20918



20665



2.0531



2.0450



2.0330



2.0273



20193



2.0122



19997



19917



19884



19867



19850



22.50



2.1943



2.1634



21352



21080



2.0934



20845



20713



20649



20560



2.0481



20341



20253



20217



20198



20179



23.00



2.2410



2.2082



2 1785



21494



21335



2 1239



21095



2.1024



2.0924



2.0837



2.0681



20586



20546



20525



20506



23.50



22876



2.2529



22217



21906



21735



21631



2 1475



21396



21286



21191



2.1019



20916



20872



20850



20829



2400



2.3342



2.2976



2 2648



22318



22134



22021



2 1853



21766



2.1646



2.1542



2.1355



2 1242



21196



21171



2 1149



24 50



2.3807



2.3422



2 3079



22728



22531



22410



22229



22135



22005



21891



2.1687



2 1567



2 1516



21490



2 1466



2500



2.4272



2.3867



23509



2.3138



22927



22798



22604



22502



22361



2 2238



22017



21888



2 1834



21806



2 1780



25 50



2.4736



2.4312



23937



23546



2 3322



23184



22978



22867



22715



22583



2.2345



22207



22149



22119



2 2092



26 00



2 5200



24756



24366



23953



2 3716



23569



23350



23230



23067



22927



22671



22523



22461



22430



22401



26 50



2.5664



25200



24793



24360



24109



23953



23720



23592



23418



23268



22994



22837



22771



22738



22707



27.00



2 6127



25643



25220



24766



2.4501



2 4336



2 4089



23953



23767



23607



2.3315



2 3149



23078



23044



23011



2750



26590



2.6086



2.5646



25170



24891



24718



24457



24312



24115



23944



23634



23458



23384



23347



23313



28.00



2 7053



2.6528



2.6072



25574



25281



2.5098



24824



24670



24460



24280



23951



23765



23687



23648



23612



28.50



27515



26969



26497



25977



25669



25478



25189



25026



24805



24614



24266



24070



23987



23947



2 3909



29.00



27977



27410



2.6921



26380



2.6057



2.5856



25553



25382



25148



24947



24579



24373



24286



24244



24205



29.50



2.8438



2.7851



2.7345



2.6781



26444



2.6234



25916



2.5736



25489



2 5278



24890



24674



24583



24538



24497



30.00



2.8899



2.8291



2 7769



2.7182



26830



2.6610



26278



26088



25829



25607



25200



24974



24878



24831



24788



34-14



PETROLEUM



TABLE



34.2-EXTENDED



SUKKAR-CORNELL



INTEGRAL



FOR



BHP



ENGINEERING



CALCULATION



HANDBOOK



(continued)



‘Pp, (zb,,)dp,r \ 02 1 + WP,,)’ Pseudoreduced



P,



1.1



0.20



0.0000



0.50 1.00



temperature 1.2



for 8 = 20.0



1.3



1.4 D.0000



1.5 6.0000



1.6 0.0000



1.7



0.0000



0.0000



0



00058



0.0056



0.0055



0.0055



0.0054



0.0054



0.0053



0.0294



0.0272



0.0262



0.0255



0.0250



0.0246



0.0243



1.8



1.9



0.0000



0.0000



0.0000



2.0



22



24



0.0000



0.0000



0.0053



0.0053



0.0053



0.0052



0.0241



0.0240



0.0239



0 0237



26



28



30



0



ooooo



0 0000



00052



00052



00052



00052



0 0236



0 0235



0 0234



0.0233



1.50



00740



0.0649



0.0610



0.0587



0.0572



0.0561



0.0551



00545



0.0541



0.0537



0.0532



00528



00525



0.0522



00520



2.00



'0.1295



0.1156



0.1077



01030



0.0998



0.0976



00958



0.0945



00937



0.0930



00918



00911



00905



00900



00895



2.50



01832



0.1712



0.1614



0.1547



0.1498



0.1465



0.1438



0.1417



0.1404



0.1393



01376



01364



01354



01346



01339



3.00



0.2350



0.2264



0.2172



0.2099



0.2040



0.1999



01964



0.1937



0.1920



01904



0.1882



01867



01853



01842



0.1832



02371



02359



3.50



02860



02801



0.2725



02657



0.2597



0.2553



0.2514



0.2484



0.2463



0.2445



02419



4.00



03365



0.3326



0.3264



0.3208



0.3154



03111



03073



03041



0.3020



0.3000



0.2972



02952



02934



02919



02906



0 355s



0.3531



03510



03492



0 3476



03462



4.50



0.3865



0.3841



0.3790



03747



0.3703



0.3664



0 3629



0 3599



0.3578



02401



0.2384



5.00



0.4360



0 4346



0.4305



04273



0.4240



0.4208



04177



0.4151



0.4132



0.4114



04088



04068



04050



0 4034



0.4021



5.50



04852



04843



0.4809



0.4787



0.4765



0.4740



0.4714



0.4594



0.4678



0 4662



0 4639



0 4622



0 4604



0 4589



0.4577



6.00



0.5341



0 5335



0.5305



0.5291



0.5279



0 5261



0.5241



0.5226



0.5213



0.5201



0.5182



05167



05151



05137



0.5125



6.50



05827



05821



0.5794



05786



0.5783



0.5771



0.5756



0.5747



0.5738



0.5729



05714



0.5703



0 5689



05676



0.5665



7.00



0.6310



0.6304



0.6277



0.6274



0.6276



0.6270



0 6261



0.6257



0.6252



0.6246



0.6236



06228



06216



06205



0.6194



750



06791



06782



0.6755



0.6754



0.6761



0.6760



0.6755



0.6756



0.6754



0.6752



0.6746



06741



06732



06722



06712



8.00



0 7269



0.7257



0.7227



0.7228



0.7238



0.7241



0.7240



0.7245



0.7247



0.7247



0.7251



0 7244



0.7237



0 7227



0 7219



8.50



0 7745



0.7728



0.7695



07696



0.7708



0.7714



0.7716



0.7725



0.7729



0 7732



0.7740



0.7735



0 7730



0 7227



07714



9.00



08219



0.8196



0.8159



08160



0.8172



0.8179



0.8184



0.8195



0.8202



0.8207



0.8218



0.8216



0.8212



08205



08198



9.50



0 8690



0.8661



0.8620



08618



0.8631



0.6638



0.8644



0.8658



0.8666



0.8673



0.8687



0.8687



0.6684



08678



08672



1000



09159



09123



0.9077



09073



09083



09091



09098



09113



09123



0.9131



0.9147



0.9148



0 9146



0 9141



09135



10.50



09626



09582



0.9530



09523



09531



09538



09545



0.9561



09571



0 9580



0.9599



0.9601



0 9599



09595



09589



11.00



10091



10039



0.9981



0.9969



0.9975



0.9980



0 9987



10002



10014



1.0023



1.0043



10045



1.0043



10039



10034



11.50



10554



10494



1.0429



1.0412



10414



10418



10423



10438



10450



10459



10479



10461



10479



10475



10470



12.00



1 1016



10946



10874



0.0851



10849



10851



1.0855



10868



10879



10886



10908



10909



10908



10903



10896



12.50



1.1476



1 1397



11317



11288



11282



1 1280



1 1282



1 1294



1 1304



1 1312



11331



1 1331



1 1328



11323



1 1318



13.00



1.1935



1 1846



1 1758



11721



1 1710



1.1706



1 1704



1 1714



1 1723



1 1730



1 1746



11745



11742



11736



1 1731



13.50



1.2392



12293



12197



1.2151



12136



12128



12122



12130



12137



12142



12156



12153



12149



12143



12136



1400



1.2849



1273s



12633



1.2579



1.2558



12547



12537



12542



12547



12549



12559



12554



12549



12542



12535



1450



1.3304



13183



13066



13005



1.2977



1 2962



12948



12949



12952



12952



12957



12949



12943



12935



12926



1500



13759



13625



1 3501



13428



13394



13375



13355



13353



13352



1 3349



13349



13339



13331



13322



13315



1550



1.4212



14067



13933



13849



13808



13784



13759



13754



13749



13743



13736



13723



13713



13704



13695 14071



1600



1.4665



14507



14363



14267



1.4220



14191



1.4150



14151



1.4142



14132



14118



14101



14090



14080



1650



1.5116



1.4945



14792



14684



1.4629



14595



14558



14544



14531



14517



14496



14475



14462



14451



I4441



1700



15567



15383



15219



15099



1.5036



1.4997



14953



14935



14916



14898



14869



14844



14829



14617



14806



1750



1.6017



1.5820



15645



15512



15441



1.5397



1.5345



1.5323



1.5298



1.5275



15238



15208



15191



15178



15166



1800



1.6467



16256



16069



15924



15844



1.5794



15735



15708



1.5678



1.5649



15603



15568



15549



15534



15522



18.50



1 6916



16691



16493



16334



16245



1.6190



16123



16090



1.6054



1.6020



1 5964



15924



15902



15887



I 5873



1900



1 7364



17125



16915



16742



16644



1.6583



1 6508



16470



16427



16388



1 6321



16275



16252



1 6235



16220



19 50



1.7611



17558



1 7336



17149



17042



1.6975



1.6847



1.6797



16752



16675



16623



16597



16579



16563



2000



1.8258



17990



1.7757



17555



17438



17364



16891 1.7271



17222



17165



17114



17025



16967



1 6938



16919



1 6902



2050



1.8705



18421



1 a176



17959



17832



17752



1.7650



1.7595



17530



17473



1 7372



1 7308



17276



17256



17238



21 00



1.9150



i 8852



18594



18362



18225



18139



1.8027



1.7965



17893



1.7829



17716



17645



17611



17589



17570



21 50



1.9596



19282



19012



18763



18616



18523



1.8401



1.6334



1.8254



1.8183



18056



17979



1 7942



17918



1 7898



2200



20041



19711



1942s



19164



19006



18906



1.8774



1.8700



18612



1.8534



16394



18310



18270



16245



18223



2250



2.0485



20140



19844



19563



19395



19288



19146



19065



18968



18882



18730



18638



16595



18568



I 8545



2300



2.0929



20568



20259



19962



19782



19668



19516



19426



19322



19229



1.9062



18963



18916



18889



18864



2350



2.1372



20995



20674



20359



20168



20047



19684



19789



19674



19573



19392



19286



19235



19206



19180



2400



2.1815



21422



21087



20756



20553



20425



20250



20149



20025



2450



2.2258



2 1849



2.1500



2 1151



20937



20801



20615 20979



2.0507 20863



20373



20256



2 0044



1 9922



1 9865



1 9832



19804



20719



20594



19916



2.0367



1.9719



20237



19605



20176



19551



20142



19521



20112



19493



2500



2.2700



22274



2.1912



21546



21319



21176



2550



2.3142



22700



22324



2 1939



21701



21550



21341



21218



21064



20930



2.0687



20549



20484



20449



20417



2600



2.3564



2.3124



22735



22332



22082



2 1923



21702



21571



21408



21265



21005



20858



20790



20753



20720



26.50



2.4025



2.3549



23145



22724



22461



22295



22062



21923



21749



21598



21321



21166



21094



21055



21020



27.00



2.4466



2.3973



23565



23115



22640



22665



22420



22274



22089



21929



21636



21471



21395



2 1355



21318



27.50



2.4907



2.4396



23964



23505



2 3218



23035



22778



22623



22428



22258



21946



21774



2 1695



21652



21614



26.00



2.5347



2.4819



2.4373



2.3895



23595



23404



23134



22971



22765



22586



22258



22075



21992



21948



21908



28.50



2.5707



2.5243



2.4781



2.4204



23971



2 3772



23409



23110



23100



22912



22566



22375



22287



22241



22200



2900



2.6226



2.5664



2.5189



2.4672



24146



24119



23848



23664



23435



23217



22873



22675



22560



22552



22600



29.50



2.6666



2.6085



2.5596



2.5060



24720



24504



2 4195



2 4008



23768



23560



23178



22967



2 2871



22822



22777



30.00



2.7106



2.6507



2.6003



2.5447



2.5094



2.4870



24547



24352



2.4100



23882



23481



23261



23161



23109



2 3063



WELLBORE



HYDRAULICS



TABLE



34-15



34.2-EXTENDED



SUKKAR-CORNELL



INTEGRAL



FOR



BHP



CALCULATION



(continued)



.PP, (z~p,,Wp,r \ 6 * 1 + w4Jpr)2 Pseudoreduced P”,



1.1



temperature 12



for 8=25.0 1.3



0.0000



14



15



16



17



18



19



0.00000.00000.00000.000000000



00000



2.0 0.0000



2 2 0 0000



24



26



28



30



0 0000



0 0000



00000



0 0000



050



0.0047



00045



0.0044



0.0044



0.0043



0.0043



0.0043



00042



0.0042



00042



00042



00042



00042



00042



00042



1.00



0.0237



0 0219



0.0211



0.0205



0.0201



0.0198



00196



00194



0.0193



00192



00191



00187



00189



00188



00187



150



0.0611



00529



0.0496



00477



00464



00454



00446



00441



00438



00435



00430



00427



00424



00422



00420



200 2.50



0.1106 0.1598



0.0961 0.1453



0.0888 0.1352



00846 0.1287



00818 0.1241



0.0798 01211



00783 01186



00771



00764



0.0758



00749



00742



00737



0 0733



00729



01168



01156



01146



01131



01121



01111



01104



0 1098



3.00



0.2079



0.1952



0.1846



0.1769



0.1711



01670



01637



01612



01596



01581



01561



0 1547



0 1534



0 1524



01515



3.50



0.2554



0.2444



0.2346



0.2267



0.2202



0.2156



02117



02087



02067



02049



02024



02007



0 1991



0 1978



01967



4.00



0.3025



02930



0 2840



0.2766



02702



02654



02613



02579



02557



02537



02508



02488



02470



02455



02442



4.50



0 3492



0.3408



0 3325



0.3260



03200



03154



03112



03078



03055



03034



03004



02982



0 2962



0 2946



0 2932



5.00



0 3957



0.3879



0 3803



0.3745



0.3693



0.3650



03610



03578



03555



03534



03503



03481



03461



0 3444



0 3429



550



04418



0.4345



04274



04223



0.4178



04139



0 4103



04073



04052



04031



04002



03980



03961



0 3963



0 3929



6.00



0.4878



0.4806



04739



04694



0.4656



0.4622



0 4589



04563



04543



04525



0 4498



04477



04458



04441



0 4428



6.50



0.5335



0.5263



05198



05158



0.5126



0.5097



0 5068



0 5045



0 5028



05012



0 4988



04969



04951



0 4935



0 4922



7.00



0.5790



0 5718



05653



05616



0 5589



0.5564



0 5539



0.5520



0 5506



0 5492



05471



05454



05437



05422



0 5409



750



0.6243



06169



0 6104



0 6069



06045



0.6024



0 6003



05987



05975



0 5964



0 5946



05932



05917



0 5902



0 5890 0 6362



800



0.6694



06618



06550



06516



0 6495



0.6477



0 6459



06447



06437



06428



0.6415



06401



06388



0 6374



850



07143



07063



0 6993



0 6960



0 6940



0 6924



0 6908



0 6899



06892



06884



06874



0 6862



0 6850



06837



900



0.7591



0 7506



0.7433



'0 7399



0 7380



0 7365



0 7351



0 7344



0 7338



0 7333



0 7325



07315



07304



0 7292



0 7282



9.50



08036



07946



0.7870



07834



07814



07800



07788



07783



07778



07774



07769



07760



07750



07739



07730



0 6826



10.00



0.8480



08384



0.8303



0.8266



08245



08231



08219



08215



08212



08208



08205



08183



08189



08178



08169



10.50



0.8922



08820



08735



0.8695



08671



08657



08645



08641



08639



08636



08635



08628



08619



08609



08600



11.00



09362



09254



09163



09120



09094



09078



09056



09063



09061



09058



09058



09052



09043



09033



09024



11.50



0.9801



09686



0.9590



0.9542



0.9514



09496



09483



09479



09477



09475



09475



09468



09459



09449



09440 09850



12.00



1.0239



1.0117



10014



0.9961



0.9930



09910



09896



09891



09889



09886



09885



09879



09869



09859



12 50



10676



1.0545



10437



10378



10343



1.0321



10304



10298



10295



10292



10290



1 0283



10273



10262



10753



1300



1 1111



1.0973



10857



10792



10753



1.0729



10709



10701



10698



10693



10689



10681



10670



10659



10650



1350



11547



1.1398



11276



11204



11161



11134



11111



11101



11095



11089



11083



11073



11062



11050



11040



1400



11979



1.1823



11693



11614



11566



11535



11509



11496



11489



11481



11472



11459



11447



11435



11425



1450



12412



1.2246



12109



12021



11968



11934



11904



11889



11879



1 1868



1 1855



1 1840



1 1827



1 1815



1 1804



1500



1.2844



1.2668



12523



12427



12368



12331



12296



12278



12265



12252



12234



12217



12202



12189



12177



15 50



13275



13089



12936



12830



12766



12725



12685



12663



12647



12631



12608



12588



12572



12558



12546



16.00



1.3705



13509



1.3347



13232



13161



2 3116



13071



13046



13026



13007



12978



12954



12937



12922



12909



16.50



14135



13928



1.3757



13632



13555



13505



13455



13426



13402



13379



13343



13316



13298



13291



13268



1700



1.4564



14346



1.4166



14031



13947



13892



13836



13803



13775



13748



13705



13674



13653



13637



13623



17.50



1.4992



14763



1.4574



14428



14336



14278



14215



14178



14145



14114



14062



14028



14005



13987



13973



18.00



1.5420



15180



14981



1.4823



14724



14661



14591



14550



14512



14476



14417



14377



14353



14334



14318



18.50



1.5847



15595



15387



1.5217



15111



15042



14965



14920



14876



14835



14767



14723



14697



14677



14660



19.00



1.6274



1 6010



15792



1.5610



15496



15422



15338



15287



15238



15192



15114



15065



15036



15015



14998



19.50



1.6700



1.6424



16196



1.6002



15879



15800



15708



15653



15597



15546



15458



15404



15373



15351



15332



20.00



17126



1.6837



16599



16392



16261



1.6176



16076



16016



15954



1 5897



15799



1 5739



1 5706



15692



1 5663



20.50



1.7551



1.7250



17001



16781



16641



1.6551



16443



16377



16308



16246



16137



16071



16035



16011



15990



21.00



17975



1.7662



17403



17169



17020



1.6924



16808



1.6736



16660



16592



16472



16400



15362



16336



16314



2150



18400



1.8073



1.7803



17556



17398



17296



17171



1.7094



17011



1.6936



1 6804



16726



15685



16658



16635



22.00



1.8824



1.8484



1.8203



17942



17775



17667



17532



1.7450



1.7359



1.7278



17134



17049



1 7005



16977



16953



2250



19247



1.8895



1.8603



18327



18150



1.8036



17892



1 7804



17705



17617



1 7460



17370



17322



17293



1 7267



23.00



1 9670



1.9304



1.9001



18711



18524



18404



18251



18156



18049



1.7955



i 7785



17687



17637



17606



1 7579



23.50



20093



1.9714



1.9399



19094



18898



18771



18608



1.8507



18392



18290



18107



18002



17949



17916



17889



24.00



20516



20122



1.9797



19477



19270



19136



18964



18856



18733



18623



18427



18315



1 8258



18224



18195



24.50



20938



20531



20193



19858



19641



19501



19318



19204



19072



18955



16744



1.8625



18565



18530



18499



25.00



2.1360



2 0938



2.0590



2.0239



2.0011



1.9864



19671



19550



19409



19285



19060



18933



1.8870



18833



18801



25.50



21761



21346



2.0985



2.0618



2.0380



2.0226



2.0023



19895



19745



19613



19373



19238



19172



19133



19100



26.00



22202



21753



21380



2.0998



2.0749



20588



20373



20239



2.0079



19939



19684



19542



19472



19431



19397



26.50



22623



2.2159



21775



21376



2.1116



2.0948



2.0723



20581



20412



20264



19994



19843



19769



19728



19692



27.00



2.3044



2.2566



22169



21754



2.1483



2.1307



21071



2.0923



2.0744



20587



20301



20142



20065



20022



19984



27.50



2.3464



22971



2..2562



2.2131



2.1848



2.1666



2.1418



2.1263



2.1074



2.0909



2.0607



2 0440



2 0359



2 0314



2 0275



28.00



2.3885



23377



2.2955



2.2507



2.2213



2.2024



2.1764



2.1601



21403



2.1229



2.0911



2.0735



20650



20603



20563



28.50



2.4305



2.3782



2.3348



2.2883



2.2578



2.2380



2.2110



2.1939



2.1730



2.1548



2.1213



2 1028



2 0940



2 0891



2 0849



29.00



2.4724



2.4186



2.3740



2.3258



2.2941



2.2736



2.2454



22276



2.2056



2.1865



2.1513



21320



21228



21178



21134



29.50



2.5144



2.4591



24132



2.3632



23304



23091



22797



22611



22381



2.2181



2.1812



2.1610



21514



21462



21417



30.00



2.5563



2.4995



2.4523



2.4006



2.3666



2.3446



2.3139



22946



22705



22496



2.2110



2.1898



21798



21744



2 1698



PETROLEUM



34-16



TABLE



34.2-EXTENDED



SUKKAR-CORNELL



INTEGRAL



'Pm I



;1 2 Pseudoreduced



pP,-



1.1



temperature 1.2



for 8=30



1.3



1.4



FOR



BHP



ENGINEERING



CALCULATION



HANDBOOK



(continued)



(z~p,rWp,r



1 + WP,J~



0 1.5



1.6



020



0.0000



0.00000.0000



0.00000.0000



0 0000



0.50



0.0039



0.0038



0.0037



0.0037



0.0036



00036



100



0.0199



0.0184



0.0176



0.0172



00168



00166



1.50



0.0521



0.0447



0.0418



0.0401



0.0390



00382



2.00 250



0.0967 0.1422



0.0823



0.0755



0.0718



0.0692



00676



01264



01164



0 1103



01060



3.00



0.1670



0.1719



0.1608



0.1531



3.50



0.2314



0.2174



0.2063



4.00



0.2756



0.2625



4.50



0.3195



0.3071



17 0 0000



1.6



1.9



2.0



2.2 ~__



28



30



0.0000



0 0000



0.0035



0.0035



0.0035



0.0158



0.0157



0.0157



0.0356



0.0355



0.0353



0.0626



0.0621



0.0618



0.0615



0.0960



0.0951



0.0943



0 0937



0.0931



0.1353



0.1334



0.1321



0.1309



0.1300



0.1292



01782



0.1765



0.1741



0.1725



0.1710



0.1697



0.1687



02242



02219



0.2199



02172



0.2152



0.2135



0.2120



0.2108



02693



02669



0.2647



02617



0.2594



0.2575



0.2559



0.2545



0.0000



2.4



0 0000



0 0000



0.0000



0.0000



00036



00035



0.0035



0.0035



0.0035



0.0035



0.0164



00162



00162



0.0161



0.0159



0.0158



00375



00371



00368



0.0365



0.0361



0.0358



00652 0 0993



00646 0 0963



0.0640 0.0974



0.0632



01033



00662 01010



0.1474



01436



01404



01381



01366



0.1980



0.1914



01869



0 1831



0 1601



0.2519



0.2436



0.2367



02318



0.2275



0.2970



0.2891



0.2823



02778



02729



2.6 0.0000



500



0.3632



0.3513



0.3416



0.3343



0.3278



03229



03186



03149



03124



03101



0.3069



0.3046



0.3025



0.3008



0.2993



550



0.4067



0.3951



0.3858



0.3789



0.3729



03683



03641



03605



03580



03558



03525



0.3501



0.3480



0.3462



03448



6.00



0.4500



0.4386



0.4295



0.4230



0.4175



04132



04092



04059



04035



04013



03981



03957



0.3937



0.3919



0.3904



6.50



0.4931



04817



0.4728



0.4667



0.4616



0.4576



04539



04508



0.4486



04465



04435



04412



0.4392



0.4374



0.4359



7.00



0.5361



0.5247



0.5158



0.5099



0.5052



0.5015



0 4981



0 4952



0 4932



04913



0.4884



0.4863



0.4843



0.4826



0.4812



7.50



0.5789



0.5674



0.5584



0.5527



0.5483



05449



05417



05391



0.5372



05355



05329



05309



0.5291



0.5274



0.5260



8.00



0.6216



0.6098



0.6007



0.5951



0.5909



0.5877



05848



05824



0.5808



05792



05767



05749



05732



0.5716



0.5703



8 50



0.6642



0.6521



0.6428



0.6372



0.6331



0.6301



0.6273



0 6252



06237



0.6223



0 6200



0.6184



0.6168



0.6152



0.6139



9.00



0.7066



0.6941



0.6846



0.6789



0.6749



0.6719



0.6693



0 6674



0.6660



0.6647



0 6627



0 6612



0.6597



0.6582



0.6570



9.50



0.7488



0.7360



0.7261



0.7204



07163



07134



07109



0.7091



0.7078



0.7066



07048



07034



07020



07006



0.6994



10.00



0.7909



0.7776



0.7674



07615



0.7573



07544



0.7520



07503



0.7491



07480



07463



07451



07436



07423



0.7411



10.50



0.6329



0.8191



0.8085



0.8024



07980



07951



07926



07910



0.7899



07888



0.7873



07861



07847



07833



07822



11.00



0.8747



08604



0.8494



08430



08384



08354



08329



08313



0.8302



06292



0.8277



08265



0.8251



06238



08227



11.50



0.9165



0.9016



0.8901



08833



06785



08754



08728



08711



0.8700



06690



08676



08664



08650



08637



06626



12.00



0.9581



0.9426



0.9306



0.9234



09183



09150



09123



09106



0.9095



09084



09070



09057



09043



09030



09019



12.50



0.9996



0.9835



0.9710



0.9633



0.9579



09544



09515



09497



0.9485



0.9474



09459



09446



0.9431



09417



09406 09787



13.00



1.0411



1.0242



10112



1.0030



0.9973



09936



0.9904



09884



0.9872



0.9860



09842



09828



09813



09799



1350



1.0824



10649



10513



10425



10364



10324



10290



10268



10254



10241



10222



10206



10191



10176



10164



14.00



1.1237



1.1054



1.0912



10318



1.0753



1.0710



10673



10649



10634



10618



10596



10579



10563



10547



10535



14.50



1 1649



11459



11310



1.1209



1.1139



1.1094



1.1054



1 1027



1 1009



10992



10966



10947



10930



10914



10901



1500



1.2060



1 1862



1.1707



1.1598



1 1524



1 1475



1.1431



1 1402



1.1382



1.1362



1 1332



11311



11293



1 1276



1 1263



15.50



1.2471



12264



1.2102



11986



1 1907



1.1855



1 1806



11774



1.1751



1.1729



1.1694



1 1670



1 1651



1 1633



1.1620



16.00



1.2681



1.2666



1.2497



1.2372



12287



12232



1.2179



1.2144



1.2117



12092



1.2052



12026



12005



11987



1 1972



16.50



13291'



13067



1.2890



12757



12666



1.2607



1.2549



1.2511



1.2481



1.2453



12407



12377



12354



12335



1.2320



17.00



1.3700



13467



1.3282



13140



13044



12981



1.2917



I.2876



1.2842



1.2610



1.2757



1.2724



12700



12680



1 2665



17.50



1.4109



1.3866



13674



1.3522



13419



13352



13283



13238



1.3200



1.3164



1.3105



1.3067



13042



13021



13005



16.00



1.4517



1.4264



1.4064



1.3903



1.3794



1.3722



1.3647



1.3596



1.3555



1.3515



1.3449



1.3407



13380



13358



1.3341



18.50



1.4924



1.4662



1.4454



1.4282



1.4167



1.4091



14009



1.3956



1.3908



1.3864



1.3789



1.3744



13714



13692



1.3674



19.00



1.5332



1.5059



1.4843



1.4661



14538



14457



1.4370



1.4312



1.4529



1.4211



1.4127



1.4077



14045



14022



1.4003



19.50



1.5738



15456



15231



1.5038



1.4908



1.4823



1.4728



1.4666



1.4608



1.4554



1.4462



1.4407



14373



14349



1.4329



20.00



1.6145



1.5852



15618



1.5414



1.5277



1.5187



1.5085



15019



1.4954



1.4896



1.4794



1.4734



1.4696



14672



1.4652



20.50



1.6551



1.6247



1.6005



1.5789



1.5644



15549



1.5440



15369



15296



1.5235



1.5123



1.5058



1.5019



1.4993



1.4971



21.00



1.6956



1.6642



1.6391



1.6163



1.6011



15910



1.5794



15718



1.5641



1.5572



1.5449



1.5379



1.5338



1.5310



1.5288



21.50



1.7361



1.7037



1.6776



1.6537



16376



16270



1.6146



16065



1.5981



1.5906



1.5773



1.5697



1.5654



1.5625



1.5601



22.00



1.7766



17431



17160



1.6909



1.6740



16629



1.6497



16410



1.6320



1.6239



1.6095



1.6013



15967



1.5937



1.5912



22.50



1.8171



1.7824



1.7544



1.7281



1.7103



16967



1 6846



16754



1.6657



1.6570



1.6414



1.6326



1.6277



1.6246



1.6220



23.00



1.8575



1.8217



1.7928



1.7651



1.7465



1.7343



1.7194



1.7096



1.6992



1.6899



1.6731



1.6636



1.6565



1.6552



1.6525



23.50



1.8979



18610



1.8311



1.8021



1.7826



17698



17541



17437



17325



1.7226



1.7046



1.6945



1.6890



1.6856



1.6828



24.00



1.9383



1.9002



1.8693



18390



1.6186



1.8053



1.7806



17777



1.7657



1.7551



1.7358



1.7250



1.7193



1.7158



I.7128



24.50



1.9786



1.9393



1.9075



1.8759



1.9546



18406



18230



18115



1.7987



1.7874



1.7669



1.7554



17494



1 7457



17426



25.00



2.0189



1.9785



1.9456



19127



1.6904



1.8756



1.8573



1.8452



1.8316



1.8196



1.7977



1.7855



1.7792



1.7754



1.7722



25.50



2.0592



2.0176



1.9637



1.9493



1.9262



1.9110



1.8915



18788



1.8644



1.8516



1.8284



1.8155



1.8088



1.8048



1.8015



26.00



2.0995



2.0566



2.0217



1.9860



1.9618



1.9460



1.9256



1.9123



1.8970



1.8835



1.8589



1.8452



1.8382



1.8341



1.8306



26.50



2.1397



2.0957



2.0597



2.0226



1.9974



1.9610



19596



1.9456



1.9294



1.9152



1.8891



1.8747



1.8674



1.8631



1.8595



27.00



2.1799



2.1346



2.0976



2.0591



2.0330



2.0159



1.9934



1.9788



1.9618



1.9468



1.9192



19040



1.8964



1.8920



1.8882



27.50



2.2201



2.1736



2.1355



2.0955



2.0684



2.0507



2.0272



2.0119



1.9940



1.9782



1.9492



1.9332



1.9252



1.9206



1.9167



26.00



2.2603



2.2125



2.1734



2.1319



2.1038



2.0854



2.0609



2.0449



2.0261



2.0095



1.9790



1.9622



1.9538



1.9491



1.9451



28.50



2.3005



2.2514



2.2112



2.1682



2.1391



2.1200



2.0945



2.0779



2.0580



2.0407



2.0086



1.9910



1.9823



1.9774



1.9732



29.00



2.3406



2.2903



2.2490



2.2045



2.1743



2.1546



2.1280



2.1107



2.0899



2.0717



2.0380



2.0196



2.0105



2.0055



2.0012



29.50



2.3807



2.3291



2.2868



2.2407



2.2095



2.1891



2.1614



2.1434



2.1216



2.1026



2.0673



2 0481



2.0386



2.0334



2.0289



30.00



2.4208



2.3679



2.3245



2.2769



2.2446



2.2235



2.1947



2.1760



2.1533



2.1334



2.0965



2.0764



2.0666



2 0612



2.0566



WELLBORE



34-17



HYDRAULICS



TABLE34.2-EXTENDEDSUKKAR-CORNELL INTEGRAL



Pseudoreduced



Pp, 020



1.1 00000



temperature 12 0.0000



for 13



00000



B=35 14 00000



FORBHPCALCULATlON(continued)



0 15 0.0000



16 00000



17 00000



18 00000



19 00000



2.0 00000



22 0.0000



24 00000



26 00000



28 00000



30 00000



0.50



0.0033



0.0032



00032



00031



00031



00031



00031



00030



00030



00030



0.0030



00030



00030



00030



000~



1.00



0.0171



0.0158



0.0152



0.0148



00145



00143



00141



0.0139



00139



00138



0.0137



00136



00136



00135



00135



150



0.0454



0.0387



00361



0.0346



00336



00329



00323



00320



00317



00315



0.0311



00309



00307



00305



00304



2.00



0.0861



0.0720



00657



0.0623



00601



00585



00573



00564



00559



00554



0.0546



00542



00537



00534



0.0531



2.50



0.1283



0.1119



0.1022



00965



00925



00900



00879



00864



00855



00847



00834



00826



00819



00813



C08tlR



3.00



0.1703



0.1538



01425



0.1350



01295



0 1259



01230



01208



01194



01182



01165



01153



01142



01134



01127



3.50



0.2120



0.1960



01644



01759



01694



01650



01613



01585



0.1567



01550



01526



0 1513



0 1499



0 1487



0 1478



4.00



02536



0.2382



02266



0.2179



02108



02059



02017



01984



01962



01942



01916



01897



01860



01866



01855



4.50



02950



0.2800



0.2688



0.2601



02529



02477



02433



02396



02372



02350



02320



02296



02279



02263



02250



5.00



0.3362



0.3216



03106



0.3023



0.2951



02899



02854



02816



02790



02766



02734



02710



02690



02672



C2658



550



0.3773



0.3630



03522



0.3442



0.3373



03321



03276



03238



03211



03187



03153



03126



03107



03089



03074



600



04183



0.4040



03934



03857



0.3791



03742



03698



03660



03634



03610



03576



03550



03529



03510



03495



6.50



04591



0.4449



04344



04270



04207



04159



04117



04080



04055



04032



03996



03972



0 3951



03932



cl3918



7.00



0.4999



04656



04752



04679



04616



04573



04532



0 4498



04473



04451



04416



04394



04373



04354



0 4339



7.50



0.5405



0 5261



0.5156



05085



0.5026



0.4983



0 4944



04912



0 4889



04867



04836



04812



04792



04774



04759



6.00



0.5810



05665



05558



05487



05431



0 5390



05352



05322



05300



0 5280



05247



05227



05206



05190



05175



8.50



0.6214



0.6066



0 5959



05686



05832



0 5792



05756



05727



05707



0 5688



D5657



05638



05619



05602



0 5588



9.00



0.6617



06466



06357



06285



06230



06191



06156



0 6129



06109



06091



0 6062



06044



06026



06009



0 5996



9.50



0.7018



06865



06753



06681



06625



06566



06552



06526



06507



0 6490



06462



06445



0 6428



06412



0 6398



10.00



0.7419



07262



0 7147



07073



0.7017



0 6978



0 6945



06919



06901



06885



06856



06842



06825



06809



0 6796



10.50



0.7818



07657



0.7539



07464



0 7406



07367



0 7334



0 7306



0 7291



0 7275



0 7250



07234



07217



07201



0 7189



11.00



0.8217



08051



0 7930



07852



07793



07753



07719



0 7694



0 7677



0 7661



07637



07621



07604



07589



0 7576



11 50



0.8614



0.8444



0.8319



08239



08177



08136



08102



08076



0 8059



08043



08019



08004



07987



07971



0 7958



1200



09011



0.6636



08707



06623



06559



06517



08461



08455



06436



06422



06396



06381



08364



06349



0 6336



1250



09407



09227



09094



09006



08939



0 8895



0 8858



08831



08813



0 8797



08771



08755



08737



08721



08708



13.00



09803



09617



09479



09386



0.9317



09271



0 9232



0 9204



09165



09168



09141



09124



09106



09069



0 9076



13.50



10197



1.0006



09863



0.9765



0 9693



0 9645



0 9604



0 9574



0 9554



0 9535



09507



09483



09470



09453



0 9439



1400



10591



10394



10246



1.0143



10067



10017



0 9973



09941



0 9920



0 9900



09869



09848



09829



09812



0 9798



14.50



10985



1.0781



10627



1.0519



1.0439



10386



10340



10305



10282



10261



10226



10205



10164



10167



10153



15.00



1 1377



1.1167



11008



10893



1.0609



10754



10704



10667



10642



10618



10580



10557



10536



10517



10503



1550



11770



1.1552



i 1388



1 1266



i 1178



1 1120



1 1066



1 1027



10999



10973



10931



1 0905



10663



10664



1 0849



16.00



1.2162



1.1937



1 1767



1.1638



1.1545



1.1484



1 1426



i 1384



1 1354



1 1325



1 1278



11249



1 1226



1 1206



1 1191



16.50



1.2553



1.2321



12144



I 2008



1 1911



1.1846



1 1784



1 1739



1 1705



1 1674



1 1622



1 1590



1 1566



1 1545



1 1529



17.00



1.2944



1.2705



12521



1.2378



1.2275



12207



12140



1 2092



12055



12020



1 1962



1 1928



1 1901



1 1860



1 1864



1750



13334



1.3087



12898



12746



12638



12566



12494



12443



12402



1 2364



12300



12262



12234



12212



12195



18.00



1.3725



1.3470



13273



13113



1.2999



1.2923



12646



1.2792



12747



12705



1 2634



12592



12563



12540



12522



1850



14114



1.3851



13648



13479



13359



13280



13197



1.3139



i 3089



1 3044



12966



12920



12889



12865



12847



19.00



1.4504



1.4232



14022



1.3844



i 3718



13634



13546



1.3484



13430



t 3380



1 3294



13245



13212



13187



13168 13485



19.50



14893



1.4613



14395



14206



14075



13968



i 3893



13826



13769



13714



1 3620



1 3566



13531



13506



20.00



1.5281



1.4993



14768



14571



14432



14340



14239



14170



1.4!05



14046



13944



13885



1 3848



13822



13800



2050



15670



15373



1.5140



14933



14766



14691



14564



14510



14440



14376



14265



14201



14162



14135



14112



21.00



16058



1.5752



1.5511



1.5294



15142



15041



14927



14849



14773



14704



14583



14515



14473



14445



14422



21.50



16446



16130



1.5862



15655



15495



15390



15269



1.5186



15104



15030



14900



14826



14782



14752



14728



22.00



1.6833



1.6509



16252



1.6014



15848



15738



15609



15522



15434



15355



15214



15134



15088



15057



15032



22.50



1.7220



1.6887



16622



1.6373



1.6199



16084



15948



15856



15762



15677



15525



15440



15391



15360



15333



23.00



1.7607



1.7264



16991



1.6732



16550



16430



16286



16189



16066



15996



15635



15744



15693



15660



15632



23.50



17994



17641



17360



17069



16900



16755



16623



16521



16413



16317



16143



16046



15992



15957



15929 16223



24.00



1.8381



1.8018



1.7729



1.7446



1.7249



17118



16959



16851



16736



16634



16448



16345



16288



16253



24.50



1.8767



18394



1.8097



1.7802



17597



17461



17294



17180



17058



16950



16752



16642



16583



16546



16515



25.00



1.9153



1.8771



18464



18158



1.7944



17803



17627



17508



17379



17264



17054



16937



16875



16837



16805



25.50



1.9539



1.9146



18831



18513



1.6291



18144



17960



17835



17696



17577



17354



17231



17165



17126



17093



26.00



1.9924



1.9522



19198



1.8867



1.8637



1.8484



18291



18161



18016



17888



17652



17522



17454



17413



17378



26.50



2.0310



1.9897



1.9564



1.9221



1.6962



1.6624



18622



16486



18333



16198



17949



17612



17740



17696



17662



27.00



2.0695



2.0272



1.9930



1.9574



1.9326



1.9163



18951



1.8810



18649



18506



18244



18100



18025



17981



17944



27.50



2.1080



2.0647



2.0295



1.9927



1.9670



1.9501



1.9280



1.9133



1.8963



1.8814



18537



18386



18308



18262



18224



28.00



2.1465



2.1021



2.0661



2.0279



2.0014



1.9838



1.9606



19454



1.9277



1.9119



16629



16670



16569



16542



16502 18779



28.50



2.1850



2.1395



2.1025



2.0631



20356



20175



1.9935



1.9775



1.9589



1.9424



19119



18953



18868



18820



29.00



2.2234



2.1769



2.1390



20963



20698



20511



2.0261



2.0094



1.9900



1.9726



19408



19234



19146



19096



19053



29.50



2.2619



2.2142



21754



2.1333



21040



2.0846



20587



20414



2.0210



2.0030



1.9696



19513



19422



19370



19327



30.00



2.3003



2.2516



2.2118



2.1684



21381



21180



20912



20732



2.0519



20331



1.9962



19791



1.9696



1.9643



19598



34-18



PETROLEUM



ENGINEERING



HANDBOOK



TABLE34.2-EXTENOEDSUKKAR-CORNELLlNTEGRALFORBHPCALCULATlON(continued)



Pseudoreduced L



1.1 0.20



0.0000



temperature 1.2 0.0000



for 8=40.0 13



0.0000



-



1.4



1.5



0.0000



0.0000



1.6 0.0000



1.7 0.0000



1.8 0.0000



1.9 0.0000



2.0 0.0000



2.2 0.0000



24



26



28



30



0.0000



0.0000



0.0000



0.0000



0.50



0.0029



0.0028



0 0026



0.0027



0.0027



0.0027



0.0027



0.0027



0.0027



0.0026



0.0026



0.0026



0.0026



0.0026



0.0026



1.00



0.0150



0.0139



00133



0.0129



00127



0.0125



0.0123



0.0122



0.0122



0.0121



0.0120



0.0119



0.0119



0.0118



0.0118



1.50



0.0403



0.0341



0.0318



0.0305



0.0296



0.0290



0.0284



0.0281



0.0279



0.0276



0.0273



0.0271



0.0270



0.0268



0.0267



2.00



0.0776



0.0640



0.0582



0.0551



0 0530



0.0517



0.0505



0.0497



0.0493



0.0488



0.0482



0.0477



0.0473



0.0471



0.0468



2.50



0.1170



0.1005



0.0912



0.0858



0.0821



0.0798



0.0779



0.0765



0.0756



0.0749



0.0738



0.0730



0.0724



0.0718



0.0714



300



0.1565



01393



01281



01208



01156



0.1122



0.1095



0.1074



0.1061



0.1050



0.1034



01023



0.1013



0.1005



00999



350



01958



0.1787



0.1666



0.1584



01520



0.1477



0.1442



0.1416



0.1398



01383



0.1362



0.1346



01335



0.1324



01315



4.00



0.2351



0.2182



02062



01973



0.1901



0.1853



0.1812



0.1780



0.1758



0.1740



0.1714



0.1696



0.1681



0.1667



0.1656



450



0.2743



0.2576



0.2457



02367



0.2292



0.2240



0.2195



0.2159



0.2135



0.2113



0.2084



0.2063



02045



0.2029



02017



5 00



0.3133



0.2969



0.2851



0.2762



0.2686



0.2633



0.2586



0.2548



0.2521



0.2498



0.2465



0.2442



0.2422



0.2405



0 2391



550



0.3523



0.3360



03244



03156



03081



03028



02980



02941



0.2913



0.2889



0.2854



0 2829



0.2808



0 2790



0 2775



6.00



03912



0.3750



0.3634



0 3549



0.3476



0.3423



0.3376



0.3336



0.3308



0.3283



0 3247



0 3221



0.3199



0.3181



0 3166



6.50



0 4300



0.4138



0.4032



0 3939



03866



03816



0.3770



03731



03703



0.3678



0.3642



0.3616



03594



0.3575



0.3560



700



0.4687



0.4525



04410



04328



04258



0.4208



0.4163



0.4124



04097



0.4073



0.4037



04011



0 3989



0 3970



0.3955



750



05073



0.4910



0.4795



04714



04646



04597



04553



04516



0.4490



0.4466



0.4431



0.4405



0 4383



0 4365



0.4350



800



0.5458



0.5294



0.5179



0.5097



0.5031



04983



04941



04905



04879



04856



0.4819



0 4797



04776



04758



0.4743



8.50



0.5843



0.5677



05560



0 5479



0.5413



05367



05325



05290



0 5266



0.5244



0.5208



0.5187



0.5166



0.5148



0.5133



9.00



0.6227



0.6059



0 5940



0.5859



0.5793



05747



05707



05673



0.5650



0.5628



0.5593



0 5573



0.5553



0.5535



0.5521



9 50



06609



0.6439



06319



06237



06171



06125



06085



06052



0.6030



06009



0.5975



0.5955



0 5936



0.5918



0.5904



0.6991



0.6818



0.6696



06612



06546



06500



0.6461



0.6429



06407



0.6386



0.6353



0.6334



0.6315



0.6298



06264



10.00 1050



07372



0.7196



0.7071



0 6987



06919



06873



0 6833



0.6802



0.6780



06760



0.6728



0.6710



0.6690



0.6673



06660



11 00



07753



0.7573



07446



07359



07290



07243



07203



07172



0.7150



0.7130



0.7099



0.7081



0.7062



0.7045



07031



11 50



08132



07949



0.7819



07729



07659



07611



0.7571



07539



07517



07496



07466



0.7448



0.7429



0.7412



07398



12.00



0.8511



0.8324



0.8190



08098



08026



07977



0.7936



07903



0.7822



07862



07830



0.7812



0.7792



0.7775



07762



0.8152 0.8507



0.8134 0.8490



08121 08476



12.50



0.8890



0.8696



08561



0.8466



08391



08341



08299



08265



0.8243



08223



08190



0.8171



13.00



0.9268



0.8931



0.8832



08755



08703



06659



0.8624



0.8602



0.8580



0.8547



0.8527



13.50



0.9645



0 9072 0.9445



0 9229



0.9196



09117



09063



09017



0.8981



0.8957



0.8935



0 8900



0.8879



0.8859



08841



08827



1400



10022



09816



0 9667



0.9559



0 9477



09421



09373



0.9335



0.9310



0.9287



0.9250



0.9228



09207



0.9188



09174



14.50



1.0396



10188



10034



0.9921



0 9835



0 9778



09727



0 9588



0.9661



0.9636



0.9596



0.9572



0.9551



0.9532



09517



15.00



1.0774



1.0558



1.0400



10282



1.0193



10133



10079



10037



1.0009



0.9982



0.9939



09914



0.9891



0.9872



0.9856



15.50



11149



1.0928



1.0765



1.0641



1 0548



10486



10429



1.0385



1.0355



1.0326



1.0279



1.0251



10228



10208



1.0192



16.00



1.1525



1.1297



1 1129



1 1000



10903



10837



10777



10731



1.0698



1.0667



1.0616



1.0586



10561



10541



10525



16.50



1 1899



1.1666



1 1492



1.1357



11255



1 1187



1 1123



11075



1 1039



1 1005



10949



10917



10891



10870



10653



17.00



1.2274



1.2034



1 1855



1.1713



1 1607



1 1536



1 1468



1.1417



1.1378



1 1341



1.1260



1.1245



1.1218



11196



1 1179



1 1958



1 1684



11811



11757



1.1714



1.1675



1.1608



1.1570



1.1541



1 1519



1.1501



1 2307



12230



12152



12095



1.2049



1.2006



1.1934



1.1892



1 1662



1 1839



1.1820



17.50



1.2648



1.2402



12217



18.00



1.3021



1.2769



12579



1.2068 1.2422



18.50



1.3395



1.3136



12940



1.2776



12655



12574



12492



12432



1.2382



1.2336



1.2256



1.2211



1.2180



12155



1.2136



19.00



1.3768



1.3502



1.3300



1.3128



13002



12918



12831



12767



1.2713



1.2663



1.2577



1.2526



1.2494



12469



12450



19.50



1.4140



1.3868



1.3659



1.3480



1.3349



13261



13168



13101



1.3042



1.2988



1.2894



1.2842



12806



1.2780



1.2760



2000



1.4513



1.4233



1.4019



1.3831



1.3694



13602



13504



13433



1.3369



1.3311



1.3210



1.3153



1.3116



1.3089



1.3068



20.50



1.4685



1.4598



14377



1.4181



1.4038



1.3942



13838



13763



1.3695



1.3633



1.3523



1.3462



1.3422



1.3395



1.3373



21.00



1.5257



1.4963



1.4735



1.4530



1.4381



14281



14171



14093



1.4019



1.3952



1.3834



1.3768



1.3727



13698



1.3675 1.3975



21.50



1.5629



1.5327



1.5093



1.4879



1.4723



1 4620



1 4503



14421



1.4341



1.4270



1.4143



1.4072



1.4028



1.3999



22.00



16001



15691



15450



1.5227



1.5065



1.4957



14834



14747



1.4662



1.4586



1.4449



1.4373



1.4328



1.4297



1.4272



22.50



1.6372



1.6054



1.5807



15574



15406



15293



15164



15072



14982



1.4900



1.4754



1.4673



14625



1.4593



1.4567 1.4860



23.00



1.6743



1.6417



1.6163



1.5920



1.5746



1.5629



15492



1.5396



1.5300



1.5213



1.5057



1.4970



14920



14887



23.50



1.7114



1.6780



1.6519



1.6266



1.6085



1.5963



1.5820



15719



1.5617



1.5525



1.5358



1.5265



1.5213



1.5178



1.5151



24.00



1.7485



1.7143



1.6874



1.6612



1.6423



1.6297



16146



1.6041



1.5932



1.5834



1.5657



1.5559



1.5503



1.5468



1.5439



24.50



1.7855



1.7505



1.7229



1.6947



1.6761



1.6630



16472



16362



1.6246



1.6143



1.5954



1.5850



1.5792



15755



1.5725



25.00



1.8226



1.7867



1.7584



17301



1.7098



1.6962



1.6797



16682



16559



16450



1.6249



1.6139



1.6078



16041



1.6010



25.50



1.8596



1.8229



1.7938



1.7645



1.7434



1.7293



1.7120



1.7000



1.6871



16755



1.6543



1.6427



1.6363



1.6324



1.6292 1.6572



26.00



1.8966



1.8591



1.8292



1.7988



1.7770



1.7624



1.7443



1.7318



1.7181



1.7059



1.6836



1.6713



1.6646



1.6606



26.50



1.9336



1.8952



1.8645



1.8331



1.8105



1.7954



1.7765



1.7634



17491



17362



1.7126



1.6997



1.6927



1.6886



1.6851



27.00



1.9705



1.9313



1.8999



1.8673



1.8439



1.8283



1.8086



1.7950



1.7799



1.7664



1.7415



1.7279



1.7207



1.7164



1.7128



27.50



2.0075



1.9674



1.9352



1.9015



1.8773



1.8612



1.8406



1.8265



1.8106



1.7965



1.7703



1.7560



1.7484



1.7440



1.7403



28.00



2.0444



2.0034



1.9704



1.9356



1.9107



1.8940



1.8726



1.8579



1.8412



1.8264



1.7989



1.7839



1.7760



1.7715



1.7676



28.50



2.0813



2.0394



2.0057



1.9697



1.9439



1.9267



1.9044



1.8692



1.8717



1.8562



1.8274



1.8116



1.8035



1.7988



1.7948



29.00



2.1182



2.0755



2.0409



2.0038



1.9771



1.9594



1.9362



1.9204



0.9021



1.8859



1.8557



1.8393



1.8308



1.8259



1.8218



29.50



2.1551



2.1114



2.0761



2.0378



2.0103



1.9920



1.9680



1.9516



1.9325



1.9155



1.8840



1.8667



1.8579



1.8529



1.8487



30.00



2.1920



2.1474



2.1112



2.0717



2.0434



2.0246



1.9996



1.9826



1.9627



1.9460



1.9120



1.8940



1.8849



1.8797



1.8754



WELLBORE



TABLE



Pseudoreduced



P,, 0.20



34-19



HYDRAULICS



1.1 0.0000



34.2-EXTENDED



temperature 12 0.0000



SUKKAR-CORNELL



INTEGRAL



FOR BHP CALCULATION



(continued)



for 8=45.0 1.3



0 0000



1.4 0 0000



1.5 0.0000



1.6 0.0000



17 0.0000



1.8 0.0000



1.9 0.0000



2.0 0



22 0



24 0



26 0



28 ooooo



30 ooooo



050



0.0026



0.0025



0.0025



0.0024



0.0024



0.0024



0.0024



0.0024



0.0024



0.0024



00023



00023



00023



00023



00023



1.00



0.0134



0.0124



0.0119



0.0115



0.0113



0.0111



0.0110



0.0109



0.0108



0.0108



00107



00106



00106



00105



00105



1.50



0.0362



0.0305



0.0284



0.0272



0.0264



0.0258



0.0254



0.0250



0.0248



00247



00244



00242



00240



00239



00238



2.00



00707



0.0576



00522



0.0494



0.0475



0.0462



0.0452



0.0445



0.0440



00436



00430



00426



00423



00420



00418



2.50



01076



00912



00823



00772



00738



00716



00699



00586



00678



00671



00661



00654



00648



00644



00640



3.00



0.1449



0.1273



0.1163



0.1093



0.1043



0.1012



00986



0.0967



0.0955



0.0944



0.0930



00919



00910



00903



00897



3.50



0.1821



0.1643



01523



0.1441



0.1378



01338



01304



0.1279



0.1263



0.1248



0.1229



01215



01203



01193



01185



4.00



0.2193



0.2015



0.1892



0.1803



01732



01685



01645



01614



01594



0.1576



01552



01534



01520



01507



01496



4.50



0.2565



0.2388



02264



0.2172



0.2096



0.2045



0.2001



0.1966



0.1942



0.1921



0.1893



01672



01855



01840



01828



5.00



0.2936



0.2760



0.2637



0.2544



0.2466



0.2412



02366



02327



0.2301



0.2278



0.2246



02223



02204



02187



02174



5.50



0.3306



0.3131



0.3009



0.2917



0.2838



0.2783



02735



02695



02667



0.2643



0.2608



0.2583



02562



02544



02530



6.00



03676



0.3501



0.3380



0.3289



0.3211



0.3156



03107



0.3066



0.3038



03012



0.2976



0.2949



02928



02909



02895



6.50



0.4045



0.3871



0.3750



0.3660



0.3583



0.3528



03480



03439



03410



0.3384



0.3347



0.3319



03297



03278



03264



700



0.4414



04239



04118



0.4029



0.3954



0.3900



03852



03811



03782



03757



0.3719



0.3692



03669



03650



03635



750



0.4782



0.4607



0.4486



0.4397



0.4323



04270



04223



04182



04154



04129



04092



0.4064



04042



04023



04008



8.00



0.5150



0.4973



0.4852



04763



0.4690



0.4638



04592



04552



04525



04500



04459



04436



04414



04395



04380



8 50



0.5517



05339



0 5216



0.5128



0.5055



0.5004



04959



0 4920



0 4893



04869



0.4828



0.4806



04785



04766



04751



9 00



05883



0 5704



0 5580



0 5492



0.5419



0.5368



05323



05286



05259



05235



0.5196



0.5174



05153



05135



05120



9.50 10.00



06248



0.6067



0 5942



0 5853



0.5780



0.5730



05686



0.5649



05623



05599



05561



0.5540



05519



05501



05486



0.6613



0.6430



0.6304



0.6214



0.6140



0.6090



06046



06009



0 5984



05961



0 5923



05903



05882



0 5864



05650



10.50



0.6978



0.6792



06664



06573



0.6498



0.6447



06404



06367



06342



06320



06283



06262



06242



06224



06210



11 00



0.7342



0.7153



0.7023



0 6930



0 6854



0.6803



0.6759



06723



0.6698



06676



06639



06619



06598



06580



06566



11.50



0.7705



07514



0.7381



07286



0 7209



0.7157



07113



0 7076



0.7051



07029



06993



06972



06952



0 6934



0 6920



1200



08068



0 7874



0.7738



0.7641



0 7562



0.7509



07464



0 7427



0 7402



07380



07343



0 7323



0 7302



0 7284



0 7270



12 50



0.8430



0.8233



0.8094



0.7994



0 7914



0.7860



0.7814



0.7776



0.7751



0 7728



07690



0 7670



0 7649



0 7680



0 7616



13.00



0.8792



0.8591



08449



0.8347



08264



08209



08161



08122



0.8097



08073



08035



08013



07992



0 7974



0 7959



13.50



0.9153



0 8949



0 8804



0.8698



0 8613



0.8556



0.8507



0.8467



0.8440



0 8416



08376



08354



08332



08313



08299



14.00



0.9514



0 9306



09157



09048



08961



0.8902



0.8851



0 8809



0.8782



08756



08715



08691



08669



08650



0 8635



14.50



0.9875



09663



09510



09396



09307



09246



0.9193



0.9150



0.9121



09094



09050



09025



09002



08983



0 8968



15.00



1.0235



1.0019



0 9863



0.9744



09652



09589



09533



09489



09458



09429



09382



09356



09332



09312



09297



1550



10595



10374



10214



1.0091



09995



09931



09872



0.9825



0 9793



09762



09712



0 9684



0 9660



0 9639



09623



1600



1.0955



10729



10565



1.0437



10338



10271



10209



10160



10125



10093



10039



1 0009



09984



0 9963



09946



16.50



1.1315



1 1084



10915



1.0782



10679



10609



10544



10494



10456



1.0422



10364



1 0331



1 0305



10283



10266



17.00



1.1674



1 1438



1.1265



1 1126



11019



1.0947



10878



1.0825



10785



10748



10685



1 0650



1 0623



10600



10583



17.50



1.2032



1 1791



11614



1.1469



1.1358



11283



11211



1 1155



11112



1 1072



1 1005



10967



10938



10915



10897



18.00



1.2391



12145



11962



1.1811



11696



11619



11542



1 1484



11437



1 1394



1 1321



1 1281



1 1250



1 1227



1 1208



18.50



1.2749



12497



12310



12153



12033



11953



11872



1 1811



11761



1.1715



1 1636



1 1592



1 1560



1 1536



11517



19.00



1.3107



12850



12658



12494



12370



12286



12200



12136



12082



1.2033



1.1948



1 1901



1 1867



1 1842



1 1823



19.50



1.3465



1.3202



1.3005



1.2834



12705



12618



12528



12460



12403



1.2350



1.2258



12207



12172



12146



12126



20.00



1.3823



1.3554



13351



1.3173



13039



12949



12854



12783



12721



12665



1.2566



12511



12474



12447



12426



20.50



1.4180



13905



13697



1.3512



13373



13279



13179



13105



13038



12978



12871



12812



1 2774



1 2746



12724



21.00



1.4538



1.4256



14043



1.3850



13706



1.3608



13503



13425



13354



13290



1.3175



13112



13071



1 3043



13020



21.50



1.4895



14607



1.4388



1.4187



14038



13937



13825



13744



13668



1.3599



13477



13409



1 3367



1 3337



13314



22.00



1.5251



1.4958



14733



1.4524



14369



1.4264



14147



14062



13981



13908



1.3776



13704



1 3660



1 3629



13605



22.50



1.5608



1 5308



15077



1.4860



1.4699



1.4591



14468



14379



14292



14215



14074



1.3997



1.3951



13919



13894



23.00



1.5965



1.5658



1.5421



1.5196



1.5029



1.4916



14788



14694



14603



14520



1.4371



1.4288



14239



14207



14181



23.50



1.6321



16008



15765



15531



15358



1.5242



15106



15009



1.4912



14824



1.4665



1.4577



1.4526



14493



14466



2400



16677



1.6357



1.6108



1.5866



1.5687



1.5566



15424



15323



15219



15127



14958



1.4865



1.4811



14776



14748



24.50



1.7033



1.6706



16451



16200



1.6015



15890



15741



15635



15526



15428



15249



15150



15094



1 5058



15029



25.00



1.7389



1.7055



16794



16534



1.6342



1.6212



16057



15947



15831



15728



15538



1 5434



1.5375



15338



15308



25.50



1.7745



1.7404



17136



16867



1.6668



16535



16373



16247



16136



16027



15826



15716



15655



15617



15585



26.00



1.8100



1.7752



1.7478



17200



1.6995



1.6856



1.6687



1 6567



1 6439



16324



16112



15996



15933



15893



15861



26.50



1.8456



1.8101



1.7820



17532



1.7320



1.7177



1.7001



16876



16741



16621



16397



1.6275



16209



16168



16134



27.00



1.8811



1.8449



1.8162



1.7864



1.7645



1.7498



17314



1.7184



1.7042



16916



1.6681



16552



1.6483



16441



16406



27.50



1.9166



1.8797



1.8503



1.8195



1.7969



1.7817



1.7626



1.7491



1.7343



1.7210



16963



16828



1.6756



16712



16677



28.00



1.9521



19144



1.8844



1.8526



1.8293



1.8136



1.7937



1.7798



1.7642



17503



17244



17102



17027



16982



16945



28.50



1.9876



1.9492



1.9184



1.8857



1.8617



1.8455



1.8248



1.8103



1.7940



1.7795



17523



17375



17297



17251



17212



29.00



20231



1.9839



1.9525



1.9187



1.8940



1.8773



1.8558



1.8408



1.8238



1.5086



17801



1.7646



17565



17518



17478



29.50



2.0586



2.0186



19865



1.9517



1.9262



1.9091



1.8868



1.8712



1.8534



1.8376



18078



1.7916



17832



17783



17742



30.00



2.0941



2.0533



2.0205



1.9847



1.9584



1.9408



1.9176



1.9016



1.8830



1.8664



18354



18184



18097



18047



18005



PETROLEUM



34-20



TABLE



34.2-EXTENDED



SUKKAR-CORNELL



INTEGRAL



ENGINEERING



FOR BHP CALCULATION



HANDBOOK



(continued)



‘PO, Wp,, Wp p, I ;, 2 1 + W/P,,) * Pseudoreduced &



11



temperature 12



for 8=50.0 1.3



14



15



16



17



18



19



2.0



2.2



2.4



2.6



28



3.0



02000000000000.0000000000.00000000000000O.OODD0.00000.0000000000.00000.00000.00000.0000 050



00023



00023



0.0022



00022



0.0022



0.0022



00021



0.0021



0.0021



0.0021



0.0021



00021



00021



0.0021



0.0021



100



00121



00111



0.0107



00104



0.0102



00100



00099



00098



00098



0.0097



00096



00096



00095



0.0095



00095



1.50



0.0328



00276



0.0257



00246



0.0238



00233



00229



00226



0.0224



0.0222



0.0220



0.0218



0.0217



0.0216



0.0215



2.00



00649



00524



0.0474



00447



0.0430



0.0418



0 0409



0 0402



0.0398



0.0395



0.0385 0.0593



0.0382



00380



0.0378



0.0587



0.0583



0.0579



00835



00827



00820



0.0814



250



00997



00835



ox)750



0.0702



0.0670



0.0650



0 0634



0 0622



0.0615



0.0608



0.0389 0.0599



300



0.1350



01173



0.1066



00998



0.0951



00921



00897



00879



0 0868



0.0858



0.0844



3.50



0 1703



01521



01402



01322



0.1261



01222



01191



01167



0 1151



0.1138



0.1119



01106



01095



01085



01078



4.00



0.2057



01873



0.1749



01660



0.1591



0.1545



01507



01477



01457



0.1440



01417



01401



01387



0.1375



0 1365



4.50



02410



02226



0.2101



02008



0.1933



01882



01839



01804



0.1781



0.1761



0.1734



01714



01697



01633



01671



5.00



02763



0 2579



0.2454



0.2359



0.2281



0.2227



0 2181



0 2143



02117



0.2094



0.2063



0 2040



02022



02006



01993



5.50



03116



0 2933



02807



0.2712



0.2632



0.2577



0 2529



0 2488



0 2461



0.2436



0.2402



02377



02357



02339



02326



6.00



03469



03285



0.3161



03066



0.2985



0 2929



0 2880



0 2838



0 2809



0 2784



0.2747



02721



02700



02681



0 2667



6.50



0.3821



03638



0.3513



03419



03339



0.3282



0 3233



03190



0 3161



0.3135



0.3097



0 3069



0 3048



03029



03014



7.00



04173



0 3990



0.3865



03772



0.3692



03636



0 3587



03544



03514



0 3488



0.3450



0 3421



0 3399



03380



03365



7.50



04525



04341



04216



04123



04044



0.3989



0.3940



0.3897



0 3868



0.3841



03803



0.3774



0 3752



03733



0.3718



8.00



0.4876



04692



0.4567



04474



04395



0.4340



0.4292



0.4250



0.4221



04194



04151



0.4128



0.4105



0.4086



0.4071



8.50



05227



05042



0.4916



04823



04745



04690



0.4643



0.4601



04573



04547



04504



0.4481



0.4458



0.4439



0.4424



9.00



05577



05391



0.5264



05171



05093



0 5039



0.4992



04951



04923



04897



04855



0.4832



0.4810



0.4791



0.4777



950



05927



05739



05612



05518



05440



05386



0.5340



0 5299



0 5271



05246



0 5204



0.5182



0.5160



0.5142



0.5127



10.00



06277



06087



05959



0.5864



05786



05732



0.5685



0.5645



0.5618



05593



05552



0.5530



05508



0.5490



05475



1050



06626



06435



06304



0.6209



06130



06076



06029



0 5990



0.5962



0 5938



05897



0 5875



05854



0.5835



05821



1100



06974



06781



06649



06553



06473



06418



06372



0.6332



0.6305



06280



06240



0.6219



06197



0.6179



06164



1150



07323



07127



0 6994



06896



0 6815



06759



06712



06672



0.6645



0.6621



0 6581



0.6559



06537



0.6519



0 6505



1200



07670



07473



07337



07237



07155



0 7099



0 7051



0 7011



0.6984



0 6959



06919



0 6897



06875



0.6857



0 6842



1250



08018



07818



07680



0.7578



07494



07437



0 7388



0 7347



0 7320



0.7295



07254



0 7232



07210



07192



07177



1300



08365



08163



08022



0.7917



0 7832



07774



0 7724



0 7682



0 7654



0 7629



07587



0 7565



07542



07523



0 7509



1350



08712



08507



08363



08256



08169



08109



08058



0 8015



0 7987



0 7960



0 7917



0 7894



07872



07852



0 7838



1400



09059



08850



08704



0.8594



08504



08443



08391



0 8347



08317



0.8290



0 8245



0 8221



06198



08178



08163



1450



09405



09193



0 9044



0 8930



0 8839



08776



08722



08576



08645



0.8617



08570



08545



0 8521



0 8502



0 8486



1500



09751



09536



0 9384



09266



0.9172



09108



09051



09004



08972



0.8942



0 8893



08866



08842



08822



0 8806



1550



10097



09878



0 9722



09601



09504



09438



09379



09331



09297



0.9265



09213



09185



09160



09139



09123



1600



10442



10220



10061



09935



09836



0 9768



0 9706



09656



09620



0.9586



0 9531



0 9501



0 9475



0 9454



0 9438



16.50



10788



10561



10399



10269



10166



10096



10031



09979



09941



0 9906



0.9847



0 9814



0 9788



0 9766



0 9749



1700



1 1133



10902



10736



10601



10495



10423



10355



10301



10260



1.0223



10160



10125



10097



10075



10058



1750



1 1477



1 1243



1 1073



10933



10824



10749



10678



10621



10578



10538



10471



10434



10405



10362



10364



1800



1 1822



1 1583



1 1409



1 1264



11151



1 1074



10999



10940



10894



10852



1.0779



10740



10709



10686



10668



1850



12167



11923



1 1745



1 1595



11478



1 1398



1 1320



1 1258



11209



11164



1.1086



1 1043



1 1012



10988



10969



1900



12511



12263



1 2081



1 1925



1 1804



11721



1 1639



1 1575



11522



1 1474



1.1390



1 1345



11312



11287



11268



1950



12855



12602



12416



12254



12129



12044



1 1957



1 1890



11834



1.1783



1.1693



1 1644



11609



11584



11564



2000



13199



12942



12751



12583



12453



12365



12274



12204



12144



12090



1 1993



1 1941



1 1905



11878



1 1858



2050



13542



13280



13085



12911



12777



1.2686



1 2590



12517



12453



12395



1.2292



12236



1 2198



1 2171



12149



2100



13886



13619



13419



13238



13100



13005



1 2905



12829



1.2761



1.2699



12589



12528



12489



12461



12439



21 50



14229



13957



13753



13565



13422



13324



13219



13140



13067



13001



1.2884



1.2810



12778



12749



1.2726



2200



14573



14295



14086



13892



13743



13643



1 3532



13449



13372



13302



13177



13108



1 3065



1 3035



13011



2250



14916



14633



14419



14218



14064



1 3960



1.3844



13758



13676



13602



13468



1.3395



1.3350



13319



1.3295



2300



15259



14971



14752



14543



14385



1.4277



1.4155



1.4066



1 3979



13900



13758



1.3680



1.3633



1 3601



13576



2350



15602



15308



15084



14868



14704



14593



1.4466



14372



14280



14197



14046



1.3964



1.3914



1 3881



1.3855



2400



15944



15646



15416



15193



15024



14908



14775



14678



14581



14493



14333



1.4245



1.4193



1 4160



1.4133



24.50



16287



15983



15748



15517



15342



15223



15084



14983



1 4880



14788



14618



1.4525



14471



14436



14408



25.00



16629



16319



16079



15841



15660



15537



1.5392



15287



1.5178



1 5081



14902



1.4803



1.4747



1.4711



14682



25.50



16972



16656



16410



16164



15978



15851



1.5700



1.5590



1.5476



15373



15184



15080



1.5021



14984



14954



2600



1 7314



16992



16741



16487



16295



16164



1.6006



1.5892



1.5772



1 5664



15465



15355



1.5294



1.5256



15225



2650



17656



17329



17072



16809



16611



16476



1.6312



1.6194



1.6068



1 5954



15744



1.5629



1.5565



1.5526



15494 15761



2700



1.7998



17665



17403



17131



16927



16788



1.6617



1.6494



1.6362



1 6243



1.6022



15901



1.5835



1.5794



2750



18340



18001



17733



17453



17243



17100



16922



16794



1.6656



1.6531



16299



16172



16103



1.6061



16027



2800



18682



18337



18063



17775



17558



17410



17226



1.7094



1.6948



1.6818



1.6574



16441



16369



16326



16291



2850



19024



18672



18333



18096



1.7872



17721



17529



17392



17240



1.7104



16849



16709



1.6634



1.6590



16553



2900



19366



19008



18722



18416



1.8187



18030



17831



17690



1.7531



1.7309



1.7122



16976



16898



16853



16815



2950



19707



1.9341



19052



18737



18500



18340



18133



1.7987



1.7821



1.7673



1.7394



17241



17160



1.7114



17076



3000



20049



1.9678



1.9381



19057



1.8814



18649



18435



18284



1.8111



1.7956



1.7664



17505



17421



17373



17333



WELLBORE



34-21



HYDRAULICS



TABLE



Pseudoreduced



34.2-EXTENDED



temperature



for B=60



P".



1.1 1.2 1.3 ______~________



0.20



0.0000



0.0000



0.0000



14 0.0000



SUKKAR-CORNELL



INTEGRAL



FOR BHP CALCULATION



(continued)



0 1.5 0.0000



16



17



18



19



20



22



0.0000



0.0000



0 0000



0.0000



0.0000



0.0000



24 0



26 0



28



30



0 0000



0 0000



0.50



00019



0.0019



0.0019



0.0018



0.0018



0.0018



0.0018



00018



00018



0.0018



0.0018



00018



00017



00017



00017



1.00



00101



0.0093



0.0089



0.0087



00085



0.0084



0.0083



00082



00081



0.0081



0.0080



00080



00080



00079



00079



150



0.0277



00232



0.0215



0.0206



0.0200



0.0195



0.0192



00189



0.0188



0.0186



0.0184



00183



0.0181



00181



00180



2.00



00559



0.0443



0.0399



0.0376



00361



00351



0.0343



0.0338



0.0334



0.0331



0.0326



00323



00321



00319



0.0317



2.50



00870



0.0715



00637



0.0594



0.0566



0.0549



00535



00524



0.0518



0.0512



0.0504



0.0499



00494



00490



0.0487



300



0.1189



01014



0.0913



0.0851



0.0808



0.0781



0.0760



00745



0.0734



0.0726



0.0714



00705



00698



00692



0.0687



3.50



01509



0.1325



01211



0.1135



01079



0.1043



01014



0.0993



0.0979



0.0966



0.0950



00939



00928



00920



00913



4.00



01831



01642



01521



0.1435



01369



01326



01291



01263



0.1245



0.1229



0.1209



01194



01181



01170



01161



4.50



02153



0.1962



01837



0.1745



01672



01624



01583



01551



01529



0.1510



0.1485



0.1466



01451



01438



01428



5.00



0.2475



02283



0.2157



0.2062



0.1984



01931



01887



01850



01826



01804



0.1775



0.1753



0.1736



0.1721



0 1709



5.50



02798



02606



02479



0.2382



0.2301



02245



02198



02158



02132



02108



0.2075



0.2051



0.2032



0.2016



02003



6.00



03120



02928



02801



0.2703



02620



02563



02515



02472



02444



02419



02383



02357



02337



02320



02306



650



03443



03251



03124



0.3026



02942



02884



02834



02791



02761



02735



02697



0.2670



0.2648



02630



02616



700



03766



03574



0.3446



0.3348



03264



03206



03156



03111



03081



03054



03015



0.2986



0.2964



0.2946



02932



7.50



0.4088



03896



0.3769



0.3671



0.3587



0.3529



03478



03433



03403



03375



0 3336



03306



03284



03265



03251



8.00



0.4411



04219



04091



0.3994



0.3910



0.3851



0.3801



03756



03725



03697



03651



0 3628



0 3605



0 3586



03572



8.50



04734



04541



04413



04316



04232



0.4174



0.4123



04079



04048



04020



03974



0 3951



0 3928



0 3909



03894



900



0.5056



04863



04735



04637



0.4554



04496



04445



04401



04370



04343



04297



04273



04251



04231



04217



9.50



0.5378



05185



05056



04958



0.4875



0.4817



04767



04722



04692



04665



04619



04596



04573



04554



04539



10.00



0.5701



05507



05377



05279



05195



0.5137



0.5087



0.5043



0.5013



04985



04940



04917



04894



04875



04861



10.50



0.6023



05828



05698



05599



05515



0.5457



0.5407



0.5363



0.5333



05305



05260



05237



05215



05196



05181



1100



06344



06149



06018



05918



05833



05775



0.5725



0.5681



0.5651



0 5624



05579



05556



05534



05515



0 5500



11 50



0.6666



06469



0.6337



0.6237



0.6151



06093



0.6042



0.5998



0.5968



0.5941



0 5896



05873



05851



05832



0 5818



1200



0.6987



06790



06656



06555



06469



06409



06359



0.6314



0.6284



06257



06212



06189



06166



06148



06133



1250



07309



07110



06975



06872



06785



0.6725



06674



06629



06599



0.6571



0.6526



0 6503



0 6480



0 6461



0 6446



1300



0.7630



07429



07293



07189



07101



07040



06986



06943



06912



0.6884



0.6838



06815



0 6792



0 6773



06756



1350



07951



07749



07611



07505



07415



07354



07301



07255



07224



07196



07149



07125



07101



07032



07067



1400



08272



08068



07929



07820



07730



07667



07613



07566



07534



07505



0.7457



07432



07409



07389



07374



1450



08592



08387



08246



0.8135



08043



07979



07924



07876



07843



07813



07764



07738



07714



07694



07679



1500



08913



08705



08562



08449



08355



08291



08233



08184



08151



08120



08069



08042



08017



07997



07962



1550



09233



09024



08879



08763



08667



08601



08542



08492



08457



08425



08371



0.8343



0.8318



08298



08282



1600



09554



09342



09195



0.9076



08978



08911



08850



08798



08762



08728



08672



08643



08617



08596



08580



1650



09874



09660



09510



0.9389



09288



09219



09156



09103



09065



09030



08971



08940



08914



08892



08876



1700



10194



09977



09826



0.9701



09598



09527



09462



09408



09368



09331



09269



09236



09208



09186



09170



1750



1.0514



1.0295



10141



10012



0.9907



09835



09767



09711



09668



09630



09564



09529



09501



09478



09461



1800



10834



1.0612



10455



10323



1.0215



10141



10070



10013



09968



09928



09858



09820



09791



09766



09751



1850



1 1153



1.0929



10769



10634



1.0523



10447



10373



10313



10267



10224



10150



10110



10080



10056



10038



1900



11473



1 1246



11083



10944



1.0830



10752



10675



10613



10564



10519



10440



10398



10366



10342



10324



1950



1 1792



1 1562



11397



11253



11137



1 1056



10976



10912



10860



10812



10728



10683



10651



10626



10607



2000



12112



1.1879



1 1711



1 1562



1.1443



1 1360



1 1277



1 1210



11155



11104



1 1015



10967



10933



1 0908



10689



20 50



1 2431



1.2195



1 2024



1 1871



1.1748



1 1663



1 1576



1 1507



1 1449



1 1395



1 1301



1 1250



1 1214



1 1188



1 1168



21.00



12750



1.2511



12337



12179



12053



1 1965



1 1875



1 1803



11741



1 1685



1 1584



1 1530



1 1493



1 1466



1 1446



2150



13069



1.2827



12650



12487



1.2357



12267



12173



12099



12033



1 1974



1 1867



1 1809



1 1770



11743



2200



13388



13143



12962



12795



12661



1.2568



12470



12393



12324



12261



12147



12086



12046



12018



‘1 1721 1 1995



2250



1.3707



13458



1.3274



13102



12964



12869



12766



12687



1.2614



12547



12427



12361



12319



12291



1 2266



2300



1 4026



13774



1.3586



1 3409



13267



13169



13062



12979



1.2902



12832



12705



12635



12592



12562



12538



23.50



14344



14089



1.3898



13715



13569



13469



13357



1.3271



1.3190



13116



1 2981



12908



12862



12832



12807



2400



1.4663



14404



14210



1 4021



13871



13768



13652



13563



13477



13399



1 3256



13179



13131



13100



13074



24 50



1.4982



14719



14521



1.4327



14173



14066



13945



13853



1.3763



13681



1 3530



13448



1 3399



1 3366



1 3340



2500



1.5300



1 5034



14832



1.4632



14474



14364



14238



14143



14048



13962



13803



13716



1 3664



13631



13604



25 50



1.5619



15349



15143



1.4937



14774



14662



14531



14432



14332



14242



14074



13983



1 3929



1 3895



13867



2600



1.5937



1 5664



15454



1.5242



15075



14959



14823



14721



14616



14521



14344



14248



14192



14157



14126



2650



16255



1 5978



15765



15547



1 5374



15255



15114



15008



14898



14799



14613



14512



14454



14417



14388



2700



16574



1.6292



16075



1.5851



15674



15552



15405



15295



15180



15076



14881



1.4775



14714



14677



14646



2750



1.6892



1.6607



1.6385



16155



15973



15847



1 5695



15582



15461



15353



15148



15036



14973



1.4935



14903



2800



17210



1.6921



1 6695



16459



16272



16143



15985



1 5868



15742



15626



15413



15296



15231



i 5191



15159



28.50



17528



17235



1.7005



16762



16570



1 6438



16274



16153



16021



15903



15678



15555



15487



15447



15413



2900



1 7846



1 7549



1 7315



1.7065



16868



1 6732



1 6563



1 6436



16300



16176



15941



15813



15742



15701



15666



29.50



1.8164



17863



1.7625



17368



1 7166



17076



16851



16722



16579



16449



16204



16070



15997



1 5954



15918



30.00



1.8462



18177



1.7934



17671



1.7463



1 7320



17139



17005



16856



16722



16465



16325



16249



16205



16168



34-22



PETROLEUM



TABLE



34.2-EXTENDED



SUKKAR-CORNELL



INTEGRAL



ENGINEERINGHANDBOOK



FOR BHP CALCULATION



(continued)



‘PO,Wp,)dppr I 1 + wP,,)* 02 Pseudoreduced



2%?!0.20



-- 1.1



temperature 1.2



0.0000



0.0000



0.50



0.0017



1.00



0.0087



1.50



for



1.3 -__---



B=70 1.4



0 1.5



1.6



1.7



1.8



1.9



2.0



2.2



~-



2.4



26



2.6



3.0



0.0000



0.0000



0.0000



0.0000



0.0000



0.0000



0.0000



0.0000



0.0000



0.0000



0.0000



0.0000



0.0000



00016



0.0016



0.0016



0.0016



0.0015



0.0015



0.0015



0.0015



0.0015



0.0015



0.0015



0.0015



0.0015



0.0015



0.0080



0.0077



0.0074



0.0073



0.0072



0.0071



0.0070



0.0070



0.0070



0.0069



0.0069



0.0068



0.0068



0.00613



0.0240



0.0199



0.0185



0.0177



0.0172



0.0168



0.0165



0.0163



0.0161



0.0160



0.0158



0.0157



0.0156



0.0155



0.0154



200



0.0491



0.0385



00345



0.0325



0.0312



0.0303



0.0296



0.0291



0.0288



0.0285



0.0281



0.0278



0 0276



0 0274



0.0273



2.50



0.0772



0.0625



00554



0.0515



0.0490



0.0475



0.0462



0.0453



0.0448



0.0443



0.0435



0.0431



0.0426



0.0423



0.0420



300



0.1063



0.0894



00799



0.0742



0.0703



0.0679



0.0660



0.0646



0.0637



0.0629



0.0618



0.0611



0.0604



0 0595 0.1010



350



0.1356



0.1175



0 1066



0.0994



0.0943



0.0910



0.0884



0.0864



0.0851



0.0840



0.0825



0.0815



0.0806



0 0599 0 0798



4.00



0.1651



0.1464



0 1346



0.1264



01202



0.1162



01129



01104



0.1087



0.1073



0.1054



01040



0.1029



01018



450



0.1947



1.1756



01634



0.1545



01475



0.1429



01391



01360



0.1340



0.1322



0.1299



0 1282



0.1268



0.1256



0.1246



5.00



0.2243



1.2050



01926



0.1833



0.1756



0.1706



01664



01629



0.1606



0.1585



0.1558



0.1538



0.1522



0 1508



01497



0.0792



550



0.2540



0.2347



02221



0.2125



0.2045



0.1991



0.1946



0.1907



0.1881



0.1859



0.1827



0.1805



0.1787



0.1772



01760



600



0.2838



0.2644



02517



0.2420



0.2337



0.2281



02233



02192



02164



0.2140



0.2106



0.2081



0.2061



0 2045



02032



650



0.3135



0.2941



0.2815



0.2716



02632



0.2574



02525



02482



02453



0.2427



0.2390



0.2363



0.2343



0.2326



02313



700



0.3433



0.3239



03113



0.3014



0.2929



0.2870



0.2820



02775



0.2745



0.2718



0.2680



0.2652



0.2630



0.2613



0 2599



750



0.3732



0.3536



03411



0.3312



0.3226



0.3167



0.3116



03071



0.3040



0.3013



0.2973



0.2944



0.2922



0.2904



0 2890 03184



800



0.4030



0.3836



03710



03611



03525



0.3465



03414



03368



03337



03309



03262



03239



03217



0 3198



850



0.4328



0.4135



04009



03909



0 3824



0.3764



0.3713



0 3667



03635



0.3607



03560



03536



0.3514



0.3495



03481



900



0.4627



0.4434



04307



04208



04122



0.4063



0.4011



0.3965



0.3934



03905



03858



03834



0.3812



03793



03779



950



0.4926



0.4733



04606



04507



04421



0.4362



0.4310



04264



04233



04204



04157



04133



0.4110



04092



04077



1000



0.5225



0.5031



04905



04805



04720



0.4660



0.4609



0.4563



0.4531



04503



0.4456



0.4432



0.4409



0.4390



04376



1050



05523



0.5330



05203



05104



05018



0.4958



0.4907



0.4861



0.4830



04801



04754



04730



04708



04689



04675



1100



0.5822



0.5629



05502



05402



05316



05256



0.5204



0.5159



0.5127



0.5099



05052



05028



05005



04987



04972



1150



0.6121



0.5927



05800



05700



05613



05553



0.5502



0.5456



0.5424



05396



0.5349



05325



05303



05284



05270



1200



0.6420



0.6226



06098



05997



05910



05850



0.5798



0.5752



0.5721



05692



05645



05621



05599



05580



05566



1250



06718



06524



0.6396



06294



06207



06146



06094



0.6047



0.6016



0.5987



0.5940



0.5916



05893



05875



05860



1300



07017



06822



06693



06591



06503



06442



06389



0.6342



0.6311



06282



06234



06210



06187



06168



06154 0.6445



1350



07316



0.7121



0.6991



06687



06798



0.6737



06683



06636



0.6604



0.6575



0.6527



06502



06479



06460



1400



07615



0 7419



0.7288



0.7183



0.7093



07031



0.6977



0.6929



0.6897



0.6867



0.6818



06793



06770



06750



0.6736



1450



0.7913



0 7717



0.7585



0.7479



0 7388



07325



0 7270



0.7222



0.7189



0.7158



0.7108



0 7062



07059



07039



0.7024



1500



0.8212



0.8014



0.7881



0.7774



0.7662



07619



07562



07513



0 7479



0.7448



0.7397



07370



0 7346



07326



07311



1550



08510



08312



08178



0.8069



0.7976



07911



07854



07804



0 7769



07737



0.7684



0.7656



0.7632



07612



0 7597



1600



08809



08609



08474



0.8363



0.8269



06203



0.8145



08094



0.8058



08025



0.7969



0.7941



07916



07896



0 7660



16.50



0.9107



08907



0.8770



0.8658



0.8562



0.8495



08435



08363



08345



08311



0.8254



0.8224



0.8198



08178



08162



1700



09406



09204



09066



0.8951



0.8854



08786



0.8724



0.8671



0.8632



0.8597



0.8537



0.8505



0.8479



0.8458



0.8442



1750



0.9704



09501



09362



0.9245



0.9146



0 9076



09013



0 8958



08918



08881



08818



0.8765



0.8758



0.8737



08721



1800



10002



0.9798



0.9657



0.9538



0.9437



0.9366



09300



09245



0 9203



09164



09098



0.9064



0.9036



0.9014



0 8997



16.50



10300



1.0095



0 9953



0.9831



0.9728



0 9656



0 9568



0 9530



0 9486



0 9446



0.9377



0.9340



0.9311



0.9289



0 9272



19.00



1.0599



1.0392



1.0248



1.0123



1.0018



0 9945



0 9874



09815



0 9769



0 9727



09654



0.9615



0.9586



0.9563



0 9545



19.50



10897



10669



1.0543



1.0415



1.0308



1.0233



10160



10099



10051



10007



0 9930



0.9889



0.9858



0.9835



09817



20.00



1.1195



10985



10837



10707



1.0597



1.0521



1 0445



10383



10332



10286



10204



10161



1.0129



10105



10087



20.50



1.1493



1 1282



11132



1.0999



1.0886



1.0808



1 0730



10665



10612



10564



10478



10432



10398



10374



10355



21.00



1.1791



1 1578



1 1426



11290



1.1175



1.1095



1 1014



10947



10692



10841



10749



10701



10666



10641



10622



2150



12089



1 1874



11721



1.1581



1.1463



1.1381



1 1297



1 1229



11170



1 1116



1 1020



10968



10933



10907



10887



22.00



12387



1.2170



1.2015



11871



1.1751



1.1667



11560



11509



11448



11391



11289



11235



11198



11171



11151



22.50



1.2685



1.2466



1 2309



1.2162



1.2039



1.1953



1 1862



1 1789



11724



1 1665



1 1558



1 1500



11461



1 1434



1 1413



23.00



1.2982



1.2762



1.2602



12452



12326



1.2236



12144



1.2069



1.2000



1 1938



1 1825



1 1763



1 1723



11695



1 1674



23.50



1.3280



13058



1.2896



12742



1.2613



1.2522



12425



12347



12276



12210



12090



1.2026



1 1984



11955



1 1933



24.00



1.3578



1.3354



1.3190



1.3031



12899



1.2807



12706



12625



12550



12482



1 2355



12287



12243



12214



1 2191



24.50



1.3876



13650



1.3483



13321



1.3185



1.3090



12986



12903



1.2824



12752



12619



1.2546



12501



12471



1 2447



25.00



1.4173



1.3946



1.3776



1.3610



1.3471



1.3374



1.3265



13180



13097



13022



12881



12805



1.2758



1.2727



12702



25.50



1.4471



1.4241



1.4069



1.3899



1.3757



1.3657



13544



13456



1.3369



13290



13142



13062



13013



12981



12956



26.00



1.4769



1.4537



1.4362



1.4107



14042



1.3940



13823



1.3732



1.3641



1.3658



1.3403



13318



13267



13235



13209



26.50



1.5066



14832



1.4655



1.4476



14327



1.4222



14101



14007



13912



13825



13662



1 3573



1 3520



13487



13460



27.00



1.5364



1.5127



1.4948



1.4764



1.4611



1.4504



1.4379



14202



14162



14092



13920



13627



1 3772



13738



13710



27.50



1.5661



1.5423



1.5240



1.5052



14895



14786



1.4656



14556



1.4452



1.4357



14178



1.4079



14023



13987



13959



28.00



1.5959



1.5718



1.5533



1.5340



1.5179



1.5067



14933



14829



1.4721



1.4622



1.4434



1.4331



1.4272



1 4235



14206



28.50



1.6526



1.6013



1.5825



1.5627



1.5463



1.5348



15209



1.510'2



14989



1.4886



1.4690



1.4581



14520



14483



14452



29.00



1.6554



1.6308



1.6117



1.5915



1.5747



1.5629



15485



1.5375



1.5257



1.5150



1.4944



1.4831



14768



14729



14698



29.50



1.6851



16603



1.6410



1.6202



1.6030



1.5909



15761



15647



1.5524



1.5412



15196



1.5079



1.5014



1.4974



14942



30.00



1.7148



16898



1.6702



1.6489



1.6313



1.6189



16036



15919



1.5791



1.5675



1.5450



15327



15259



15218



15165



WELLBORE



HYDRAULICS



34-23



The integral function on the left side of Eq. 34 can be evaluated by use of Table 34.2 from Ref. 8. These tables were prepared by using an arbitrary reference point of ppr of 0.2. Evaluation of the integral is based on the following relationships:



(P,,)



-



1 +&/P,,)*



~



1



(pv) I WP,,)dp,,



(pw) I (Z/P,,)dp,,



5



1 +wP,,)*



= [i0.2



1



(pd2 (Z~Pprm,, = 11 1 +~(z/p,r)* 0.2



0.01877ysL T



. . (35) ..’



Since the tables and charts provide numerical values for the bracketed terms in Eq. 35, a calculation ojflowing BHP can be obtained directly, with only simple rnathematits being involved. In the previous and subsequent calculation procedures, the diameter of the flow string enters into the calculations as the fifth power. It is important, therefore, that the exact dimensions of the flow string be used rather than nominal flow-string sizes. Table 34.3 lists the pertinent information on various flow-string sizes. The effect of assuming a constant average temperature over the entire gas column in Eqs. 17, 21, and 35 can be mitigated by taking only small increments of depth from top to bottom and using a constant temperature for each increment of depth. Assuming a linear temperature gradient, the average temperature for each depth increment can be calculated. The larger the number of depth increments taken in calculating the pressure traverse, the closer one approximates the rigorous integration of the equations.



Example Problem 3. 6 Calculate the BHP of a flowinggas well. Given: length of vertical pipe, L = 10,ooO ft, tubing ID, dti = 2.00 in., gas-flow rate, qg = 4.91X106 cu MD, flowing wellhead pressure, p2 = 1,980 psia, average flowing temperature, !? = 636”R, gas gravity (air=l.O), yg = 0.750, = 660 psia, PPC TpC = 4OO”R, and f= 0.016. Solution. 1. Calculate



B.



B=66V3,2~2



=



(667)(0.016)(4.91)2(636)2 (2.00)5(660)2



dri 5Pp~*



2. Calculate



O.O1877y,L T



O.O1877y,L T



=7.48,



=



.



(0.01877)(0.750)(10,000) 636



=0.2213.



TABLE



34.3-FLOW API Ratln( m 1



STRING



WEIGHTS



Nelght per Fool



AND SIZES



ID OnI



OD



(Ibmltt)



(In) I I 2 1 2



660 900 375 375 875



I 380 I 610 041 1 995 2 469



6 25~ 6 5 7.694 a 50 9 30 to 2



2 3 3 3 3



a75 500 500 500 500



1 3 3 2 2



441 068



9.26 or 9 II 00 IO 98 II 75 12.75



4 4 4 4 4



000 000 500 500 500



3 3 4 3 3



548 476 026 990 958



16.00 16 50 12 85 13 00 15 00



4 4 5 5 5



750 750 000 000 000



4 4 4 4 4



062 070 500 494 408



I8 00 21 00 I6 00 17 00 20.00



5 5 5 5 5



000 000 250 500 500



4 4 4 4 4



27b I54 648 892 778



14 00 17.00 19 50 22 50 20 00



5 5 5 5 6



750 750 750 750 000



5 5 5 4 5



190 190 090 990 350



10 00 14 00 26 00 28 00 29.00



6 6 6 6 6



625 625 625 625 625



6 5 5 5 5



049 921 855 791 761



20.00 22 00 24 00 26 00 28.00



7 000 7 000 7 000 7.000 7 000



6.456 6 398 6 336



30 34 26 28 32



7 000 7 615 8 000 8 125



6 6 7 7 7



8 8 8 0 8



7.281 7. I85 7 I25 8.097 8 017



2 3~2.4 2 9 or 2 748 4 00 4.5or4 7 5.897



50



00 00 00 00 00



35 50 39. 5 4.277 in. Values of F, are presented in Table 34.4 for various tubing and casing sizes.’ The right side of Eq. 36 may be integrated numerically by employing a two-step trapezoidal integration:



and



(Pm-P2)Um



18.75y,L= 4. For T,, = 1.59, read from Table 34.2



(PP,) 2 (zJp,,)dp,, s 0.2



-cJ2)



+ (PI



-P,n)U,



...........



=0.4246.



+I,,)



2



2



.........



’



. . . . (40)



1 +&z&A2 where O.O1877y,L



5. Add



to T



(P/j,) 1 Wp,,)Q,, I=



1 +fqz&J2



02



PUZ)



F* +O.O01[pl(T~)]~



0.4246+0.2213=0.6459. and 6. From Table 34.2 find the pseudoreduced corresponding to (p,r),



(zb,r)dp,r



s



Eq. 40 may be separated into two expressions, each half of the flow string. 18.7Sy,L=(p,,



-p2)(lm



+fz)



by ppc to obtain BHP.



(p,,)



pl =4.358x660=2,876



1875y,L=(p,



psia.



Another procedure for calculating the BHP of flowing gas wells that has found widespread use since its adoption by various state regulatory agencies is that of Cullender and Smith.7 The method avoids the assumption of a constant average temperature by including the temperature within the integral.



-p,)(/,



where *)ldi5,



.



.



(37)



ff is the Fanning friction factor and is equal to ff=f/4, and f is the Moody friction factor from Fig. 34.2 Eq. 37 can be simplified by using the Nikuradse friction factor equation for fully turbulent flow and for an absolute roughness of 0.0006 in.:



F= F,q,



=



O.l0797q, d 2,6,2 1



,



..



. .



.



.



+I,)



. (42)



for the lower half. By trial and error, pm is calculated from Eq. 41, p r then is calculated in a similar manner by using the value of I, from Eq. 41 and substituting in Eq. 42. Simpson’s rule then is employed to obtain a more accurate value of the BHP.



(I2 +41,



F2 =(2.6665ffq,



(41)



for the upper half, and



=4.358.



7. Multiply



one for



=0.6459.



1-tB(z21ppr2)



0.2



(p,,,



pressure



+I,).



. .



(43)



Rather than using the two-step trapezoidal integration to make the first estimate of the BHP, Simpson’s rule may be used directly and the BHP calculated by trial and error. As this indicates, the Cullender and Smith method involves tedious trial and error solution if hand calculated. The method is best solved by computer. Quoting Ref. 8. Because the Cullender and Smith method considers both temperature and Z to be functions of pressure, it might appear that this method is somewhat more accurate than the Sukkar-Cornell approach. This is only an apparent advantage. If temperature IS known in the gas column, it is possible to break the depth into several increments, each with one appropriate mean temperature.



This was alluded to previously. The Sukkar-Cornell method is an accurate, fast hand calculation procedure that avoids trial and error calculations. It is also amenable to computer solution.



WELLBORE



34-25



HYDRAULICS



Example Problem 4.’ Calculate the flowing BHP by the method of Cullender data:



and Smith from the following



well



gas gravity, yfi length of vertical pipe, L wellhead temperature, T2 formation temperature, T, wellhead pressure, pz flowrate, qr



= 0.75. = 10,000 ft, = 570”R, = 705”R, = 2,000 psig. = 4.915x 106 cu ft/D, tubing ID, d,, = 2.441 in., Tpc = 408”R, and ppr = 667 psi.



pseudocritical temperature, pseudocritical pressure,



~ TI+Tz



TX-----C 2



T,,, = $



wellhead



=



1,397,



T 638 Tpr =-z-=1.564, T,,,. 408 Tpr = $



bottom



=g



= 1.728,



P’ 2,000 =2.999, ppr = E = __ P&l< 667



wellhead



F= (0.10797)(4.915)



=0.05158,



(2.441)2.6’2 and F2 =0.00266. Left side of Eq. 36, 18.75 y,L=(l8.75)(0.75)(10,000) = 140.625. Calculate 12. From the compressibility Chap. 20) ~2 =0.705. Therefore,



zoo0



P2 -=



T2z2



(570)(0.705)



factor chart (see



=4.977



and 4.977



12=



~181.44.



0.00266+0.001(4.977)2 Assume



11 =I,.



Solving



Eq. 41 for pm,



l40,625=(p,-2,000)(181.44+181.44), pm =2,388



psia.



OD



ID



(InI



lbmilt



tin1



1315 1660 1990 2 375 2 875 3 500 4 000 4 500 4 750 4 750 5000 5000



I 80 240 2 75 4 70 6 50 9 30 11 00 12 70 16 25 18 00 1800 21 00



1 043 1380 1610 1 995 2441 2 992 3 476 3 358 4082 4 000 4276 4 154



5000 5000 5 500 5 500 5500 5 500 5 500 5 500 6000 6000 6000 6000 6000 6625 6625 6625 6625 6625 6625 6625 7000 7000 7000 7000 7 000 7000 7000 7625 7625 7625 7625 7625



1300 1500 14 00 1.500 1700 20 00 23 00 25 00 1500 1700 20 00 23 00 26 00 20 00 22 00 24 00 26 00 28 00 31 80 34 00 2000 2200 2400 26 00 28 00 30 00 4000 26 40 29 70 33 70 38 70 4500



4494 4406 5 012 4976 4892 4778



8000 8125 8125 8125 8125 8625 8625 8625 8625



0 095288 0046552 0031122 0017777 0010495 0 006167 0 004169 0 002970 0002740 0 002889 0002427 0002617



=638”R,



PC



midpoint



34.4-VALUES OF I=r FOR VARIOUS TUBING AND CASING SIZES



570+705 2



= z



TABLE



8625 8625 8625 '3625 9000 9000 9000 9000 9625 9625 9625 9625 9625 9625 10000 10000 10000 10 750 10 750 10750 10 750 10 750 10 750



4670 4 580 5524 5450 5352 5240 5 140 6049 5 989 5 921 5855 5 791 5675 5595 6456 6398 6 336 6276 6 214 6 154



00021345 00022437 00016105 00016408 00017145 00018221 0 0013329 00020325 0 0012528 00012972 00013595 00014358 0 0015090 0 0009910 00010169 0 0010473 0 0010781 0 0011091 00011686 00012122 0 0008876 00008574 0 0008792 0 0009011 0 0009245 0 0009479



2600 2800 3200 3550 3950 1750 2000 24 00 2600



5836 6969 6875 6765 6625 6445 7386 7485 7385 7285 7 185 8 249 8 191 8 097 8003



00010871 00006875 00007121 00007424 00007836 00008413 00005917 00005717 0 0005919 00006132 00006354 00004448 00004530 00004667 00004610



3200 3600 3800 43 00 3400 3800 4000 4500 3600 4000 43 50 4700 53 50 5800 33 00 55 50 ,61 20 32 75 35 75 4000 45 50 4800 5400



7907 7825 7775 7 651 8 290 8 196 8 150 8032 8 921 8835 8755 8 681 8535 8435 3 384 8 908 8 790 10 192 10 136 10050 9 950 9 902 9784



0 0004962 0 0005098 00005183 00005403 0 0004392 00004523 0 0004589 00004765 00003634 00003726 00003814 0 0003899 00004074 00004200 0000416! 00003648 00003775 00002576 00002613 00002671 00002741 00002776 00002863



PETROLEUM



34-26



2,388 ~ =3.580, 667



Pm PPC zm =0.800



HANDBOOK



and



Second trial:



ppr=-=



ENGINEERING



at ~,,=1.564,



Pt?!



2,388



Tmzm



(638)(0.800)



4.481 I, =



= 197.06. 0.00266+0.001(4.481)*



p,,=3.580,



Solving Eq. 42 for p I1 l40,625=(p,



=4.679,



p t =2,739



-2,377)(197.06+191.21), psia.



and Third trial:



4.679 = 190.57



I, = (0.00266)+0.001(4.679)*



PI ppr=-=



Eq. 41 for pm,



Solving



l40,625=(p,-2,000)(190.57+181.44)



PPC



2,739 ~ =4.106, 667 at TPr = 1.728, ppr =4.106,



z 1 =0.869



and pm =2,378



psia.



PI



2,739



T, z,



(705)(0.869)



=4.47 1)



Third trial: and 2,378 =3.565, 667



Pm ppr=-= PPC



z,=O.800



at T,,=1.564,



4.471 = 197.40.



I, = 0.00266+0.001(4.471)2



p,,=3,565,



Solve Eq. 42 for p 1 2,378



-=Pm Td,



=4.659, l40,625=(p,



(638)(0.800)



p I =2,739



and 4.659



-2,377)(197.40+191.21), psia



=191.21.



I, = 0.00266+0.001(4.659)*



Using Simpson’s



Solving Eq. 41 for pm,



lLj.0625 =



rule from Eq. 43,



(‘I -“I



l40,625=(p,-2,000)(191.21+181.44),



x[181.44+4(191.21)+197.40],



6



therefore pm =2,377



p I -p2



psia.



For the lower half of the flow string assume It =f,,, = 191.21. Solving Eq. 42 forpt, l40,625=(p, p, =2,745



PI



z, =0.869



and pI =738+2,000=2,738



psia.



-2,377)(191.21+191.21), psia.



Second trial:



ppr=-&=



=738,



2,745 -=4.115, 667



at T,,=1.728,



PI



2,745



T, z,



(705)(0.869)



p,,=4.115,



=4.481



A simplified method for calculating flowing BHP of gas wells results if an effective average temperature and an effective average compressibility are used over the length of the flow string. Low-pressure wells at shallow depths or wells where pressure drop is small are especially well suited for this method. With the usual assumptions that kinetic energy is negligible, g/g, equals unity, etc., the following equation for vertical gas flow has been developed by Smith”:



Phh2--esPth2=



25fq, 2 T2T2(e” - 1) 0,0375d;5



’ ““”



. (44)



WELLBORE



34.27



HYDRAULICS



where dci = inside diameter of casing, ft, d,, = outside diameter of tubing, ft, and rH = hydraulic radius, ft.



where Pbh



=



Prh



=



.f= 9g



=



s=



BHP, psia, tophole pressure, psia, friction factor, dimensionless, from Fig. 34.2, gas flow rate, IO6 cu ft/D referred to 14.65 psia and 60”F, exponent



of e=



The diameter be



O.O375y,L ~ TZ ’



d,, =dci -d,,.



gas gravity (air = 1 .O), length of vertical flow string, ft, average temperature, “R, average compressibility of gas, dimensionless, di = internal diameter of flow string, in., and e= natural logarithm base=2.71828.



Yg



=



L= TX z=



The method using Eq. 44 is also a trial and error procedure. In evaluating the friction factor for commercial pipe, Smith lo and Cullender and B’inckley ’’ have shown from an analysis of flow data that average absolute values of roughness, 0.00065 and 0.0006 in., respectively, are the correct values to use for clean commercial pipe. For an absolute roughness of 0.0006 in., Cullender and Binckley ” derived an expression for the friction factor as defined in Fig. 34.2, as a power function of the Reynolds number and pipe diameter. In terms of field units, -0.065d;



f=30.9208x



10-j



-0.058



qK PK



. .......



YK



q.8 = gas flow rate, lo6 cu ft/D, d; = internal diameter of flow string, YK = gas gravity (air= 1 .O), and p‘v = gas viscosity, lbm/ft-sec.



Flow Through a Tubing-Casing



....



(45)



rH=



s(d,., +d,,, 1



di5 =(d,;+d,,)2(dci-d,,)3.



(49)



Gas/Water Flow The effect of water production on calculated pressure drop for gas wells operating in mist flow can be included by using an average density assuming zero slip velocity and by using total rate in the friction loss term. The volumetric average density can be calculated as



where p is the average density at flowing conditions and q is the volumetric flow rate at flowing conditions. To include the effect of water in the Cullender and Smith calculation, modify the integrand, I, as follows (see Page 24):



+0.001[pi(Tz)12(Pl~



(46)



d,.; -d,, 4



Modification of Eq. 32 for annular flow involves only substituting d,, for di. Likewise d,, replaces dj when determining friction factor (from the Reynolds-number plot, Fig. 34.2). However, the simplification of Eq. 32 includes velocity expressed as a function of diameter and volumetric flow rate, and so di 5 in B of Eq. 33 and in Eq. 44 becomes



[PQTz)I(PIP~)



annulus, 21



. (48)



KI2



Annulus.



.



(ai4)(d,.; * -d,,



.. ....



ft,



The flow equations that relate to flow through a circular pipe, when properly modified, can be used for conditions where flow is through an annular space. This modification involves determining the hydraulic radius of the annular cross section and using the friction factor obtained for an “equivalent” (i.e., having the same hydraulic radius) circular pipe. The hydraulic radius is defined as the area of flow cross section divided by the wetted perimeter. For a circular pipe,



For a tubing-casing



...



pipe, thus, would



-0.065



. .. ...



di T.



.



circular



-0.065



where



*d,2f4 rH=-= ad;



of an equivalent



’ .“““.’



(47)



Gas-Condensate Wells Calculation of BHP. Calculations



of BHP on gascondensate wells are based on equations previously presented for gas wells. The application of these equations may be limited somewhat by the amount of liquid present in the flow string. Upon shutting in a gas-condensate well, part of the liquids that were being carried in the flow stream may fall back and accumulate in the bottom of the wellbore. For this reason, it is advisable to determine whether or not such a static liquid level exists in a gas-condensate well before relying on a BHP calculated from surface measurements. When the location of the static liquid level is known, the gas calculations can be used to determine the pressure at the gas-liquid interface and the length of the liquid column. An estimated liquid density will provide the additional pressure needed to determine pressure at formation level.



34-28



PETROLEUM



GRAVITY STOCK



TANK CIOUID



ENGINEERING



HANDBOOK



Liquid Injection Calculation of Injection BHP. For isothermal



flow of incompressible fluid, assuming gig, = 1, and integrating between limits of the top and bottom of the hole, Eq. 30 may be written as follows:



f!f -tAz-cE,=O.



.. ...



. . (51)



P



(Since the datum plane is at the surface, AZ will be a negative number.) Then 0



20 40 60 80



BARRELS



Fig.



OF



lIXl20



140 160 180200220240260280xx)



CONDENSATE



PER



MMSCF



34.4-Gas/gravity ratio vs. condensate/gas tlon of condensate gravity.



OF



GAS



ratio as a func-



p* =p, -Azp--Et/I, since



-AZ=D,



.



the depth.



p* =p l , eDp-E,p. In the flow equations for gas, the gas gravity is the flowstream gravity. This is calculated for condensates from the following I2 :



y = (Yg)sp +(4,59lyfIR,L) 1 +(1.123,R,L) R



,



. . . . . . (50)



and



Nisle and Poettmann I3 published a simple correlation based on field data (Fig. 34.4) that can be used to calculate the flow-stream gravity of the entrained mixture such as occurs in the case of a flowing gas-condensate well. Accuracy of the flow equations for gas, as modified for gas-condensate wells, is influenced by the amount of liquid in the flow stream. The higher the gas-liquid ratio, the more accurate the calculated results will be.



p2=p,



where p2 p, D p f v d; g,



..



.



(53)



.



.



(54)



(Fig. 34.2),



p2 =p, +Dp-‘2 2g,di.



pressure



..



units to pounds



per square inch,



+Dp-fv’ 144 Dp 288g,di,



= = = = = = = =



. . . (52)



Therefore, .



Since Et=fi2D/2g,di



Converting where (Y~).~~ = separator gas gravity (air= l), yL = specific gravity of condensate, R KL = gas-liquid ratio, cu ftibbl.



.......



.



(55)



bottomhole pressure, psia, at depth D, surface pressure, psia, depth of well, ft, density of injected fluid, lbm/cu ft, friction factor (Fig. 34.2), fluid velocity, ft/sec, internal diameter of pipe, ft, and 32.2 conversion factor.



Injection Wells Petroleum-production operations often involve the injection of fluids into the subsurface formation, as is the case in waterflooding, pressure maintenance, gas cycling, and designing gas lift installations. Therefore, it becomes desirable to have a means of predicting the variation of pressure with depth for the vertical downward flow of fluids. Eqs. 29 and 30, previously discussed, form the basis of any specific fluid-flow relationship. They contain no limiting assumptions other than those arrived at in deriving Eq. 30 from Eq. 29. The only difference in applying Eq. 30 to vertical downward flow when compared with upward flow is that the integration limits are changed; that is, the sign of the absolure values of potential energy then changes and, depending on the rate of injection in the case of gas injection, the absolute value of the compressional energy change may vary from positive to negative. In other words, at low flow rates. the BHP is greater than the surface pressure; whereas. at high flow rates, the BHP is less than the surface pressure.



Eq. 55 reveals that the BHP for the case of incompressible flow as assumed for liquid injection into a wellbore is simply the surface pressure plus the pressure from the “weight of the liquid column” minus the pressure drop caused by frictional effects. For no flow, it reduces to the well-known expression for a static-fluid column



,,=,,+z.



. ..____................,,..



Gas Injection Calculation of Injection BHP. Starting with the general differential equation, Eq. 30, Poettmann’ derived an expression for calculating the sandface pressure of flowinggas wells in which the variation of the compressibility factor of the gas with pressure is taken into consideration. The same integral factor as given in Table 34.1 is employed for the calculation of static BHP in Table 34.5.



WELLBOAE



HYDRAULICS



34-29



By following the same reasoning as in the previous section, the equation can be rearranged so that the pressure traverse for vertical flow downward can be calculated as follows:



D.,



D=



di 5 =(dci +dt,)2(d,.; -d,J3



{0.9521x10-61fq,‘y,~D,~21d,,5(A~)’]}-l’



,,..................~



The nomenclature is the same as used in the corresponding Eq. 44. In the case of gas injection down the annulus of a well, d,i5 of Eq. 57 (or d; 5 of Eq. 60) is replaced as defined in Eq. 49; that is,



(57)



In the case of annulus injection replaced as follows: d,s05X=(d~.,+d,o)‘035(d~;-d,,,)3



where D = depth of well,



using Eq. 58. d,, 5.058 is



‘*j.



.(61)



ft,



Ap = p2-PI, psia, d,; = ID of tubing, ft, qx = gas flow, lo6 cu ft/D at 14.65 psia and 60°F. f = friction factor (Fig. 34.2), and D,, = D under static conditions (static equivalent depth for pressures encountered at flowing conditions) 53.2417



Using the expression for the friction factor as derived by Cullender and Binckley ” (Eq. 45) and substituting in Eq. 57 gives



Eqs. 57 through 60 provide a basis for calculating the BHP in a gas-injection well. In solving Eqs. 57 and 58, the calculating procedure is to assume a pressure pl and solve for the corresponding depth, D. The depth, D. so found will be the depth at which pressure p2 occurs. By calculating several such points, a pressure-depth traverse can be plotted from which the pressure at the desired depth can be determined. It is apparent that BHP during gas in,jection can be either greater or less than tophole presaurc dcpcnding on the energy losses encountered. At low rates of flow. the pressure gradient is positive, whereas at high flow rates. the pressure gradient is negative. This is because. as flow rate increases, energy or frictional losses incrcasc and they can be overcome only by a dmm~.s~~in the (./IMI,~Po/‘M?Iprcxsior~ energyor pV energy of the system. The decrease in potential energy resulting from elevation is constant and the change in kinetic energy is negligible. This can be illustrated by examining and rearranging Eq. 4 and considering the kinetic energy negligible.



D=



I’I



C’dp+E,=-KilZ. CS,,



(62)



For low flow rates,



,,,..,....,..................



Cullender to calculate follows:



(58)



and Smith’s Eq. 36 also can be rearranged the BHP for the case of gas injection as



.



.



(59)



-F’



The solution of this equation is identical to that previously described for flowing gas wells. D, depth of well, can be used interchangeably with L, length of flow string, when the well is vertical. Similarly, by considering the downward flow of gas, the simplified equation developed by Smith lo for upward flow (Eq. 44) can be rearranged so that the pressure traverse for vertical flow downward can be calculated.



eSPth



2-pbh2=



25fq, * T2z(eS - 1) (),fJ375di5



‘.‘....



(60)



[“‘Vdp



is positive and Eta is always positive; thus, the sum of the compression energy and energy losses must equal the change in potential energy, which for a given depth is constant (the absolute value of -AZ is positive for gas injection since the absolute value of AZ is negative). As E,, increases with flow rate. the



must decrease for the sum to remain constant. When E,, is equal to (g/g(.) AZ, the pressure at the top and bottom of the hole is the same. This means that the decrease in potential energy is equal to the frictional losses. As E,, further increases, the added energy to overcome friction losses must come from the compressional energy since -(g/g:,.) AZ is constant. This then means that the pressure gradient is negative.



34-30



PETROLEUM



TABLE



L



L 680 700 720 740



20 20 20



(6)



1.015 1045 1.074 1.104



1.586 1611 1636 1662



the pressure



0.025 0.025 0.026



at 4.000 ft



well. Given:



tubing ID, d,, = 0.1663 ft. gas flow rate, qs = 0.783~10"



cu



HANDBOOK



CALCULATIONS



a-



Example Problem 5. Calculate in a gas injection



34.5-SAMPLE



ENGINEERING



1,276 1.278 1.329



(7)



(8)



- 1.460 - 1,460 - 1,532



0 1,460 2.920 4,452



3. Assume values for Ap and solve for D (Table 34.5). 4. From plot of Cal. 2 vs. Col. 8 read pressure at 4.000 ft to be 734 psia.



Oil Wells Inflow Performance



, r average temperature, T wellhead injection pressure, p, gas gravity, yY gas viscosity, p”c



Solution. 1. Substitute



D=



= = = =



0 60!‘“,: 680 psia, 0.625. and 8.74~10~~ Ibmift-sec.



The simplest and most widely or backpressure equation used pseudosteady-state flow at any by the productivity index (PI) y. =J(pR -P,,.~). In terms of measured



given values in Eq. 58.



D,



J=_--,



-1



(4)'



+b,,, Pw



D,=



data the PI is represented



as



(64)



wf



where J= stabilized productivity index. STBID-psi. Yo = measured stabilized surface oil flow rate, STB/D. P l1.f = wellbore stabilized flowing pressure, psia, and average reservoir pressure, psia. PR =



D, (3.00x10-')D,,'



-'



(63)



.



P R -P



2.944x10~R(0.783)'9.7s(0.625)'93sD,2 (0.1663)5058(8.75x10-h)-"ohs(~p)~



used inflow performance to determine stabilized or backpressure pl,f is given equation as



J is defined specifically as a PI determined from flow rate and pressure drawdown measurements. It normally varies with increasing drawdown (i.e., is not a constant value). In terms of reservoir variables, the stabilized or pseudosteady-state PI J* at zero drawdown or asp ,s-f’-+pR can be written as 7.08kh J*=



2. Determme



p,,< and T,,, (Fig. 34.3)



p,, 1,000 md. In all cases, oilwell backpressure curves were found to follow the same general form as that used to express the rate-pressure relationship of a gas well: Y~,=J'(F~~-~,,~~~)~I.



. . . . . . . . . . . . . . . . . . . ..(69)



For the 40 oilwell backpressure tests examined, the exponent n was found to lie between 0.568 and 1 .OOO-that is, within the limits commonly accepted for gas well backpressure curves. In terms of measured data, J' is defined by



(67)



where pea is the reservoir ary, psia. and



pressure



at the external bound-



Calculations using Eq. 67 with typical reservoir and tluid properties indicated that PI at a fixed reservoir pressure l>,, decreases with increasir,g drawdown. This apparently complex form of an inflow-performance-relationship (IPR) equation found littlc use in the field. In a computer study by Vogel. ” results based on twophase flow theory were presented to indicate that a single empirical IPR equation might be valid for most solution-gas-drive reservoirs. He found that a single dimensionless IPR equation approximately held for several hypothetical solution-gas drive reservoirs even when using a wide range of oil PVT properties and reservoir relative permeability curves. The fact that his study covered a wide range of fluid properties and relative permeability curves to obtain a single reference curve cannot bc ovcremphasiLcd. Vogel proposed that his simple equation bc used in place of the linear PI relationship for solution-gas-drive rehcrvoirs when the reservoir pressure is at or below the bubblcpoint pressure. The proposed equation (IPR) in dimensionless form was given as



(70)



where J’ is the stabilized PI, STBiD (psi ‘)‘I. The exponent n usually is determined from a multipoint or isochronal backpressure test and is an indicator of the existence of non-Darcy flow. If n = I, non-Darcy flow is assumed not to exist. With PI expressed in terms of pressures squared. jR 2 and P$,



J’=J”.



. . . . . . . . . . . . . . . . . . . . . . . . . . . . . ..(71)



%R



Expressing the pseudosteady voir variables.



state J’ in terms of reser-



7.08kh



J’= 2FR[,n(ry+s]



w,,n.



,,..(72)



or



7.08kh qiJ= [ln(r?)



+s]



w,>R



(73)



PETROLEUM



34-32



Expressed in a form with reservoir variables and a nonDarcy flow term. Fn,,, where the resulting n would be less than 1.0 and a function of FD,,, 7.08kh



1



. (Pi?? -P,/) 2pR



.



...



....



....



(74)



When pR is equal to or less than the bubblepoint pressure ph and n is less than I, a non-Darcy flow factor, F m, is indicated. When FDc, =O, n= 1. The term FL,,, normally is developed from multipoint test data. As shown in a later example, it is possible to have For, =0 and tz less than I .O for undersaturated wells producing at llowing pressures below the bubblepoint pressure. (See Fig.8 of Ref. 16.) This is strictly a result of the shape of the k,,,i(l,,B,,) pressure function. Expressing the backpressure form of the IPR equation in terms similar to that of Vogel’s equation (instead of Vogel‘s equation in terms of the backpressure curve), we have. from Eq. 69,



ENGINEERING



Example Problem 6 (IPR). The following



example illustrates the various possible methods of computing inflow rates. An oil well is producing at a stabilir.ed rate of 70 STBiD at a flowing BHP paf = 1,147 psia. The average reservoir shut-in static pressure, PR = 1,200 psia. Calculate the maximum possible flow rate, y(,, at 0 psig, and the producing rate if artificial lift were installed to lower the flowing BHP to 550 psia. Make the calculations using the PI Eq. 63. Vogel’s method, and the backpressure curve method with n= I .O and n=0.650. (The data are from an actual IPR test reported in Ref. 16.) Productivity



Index (PI) 70



J=



= 1.32 STBiD-psi: 1,200-I.147



q,, (15 psi)=J(FR-pLL~~) =I.32



(1,200-15)=1,564



STBID;



q,, (550 psi)= 1.32(1,200-550)=858 Vogel IPR



y,, =J’(pR 2 -p,,t2 I” q



and



=70



B(,pD.



pd’=



40,111;,\,



(,)K21,,



Substituting



(75)



and rearranging



yields = l-0.191



For tI = I , we have the simplest possible form of a multiphase IPR equation based on results obtained front actual field data: YII ---=IYocmax)



~“147 =0.9558; I.200



PI,j 1=0.9136; (4PR



or J’=-



0



’ PR



=J’(p,?)” 40,111,,x,



( I’



I’R



7 > .,,.,.,,,...........



Comparing Eq. 77 to Vogel’s Eq. 68. which was derived only from computer sitnulation data. we see that the coefficient for ~,,,/j~ is 0. and the coefficient for (P,,~/ pK)? is equal to 1. This results in an IPR Eq. 77 that yields a slightly more conservative answer than given by Vogel’s original equation. (Actually, Vogel’s Fig. 7 show\ computer model calculated IPR results less than obtained from his reference equation. ‘“) Not included in any of Vogel’s simulation runs were cffccts of non-Darcy 110~ in the reservoir or perforation restrictions. which in the field result in II values less than I .O and an even more jevcrc IPR rate reduction relationship.



16-0.73088=0.07796:



and y,, at p,,~ = 15 psia. 4,,(15 psi)



(77)



HANDBOOK



15 __ ( 1,200 >



=I-0.20



4 idImax)



=0.99738;



4,,(15 psi)=y,,,,,,,,(O.99738) =898(0.99738) yi) at pl,,=550 y,,(550



= 896 BOPD: psia.



psi) =I -0.20



4,,l,,l~~X,



550 ~ ( I .200 >



STBID.



WELLBORE



HYDRAULICS



550 ~ I .200



( i



-0.80



34-33



?



70



J’=



=0.740277;



y,,(SSO psi)=q,,,,,,,,(O.740277)



=0.0341580



=X98(0.740277)=665 BOPD.



q(, (15 psi)=J’(jjRz



Backpressure



Curve



2.049.3



STBiD-psi*“; -pb!fZ)c.hsO



=0.0341580(1,440,000-225)”



(n= 1 .O) IPR



70



=-



(l,440.000-l,315.609)0~h50



6s0;



q,, (15 psi)=0.0341580(10,066.8)=344



BOPD;



q,, =70 BOPD; FR’ =( 1,200)’ = I ,440,OOO: q.



(5.50 psi)=0.0341580(l.440,000-302.500)”~6so



p,,f.2=(l,147)~=l.315,609: =295



BOPD



70



J’:



(1.200)



-(1.147)’



70 =~=0.00056274 124.391 (/,,(I5



STB/D-psi’:



psi)=J’(pR



=0.00056274



Using the dimensionless



backpressure



Of



and



4o/q,,(,,,,,,



> P&JR,



curve form in terms



n=0.650,



y,?,f~,~ with tl= 1.O.



y,, = 70 BOPD;



(z)‘=



(~)‘=0.9136:



q,, at pI,f =550



=344



BOPD:



psia.



4,,(550 psi) q,,,,,,,,,) = [t - ( j=)



y,, =344(0.857892)=295 40



= I -0.9136=0.0864;



Y,,~,,,,,] 70 4~~lnu~,= ~o,0864



y,, at p,,f=s50



=8lO



BOPD;



psia.



‘1 “‘h.50=0.857892:



BOPD.



Again. this example is based on field data where several rates were measured to establish the real IPR relationship of the well. The real absolute open flow of the well was 340 BOPD. This is 38% of the rate predicted by Vogel’s IPR equation and 42% of the rate predicted by the backpressure equation with n = 1. A value of tz=0.650 as illustrated in this example is required to match the field data. A non-Darcy flow factor FD,, is indicated for this test.



y. (550 psi)



Single-Phase and Two-Phase IPR Equation. Fetkovicht6 gives a general equation that treats flow both above and below the bubblepoint pressure for an undersaturated oil well.



4i,(ln;,rl = 1-O. 168056=0.78993:



[/,,(550 psi)=81 O(0.78993) =640 Backpressure



Equation



BoPD



(t1=0.650)IPR



4,~ =l*(PR



(78)



-Ph)+J’(/J/>‘-/J,,,.‘).



where



L/,, =70 BOPD; pK =(I ZOO)’ = 1.440.000; /~,,,=(1,147)‘=1,315.609:



J’=J*(~,,BI,),,R,,,,,



(



1



)



34-34



PETROLEUM



Assuming (p(,B,,) is a constant value blepoint pressure equal to (pLoBo)h (the stant PI assumption for flow above pressure, oh), then a1 = l/[Ph(~~,B,~)h] Ref. 16).



above the bubbasis of the conthe bubblepoint (see Appendix of



9() at p1,f=550



9J550



ENGINEERING



HANDBOOK



psia



Ps9=J*(PR-pb)+&(pb2



=0.045454(3,200-



-p,J)



1,800)



Then 0.045454 J”(c(n~o)h



J’=



+



J* =2p,T,



..,



.



2Phh43,~)h



(1,800* -5502), 2( 1,800)



(80)



=64+0.000012626(3,240,000-302,500), Substituting Eq. 80 into 78 we obtain the final form of the single-phase and two-phase IPR equation: J* y,, =J*(I-‘R-P,,)+-(P/,*



-p&.



2Ph



(81)



Example Problem 7. The following example illustrates the method of computing inflow rates for flows both above and below the bubblepoint pressure of an undersaturated oil well. An oil well is producing at a rate of 50 STB/D at a flowing BHP of 2,100 psia. The reservoir average shut-in pressure is 3,200 psia with a bubblepoint pressure of 1.800 psia. Calculate the maximum possible flow rate, 9,. at p,!f=O psig and the producing rate at 5.50 psia flowing BHP. (For flows above I>/,, J=J*.) 90



J=J*=



=64+37=



101 BOPD.



The additional 535-psi pressure drop from 550 psia to 15 psia results in only 4 BOPD increase. It is significant to point out that if several flows, all with flowing pressure p ,f below the bubblepoint pressure pb, were calculated usmg the above equation and example and then plotted as a backpressure curve but with pR’ -~,,f’, it would indicate a value of n =O. 820. We would have an indicated n less than 1 .O without a non-Darcy Bow term Fo, With the uncertainty involved in really knowing the true bubblepoint pressure of a particular well, we could obtain test n values less than 1 .O without non-Darcy flow existing. To illustrate more clearly a case of drawdown data obtained at flowing pressures below the bubblepoint pressure to obtain J*, we will use the 550 psia rate obtained above and the previously specified data. Actual unrounded calculated rate is 100.73 BOPD.



GR -PM/) ’ therefore,



50



J*=



50



(3.200-2.100)



90



J*=



(pR-Ph)+



1,100



(Ph27hf2)



@h =0.045454



1 ’



STBiD-psi 100.73 (3,240,000-302,500)



and



(3,200-1,800)+ 2( 1,800)



I* 9(,(15 psi)=J*(PR-ph)+~(pb’-p,,.i2), %Jh



100.73



zz-



= (l,400+816) =0.045454(3,200-



1 ’



100.73 2,216



1,800) =0.045450



STBiD-psi



(good check)



0.045454 +



(1,800’-15’) 2( 1,800)



Future Inflow Performance. Standing ” presented a method for adjusting IPR by using Vogel’s equation from a measured condition to a future reservoir pressure pR, It is based on the fact that PI can be defined uniquely only at a zero drawdown, pl$-‘pR.



=64+0.000012626(3,240,000-225). =64+41=



105.



This compares to 145 BOPD if the regular is assumed valid to 15 psia.



PI equation



J*=



lim J. Ap+O



.



.



(82)



WELLBORE



Applying yielded



J*=



HYDRAULICS



the limit



34-35



condition



1.89,>cmaxi pR



Using the same approach and II= I.



using



Vogel’s



.



(83)



with the backpressure



PI=f(Ap)assumpmn



equation -kro 40



40 -= %(max) -



equation Fig. 342%Simple n=l.



P



pressure function for Ap2 relationship and



400 = [1+830)*]=lJ~~BoPD~



which yields



J*=-.



...........................(84) PR



240,max) 2(1,152)



J*=-m--=



If we define 90*(max, as that absolute open flow potential we would obtain. assuming conventional Ap PI were used. qo~max) =J*(PR -0)



-=1.017. 2,265



PR



~P,,B,,~~=1,0170.1659 -=0.755.



J*.f=J*P



0.2234



and qo*(maxl=J*jTR =2qorm3rr.



.



(85)



Note that the “real” qocrnaXj is % that assuming a Ap productivity index relationship. This is more clearly seen from Fig. 34.5 and Eq. 86. In terms of the EvingerMuskat equation,



J*&jR) 9omlax)~= L 2



Example Problem 8. Using Standing’s example data we will (1) calculate present J*,, from present flow data, (2) adiust J*, to a future J*f, and (3) calculate a future rate at p ,,f = i:200 psig ’ The following was given in Standing’s example. I7 The present PI, J, was determined to be 0.92 at a flow rate of 400 BOPD with pIIf= 1,815 psig. Average reservoir pressure. pR, at this time is 2,250 psig. Future reservoir pressure jR will be 1,800 psig. k,,/(pr,B,,)=0.2234 present and 0.1659 future.



qdmax’ []- ($2, =



2



=685 BOPD,



[l+L)2] Psk)=q,~,,,,~~ =685[1-(%)‘I



under curve.



For the n = 1 .O IPR relationship, the area under the curve (A, C, D) is exactly 1/2that area (A, B, C. D) assuming Ap PI relationship when p,,l=O.



0.755( 1,800+ 15)



and



90f(1,200



where A,.=area



=



=378



BOPD.



Multiphase Flow Introduction Much has been published in the literature on the vertical simultaneous flow of two or more fluids through a pipe. The general problem of predicting the pressure drop for the simultaneous flow of gas and liquid is complex. The problem consists of being able to predict the variation of pressure with elevation along the length of the flow string for known conditions of flow. The ability to do this in the case of flowing oil wells provides a means of evaluating the effects of tubing size, flow rate, BHP, and a host of other variables on one another. In the case of gas lift installations in oil wells, it would be particularly useful in designing the installation and providing such information as the optimum depth, pressure, and the rate at which to inject the gas, the horsepower requirements to lift the oil, and the effect of production rate and tubing size on these quantities. In other words, a means of systematically studying the effects of the different variables upon one another.



34-36



PETROLEUM ENGINEERING HANDBOOK



Theoretical



Considerations



As discussed in the Theoretical Basis section. the basis of any fluid-flow calculation consists of an energy bolancc on the fluid flowing between any two points in the system under consideration. The energy entering the system by virtue of the flowing fluid tnust equal the energy leaving the system plus the energy interchanged between the fluid and its surroundings. The pressure drop in a vertical pipe associated with either single- or tnultiphase flow is given by 7,dD + KP dD+ X’ ~ p-1,. 144 144g,. 144g,



-dp-



(87)



where Fig.



34X-Flow regime classifications for vertical two-phase flow.



Multiphase flow may be categorized into four different flow configurations or flow regimes, consisting of bubble flow. slug flow, slug-mist transition tlow. and mist flow. In bubble flow, the liquid is continuous with the gas phase existing as bubbles randomly distributed (Fig. 34.6). The gas phase in bubble flow is small and contributes little to the pressure gradient except by its effect on the density. A typical example ofbubble flow is the liberation of solution gas from an undersaturated oil at and above the point in the flow string where its bubblepoint pressure is reached. In slug flow, both the gas and liquid phases significantly contribute to the pressure gradient. The gas phase in slug flow exists as large bubbles almost filling the pipe and separated by slugs of liquid. The gas bubbles arc rounded on their leading edge, fairly flat on their trailing edge. and are surrounded on their sides by a thin liquid film. Liquid entrainment in the gas phase occurs at high flow velocities and small gas bubbles occur in the liquid slug. The velocity of the gas bubbles is greater than that of the liquid slugs. thereby resulting in a liquid holdup that not only affects well friction losses but also flowing density. Liquid holdup is defined as the insitu flowing volume fraction of liquid. Slug flow accounts for a large percentage of two-phase production wells and, as a result, a good deal of research has been concentrated on this flow regime. In transition flow, the liquid slugs between the gas bubbles essentially disappear, and at some point the liquid phase becomes discontinuous and the gas phase becomes continuous. The pressure losses in transition flow are partly a result of the liquid phase, but are more the result of the gas phase. Mist flow is characterized by a continuous gas phase with liquid occurring as entrained droplets in the gas stream and as a liquid film wetting the pipe wall. A typical example of mist flow is the flow of gas and condensate in a gas condensate well. Complete sets of pressure traverses for specific flow conditions and oil and gas properties have been published by service companies and others. These pressure gradient curves can be used for quick hand calculations.



p = pressure. psia. Ibfisq ft-ft. = friction loss gradient. D= depth, ft. of gravity. ftisec’. h’= acceleration gravitational constant, (ft-Ibm)/(lbf SC = fluid density. Ibm/cu ft. and P= \’ = fluid velocity, ftiscc. Tf



SW’),



Eq. 87 states that the fluid pressure drop in a pipe is the combined result of friction. potential energy. and kinetic energy losses. The friction loss gradient and average density term for multiphase flow are evaluated using specific relationships for each flow regime. The kinetic energy term is usually small except for large flow rates. Duns and Roa Ix have shown that for two-phase flow the kinetic energy term is significant only in the mist flow regime. Under this flow condition. 1*$B 1’1.. and the kinetic energy term can be expressed as



p”dlr= Kc,



-5%. I



(88)



where A = pipe area. sq ft, M’, = total mass flow rate, lbmisec, and 4x = gas volumetric flow rate. cu ftisec. Eq. 87 now can be written in difference form for any depth increment, i, by assuming an average temperature and pressure exists over the increment. Making this assumption we have



AP,=&(,-“:;~,~ )AD;s 4637A’j where p Ap; p ADi



= = = =



average fluid density, lbmicu ft. pressure drop for increment i, psi. average pressure, psia. and the ith depth increment. ft.



(89)



WELLBORE



34-37



HYDRAULICS



Eq. 89 can bc solved incrementally either by settrng -$, and solving for AL), or by setting ;1D, and solving for Al>, Since pressure usually has more effect on average fluid properties than temperature and since rempcraturc can be expressed as a function of depth. &I, should be set and AD, calculated. The calculation procedure described here is an iterative process for each section and generally is programmed for solution on a computer.



Correlations Since the original work in this area, which was presented by Poettmann and Carpenter.3’ several studies have been undertaken to collect additional experimental multiphase Bow data and to develop new multiphase pressure drop correlations. I’~“) Also. various statistical studies have been performed comparing recent multiphasc flow correlations3”~iZ for large sets of flowing and gas lift cases. Espanol et cl/. ‘(’ selected the Hagedorn and Brown.” Duns and Ros. Ix and Orkiszewski” methods as three of the beat correlations for calculating multiphase pressure drops. An analysis of results calculated on 44 wells was used to determine the best overall correlation. This work concluded that the Orkiszewski correlation was the most accurate method over a large range of well conditions and it was the only correlation of the three considered suitable for evaluating three-phase flow for wells producing significant quantities of water. Lawson and Brill”’ point out that the Poettmann and Carpenter method is still a base line for comparing new multiphase flow correlations. Their original work is based on flow conditions similar to those found in many gas lift conditions and, therefore, is briefly discussed



Poettmann and Carpenter.>” Poettmann and Carpenter used data on flowing and gas lift wells to correlate the combined energy losses resulting from liquid holdup. frictional effects caused by the surface of the tubing, and other energy losses as a function of flow variables. No attempt was made to evaluate the various components making up the total energy loss. The flowing tluid was treated as a single homogeneous mass. and the energy loss was correlated on this basis. A total flowing density or specific volume was used rather than an in-situ density or specific volume. That is, the energy of the fluid entering and leaving the tubing is a function of the pressure-volume properties of the total fluid entering and leaving the tubing, and not of the pressure-volume properties of the fluid in place, which would be different because of slippage or liquid-holdup effects. Lastly. in calculating flowing density or flowing specific volume, mass transfer between phases as the tluid flows up the tubing was taken into consideration, as well as the entire mass of the gas and liquid phases. Viscosity as a correlating function was neglected. The degree of turbulence is of such a magnitude, in general. for a two-phase flowing oil well that the portion of the total energy loss resulting from viscous shear is negligible. This is not surprising since it is also true for singlephase turbulent flow. There the energy loss is independent of the physical properties of the flowing fluid. A



number of others* working on the same problem of multiphase flow have made the same observation. Baxendell extended Poettmann and Carpenter’s correlation by using large-volume Bow data from wells on casing flow,. ” A detailed discussion of the Poettmann and Carpenter development can be found in the original 1962 edition of this handbook and in Ref. 33. The Poettmann and Carpenter correlation has served as the take-off point for many of the newer multiphase flow correlations.



Orkiszewski. To obtain a set of calculation procedures covering all flow regime:; in two-phase flow. OrkiszcwskiZs made a thorough review of the literature. tested various methods against a few sets of experimental data by hand calculations. and then selected the two methods, Griffith and Wallis ” and Duns and Ros. Ix for his final evaluation. Orkiszewski programmed both methods and tested them against data from I48 wells. Neither method was accurate over the entire set of flow conditions, Griffith and Wallis’s method. however. appeared to provide the better foundation for a general solution in slug flow, and, thus. Orkiszewski clccted to modify their work. Orkiszewski called his calculation procedures the Modified Griffith and Wallis method since their work was involved strictly with fully developed slug flow and since 95% of the 148 wells used by Orkiszewski in developing his method were in slug flow. Duns and Ros’ method was used for mist flow and partly for transition flow since it appeared to be more fundamental than the Lockhart and Martinell?j method recommended by Griffith. Orkiszewski’s method essentially establishes which tlow regime is present and then applies (1) Griffith’s proccdure for bubble flow, (2) Griffith’s procedure modified by a liquid distribution coefficient parameter based on field data for slug flow. (3) a combination of the modified Griffith method and the Duns and Ros method for transition flow. or (4) Duns and Ros’ method for mist flow. Accuracy claimed for this correlation is about k 10% for a wide range of flow conditions. The determination of which flow regime applies for a given pipe segment is accomplished by checkmg the various dimensionless groups that define the boundaries of each flow regime (Fig. 34.7). Griffith and Wallis are responsible for defining the boundary between the bubble and slug flow regimes. Duns and Ros have defined the boundaries between the slug and transition tlow regimes and between the transition and mist flow regimes. These boundaries are given by the inequalities listed below. I. For the bubble flow regime, the boundary limits are Y&ILB.



the boundary



limits arc



l’,Dtx,~~>Ls. 4. For the mist flow regime, the boundary limits are “,yr,>LM. In these equations the subscripts 5, M, and S indicate bubble. mist, and slug flow. respectively. ‘Earlyinvesllgatorsof lhtsproblem were T.V Moore and H D WildeJr, “ExperlmenfalMeasurement of SltppageI”Flow Through Vertical Popes,”Tram, AIME (1931j 92, 296-313; and TV Moore and R.J Schllthuls. “Calculation of PressureDrops \n FlowingWells.”Trans AIME (1933)103, 170-86.



PETROLEUM



34-38



ENGINEERING



HANDBOOK



Bubble Flow. The average flowing density in bubblej~w is calculated volumetrically



from the following equation, which weights the gas and liquid densities.



P=PgfK+f(l-fg)PL. The flowing



..(..........(....,...



(94)



gas fraction, fg, in bubble flow is given by



/



0 3 2



I



&=~[l++p$jg,



:’



4 z 5 0 ci I5 > z 0.110 ,ol



:,a1*e :, *#‘,-, A:: : .*;/ ;A: I::‘**l:: I PL”G FLOW ^. 2



5



,



2



5



DIMENSIONLESS



1.



2



5



,$



2



I



,$



where the slip velocity, v, , is the difference between the average gas and liquid velocities. Griffith suggests the use of an approximate value of v,=O.8 ft/sec for bubble flow.



GAS VELOCITY. V,,b,lga)ozs



Fig. 34.7-Flow



. . ..(95)



The friction loss gradient single-phase liquid flow,



regime map.



for bubble flow is based on



2



These dimensionless set of equations.



groups are given by the following



7f= 2g,.d” fp L”L cos*)



..



.



(94)



. .



where



v 8



( . . . . . . . . . . . . . . . . . . . . (90)



A



4L . . . . . . .



YL=A(l-fg).



at the bubble-slug



. . . .



. . . . . . . .



.



. .



. (97)



boundary



0.2218v,* )



Lg=1.071-



. . . . . . . . . . . . . . . . . . (91)



dH



The friction factor, f, in Eq. 96 is the standard Moody * friction factor, which is a function of Reynolds number and relative roughness factor. The Reynolds number that is used for bubble flow is the liquid Reynolds number.



but L,rO.13, at the slug-transition



Ls =50+



1488PLdHvL ccL



NR~=



boundary



36VgD4r.



. . . . . ..I..............



(92)



.



Slug Flow. The average density term for sIugflow pressed as



. (98)



is ex-



boundary



p’ “‘I +PLVd LM=75+84(VgD4L)‘.“, \ 9g ’



..



where dH is the hydraulic pipe diameter (4Alwetted perimeter), ft, and hL is the-liquid viscosity, cp.



4a and at the transition-mist



) .. .. .



. . . . . . . . . . . . . . (93)



where vgD = dimensionless gas velocity, V t= total fluid velocity (9,/A), ft/sec, pi = liquid density, lbm/cu ft, u = liquid surface tension, lbm/sec*, L = flow regime boundary, dimensionless, dH = hydraulic pipe diameter, ft, qg = gas flow rate, cu ftisec, g = acceleration of gravity, ftisec2, and A = flow area of pipe, sq ft. The average density and friction loss gradient is defined later for each of the four possible flow regimes. These terms are evaluated for each pipe segment and are then substituted into Eq. 89 to calculate the pressure drop over the segment.



+6pL.



.



. . .



9t +vbA



. . . . . . (99)



Eq. 99, with the exception of its last term, is equivalent to the average density term derived by Griffith and Wallis. The last term of Eq. 99 was added by Orkiszewski and contains a parameter, 6, that was correlated from oilfield data. The slip or bubble rise velocity, vb, for slug flow was correlated by Griffith and Wallis and is given by vb=c,c&&.



.. .. . .. . .



. . . . . . . . . (I@,)



The coefficient Ct is the bubble-rise coefficient for bubbles rising in a static column of liquid. Values of Ct have been determined theoreticallv bv Dumitrescu 36 and experimentally by Griffith and Wajlis l9 as a function of bubble Reynolds number, Fig. 34.8, where 1488pLdHvb . . . . . . . . . . . . . . . . . . . . .(101)



NR~, = CLL



WELLEORE



34-39



HYDRAULICS



The coefficient C2 is a function of liquid velocity and, when multiplied by Ct , represents the bubble-rise coefficient for bubbles rising in a flowing liquid. The coefficient C2 has been determined experimentally by Griffith and Wallis I9 and is correlated as a function of both bubble Reynolds number, NReh , and liquid Reynolds number (Fig. 34.9), where 1488pLdHv, . . . . . . . . . . . . . . . . . . . . . (102)



NR~ =



When Reynolds numbers larger than 6,ooO are encountered, vh can be evaluated from the following equations, which were developed by Orkiszewski and based on the work of Nicklin et al.” For bubble Reynolds numbers, NRC,, . less than 3,000, ,,i,=10.546+8.74(10-6)NR,jJgdH.



BUBBLE



REYNOLDS



NUMBER



N,,



= ~ PL



.(tO3) Fig. 34.8-Bubble-rise



When bubble 8,000,



Reynolds



number



is between



3,000 and



where _. .



r~,,,=[0.251+8.74(10-6)N,,]v&. For bubble



Reynolds



numbers



greater



(105)



than 8,000.



,,,,=[0.35+8.74(10~6)NR~]~.



.(106)



The friction loss gradient term for slug flow is the result of Orkiszewski’s work and is given by



T.f=



fpL”i2 (“‘““+A). 2g,dH cos0 q,+\‘/,A



0



1,000



,..,....



(107)



2,000 REYNOLDS



Fig. 34.9-Bubble-rise



coefficient for bubbles rising in a static liquid column vs. bubble Reynolds number.



The friction factor in Eq. 107 is a function of relative roughness and the Reynolds number given by Eq. 102. Orkiszewski defined the parameter 6, which appears in Eqs. 99 and 107 as a liquid distribution coefficient. This coefficient implicitly accounts for the following physical phenomena. 1. Liquid is distributed not only in the slug and as a film around the gas bubble but also as entrained droplets inside the gas bubble. 2. The friction loss has essentially two contributions, one from the liquid slug and the other from the liquid film. 3. The bubble rise velocity approaches zero as mist flow is approached. Liquid distribution coefficient, 6, was correlated as a function of liquid viscosity, hydraulic radius, and total velocity and may be evaluated by one of the following empirical equations.



3,000 NUMBER



4,000



5,000



#Re = 1’488Aq:PHp’



coefficient accounting for bubbles rising in a flowing liquid vs. Reynolds number.



6,000



34-40



PETROLEUM



0.0127 6= ,,log(/.q+l)-0.284+0.167 dH



log V,



fg2L qg+qL.’ log dH,



. .



...



.



Average P=(l



0.0274 ,,37, log(fiL + l)+o. dH



-log



l’,



161+0.569



log dH



.



is given by



-fg)pL+fgpg.



..



..



(115)



and f is a func-



where vKs is the superficial gas velocity tion of the gas Reynolds number,



NRC = 1488



PXdHVRs



. ... ....



. . . . . . (117)



px log vI -0.428



log dH.



(110)



hf



i/H.



and a modified relative roughness factor, cldH, which was developed by Duns and Ros. The roughness factor for mist flow is a function of the liquid film wetting the pipe walls and is given by the following set of equations and constraints. Let



162 log v, _.



~=~.~~(~~~‘)(v~~~~/u)~(P~IP~),



. ..(lll)



Eqs. 108 through II 1 are constrained by the following limits. which eliminate pressure discontinuitics between tlow regimes. When \*,< IO. 62 -0.065\*,, and when \‘, > 10.



6r-



density



.(109)



PL -0.681



0.045 o,799 log pLL-0,709-O. dH -0.888



flowing



(I 16)



When V, > 10,



6=-



(114)



+ 1)



Continuous Water Phase. When vy < 10.



+0.232



.. .....



log d,c/



I



0.013 6= -log dH



... .....



The friction loss gradient for mist flow is primarily a result of the gas phase and is given by



0.01 ~WPL dti



+0.397+0.63



...



. ..(108)



When v, > 10,



6=-



HANDBOOK



Mist Flow. In mistjbw the slip between the gas and liquid phases is essentially zero. The fraction of gas flowing can be expressed, therefore, as



Continuous Oil Phase. When 1’, < 10,



+O.l13



ENGINEERING



v,,A(l --P/p,) q, + I’d



Transition Flow. The Duns and Ros method for calculat-



where N is a dimensionless



t



-= d,



number.



(118)



Then for N0.005, -5



-= dti



174.8~(N)‘-~‘* 2dH PKVRT



..



(120)



Eqs. I 19 and 120 are limited by upper and lower bounds for E/dH of 0.001 and 0.05. Camacho3’ studied 111 wells with high gas/liquid ratios and concluded that Orkiszewski’s method performed better when mist flow calculations were used for gas/liquid ratios greater than 10,000. Obviously, if this approach is taken, an appropriate transition zone between slug and mist flow should be used to avoid abrupt pressure gra.....(112) dient changes. In another study, Gould er a1.27also indicate that the onset of mist flow should occur at lower dimensionless gas velocities, especially for dimensionless liquid velocities less than 0.1. where subscripts M and S are mist and slug flow conditions, respectively. Similarly, the friction loss gradient Continuous-Flow Gas Lift Design Procedures is defined as Gas liftZ8,33.37 is a method of artificial lift that uses the compressional energy of a gas to lift the reservoir fluid (see Chap. 5). The prime requisite is an adequate source of gas at a desired pressure and volume. ing average flowing density and friction loss gradlent in r,nrz.sition,fk,,c, is used. They evaluated p and 7/ by linearly weighting the values obtained from slug and mist flow wsith dlmensionless gas velocity, v,~, , and the dimensionless boundaries defining transition flow, L,v and Ls. The average density term is defined as



j=(yps+(~)&,,



WELLBORE



HYDRAULICS



Wells having high water/oil ratios (WOR) and high productivity indices (that is, producing large volumes of fluid with high sustaining reservoir pressures) can be efficiently gas lifted through the tubing or the well annulus. Quite often it is necessary to produce very large volumes of water to obtain economic rates of oil production. Situations are known where it is possible to gas lift economically as much as 5,000 to 10.000 B/D total fluid, with the oil present being I % of the total fluid produced and the rest being water. In applying the correlations to gas lift design calculations, the following procedure is recommended. 1. Establish the flow characteristics of the well-i.e., productivity index, WOR, gas/oil ratio (GOR), fluid prop erties, tubing size, etc. 2. Calculate the pressure traverses below the injection point for the range of flow rates. 3. Calculate the pressure traverses above the point of injection for different injection GOR’s, holding the surface tubing or casing pressure constant. From these three steps, as illustrated in Fig. 34.10, the horsepower requirements, pressure at injection point, depth of injection, and injection GOR’s for a given rate of production, tubing size, and tubing or casing pressure can be calculated. For a given set of well conditions and fluid production, there is an optimum depth and injection pressure that result in minimum horsepower requirements. In some cases, the optimal injection depth will be at the total depth of the well. There are two ranges of operation in gas lifting a reservoir fluid. One is an inefficient range characterized by high GOR and high horsepower requirements, and the other is an efficient range characterized by low GOR and low horsepower requirements. A plot of GOR vs. mjection pressure is shown in Fig. 34.11. In the inefficient range of operation, gas literally is “blown” through the flow string. The efficient range is to the left of the minimum injection pressure, and the inefficient range to the right. Inefficient and efficient ranges of operation have been observed in the laboratory on experimental gas lift involving short lengths of tubing. 3840 One investigator used a large amount of field data from a California field to develop empirically curves similar to those shown in Fig. 34.11 but had no way of predicting these curves for other fields where the physical properties of the fluids and the production data were different. 4’ In a plot of horsepower requirement vs. injection pressure (Fig. 34.12) the horsepower generally passes through a minimum value, which represents the maximum efficiency of the operation. Another interesting result of these gas lift calculations has been to show that the lower the surface pressure of the flow string that can be maintained consistent with efficient surface operations, the less will be the horsepower required to lift the reservoir fluid. The use of the calculation procedure can best be expressed by use of a typical example problem.42



Example Problem 9.



It is desired to gas lift a well by flowing through the annulus. The well has a productivity index of 10.0 bbl total liquid per day per psi pressure drop. The static reservoir pressure is 3.800 psia at a well depth of 10,000 ft. The WOR is 18.33. Other pertinent information is as follows.



34-41



I,



(



f



DEPTH



Fig. 34.10-Pressure



traverse



in gas-lift well.



PRESSUR_E_ CONSTANT :



OIL RATE TUBING PRESSURE TUBING SIZE WATER-OIL RATIO



ki!



2 2 E



is



Fig.



INJECTION 34.1 l-Effect



GAS-OIL



RATIO -



of injection pressure



on injection GOR.



Tubing ID (2% in. nominal, 6.5 lbmift)=2.441 in.; tubing OD (2% in. nominal, 6.5 lbm/ft)=2.875 in.; casing ID (7 in. nominal, 26 lbm/ft)=6.276 in.; casing pressure= 100 psia; average flowing temperature in annulus above injection depth= 155°F; average flowing temperature in annulus below injection depth= 185°F; average flowing temperature in tubing= 140°F; gravity of stock-tank oil at 60”F=0.8390; gravity of separator gas (air= 1.0)=0.625; gravity of produced water= 1.15; 8=0.0000723p+ 1.114; R, =O. 1875p+ 17; and R=600 cu ft/bbl oil.



34-42



PETROLEUM



I CONSTANT: OIL RATE TUBING PRESSURE TUBING SIZE WATER-OIL RATIO



INEFFICIENT



5 -25



I-----t



EFFICIENT



RANGE



RANGE



% kc! P



-



INJECTION



Fig. 34.12-Effect



PRESSURE



of injection quirements.



pressure



on horsepower



re-



ENGINEERING



Calculate the variation of injection GOR with injection pressure and injection depth for a total liquid production rate of 4,000 B/D. Calculate the horsepower requirements to lift the oil as a function of injection pressure. The solution of the problem involves the following steps. 1. Calculate the pressure traverse below’the point of gas injection. 2. Calculate the pressure traverses above the point of gas injection for various GOR’s. 3. Solve 1 and 2 simultaneously to determine the depth of injection for various injection GOR’s and a casing pressure of 100 psia. 4. Calculate the theoretical adiabatic horsepower required to compress the gas from 100 psia to the injectionpoint pressure. The first step in the solution of this problem is the calculation of the flowing density of the three-phase fluid produced into the well as a function of the pressure. Using Fig. 34.13, the differential pressure gradients were determined as a function of fluid der$ty and, therefore, pressure. These calculations are illustrated in Table 34.6. These results then were placed on a plot of dDldp vs. p. The depth traveled by the fluid flowing from the BHP to any lower pressure was determined by integrating this curve. In this way, Curve A in Fig. 34.14 was determined. The second step of the solution was carried out mechanically the same as the first step, with the exceptions that the fluid densities were calculated for injection GOR’s of 3,000, 3,500, 4,000, 5,000, and 7,500 scfibbl, and that the integrations were carried out from the wellhead casing pressure of 100 psia to the pressures farther down the casing. The results of these calculations are shown in Fig.



20



dpldD, psilft



Fig.



HANDBOOK



34.13-Calculation of pressure traverses for flow in annulus Tubing size is 2% in. nominal (6.5 Ibmlft, 2.441-In. ID, 2.675in. OD). Casing size IS 7.0 in. nominal (26 Ibmlft, 6.276-in. ID).



WELLBORE



34-43



HYDRAULICS



TABLE 34.6-CALCULATION OF THE PRESSURE TRAVERSE BELOW THE POINT OF GAS INJECTION 4.000 ~ 19.33



40=



=206.9



q,m=l.594x106



BID



lbm/D



p=m,



7701.5 lbmlcu f!



V,



18.2W’O-



5.618+



R,)



+ 1o2,8



P/2 Flowing BHP = 3,400



psia



Establishing p vs. l/dp/dD P B 3,4001.339 3,000 1.331 2,000 1.259 1,000 1.286 500 1.150



P/Z



R, 588 392 205 110.8



P



3,800 3,440 2,270 1,078 520.8



dPldD



1 ldP/dD



69.80.487 69.8 0.487 69.0 0.481 66.3 0.460 60.9 0.425



2.053 2.053 2.079 2 174 2.353



I



! !



!



0



DP,



3,400 3.000 2,500 2,000 1,500 1,000 500



-DP,



0 821.2 - 1,849.7 - 2,884.7 - 3.933.2 - 5,004.7 -6,125.7



m7000



a



0



!



0



I



!



!



!



!



2 3 4 5 DEPTH,THWSANDS



Fig. 34.14-Pressure



!



! !



! I



I



3500



GOR vs. injection pressure.



34.14 as curves B, C. D, E, and F. The intersections of these curves with Curve A represent the injection points for these flow rates and injection GOR’s. The injection GOR is plotted as a function of the injection pressure at injection depth in Fig. 34.15. For the conditions of this example problem, it will be noted that the injection pressure continually decreases as the GOR is increased from 3,000 to 7,500 scfibbl. Fig. 34.16 shows the relationship between injection depth and injection GOR. This plot shows that. as the injection GOR is decreased, the point of injection is moved down the hole.



10,000 9,179 6,150 7,115 6,066 4,995 3,874



2500 I



! !



AD



--



821.2 - 1,028.5 - 1,035.o - 1,048.5 - 1,071.5 -1,121.0



! !



500 1000 1500 2000 2500 xxx) INJECTION PRESSURE,PSIA



Fig. 34.15-Injection



P



!



01 1 1 1 1 1 1 1 1 1 1 1 1 1 I



6



1



7 8 OF FEET



vs. depth.



9



I



’



’



’



’



m \



IO INJECTION DEPTH,THOUSANDS Fig. 34.16-Injection



OF FEET



depth vs. injection GOR.



34-44



PETROLEUM



ENGINEERING



HANDBOOK



El24 &22 $20 $118 =I16 kg; IpO JlO8 4106 El04 El02 $00 +



0



500



1000



1500



INJECTION



2000



2500



3ooo



PRESSURE



Fig. 34.17-Horsepower



vs. injection pressure



Fig. 34.18-Equipment



Fig. 34.17 shows the theoretical required to compress the injected



adiabatic horsepower gas from the surface



pressure to the injection



For the conditions



pressure.



of



this problem. the minimum horsepower is required when the injection point is at the bottom of the well, although. as pointed out in the earlier possible to obtain minimum



discussion, it is theoretically horsepower requirements at



points other than at the bottom of the hole. The literature reports an interesting series of well tests in which



curves



above completely



calculated



by the procedure



characterize



the gas lift



described



performance



arrangement.



of the well tested. ” Fig. 34. I8 shows the physical installation of the well tested. Tests were conducted at two points



of gas injection,



descriptions



3.800



and 4.502



of the tests are available



from



ft.



Detailed



Ref. 43.



Figs. 34. I9 and 34.20 show a comparison of the observed and calculated pressure traverses above the point of gas injection. The comparison indicates good agreement, Fig. 34.21 shows a comparison of observed data with curves calculated for average well conditions of total liquid



flow vs. rate of gas injection.



2800! c



2600



-



2600



-



2400



-



2400



-



2200



-



2200



-



2000



-



2000



-



o CALCULATED l OBSERVED



1800



-



1600



-



a 1600 -



5 1400



-



z n 1400



2 w 1200



1800



-



W



-



“3 1200-



5 u-j IOOO-



2 IJJ IOOO-



w” g 800-



8i



600



-



n o CALCULATED l OBSERVED



-



800



-



600



-



0’ DEPTH-



500



FEET



PER DWISION



Fig. 34.19-Calculated and field-measured pressure traversesinjection depth is 4,502 ft.



’ ’ ’ I I c ’ ’ ( ’ ’ ’ ’ DEPTH-500FEET



Fig. 34.20-Calculated



’ I ’



PER DIVISION



and field-measured pressure traverses-



injection depth is 3,810



ft.



WELLBORE HYDRAULICS



34-45



WATER-OIL RATIO 41.5 FORMATN)FJ GAS-TOTAL LIOUID RATIO 85.0 CU FT/E!ARREL TUBING PRESSURE IOOPSIA GRADIENT BELOW POINTOF INJECTION 0453 PSI PER FOOT TUBING SIZE ZINCH (4.7LB/FT-I 9951NCHES ID)



0.030



0



THOUSANDS OF CUBIC FEET OF GAS INJECTED PER DAY Fig: 34.21-Total



TOTAL BARRELSOF



liquid flow vs. rate of gas injection.



Fig. 34.22 is an example of a very useful type of plot that can be calculated for the optimum conditions of lift. It is a plot of ideal adiabatic horsepower per barrel per day of total fluid produced vs. total barrels of fluid produced per day under the conditions as indicated. Horsepower as used here is the horsepower required to compress the injected gas between the tubing pressure and injection pressure.



Flow Through Chokes A wellhead choke or “bean” is used to control the production rate from a well. In the design of tubing and well completions (perforations, etc.), one must ensure that neither the tubing nor perforations control the production from the well. The flow capacity of the tubing and perforations always should be greater than the inflow pert’ormante behavior of the reservoir. It is the choke that is designed to controi the production rate from a well Wellhead chokes usually are selected so that fluctuations in the line pressure downstream of the choke have no effect on the well flow rate. To ensure this condition, flow through the choke must be at critical flow conditions; that is. flow through the coke is at the acoustic velocity. For this condition to exist, downstream line pressure must be approximately 0.55 or less of the tubing or upstream pressure. Under these conditions the flow rate is a function of the upstream or tubing pressure only. For single-phase gas flow through a choke. the following equation is used:



Ye’



CP Jr,r,



,.....___.



.._



(121)



where p = 7,s = T = C = 4,s =



upstream pressure. psia. gas gravity. upstream or wellhead temperature. “R. coefficient, and flow rate measured at either 14.4 or 14.7 psia and 60°F. lo3 cu ft/D.



Fig. 34.22-Horsepower



LIOUID PRODUCED PER



requirements vs. total fluid produced.



The coefficient, C, will vary depending on the base pressure. Table 34.7 presents values of C taken from Rawlins and Schellhardt. 44 These values are for a standard pressure of 14.4 psia. Rawlins and Schellhardt did not make corrections for deviation from ideal gas. Correction can be made to Eq. 121 by multiplying the right side of the equation by ,&, where I is the compressibility factor of the gas at the upstream pressure p and temperature T. In the case of multiphase flow, Gilbert developed the following empirical equation based on data from flowing wells in the Ten Section field of California relating oil flow, GOR, tubing pressure, and choke size.4”



Ptf=



435R,, o.546q, sl,89 , ..



where ptf = R .SL = y, = S = Gilbert’s p,f=Aq,,



.



..



.



tubing flowing pressure, psig. gas/liquid ratio, IO1 scfibbl. gross liquid rate (oil and water), choke size in 1164 in. equation



may be written



BID, and



in the form:



..



(123)



TABLE 34.7COEFFICIENTS FOR CHOKE NIPPLE Orifice size (in.) 118 0.125 3116 0.188 l/4 0.250 5116 0.313 318 0.375 7116 0.438 112 0.500 5/8 0.625 3/4 0.750



.(122)



C 6.25 14.44



26.51 43.64 61.21 85.13 112.72



179.74 260.99



34-46



where A =435R,~,~0.5’6/Si.Xy and where the tubing pressure is proportional to the production rate. This is true only under conditions of acoustic flow through the choke. At low flow rates. the rate is also a function of the downstream pressure and Eq. 123 no longer holds. Ros presented a theoretical analysis on the mechanism of simultaneous flow of gas and liquid through a restriction at acoustic velocity. “.” The result was a complex equation relating mass flow of gas and liquid, restriction size. and upstream pressure. Ros’ equation was checked against oilfield data under critical flow conditions with good results. However. the equation is expressed in a form not really amenable to use by oilfield personnel. Using Ros’ analysis. Poettmann and Beck converted Ros’ e uation to oilfield units and reduced it to graphical form.’ 1 The result was Figs. 34.23 through 34.25 for oil gravities of 20. 30. and 40”API. The 20” gravity chart should be used for gravities ranging from I5 to 24”APl: similarly. the 30” chart should be used for gravities ranging from 25 to 34”. and the 40” chart for gravities ranging from 35” on up. The charts are not valid if there is appreciable water production with the oil. The charts can be entered from either the top or bottom scale. When entering from the GOR scale, go first to the tubing pressure curve and then horizontally to the choke size curve and then read the oil Bow rate from the top scale. Conversely, when entering the chart at the oil tlow rate scale. the reverse order is followed. Reliable estitnates of gas rates, oil rates. tubing pressures. and choke sizes can be made by using these charts. Chokes are sub.ject to sand and gas cutting as well as asphalt and wax deposition. which changes the shape and size of the choke. This. then. could result in considcrable error when compared to calculated values of flow for a standard choke size. A small error in choke size caused by a worn choke can effect a considerable error in the predicted oil rate. Thus. a cut choke could result in estii mated oil rates considerably lower than measured. From the inflow performance relationship of a well and by knowing the tubing size in the well, the tubing pressure curve for various flow rates can be calculated. The intersection of the choke performance curve for different choke sizes with the tubing pressure curve then gives one the wellhead pressures and flow rates for any choke size. as illustrated in Fig. 34.26.



Example Problem 10. a I. Determine the flow rate from a well flowing through a %,-in. choke at a flowing tubing pressure of 1,264 psia and a producing GOR of 2,2SO cu tiibbl. Stock-tank gravity is 44.4”. From Fig. 34.25, the solution is 60 B/D oil. 2. For this example. estimate the free gas present in the tubing. The solution gas at a tubing pressure of I .264 psia frotn Fig. 34.25 is R, =310 cu ftibbl. Then, the free gas present is R-R, =2.250-3 IO or I.940 cu ft/bbl of oil at the wellhead. 3. It is desired to produce a well at 100 BID oil. The producing GOR is 4,000 cu ftibbl. At this rate the tubing pressure is 1.800 psia. Estimate choke size. All three charts show estimated choke size to be %, in. Gilbert‘s charts also give Xj m.J A number of other choke design correlations have been suggested. However. Poettmann and Beck’s adaption of the Ros equation is recommended when no water is pro-



PETROLEUM



ENGINEERING



duced with the oil, and Gilbert’s when water is present.



equation



HANDBOOK



can be used



Liquid Loading in Wells Liquid loading in wells occurs when the gas phase does not provide sufficient transport energy to lift the liquids out of the well. This type of well does not produce at a flow rate large enough to keep the liquids moving at the same velocity as the gas. The accumulation of liquid will impose an additional backpressure on the formation that can affect the production capacity of the well significantly. Initially, the occurrence of liquid holdup may be reflected in the backpressure data obtained on a well wherein at the lower flow rates its performance, expressed as a backpressure curve, is worse than expected. Eventually, the well is likely to experience “heading” (fluctuating flow rates) followed by “load up” and cease to produce. Methods sometimes used to continue production from “loading” wells are pumping units, plunger lifts. smallerdiameter tubing, soap injection. and flow controllers. This section is directed mainly toward relating loading to flow conditions within the well. In the simplest context, loading. as reflected on a deterioration of flow performance at lower Bow rates on a backpressurc curve. is related to the superficial velocity of the gas in the conduit at wellhead conditions. Duggan’” found that a velocity of 5 ft/sec would keep wells unloaded whereas Lisbon and Henry” found that I .OOOftimin (16.7 ftisec) could be required. R.V. Smith”’ reported that experience with lowpressure wells in the West Panhandle and Hugoton fields showed that a velocity of 5 to IO ftisec is necessary to remove hydrocarbon liquids consistently and a velocity of 10 to 20 ft/sec is required for water. Turner er al. 5’ analyzed the problem of liquid holdup on the basis of two proposed physical models: (I) liquid film movement along the walls of the pipe and (2) liquid droplets entrained in the high-velocity core. They concluded, on the basis of comparisons with field data, that the entrained drop movement was the controlling mechanism for removal of liquids. Their results indicated that in most instances wellhead conditions were controlling and the fluid velocity required to remove liquids could be expressed by the-following equation.



l’, =



20.4&‘“(pL px



-p,q)“.2” 0.5



,



(124)



where \‘I = terminal velocity of free-falling particle. ftisec. u = interfacial tension. dynes/cm. P,Y = gas phase density, Ibm/cu ft. and 0~ = liquid phase density. lbmicu ft. Using simplifying assumptions with respect to gas. condensate, and water properties as given in Table 34.8, Eq. 124 can be expressed for water as 5.62(67-0.003Ip)“~” I’$,, =



..,



(0.003 ljIqCJ5 (continued



(125)



on Page 34-50)



WELLBORE



HYDRAULICS



34.47



34-48



PETROLEUM



ENGINEERING



HANDBOOK



FLOW



RS



-



GAS



RATE



OIL



-



RATIO



Fig. 34.25-Simultaneous



BARRELS



-



CUBIC



PER



DAY



FEET



PER



gas/oil flow through chokes.



BARREL



PETROLEUM



ENGINEERING



HANDBOOK



Nomenclature



Tubing Performance Curve



a,b = constants A= flow area of conduit A, = area under curve



B=



667s g 2T2



(see Eq. 33)



di 5Ppc 2



Production Fig. 34.26-Tubing



and for condensate vgc =



c, = bubble-rise coefficient c2 = coefficient, function of liquid velocity d,i = inside diameter of casing 4, = diameter of an equivalent circular pipe dH = hydraulic pipe diameter d,; = ID of tubing dto = OD of tubing pi = the ith depth increment D, = D under static conditions (static equivalent depth for pressures encountered at flowing conditions) energy losses El = irreversible f= friction factor (Fig. 34.2) ff = Fanning friction factor



Rate



and choke performance



curves



as



4.02(45-0.0031P)“.25 (o,oo31p)*~5



,



. ..



. .



(126) O.l0797q,



F= where Vgn = gas velocity for water, ftisec, vKc = gas velocity for condensate, ftisec, p = pressure, psi.



3.06pvgA Tz



. . . . . . . . . . . . . . . . I.......



and



FD,



(127)



where q8 = gas flow rate, lo6 scf/D, A = flow area of conduit, sq ft, T = temperature, “R, and z = gas deviation factor.



34.8-GAS, WATER



CONDENSATE, PROPERTIES Gas



interfacial tension, dynes/cm Liquid phase density, lbmlcu ft Gas gravity Gas temperature, OF



0.6 120



=



F,



d 2.612



non-Darcy



= &e q8



flow term



Eq. 38)



P/( Tz) F2 +O.OOl[pl(



AND



Condensate



(see Eq. 38)



F, = function of Reynolds number F2 = function of Reynolds number roughness &i-c= conversion factor of 32.174



I=



Tek et ~1.~~ introduced a concept called “the lifting potential” to explain loading, unloading, heading, and dying of wells. Further, the concept relates the inflow behavior of the well with the multiphase flow in the well. Accordingly, it appears possible to address engineering considerations directed toward performance analysis or design of well equipment. Calculation procedures described earlier in this chapter with respect to well inflow performance and multiphase flow in the well should be adaptable to use the lifting potential concept.



TABLE



=



I



Further, a minimum flow rate for a particular set of conditions (pressure and conduit geometry) can be calculated using Eqs. 125 through 127. qg=



F,q,



Water



20



60



45



67



Tz)12



and relative



(see Eqs. 40-43)



J* = ;tabilized PI at zero drawdown -1’= ;tabilized PI from J*j = ;tabilized PI at zero drawdown, future flow data from J*p = stabilized PI at zero drawdown, present flow data J*, = I transient form of the flow coefficient L= 1ength of the pipe string (subscripts B, M, and S indicate bubble, mist, and slug flow) L= 1Bow regime boundary, dimensionless n= :xponent, usually determined from multipoint or isochronal backpressure test number NR~, = rubble Reynolds ,ubblepoint pressure Pb =



WELLBORE



HYDRAULICS



34-51



BHP Pe = reservoir pressure at the external boundary Ap; = pressure drop for increment i Phh



=



Sl. (P IN,



z



-dp,,



s



PI+-



Pi



+P2



p = kPa, L = m, and T = “K.



Pf = tubing flowing pressure = tophole pressure Pl = surface pressure P? = bottomhole pressure at depth D 9of = future oil rate producing rate at p,,f=O 4oCmax) = maximum heat absorbed by system from Q= surroundings radius rH = hydraulic R RL = gas-liquid ratio s = skin effect, dimensionless exponent of S= Pth



S=



T LM



=



T,,T2



= U= b’h =



l.‘,&,c = L’#D = 1’$,,’ = 1’,p,. = \‘L., = $3, = I’,



=



w,



=



z=



Z= (-(s).sp



=



YL = 6= t= lJ= ?f



=



Eq. 28 Customary. OOI877y,LI(?zi~



PI=P2e



SI. O.O342y,L/(TT)



P I =P2e



O.O375y,L TY



(see Eq. 44)



choke size in & in. log mean temperature respectively, bottomhole and wellhead temperatures internal energy slip or bubble rise velocity gas velocity for condensate dimensionless gas velocity superficial gas velocity gas velocity for water superficial liquid velocity terminal velocity of free-falling particle total fluid velocity (q,/A) total mass flow rate compressibility factor or gas deviation factor difference in elevation separator gas gravity (air= 1) specific gravity of condensate liquid distribution coefficient absolute roughness liquid surface tension friction loss gradient



where p = kPa, L = m, and T = “K.



Eq. 35 Customary.



(P VI ,



(P/Jr):



! 0.2



zz



s 0.2



O.O1877y,L T



’



Sl.



(p



P’) 1



! 0.2



=



O.O342y,L T



Metric Conversion for Key Equations



B=’



1 354fq



*T2 K d5ppc2 ’



Eq. 21 where



Customary.



.(P,‘,I,; -dp,,r I



6.2



PV



(P/M? s o,2



where



P2



e=



+



29.27T



PFr



0.2



Pm =



= w Ly,



=-



Ly,q 53.241T



+



(PP,, s o,



z



-dp,,r. Pp,



9x T d ppr



= = = =



lo6 m3/d, “K, m, and kPa.



z



--dp,m PPr



PETROLEUM



34-52



Eq. 36*



Eq. 56



Customary.



Customary.



P2=PI



bUz)ldP



18.75~,~L=



+t.



ENGINEERING



DP



\“I ,;: F2 +O.OOl[pI(Tz)]’



SI. SI. p2 =p I +9.8x 34.4704y,yL=



10-3Dp,



WV:)ldp



\“’ ;,, F’+O.OOl[p/(T:)]’



where p = kPa, D = m, and p = kg/m3.



Eq. 37* Customary.



Eq. 65 Customary.



F’ =(2.6665ffq;)ld,’



7.08kh J*= [ln(;)



SI.



-i+q.



km (PPJpn



SI. where J, = = = = = =



4s T p d, L



Fanning friction lo6 m’id, “K, kPa. m, and m.



factor,



dimensionless,**



0.0005427kh



J*=



where J* = m’id-kPa, h = m, and PC1 = I?a.s.



Eq. 44 Customary.



2 p /,I! -(J’p;,



=



25&‘T’+~‘-1) 0.0375d;"



Eq. 66 Customary.



SI.



I),,,, 2 -e”p,,, ? =



1.354fq,‘T’$(r’-1) d,’



SI. where p = = = = =



4: f T d



S=



kPa. lo6 m’id, from Fig. 34.2, “K, m,



J**



(I)



0.000.5427kh



=



ChbJz



+s j[



where



O.O683y,L . and 7-Z



L = m. ‘Inusmg SI ““IIS Table 34 4 and Eqs 38 and 39 ate not appkable ’‘f,ISthe Fanning frlclion factor equallo f, =f/4. where I ISthe Moody frlctlon factor from Fig 34 2



h = t = p = CI = r,,. =



m. d. Pa-s, l/kPa, m.



and



HANDBOOK



WELLBORE



34-53



HYDRAULICS



Eq. 87



Eq. 91



Customary.



Customary.



-dp=



r+dD ---+



L!!!LdDf



144



14483,



X”&>. 144g,



0.2218v,* Lfj=1.071du



SI.



SI. IOOOpv -----dv, cs,



1ooogp -dD+ sc



-dp=T,dD+



where



0.7277v,’ Lg=1.071du where



p = T., = D = p = g = ,y(. = I’ =



kPa, kPa/m, m, g/cm3. 9.80 m/s’, 1000 kglm.kPa.s’, m/s.



V , = m/s and dH = ITl.



Eq. 98 and



Customary.



1,488PLduvL



Eq. 89



NR~= PL



Customary. SI.



lO~P&uVL NRe= PL



where



SI.



PL



9.806p-t7, Ap,= I--



“‘,fl,q l OOOA ‘p



where 11‘) = hgis. f/ ” = d/s. A = Ill2 Eq. 90



=



g/m3,



dH = m, vL = m/s, and ,uL = Pa.s.



AD,,



Eq. 101 Customary.



and



1>488PLduvb NRC=



PL



SI.



Customary. 1000/)LdHVb



NRC= PL



Eq. 102 SI.



Customary.



Nue =



1&%Lduv, PL



SI. lOOOp,d,v, NR~ = PL



.



34-54



PETROLEUM



Eq. 117



ENGINEERING



where a = g/s*, VRS = m/s, and PR = g/cm3.



Customary. 17488PgdHVgs



NR~ = PR



Eq. 121 Customary.



SI.



CP ‘s= m



loo0 PgdHVgs NR~ = p"R



SI.



Eq. 118 Customary.



3.0169Cp % =



JP



1



where qx = m’/d, T = “K. and p = kPa.



SI. N= lo6 (~)p)



(



Eq. 122 Customary. where vgr = m/s, pi = Pa*s, and u = g/s*.



PI/ =



Customary. Ptt = t



’



2.50R,vLo.5”6q, si.89



34u



-=



P,q “#I ‘d//



where p+ = kPa, R .qL = m”/m3.



SI.



t



1.115(10-~)a



l/H



P,y”p., ‘(1”



-zz



~I.89



SI.



Eq. 119



‘IH



435R,yL0.546q,



qr = m”/d, S = cm.



and



Eq. 125 where



Customary.



a = gls’. 1’q.r = m/s. and P (8 = g/cm>.



1’ ,&,,I’ =



Eq. 120



du



(0.003 1pp5



SI.



Customary.



6



5.62(67-0.0031p)“~‘5



Pg vg.\*dH



l.713(67-0.00045p)o~~”



1’C,,’=



174.8~(N)‘.~~*



(o.ooO45p)o~”



’



Eq. 126 SI.



Customary. E



-=



dn



5.735( 10 -4)c?(Npo2 2dH PRVR-’



4.07(45-0.003 i ’



l’,q(.=



1P )‘).2s



(0.003 lp)“.”



.



HANDBOOK



WELLBORE



HYDRAULICS



34-55



SI. I .225(45 -0.00045p)“-25



VKC =



(o.ooo45p)“~”



’



where p = kPa and Vg = m/s.



Eq. 127 Customary.



9,sj=



3.06pv,A Tz



SI.



9g=



0.24628*pv,A Tz



’



where p = “K = A = T = qR = ‘Based



cm standard



kPa, m/s, m*, “K, and lo6 m3/d. conditvms



of 520°R



and



14.7



psia.



References I. Brown, G.G. ef al.: (init Operarions, John Wtley & Sons Inc., New York City (1950). 2. Moody, L.F.: “Friction Factors for Ptpe Flow,” Trans., ASME (1944) 66. 671. 3. Fowler, F.C.: ‘*Calculations of Bottom Hole Pressures.” Per. Eng. (1947) 19. No. 3, 88. 4. Poettmann, F.H.: “The Calculation of Pressure Drop in the Flow of Natural Gas Through Pipe,” Trans., AIME (1951) 192.317-24. 5. Rzasa, M.J. and Katz, D.L.: “Calculation of Static Pressure Gradients in Gas Wells,” Trans., AIME (1945) 160, 100-06. 6. Sukkar, Y.K. and Cornell, D.: “Direct Calculation of Bottom Hole Pressures in Natural Gas Wells,” Trans., AIME (1955) 204,43-48. 7. Cullender, M.A. and Smith, R.V.: “Practical Solution of Gas-Flow Equations for Wells and Pipelines with Large Temperature Grad&s,” J. Par. Tech. (Dec.. 1956) 281-87;~Trans. ,. AIME, 207. 8. Messer, P.H., Raghaven, R., and Ramey, H. Jr.: “Calculation of Bottom-Hole Pressures for Deep, Hot, Sour Gas Wells,” J. Per. Tech. (Jan. 1974) 85-94. 9. 77znteory and Practice ofthe Testing r$Gos Wells, third edition, Energy Resources and Conservation Board, Calgary, Alberta, Canada (1978). IO. Smith. R.V.: “Determining Friction Factors for Measuring Prcxluctivity of Gas Wells,” Trans., AIME (1950) 189, 73. 1 I. Cullender. M.H. and Binckley, C.W.: Phillips Petroleum Co. Report presented to the Railroad Commission of Texas Hearing, Amarillo (Nov. 9, 1950). 12. Back Pressure Test for Natural Gas Wells, Railroad Commission of Texas, State of Texas. 13. Nisle, R.G. and Poettmann, R.H.: “Calculation of the Flow and Storage of Natural Gas in Pipe,” Pet. Eng. (1955) 27, No. I, D-14; No. 2, C-36; No. 3, D-31. 14. Evinger, H.H. and Muskat, M.: “Calculation of Theoretical Productivity Factor,” Trans., AIME (1942) 146, 126. 15. Vogel, J.V.: “Inflow Performance Relationships for Solution-Gas Drive Wells,” .I. Per. Tech. (Jan. 1968) 83-92.



16. Fetkovich, M.J.: “The lsochronal Testing of 011 Wells,” Prmsure Iiunsirnr Tesfing Metho&, Reprint Series, SPE, Richardson (1980). 17. Standing, M.B.: “Concerning the Calculation of Inflow Performance of Wells Producing From Solution Gas Drive Reservoirs,” J. Pet. Tech. (Sept. 1971) 1141-50. 18. Duns, H. Jr. and Ros, N.C.J.: “Vertical Flow of Gas and Liquid Mixtures from Boreholes,” Proc., Sixth World Pet. Congress. Frankfurt (June 19-26, 1963) Section II, Paper 22.106. 19. Griffith, P. and Wallis. G.B.: “Two-Phase Slug Flow.” J. Hear Transfer (Aug. 1961) 307-20, Trans., ASME. 20. Nicklin, D.J., Wilkes, J.O., and Davidson, I.F.: “Two-Phase Flow in Vertical Tubes,” Trans., AlChE (1962) 40. 61-68. 2 I. Baxendell, P.B. and Thomas, R.: “The Calculation of Pressure Gradients in High-Rate Flowing Wells,” J. Pet. Tech. (Oct. 1961) 1023-28. 22. Fancher, G.H. Jr. and Brown, K.E.: “Prediction of Pressure Gradients for Multiphase Flow in Tubing,” SoP t



cpc,;.



.



..,.



.



(3)



This equation shows that the conditions of homogeneity are not necessarily met. The concepts of total mobility, (k/p), , and total compressibility, ct, are introduced. The total mobility is the sum of the individual phase mobility as follows.



-k 0P



TABLE



ko kg kw



=-+-+-, f PO Pg



. . . . . . . . . . . . . . . . . (4)



Pw



3&l-ANALOGIES OF SINGLE-PHASE MULTIPHASE EQUIVALENT Single-Phase Value w



C



98



VALUE



TO



qg R, B,



- total reservoir flow rate, STB/D, I total formation volume factor, RB/STB, = oil flow rate, STBID, = oil formation volume factor, RBISTB, = gas flow rate, Mscf/D, = solution gas-oil ratio, scf/STB, = gas formation volume factor, res cu ftlscf water flow rate, STBID, and water formation volume factor, RB/STB.



Martin’s equation is a nonlinear partial differential equation. Therefore the general case does not have analytical solutions. However, for practical purposes, Eqs. 3 through 6 can be used for most well performance equations if the meaning of the mobility, compressibility, and flow rate are taken in this general three-phase sense. The single-phase solutions of Eq. 1 can be applied to the multiphase case by using the analogies given in Table 35.1.



Oil Well Performance Well Pressure Performance-Closed



Multiphase Equivalent WI4 t Ct 9&3,



Reservoir



The performance of a constant-rate well in a closed reservoir (of any geometry or heterogeneity) has the general form shown in Fig. 35.1. The lower curve of Fig. 35.1 shows that the wellbore flowing pressure, p 4, goes through a rapid pressure drop



WELL



PERFORMANCE



EQUATIONS



35-3



at



early (transient) times and then flattens out until it reaches a constant slope. On this coordinate plot, the closed-reservoir, constant-rate case has the properties aP, --co



at



and a*Pwf >O at*



-



log t



.



When p of reaches a straight line on the coordinate plot, the period of pseudosteady state has been reached. Every pressure point in the reservoir declines at the same constant rate of depletion after that time. Of particular importance is the decline of the average reservoir pressure, j?~, which assumes the pseudosteady-state depletion rate from the very beginning of production. The constant elope of Fig. 35.1 is valid only for constant-compressibility single-phase fluid. However, the general concept of the transient period and the pseudosteady-state period is the same for a multiphase flow with changing compressibilities. The PR slope would be changing according to the changes in compressibility, and the pR curve after a pseudosteady-state would not be exactly parallel to the p,,,f curve. This nonideal behavior would be typical of a solution gas drive reservoir or a dry gas reservoir where the compressibility and mobilities are continually changing. The infinite-acting solutions and the pseudosteady-state solutions to follow are still ap-



Fig. 35.2-Typical



constant-rate



drawdown



test graph.



plicable for the multiphase flow case by using the analogies in Table 35.1. The value of pR, however, must be calculated by the material balance method that applies for this case. Infinite-Acting



Solution (MTR)



The pressure behavior of constant-rate flow in a closed reservoir goes through several periods: the early-time region (ETR), middle-time region (MTR), and late-time region (LTR). These periods are illustrated on a semilog plot ofp$ vs. log t in Fig. 35.2. The MTR solution is discussed first. Eq. 1 can be solved for the infinite-reservoir case, which is useful for application at early times. The solution applies to a well producing at constant rate, beginning at t=O, and a homogeneous reservoir of constant thickness.



PO=



10



I



IO



102 tDr



Fig. X.3-Dimensionless



= tD/rD



pressure for a single well in an infinite system,



IOJ



IO’



2



no wellbore



storage,



no skin. Exponential-integral



solution.



PETROLEUM



35-4



There are two important solutions for the intinitereservoir case. One solution8 assumes that the wellbore has a finite radius, r,. This solution is used mostly for aquifer behavior with the oil field being the inner radius rather than a wellbore. This solution is given in Chap. 38 for the infinite-aquifer case. A simpler solution applies for well behavior. This solution, called the “line-source” or “exponential-integral” solution, assumes that the wellbore radius, rw, approaches zero. This solution has the form



ENGINEERING



HANDBOOK



Skin Effect The solutions to Eq. 1 are modified to account for formation damage near the wellbore. The damage near the wellbore can be considered concentrated into a very thin radius around the wellbore such that the thickness of the damage is insignificant but a finite pressure drop results from this damage. Fig. 35.4 shows a sketch of the physical concept of the damaged region and Fig. 35.5 shows the pressure profile resulting from this damage. The magnitude of the pressure drop caused by the skin effect Ap, is



Ap,=O.87ms,



.....



.... ....



. . (10)



where po



rD tD h pi rw



= kh(pi-p)l(141.2



where s is the skin effect, defined in terms of dimensionless pressure such that it would have the following effect on Eq. 8.



q&)=dimensionless pressure, = r/r,,, =dimensionless radius, = (O.O00264kt)l$+c,r,.’ =dimensionless = formation thickness, ft, = initial pressure, psi, and = wellbore radius, ft.



time, pD=%



The exponential-integral function, Ei, is a special function that results from the solution of the line-source problem. A more practical solution to the problem is the plot of the dimensionless pD vs. t&rD2, which is shown in Fig. 35.3. The tDr term is the dimensionless time based on external radius, re. Fig. 35.3 can be used to determine the pressure at any time and radius from the producing well. This solution is valid as long as the radius at which the pressure is calculated is greater than 20 r,+ or at the wellbore of the producing well (at r,v) at a value of fo/rD * > 10. Fig. 35.3 is used mostly to determine the pressure at distances away from the well such as at a nearby well location during an interference test. The more common solution of the exponential integral solution is the “semilog straight line solution,” which applies after to is greater than 100. After this time, Eq. 8 applies at the wellbore: pD=%



.....................



hl t,+0.406.



(8)



In customary oilfield units, this equation has the form



pKf=pj -In



log



kt



+crrw2



-3.23



>



, .



.



(9)



where m equals (162.6qBp)lkh and p,+f is the flowing bottomhole pressure, psi. This equation results in a semilog plot of p,,f vs. log t with a slope of -m psi/cycle (the MTR of Fig. 35.2.) Eqs. 7 through 9 are used for infinite-acting solutions before the effects of boundaries affect the pressure transient behavior. When the closest boundary begins affecting the behavior at the wellbore, this time is the end of the semilog straight line, t,,d . The last column in Table 35.2 shows tend for various drainage shapes (shape factors).



ln tD+o.@ts+s.



.. ...



. . . . . .(ll)



The value of the skin effect is calculated from transient well test data such as a buildup test or a drawdown test. The exact nature of the cause of the skin effect might not bc known but might be caused by a combination of several factors. Some of these factors are (1) mud filtrate or mud damage near the wellbore, (2) the cement bond, (3) limited perforations through the casing and cement bond, and (4) partial penetration (completion). On the other hand, the value of the skin effect, s, might be negative. This would indicate an improved wellbore condition, which might be caused by (1) improved permeability in the vicinity of the wellbore because of acidizing or other well treatments, (2) a vertical or horizontal hydraulic fracture at the wellbore, or (3) a wellbore at an angle rather than normal to the bedding plane. The determination of the skin effect is important in determining the need for a workover or the benefits of a workover. The effect of the skin can be stated as a modification to the wellbore radius by calculating an effective wellbore radius, r’,,,, calculated by r’w=r,e



- s .............................



.(12)



This effective wellbore radius, rlw, can be considered the equivalent wellbore radius in an undamaged or unimproved formation, which would have the same flow characteristics as the actual well with the skin effect. Wellbore Storage Effect (ETR) At very early times the fluid production tends to come from the expansion of the fluid in the wellbore rather than the formation. This tends to delay the production rate from the formation. The relationship between the surface production rate, the expansion of the wellbore fluids, and the formation production rate are shown in Eq. 13:



q$=q+L+



24C. Lb B



at



.



.



. (13)



WELL



PERFORMANCE



EQUATIONS



TABLE 35.2-SHAPE



35-5



FACTORS FOR VARIOUS CLOSED SINGLE-WELL DRAINAGE AREAS ftDA)end



In Bounded



Reservoirs



0 0 A n



Exact For tDA >



Less Than 1% Error For t, >



cA



In CA



31.62



3.4538



- 1.3224



0.1



0.06



0.10



31.6



3.4532



- 1.3220



0.1



0.06



0.10



27.6



3.3178



- 1.2544



0.2



0.07



0.09



27.1



3.2995



- 1.2452



0.2



0.07



0.09



21.9



3.0865



- 1.1387



0.4



0.12



0.08



0.098



- 2.3227



f 1.5659



0.9



0.60



0.015



30.8828



3.4302



- 1.3106



0.1



0.05



0.09



12.9851



2.5638



- 0.8774



0.7



0.25



0.03



4.5132



1.5070



- 0.3490



0.6



0.30



0.025



3.3351



1.2045



-0.1977



0.7



0.25



0.01



21.8369



3.0836



-1.1373



0.3



0.15



0.025



10.8374



2.3830



- 0.7870



0.4



0.15



0.025



4.5141



1.5072



- 0.3491



1.5



0.50



0.06



2.0769



0.7390



+ 0.0391



1.7



0.50



0.02



3.1573



1.1497



-0.1703



0.4



0.15



0.005



0.5813



- 0.5425



+ 0.6758



2.0



0.60



0.02



0.1109



-2.1991



+ 1.5041



3.0



0.60



0.005



5.3790



1.6825



- 0.4367



0.8



0.30



0.01



2.6896



0.9894



- 0.0902



0.8



0.30



0.01



0.2318



- 1.4619



+I.1355



4.0



2:oo



0.03



q3



0.1155



-2.1585



+ 1.4838



4.0



2.00



0.01



c&ID



2.3806



0.8589



- 0.0249



1 .o



0.40



0.025



2.6541



0.9761



- 0.0835



0.175



0.08



Cannot



use



2.0348



0.7104



+ 0.0493



0.175



0.09



Cannot



use



,&



1.9988



0.6924



+ 0.0583



0.175



0.09



Cannot



use



,@



1.6620



0.5080



+0.1505



0.175



0.09



Cannot



use



,&



1.3127



0.2721



+ 0.2685



0.175



0.09



Cannot



use



>@



0.7887



- 0.2374



+ 0.5232



0.175



0.09



Cannot



use



In Vertically-Fractured IO ,m



In Waterdrive



Reservoirs* x”xe



Reservoirs



0 In Reservoirs Production



‘Use (xJx,)’



@DA)pss



Use Infinite System Solution With Less Than 1% Error For t,
pss= --$



-0.234qB “~‘pCt



. ....



(15)



During pseudosteady-state behavior, wellbore pressure is related to the average reservoir pressure, PR, by a productivity index (PI), J, as follows. q=J(pR



-p,j).



.......



. . . . . . . . (16)



This PI equation relates the pressure drawdown to the production rate. For a circular drainage area we can write out the complete expression for the PI equation as



1



7.08x 10 -3khl(B/t) 4=



In r,/r,



_ (Pi?-Pwj),



-0.75s~



‘. . . . . . (17)



where re is the exterior boundary radius, ft. Note that the quantity in brackets is equivalent to J in Eq. 16 for the circular drainage area. J is a constant if the viscosity and formation volume factor of the producing fluid are constant. If these fluid properties are not constant, Eqs. 16 and 17 still apply but the PI value changes with the changing fluid properties. For multiphase flow these equations still can be used by substituting the definition in Table 35.1 into Eqs. 16 and 17. Eq. 17 has to be modified if the drainage area is not circular with the well in the center. A general form of the pseudosteady-state equation has been worked out by Dietz l1 and has been cited by other authors. I-5 The generalized pseudosteady-state equation has the form



7.08x 10-3khl(Bp) 4=



CA



rw



1 (PR



A



2.2458 ‘15ln--



2



+s



-pwf),



. . . (18)



WELL



PERFORMANCE



EQUATIONS



35-7



Fig. 35.6-Dimensionless and a finite



pressure for a single well in an infinite reservoir skin-composite reservoir.



where A is the drainage area, sq ft. and CA is the shape factor (Table 35.2). This equation can be applied by using the values for CA in Table 35.2 or by moving the terms in the denominator to the form



2.2458



% hl-



@R



A



+% lnT+S



-Pwfh



...



storage



+s,c,



=[3.0+(0.75)(8.5)+(0.25)(3.2)]



I



rw



CA



wellbore



Calculate the bottomhole pressure (BHP), pwf, after 12 hours and after 120 days for a constant oil production rate of 80 STB/D. Solution. From Eq. 5, Cr=CffS,C,



7.08 x 10 -3khl(&) 9=



including



x 1O-6



=10.2X 10e6 psi-‘. . ..



. .. ..



(19)



This form is easier to use because the first term of the denominator also is tabulated in Table 35.2. In Table 35.2, x, is the distance from the well to the side of the square drainage area, and xf is the distance from the well to either end of the vertical fracture. Table 35.2 also shows the dimensionless time, tom, at which the infinite-acting solution ends, and also the time at which pseudosteady state begins, (t~~)~~,r. Example Problem 1 (Transient and Pseudosteady State). A well is centered in an approximately square drainage area. The following data are given.



Calculate the time required to reach pseudosteady state. From Table 35.2, O.O00264(45)t,,, (tDA)pss=O.l=



(O.18)(1.5)(1O.2x1O-6)(1.74x1O6)’



where tpssis 40.3 hours. So the well is infinite acting after 12 hours. By using Eq. 11, p~=‘h



h tD+o.do&i+s.



By using the definitions of pD and tD in Eq. 8, we have



A = 1.74~ lo6 sq ft (40 acres),



h = 21 ft, s = 1.6, rw = 0.25 ft, k, = 45 md, PO = 1.5 cp, $fJ= 0.18, cc7 = 8.5~10~~ psi-‘, CW = 3.2~10~~ psi-‘, cf = 3.0X10p6 psi-‘, S, = 0.25, B, = 1.12, and pi = 5,100 psi.



WKWW~-p,vf) 141.2(80)(1.12)(1.5)



=% In



0.000264(45)( 12) (O.18)(1.5)(1O.2x1O-6)(O.25)2



0.0498(5,100-p,,&=%



In (8.28~10~)+0.4045+1.6;



5,100-p,,=(8.82)/(0.0498)= p,f=4,923



+0.4045+1.6;



177; and



psi at 12 hours.



35-8



PETROLEUM



4,199-p,,f=



ENGINEERING



HANDBOOK



178; and



p,,=4,021



psi at 120 days.



Production Rate Variation (Superposition)



0



tl



12



t3



FLOW



t N-I



t N-2



t4



TIME,



t,



HOURS



Fig. 35.7-Schematic representation rate schedule.



of a variable



production-



At 120 days, the well is in pseudosteady state (greater than 40.3 hours). First, calculate PR. Using Eq. 15, the rate of pressure decline can be calculated. aP



(-> at



P==



These solutions have included only the constant-rate case. Of general interest, of course, are the cases where rate changes with time. These cases are best handled by using the principle of superposition. The principle of superposition amounts to dividing the production history into a sequence of rate changes such as that shown in Fig. 35.7. The total effect of the production on the pressure response, Ap, is the additive effects of each of the rate changes. In Fig. 35.7, rate q1 applies from t=O to the current time. At t, the rate increases to q2. The effect of this rate change can be viewed as an incremental rate, q2 -91, which has been in effect for a period of time t-t l . Then q3 would also be seen as a rate change, q3 -92, which has been in effect for a period of time t- 12. The effect of all these rate changes is computed by superposing the solutions that applied to each rate change and its corresponding time that it has been in effect. The equation for computing the total pressure drop, Ap,, is



-0.234qB



N



“pc,



p; -p,#=



c



(qj -qj-,)f((t-tr-,)



,



.



.



(20)



i=l



-0.234(80)(



1.12)



= (21)(O.18)(1.74x1O6)(1O.2x1O-6)



= -0.313



psi/hr.



p,=5,100-0.313(120)(24) =4,199 psi. Now, using Eq. 19,



7.08x 1O-3 khl(&) 90 =



A 2.2458 +% In-+s % In-



1



rM



CA



7.08x10-3(45)(21)!(1.12x1.5) (80) = - 1.3224+ % In



1.74x 10-6



when qieI =0 when i=l. The functionf(t) can be called the unit responsefinction. The unit response function is the pressure drop, pi -pKf, which occurs at time f for a unit production rate (q= 1). The unit response functions may be quantified by the cases described such as the wellbore storage equation at early times (ETR), the semilog straight line solution at MTR, and finally the pseudosteady-state solution at later times (LTR). For example, if q 1 had been in effect for a time longer than tpss, its contribution to the pressure drop at time t would be calculated from the pseudosteadystate equations, which would comprise the calculation of the reduction in p from Eq. 15 and the pressure drop from p R to pwf in Eq. 16. The effect of the second rate might be still in the transient period, which would call for Eq. 11 to be applied. Note that the calculation of the pressure decline of p R can be calculated with Eq. 15 only for the constantcompressibility case. For the general case, such as a solution gas drive reservoir, the appropriate material balance equations would be applicable to calculate PR. If the last rate change has been in effect for a time greater than tP,rSand the system has constant compressibility, the following simplification can be made for Eq. 15.



+1.6 1



(0.25)*



5.615 NpB, PR’Pi-



VpCr



.



.



.



.



.



(21)



*(4,199-p++&



(80) =



3.982 -1.3224+8.571+1.6



1



(4,199-p!&



The following example problem shows how superposition can be applied for a case where both pseudosteadystate and transient pressure drops are added.



WELL



PERFORMANCE



35-9



EOUATIONS



Exynple Problem 2 (Superposition). The well in Example Problem 1 produces according to the following schedule. time (hours) Oto2 2 to 8 thereafter



so the values off(l2),



f(lO),



5,100-p,,=(300)[0.1256



-(180)[0.1256



300 120



ln(6.9x lo4 x 12)+0.504]



ln(6.9x104



x 10)+0.504]



80



Calculate p,,,, at 12 hours and at 120 days. So&ion. As we observed in Example Problem 1, the well was infinite acting after 12 hours, so we use Eq. 20.



ln(6.9~10~~4)+0.504]



=(300)(2.22)



-(180)(2.19)



N C i=



are used, giving



(SI%D)



-(40)[0.1256



pi-Pwf=



andf(4)



(4i-qi-Of(f-ti-1) I



-(40)(2.08) We first needf(t), the unit response function. We can use Eq. 11 to find Ap in terms oft for q=l: pD=%



= 189;



In tD +0.4@5+3,



p,,=4,911



psi at 12 hours.



At 120 days, the well has a cumulative production of 141.2(1)(1.12)(1.5) N, =300 STB/D x (2/24 days) 0.000264(45)?



=% In



(0.18)( 1 .S)( 10.2 x 10 -6)(0.25)2



+ 120 STBlD x (6/24 days)



+0.4045+1.6,



t80



3.98Ap= 1/2In 6.90x 104t+2.004,



STBiDx(l19.5



days)



and =9.615 STB.



Ap=O.1256 ln(6.90x



lO”t)+0.504,



Using Eq. 21,



so



5.615NpB, pREpi-



f(t)=Ap=O.1256



vpct



ln(6.90x104t)+0.504.



Substituting into Eq. 20,



5.615(9,615)(1.12) =(5,100)-



(21)(0.18)(1.74x106)(10.2x10-“)



=5,100-901=4,199. Using Eq. 19 (the same as Example Problem l), we calculate +@I3



-921f(t--12);



and again, pwf=4,199-178=4,021 +(120-3OO)f(12-2)



+(80-12O)f(12-8),



psi at 120 days.



The effect of the early rate variation is “forgotten” after the rate is constant for tpss=40.3 hours, except for the slight increase in cumulative barrels ( 15 STB), which is negligible in this case.



35-10



PETROLEUM



Gas Well Performance The performance of gas wells is similar to oil wells (liquid reservoirs) except for two major differences: (1) the fluid properties of gas change dramatically with pressure and (2) flow can become partially turbulent near the wellbore, resulting in a rate-dependent skin factor. These two factors are discussed and alternative forms of gas performance equations are presented. The application of these principles to gas flow is only slightly more complicated than to liquid flow, but there is often much confusion about gas wells. There are several reasons for this. One reason is that there are many versions of gas flow equations in the literature. Some are in terms of p, some in terms of p2, and some in terms of a real gas pseudopressure, m(p). All these equations can be used and are valid forms. Another reason for confusion is the different coefficients in the equations, which sometimes arise from the assumed temperature and pressure base of a standard cubic foot of gas. The following equations use only the symbols T,, and psC, since the pressure base in different areas does vary significantly. Still another reason for confusion is that deliverability testing has been customary with gas wells because of government requirements. Deliverability testing, based onalog(pR2 -pwf2) vs. log qg plot, is largely an empirical approach. The deliverability plot approach was developed mainly for low-pressure gas wells and does not work well with the deeper, higher-temperature, and higher-pressure wells that are more common today.



The gas compressibility, of 2 as 1 cg=----.



Id.2



P



ENGINEERING



HANDBOOK



cg , can be expressed in terms



.......



.



.



____ __ (25)



zQ



For practical purposes, however, Eq. 23 can be taken as a linear differential equation in terms of m(p). This was confirmed by the result of computer simulations performed by Wattenbarger and Ramey. l3 They showed that the pressure transient equations can be used, with very good approximation, in terms of m(p). After pseudosteady-state, PI equations similar to Eqs. 16 through 19 can be used. The application of the m(p) solutions is not difficult. the values of m( p) vs. p can be determined by graphical integration or can be calculated with computer programs that use built-in correlations to estimate the variation of z and p with pressure. Since our equations and graphical techniques depend on equations of a straight line of p either on a linear plot or a semilog plot, it is worth analyzing how the slopes of m(p) are related to the slopes of p plots, or p2 plots; we can show that the derivative of m(p) with respect to, for example, log t is as follows.



am(p) --=c$&= . Eq. 33, when put in more practical form, can be expressed in terms of m(p), p, or p2, as



where



m(pi)-m(pwf)



2.303 =-log 2



1



0.000264kc (4P41~W2



and = dimensionless m(p), tD = dimensionless time, T,, = standard condition temperature, “R, pressure, psia, PSC = standard condition TR = reservoir temperature, “R, m(pi) = m(p) at initial pressure pi, psia2icp, and m(pWf) = m(p) at wellhore flowing pressure pWf, psia2/cp. mD



The value of TV is evaluated with &LC evaluated at the initial pressure.



+0.4045+~+F~,



1.987x 10 -5



t



(p 2.303 =-log 2



( qn / ,



>



. . . . . . . (35a)



(Pi-P&$) P



0.000264kt (4W)



+0.4&t5+S+F,



ir w



2



( qe 1 ,



.. ....



. . (35b)



35-12



PETROLEUM



0.08



Pseudosteady-State



GAS GRAVITY



HANDBOOK



Solutions (LTR)



The pseudosteady-state solutions are analogous to the liquid solutions and can be put in essentially the same form. The only changes are to allow for the changes of fluid properties with pressure and non-Darcy flow. The inclusion of these effects is the same as discussed above. The result is the following form of the pseudosteady-state equations, in terms of m(p), p, and p*.



= 0.7



REDUCED TEMPERATURE=



ENGINEERING



17(195’F)



0.06



kh 2.2458A



% ln----



+~+FD,



kg



I



CAT,'



* m(p)-m(p,f) [



0



2,000



4,000



6,000 P,



Flg. 35.9-Typical



8,000



10,000



1 ,



_. _. _.



_.



(364



where m@)=m(p) at p R, psia’/cp, and CA =shape factor from Table 35.2.



psla



variation of m(p) and zp with pressure.



kh 2.2458A



Vi In-



C,4rw2



+s+FDa I qgI



1.987x 1O-5 (PR-p,,,,), 2.303



O.ooo264kt



=-log 2



(4Pc)irw



+0.4045+s+FDa



...



. ....



. . . (36b)



P and



2



I qg I ,



.



. . ...



(35~)



where (@PC);=&LC evaluated at pi. Eq. 35 can be used to predict p,,f for the infinite-acting period (MTR) between the wellbore storage period and the beginning of pseudosteady state. Fig. 35.9 shows a typical relationship of zp with pressure. The value of Z,Uis almost constant when p is below 2,000 psia. This makes the p2 type of equation fairly accurate below 2,000 psi because Z,Ucan be taken out of the integral in Eq. 23 if zp is constant. p2 plots and equations tend to work well in reservoir pressures less than 2,000 psia. Fig. 35.9 also shows that m(p) tends to be linear with pat higher pressures (above 3,000 psia). This means that p plots and equations tend to work well for higher-pressure reservoirs. If there is a doubt about whether these p* or p simplifications apply to a particular reservoir, then m(p) plots and equations should be used.



kh



l/z In-



2.24584 +~+FDcI



I qs



I



c*rw2



Eqs. 36 have general application for pseudosteady-state gas flow. Note that these forms of the pseudosteady-state equations are considerably different from the gas deliverability approach that is used extensively. The gas deliverabili approach is empirical and based on a log-log plot ofp 9 -p,,,,’ vs. qg. The comparison between Eqs. 36 and the deliverability plot approach is discussed by Lee. 5



WELL



PERFORMANCE



EQUATIONS



35-13



From Table 35.2,



Long-Term Forecast Long-term forecasting can be accomplished in a fairly straightforward manner using Eqs. 36 along with a p R/z plot. The CR/z plot, of course, is simply a material balance for a closed gas reservoir. Through this plot the value of P.Q can be determined for any value of cumulative production, G,. Given this value of p R, one of the forms of Eqs. 36 then can be used to determine qx. Note that in deep, high-pressure reservoirs, the influence of formation and water compressibility can become important compared with gas compressibility. At these high pressures, greater than about 6,000 psig, the p R/Z plot should be modified to account for the formation and water compressibilities. A technique for this modified p,& plot is presented by Ramagost and Farshad. tJ A complete forecast of production rate vs. time can be generated by converting the cumulative production to a time scale. The value ofp%f might be fixed as a condition of the production forecast, or it may be solved simultaneously with wellbore hydraulic relationships, such as given in Chap. 34. Example Problem 3. A gas well produces from a drainage area that approximates a 4: 1 rectangle with the well in the center. The following data apply.



CA z5.3790.



Eq. 36b is



kh 2.24584



l/2 In T+S+FDO



hi:



2P



(-> z/J



(PR-Pwf);



p



(520)



q,=1.987x10-5



(14.7)(670) (0.52)(34) % In



2.2458(6.96x



106)



(5.379)(0.23)* A = 6.96x lo6 sq ft (160 acres), h = 34 ft, s = 2.3, = 0.0052 (lo3 cu MD)-‘, FD, rw = 0.23 ft, k, = 0.52 md,



= see Fig. 35.9, 4 = 0.11, TR = 210”F+460=670”R, T,, = 6WF+460=520”R, pSc = 14.7 psia, and j?~ = 4,150 psia.



+2.3+0.0052



17.68



= 1.987x 10 -5(0.0528)



8.91+2.3+0.0052 .(3.42x



1.68~10~ ( qg 1



rate, qg , if pWf= 1,500 (11.21+0.0052



1qg I)q,=1.68x104.



simplest form of the This equation can be solved as a quadratic equation, or simply by trial and error, by using estimates of I qx I starting with I qg 1 =0:



(11.21+O)q,=1.68x104 4,: = 1,499.



=2,825 psia. Next, try From Fig. 35.9, we estimate ~~~ at 2,825 psia as (11.21+0.0052x1,499)q,=1.68x104; zpR =0.0165



qg =884.



Next, try



0.0165



1qR 1



105)(2,650)



11.21+0.0052



= (4,150 + 1,500)/2



2(2,825) =-=3.42x



1 qK 1



*(3.42x105)(4,150-1,500)



ZPR



Calculate the pseudosteady-state psia. Solution. Use Eq. 36b-the pseudosteady-state equation.



1



CArw



lo5



(11.21+0.0052x884)q,=1.68x qR = 1,063.



104;



35-14



PETROLEUM



970



k,h=



-



lf=%oBo~o m



; 950 0. E



E P ?I ii



, . . . . . . . . . . . . . . . . . . . (374



and for gas wells,



940



930



k,h=



-5.792



x 104q,(p,,TR/Ts,)



. . . . . . . . Wb)



m*



920



where m* is the slope of m(p) plot,



910



k,h= FLOW Fig. B&10-Semilog



HANDBOOK



For oil wells,



.(I, 960 8



2 2



ENGINEERING



TIME,



-5.W!X



~04q,(p,,TR/Ts,) m’



t, hours



data plot for drawdown



wb ’



. . . . . . . . . . . . . . . . . . . . . . . . . . . (37c)



test.



where m’ is the slope of p plot, or Next, try k,h=



-5.792x



104q,(p,,TR/T,,)



qg=1m4 until the solution converges at qg = 1,018 x lo3 cu ft/D.



Drawdown Test The drawdown test is accomplished simply by putting a well on a constant production rate after the well has been shut in. Variations of the drawdown test involve analysis of variable rates, but only the constant-rate case is covered here. The analysis is based on the infinite-acting solution (MTR). The data are plotted on a pressure vs. log time semilog plot and the slope of the plot, m, is determined graphically in units of psi/cycle (see Fig. 35.10). The equations for determining w1 for an oil well or a gas well are as follows.



Wd)



where m” is the slope ofp* plot and subscript wb refers to wellbore. The values of zpl2p in Eq. 37c and zp in Eq. 37d are evaluated at pW, rather than’(pR+p,,)/2, which is used in the pseudosteady-state equations. The value of the skin effect, s, is determined from one of the following equations for oil and gas wells. For oil wells,



Transient Well Test Analysis The subject of transient well test analysis can be very complicated and has been covered very thoroughly in the literature. I-5 These references show not only the straightforward cases of transient well test analysis but also go into many exceptions, alternative techniques for analysis, and other complications. It is the intent here to cover only the most straightforward and routine methods for analysis of oil and gas wells. The most common values to calculate from a transient well test analysis are kh, s, and PR. With these three values plus a knowledge of the drainage area and shape of the drainage area (values of CA and A), the flow rate can be calculated or forecast for a particular BHP, p,,,f, by using the pseudosteady-state equations. The method of analyzing kh and s for a drawdown test and a buildup test are summarized now.



(z~~)wb.



nP



(11.21+0.0052x1,063)q,=1.68x104;



x=1.151



Pi-P1



112



I



k



-log-



~wtr,2



. . . . . . . . . . . . . . . . . . . . . . . . (384 where p 1 is the pressure at AZ= 1 hour; and for gas wells,



s=1.151



k m(pi)-m(pl) -logm*



+crr,2



. . . . . . . . . . . . . . . . . . . . . . . . . . . (38b)



I



-log---



k



hc,r,2



. . . . . . . . . . . . . . . . . . . . . . . . . . . (38~) or



. . . . . . . . . . . . . . . . . . . . . . . . . . . (3W The disadvantage of this equation (compared to buildup testing) is that pi must be known to calculate S.



WELL



PERFORMANCE



EQUATIONS



35-15



It is important to evaluate the proper semilog straight line. In many cases it is difficult to tell whether an apparent semilog straight line is in the MTR solution or is still being affected by wellbore effects (ETR) . It is often helpful to make a log-log plot of Pi -pwf vs. flowing time, t, to analyze when the wellbore effects are finished. A straight line with a slope of unity on this log-log plot indicates that the pressure behavior is being totally dominated by wellbore storage. The semilog straight line then can be expected to begin at about 1.5 log cycles after the data points leave the log-log straight line of unity slope.



3350 = 3317



Buildup Testing Buildup testing is more common than drawdown testing. The main reason for this is that the well rate is known when the well is shut in (q=O). The analysis of a buildup test is based on the assumption that a constant flow rate is maintained for a producing time, tp , and then the well is shut in. Variations of the buildup test include analysis of variation in production rate before shut-in, but only the constant-rate production period is covered here. The pressure, p$ (At=O), is measured just before shut-in and then at different shut-in times, A?, after the time of shut-in. A plot is made of the shut-in pressures, PDF, vs. a time scale based on the shut-in time, At. The time scale is either log At or log (I,, +At)iAt. The first of these plots (Fig. 35.11) is called an “MDH plot” (Miller, Dyes, and Hutchinson 15). The second type of plot (Fig. 35.12) is called a “Homer ~10~“~~ Both plots give the same semilog straight line slope, which is also the same as measured in the drawdown test. The kh for an oil or gas well can be determined from the slope of this semilog straight line by the following equations (identical to Eqs. 37, except for the sign). For oil wells, k h= 162.6qoBofio , 0 m



. . . . . . . . . . . . . . . . . 094



and for gas wells,



!i 30000



,454 6 I



10-I



IO



SHUT-INTIME, At, hours Fig. 35.11 -MDH



plot for buildup test.



SHUT-IN-TIME,At, hours



01 ‘E3300



PI, * 3266 P



Pi



-40 OF STORAGE



3 2 u3200 h



43 2 8



654



3



2



86543 2



IO’



. .



plot of pressure buildup data from Fig. 35.11.



(39c)



m’



and for gas wells,



wb ’



or s=1.151 hg)wb.



WW



Note that the signs are reversed for the Homer plot. The skin factor, s, can be determined from one of the following equations. For oil wells, -log



a IO’



(39b)



Fig. 35.1 P-Horner



5.792x 104q,(p,,WW



k h= 5.792 x 104q,hJ-dW g m"



2



(tp +At),A:’



k h= 5.792x104q,hJ’/dL) g m* k,h=



PS/o/CYCL



-3250 2



ko



4ihctr?



. . . . . . . . . . . . . . . . . . . . . . . . . . . (404



-log-



(I kg



m(pl)-m(p,f) I m*



+3.23



4ClgCt



s=l.151



>



Pl -Pwf mr



, .. ...



. . . . . . . . . (40b)



I -log kg hsctrw2



. . . . . . . . . . . . . . . . . . . . . . . . . . . (4Oc)



PETROLEUM



35-16



2 ,300 2 z Kcr a,



w



QIZOO



d ; IQ 8 II00 I= 8 IO00 343



2



86543



2



82.3.1



I02



2



IO



(to + Af)/U Fig. 35.13-Horner plot of typical pressure buildup data from a well in a finite reservoir.



or



s=l.l51



P2 I -P2 wf



(I



m”



kg



-log



CbgCd



. ........ . .....



+3.23



>



.



. (404



The slope refers to the corresponding semilog straight line. prr,f is the last pI(,f at At=O. These equations are based



on the equation of the semilog straight line. Therefore, if p ws does not fall on the extrapolated semilog straight line at At= 1 hour, then p I is read on the semilog straight line rather than at the actual data. Again, be reminded that transient well test analysis can be very complicated and can depart in many ways from the simple analysis presented here. These equations are presented only for quick reference and to show the proper interpretation of the real gas formulas for the normal cases. The reader should refer to Refs. 1 through 5 for more details and explanation of departures from these simple cases. of p 8 The value of PR represents the average reservoir pressure in the drainage area of the well. It is important to determine PR from a buildup test so that PR can be used for material balance calculations, history matching in reservoir simulation, or in pseudosteady-state perform-



ENGINEERING



HANDBOOK



Asymptotically, the data approach the correct value of PR as At approaches infinity. Since our shut-in time normally is limited, the MBH method is based on extrapolating the semilog straight line to At= 03, or (fp +At)lAt= 1 .O. This value is called p*. The method then provides a correction to calculate the correct value of j?~ from the extrapolated value of p*. The MBH method assumes that the well flowed at a constant rate for tp and that the drainage area A is known for the well. The dimensionless producing time, tpDA , is calculated. If tpDA is greater than (tp~A)psJ, the later value can be used as tpDA . In other words, it is not important what the rate history was before pseudosteady state was achieved. Now that p* has been extrapolated from the data and tpDA has been calculated, then the correction between p* and jYR is made by using the MBH correction curve that best represents the drainage shape. The MBH correction curves are presented in Figs. 35.14 through 35.17. A stepwise procedure to determine p.8 can be summarized as follows. 1. Make a Horner plot. 2. Extrapolate the semilog straight line to the value of p* at (tP +At)lAt= 1.0. 3. Evaluate m, the slope of the semilog straight line. 4. calculate tpDA =(o.ooo264kt,)/~pcr~. 5. Find the closest approximation to the drainage shape in Figs. 35 _14 through 35.17. Choose a correction curve. 6. Read the value of 2.303(p*-jY~)lrn from the correction curve at t,~~. 7. Calculate the’value of 5 R. This procedure gives the value of p R in the drainage area of one well. If a number of wells are producing from the reservoir, each well can be analyzed separately to give a j?~ for its own drainage area. This is done, assuming that all wells are producing in pseudosteady state, by dividing the reservoir up into drainage areas for each well by constructing no-flow boundaries between the wells. Fig. 35.18 shows an illustration of such a segmentation of a reservoir. These no-flow boundaries represent the “watersheds” of the different drainage areas. The drainage areas are calculated so that each drainage area has the same reservoir flow rate compared to its PV. Thus,



Determination



ance



equations.



There are several methods for determining Jo from a buildup test but the most general is the MBH (Matthews, Brons, and Hazebioek I’). This method is generally applicable because a number of different reservoir drainage area shapes are available for analysis. These reservoir shapes are the same as those used for evaluating shape factors in Table 35.2. Fig. 35.13 shows how the method is applied. The buildup test has a semilog straight line, which begins bending at the later shut-in times because of the effect of the boundaries. The data normally will bend down and become flat from this curve, but for unusual cases the data actually can bend up from the semilog straight line before it eventually becomes horizontal.



(qr/Vp)



1 =(qr/Vp)2



=(q,lvp)3=(qtlvp)i.



. . .



C41)



This relationship divides the drainage area (or PV) according to the producing rate of the well. As the well’s rates change, then the drainage area changes for the well. If q=O, for example, then no area would be allocated to that well. This procedure of calculating the drainage area and approximating drainage shape is repeated at the time of each pressure survey. The drainage areas and shapes keep changing as rates change. There is often confusion about the meaning of p* in the Horner plot. The value of p* has no physical meaning except in the special case of an infinite-acting well (T?=w). This is the case that Horner16 originally addressed in determining the initial pressure, pi ,-in a newly discovered well. In this special infinite-acting case, p*= p R =pi. Otherwise, p* has no physical meaning.



P meH =2.303(p*-pR)/m



PansH =2.303( p’-fn)lm N



Y



h



u



I! ”



0



ojs, E P r



PETROLEUM ENGINEERING HANDBOOK



,-



I-



I



-



DIMENSIONLESS



Fig. 35.16-MBH



dimensionless



pressure



PRODUCTION



for different



TIME.



welt locations



tCD.



in a 2: 1 rectangular



,o-



dimensionless



area.



I DIMENSIONLESS



Fig. 35.17-MBH



drainage



pressure



for different



PRODUCTION



TIME.



well locations



tpo4



in 4: 1 and 5: 1 rectangular



drainage



area



WELL



PERFORMANCE



35-19



EQUATIONS



TABLE



t,



At



(hours)



Fig. 35.18--Reservoir map boundaries.



showing



approximate



no-flow



Example Problem 4 (Pressure Buildup Analysis) (after Earlougher 2). Pressure Buildup Test AnalysisHomer Method. Table 35.3 shows pressure buildup data from an oil well with an estimated drainage radius of 2,640 ft. Before shut-in the well had produced at a stabilized rate of 4,900 STBiD for 310 hours. Known reservoir data are D = 10,476 ft,



rw = (4.25112) ft, psi-‘, C - 22.6~10~~ 4; i 4,900 STB/D, h = 482 ft,



pdAt=O)



= 2,761 psig, PO = 0.20 cp, c#l= 0.09, B, = 1.55 RBISTB, casing di = (6.276/12) ft, and rp = 310 hours.



k,=



0.0: 0.10 0.21 0.31 0.52 0.63 0.73 0.84 0.94 1.05 1.75 t .36 1.68 1 .ss 2.51 3.04 3.46 4.08 5.03 5.97 6.07 7.01 8.06 9.00 10.05 13.09 16.02 20.00 26.07 31.03 34.98 37.54



+At



BUILDUP TEST =310 HOURS PW



Pwn-Pwt



At



(Psk3)



(PW 296 392 473 480 495 499 502 505 506 507 510 513 515 519 522 525 528 532 536 536 539 542 544 545 549 552 556 559 561 562 562



-



310.10 310.21 310.31 310.52 310.63 370.73 370.84 37 0.94 311.05 371.15 37 1.36 311.68 311.99 312.51 313.04 313.46 314.08 315.03 315.97 316.07 317.01 318.06 319.00 320.05 323.09 326.02 330.00 336.07 341.03 344.98 347.54



3,101 1,477 1,001 597 493 426 370 331 296 271 229 186 157 125 103 SO.6 77.0 62.6 52.9 52.1 45.2 39.5 35.4 31.8 24.7 20.4 16.5 12.9 11.0 9.9 9.3



2,761 3,057 3,153 3,234 3,249 3,256 3,260 3,263 3,266 3,267 3,268 3,271 3,274 3,276 3,200 3,283 3,286 3,269 3,293 3,297 3,297 3,300 3,303 3,305 3,306 3,310 3,313 3,317 3,320 3,322 3,323 3,323



Ap, =0.87(40)(8.6)=299. Average Drainage-Region Pressure-MBH. We use the pressure-buildup test data of Table 35.3. Pressure buildup data are plotted in Figs. 35.12. Other data are A= ?rre2 =a(2,640)2



sq ft.



To see if we should use tp = 310 hours, we estimate tpss using @DAlpss =O.l from Table 35.2.



=12.8 md.



WWW



Skin factor is estimated from Eq. 40a using p ,hr = 3.266 psig from Fig. 35.12: 3,266-2,761



s=1.1513



40



(12.8)(12)2 (0.09)(0.20)(22.6x



10 -6)(4.25)2



1



tpss =



10 -6)(7r)(2,640)2(0.



I



=8.6.



1)



(0.0002637)(12.8)



=264 hours. Thus, we could replace tp by 264 hours in the analysis. However, since tp is only about l.l7t,,,, we expect no difference in j?~ from the two methods, so we use t,=310 hours. As a result, Fig. 35.12 applies. Fig. 35.12 does not show p* since (t,, +At)lAt does not go to 1.0. However, we may compute p* from pws at (tp +At)lAt= 10 by extrapolating one cycle: p* = 3,325 + (1 cycle)(40 psi/cycle)



+3.2275



DATA



(At, + At)



(hours)



(0.09)(0.2)(22.6x



-log



4,i,



We can estimate Ap across the skin from Eq. 10:



Solution. The Horner plot is shown as Fig. 35.12. Residual wellbore storage or skin effects at shut-in times of less than 0.75 hour are apparent. The straight line, drawn after At=0.75 hour, has a slope of -40 psigicycle, so m=40 psiglcycle. Eq. 37a is used to estimate permeability: 162.6(4,900)(1.55)(0.20)



35.3-PRESSURE FOR EXAMPLE



=3,365



psig.



35-20



PETROLEUM



(0.0002637)(12.8)(310) 10 -6)(a)(2,640)2



=0.117. From the curve for the circle in Fig. 35.14, poMnn(~~D,+, =O. 117)= 1.34. Then, from our step-wise procedure,



HANDBOOK



tpss = time required to achieve pseudosteady state u = macroscopic (Darcy) fluid velocity V, = volume of the wellbore xe = distance from well to side of square drainage area xf = distance from well to either end of a vertical fracture



Using the definition of tpDA:



rpDA= (0.09)(0.20)(22.6x



ENGINEERING



Subscript wb = wellbore



pR=3,365-



p(1.34) 4o 2.303



Key Equations in SI Metric Units



This is 19 psi higher than the maximum pressure recorded.



Nomenclature A = drainage area of well



cfi cWf CA C, f(t)



= = = = =



F Da= F, = m =



mD =



total compressibility evaluated at p; wellbore fluid compressibility shape factor from Table 35.2 wellbore storage constant unit response function non-Darcy (turbulence) factor turbulence factor (162.6qBp)lkh dimensionless m(p)



m(p) = 2jPtdp=



1



v2p=



=3.342 psig.



46 -k



3,557x10-9



ap at’ .......,,......



where p is in kPa, 4 is a fraction, p is in Paas, c, is in kPa-t, k is in md, and t is in hours. 4t4



=qoB,



+(s,



-R,q,)B,



+q,B,,



(6)



where qo,qr,qw are in std m3/d, B,,BI,B, are in res m3/std m3, qg is in std m3/d, and B, is in res m3/std m3,



real gas pseudopressure



, ..... .. ..



0 m(p)



(1)



(7)



= m(P) atpR



m( pi) = m(p) at initial pressure pi m(p,,,f) = m(p) at wellbore flowing pressure p,,,f m* = slope of m(p) plot m’ = slope of p plot m ” = slope of p* plot p* = MTR pressure trend extrapolated to infinite shut-in time po = kh(pi -p)/( 141.2qBp) =dimensionless pressure PDMBH = 2.303(p*-pR)lm, dimensionless pressure, MBH method Aps = additional pressure drop across altered zone (qg 1 = absolute value of gas rate qsf = flow rate at the sandface = r/rw =dimensionless radius rD re = external drainage radius rw’ = effective wellbore radius s’ = effective skin effect tD = dimensionless time tDA = dimensionless time based on drainage area, A = time required to reach pseudosteady bDA)Pss state, dimensionless t end = end of MTR in drawdown test = dimensionless producing time tpDA



where PD



=



[kh(pi



-~YW-W%41,



r rD = -, rw



tD =



3.557 x 10 -9kt 4wrrw 2



h,r,rw are in m, k is in md, p,pi are in Pa, q is in m3/d,



I3 is in res m3/std m3, p is in Pa*s, t is in hours, 4 is a fraction, and c,is in kPa-*. pwf =pi -m



kt log choir,



2 -8.10



where m=2.149~lO”qB~/(kh). units.



ap (-> at PSS



-4.168~1O-~qB



VpC,



>



,



. . . . . . . . . (9)



See Eq. 7. for other



)



. . . . . . . . . . . . (15)



WELL



PERFORMANCE



35-21



EQUATIONS



where VP is in m3, See Eq. 7 for other units.



See Eqs. 7 and 9 for units.



5.356x10p1E BP



4=



>



In T’ -0.75+s



(PR



-Pw&



. .



. . . (17)



k,h=



. . . . . . . . . . WW



m*



rw where re =m, s is dimensionless, and p~,p~f are in kPa. See Eq. 7 for other units.



vpc,



pR=pi--



)



where m* is in kPa2/Pa* s-cycle. See Eq. 33 for other units.



s=1.151 (I?



.. . . .. .. . .. .. . .... . . ..



. . . (21)



1



4Wg am(p) ~k



at



’ ....I



. (24)



where m(p) is in kPa2 and cg is in kPa-’ . See Eq. 7 for other units. h, t,+o.4@,5+S+FD,IqgI,



. . . . . . . (33)



where mD = 2.708x10-”



tD =



3.557x



.,.lO),



(384



See Eq. 7 for other units.



References



3.557x10-9



,,lj,=%



,......................... where m is in kPa/cycle.



where Np isinm3, VP is in m3, B, is in res m3/std m3, c, is in Wa-‘, and p~,p; are in kPa. V2m(p)=



) -log4pc;rw,2



10-9kt 2



dw, r,,’



’



s is dimensionless, FD, is dimensionless, qg is in m3/d, T,,.,TR are in K, prc is in kPa, k is in md,



h is in m, and m(p;),m(p,j) are in kPa2/Pa.s.



See Eq. 7 for other units. k h= _ 2.149x 106qoB,~o 0 .._ m



. .



(374



I. Matthews, C.S. and Russell, D.G.: Pressure Buildup and Fknv Tests in Wells, Monograph Series, SPE, Richardson, TX (1967) I. 2. Earlougher, R.C. Jr.: Advances in Well Test Analysis, Monograph Series, SPE, Richardson, TX (1977) 5. 3. Dake, L.P.: Fundmmntals ofReservoir Engineering, Elsevier Scientific Publishing Co., Amsterdam (1978). 4. Gas Well Testing-Theory and Practice, fourth ed., Energy Resources and Conservation Board, Calgary, AIL, Canada (1979). 5. Lee, John: Well Testing, Textbook Series, SPE, Richardson, TX (1982). 6. Pressure Analysis Methods, Reprint Series No. 9, SPE, Richardson. TX (1967). 7. Pressure Transient Testing Methods, Reprint Series No. 14, SPE, Richardson, TX (1980). 8. van Everdingen, A.F. and Hurst, W.: “The Application of the Laplace Transformation of Flow Problems in Reservoirs,” Trans. AIME (1949) 186, 305-24. 9. Martin, J.C.: “Simplified Equations of Flow in Gas Drive Reservoirs and the Theoretical Foundation of Multiphase Pressure Buildup Analyses,” Trans., AIME (1959) 216, 309-l 1. 10. Wattenbarger, R.A. and Ramey, H.J. Jr.: “An Investigation of Wellbore Storage and Skin Effect in Unsteady Liquid Flow: II. Fimte Difference Treatment,” Sot. Pet. Eng. J. (Sept. 1970) 291-97; Trans., AIME, 249. 11. Die& D.N.: “Determination of Average Reservoir Pressure From Buildup Surveys,” .f. Pet. Tech. (Aug. 1965) 955-59; Trans., AIME. 234. 12. Al-Hussainy,R., Ramey,H.J. Jr., and Crawford, P.B.: “The Flow of Real Gases Through Porous Media,” J. Pet. Tech. (May 1966) 624-36; Trans., AIME, 237. 13. Wattenbarger, R.A. and Ramey, H.J. Jr.: “Gas Well Testing With Turbulence, Damage and Wellbore Storage,” J. Pet. Tech. (Aug. 1968) 877-87; Trans., AIME, 243. 14. Ramagost, B.P. and Farshad, F.F.: “p/z Abnormally Pressured Gas Reservoirs,” paper SPE 10125 presented at the 1981 SPE Annual Technical Conference and ExhibItion, San Antonio. Oct. 4-7. 15. Miller, C.C., Dyes, A B., and Hutchinson, C.A. Jr: “The Estimation of Permeability and Reservoir Pressure From Bottom Hole Pressure Build-Up Characteristics,” Trans., AIME (1950) 189, 91-104 16. Homer. D.R.: “Pressure Build-Up in Wells,” Proc.. Third World Pet. Gong., The Hague (1951) Sec. II, 503-23. 17. Matthews.C.S., Brons, F., and Hazebroek, P.: “A Method for Determination of Average Pressure in a Bounded Reservoir,” Trans., AIME (1954)201, 182-91



Chapter 36



Development Plan for Oil and Gas Reservoirs Steven W. Poston,



Texas A&M u.*



Introduction The following discussion on the determination of the proper development plan for oil reservoirs or gas reservoirs is a summation of the current thinking in the oil industry. Conditions have changed dramatically since R.C. Craze wrote this chapter for the original book in 1962. At that time, the price of crude oil and gas was so low that the industry was concerned mainly with recovering the grass reserves fmm a field. Today’s economics have changed our outlook to such a degree that the need for a logical and efficient plan for the orderly development of an oil or gas field is of utmost importance. The bidding competition for reserves often has caused successful field development to be at least partially dependent on getting the most out of the ground with the minimum number of wells. The oil business was originally an endeavor that allowed one to explore for hydrocarbons in relatively unexplored areas. The probability of finding large fields was quite high, and an excellent return on investment resulted when a new field was found. A majority of the large oil and gas fields have been found after 25 years of intensive exploration. The number of companies searching for hydrocarbons has increased while at the same time the fields are harder to locate. Now we are a very competitive industry in which there is little room for ermr. In other words, the rules of the game have changed. New technology and thinking about logical field development has evolved during the last 20 years. Continuity of producing intervals between wells is now known to be much more important than previously thought. Advances in well test analysis have allowed the engineer and geologist to estimate reservoir size and intrawell continuity. Improved seismic techniques have allowed geophysics to play an increasingly important role in allocating well locations for efficient reservoir drainage. ‘Author



of the ongmal chapter on this topic I” the 1962 edition was &pert



C Craze.



A person interested in developing an oil or a gas field must use a basic understanding of geology, engineering, and economics. Other, more sophisticated techniques may have to be used at times to arrive at a realistic development plan. However, when one begins to develop a field, a number of questions need to be mulled over and should be discussed with colleagues. The thinking process occurs as follows.



Is the Well Being Drilled to Develop Proved, Probable, or Possible Reserves? The drilling of a development well in the middle of a field for proved reserves is considerably different than drilling an outpost well to help define the field limits. Greater reserves must be assigned to well questing for probable or possible reserves than for an infield development well. The drilling for known reserves often allows for a low return on investment. However, the reward must be greater if the risk of drilling and not finding the hydrocarbon accumulation increases. The benchmark for the go/no-go decision for the drilling of a well is a function of not only the return on investment but also the degree of risk to be incurred. Answering these questions requires a combination of all disciplines in the petroleum industry. The greater the certainty of the reserves, the less the need for geological and engineering opinions.



What Are the Reservoir Rock and Fluid Characteristics? Field development is conducted far differently in a clean, well-developed sand than it is in a place such as the low-porosity and low-permeability Austin chalk region of Texas. High porosities and permeabilities and low oil viscosities permit high offtake rates and wide well spacing. These large “per well” recoveries often preclude the need for the serious study of the minimum economic reserves requirements.



36-2



Development drilling will continue at a different pace for a continuous and homogeneous sand than for a field composed of a series of productive intervals sandwiched between shale layers of unknown lateral extent. A well completed in a series of sand stringers of uncertain area1 extent should be placed on production for a time to see how much it actually will produce. Extensive drilling in such a field should wait until the economic worth of the total effort is determined from field production figures. Any knowledge concerning the geology of the prospect attained before the well is drilled would furnish insight into the probable number of completion zones and where the completion intervals should be. The proper well spacing would be predicated on this knowledge. The type of drive mechanism often will predicate the placement of the development wells. If a water drive is expected, the wells should be placed in the most updip locations possible. However, the updip placement of the wells would be a disaster if there is an expanding gas cap drive. The information is derived from reservoir engineering evaluations.



What Is the Surface Environment?



Development considerations are completely different when drilling in a shallow well in west Texas or a Jurassic well in the North Sea. Platform rigs often are used to drill offshore wells. The number of drilling slots is limited, and, once the rig is moved off, it is often prohibitively expensive to move back on if new ideas arise.



What Surface Production Facilities Are Required? There is no sense in drilling an offshore development well if there are no facilities available for production hookup. The production facilities could cost much more than the value of the reserves. Drilling on land in an area where costs may be reduced considerably could allow the production facility costs to be only a fraction of the reserves’ worth.



By What Method is the Product to be Sold? Gas must be transmitted by pipeline, whereas oil must be trucked or lightered to a receiving facility. For an oil well, revenue usually begins upon completion, while a gas well must wait for the installation of a pipeline. The cash flow situation for development of either an oil or a gas field is usually different because of the type of product. What Is the Relationship Between the Costs and the Profit Margin? The margin of profit for an operator will vary considerably according to geographical location and the type of lease. Also, overhead costs may be greater for a large company than for a smaller company. The cost of money may be less for a large company because of a significant and established cash flow. Foreign profit margins are generally much less than margins from U.S. oil and gas sales. Readers will see other areas of uncertainty in addition to those discussed here. However, the following discussion will shed light on some of the more important points that one should remember concerning the formulation of a development plan for either an oil reservoir or a gas reservoir. There are no handy formulas to use nor are there any tried-and-true rules to follow. Proper field development for a particular set of conditions requires a combination of a variety of oil field disciplines.



PETROLEUM ENGINEERING



HANDBOOK



Oil and Gas Differences Method of Sales Development plans for oil or gas reservoirs generally follow different paths not only because of the optimal depletion characteristics but because of the method of sales. Crude oil is a reasonably stable substance and, being liquid, may be loaded easily into some type of container for transportation to a sales point. The container is often at or very near atmospheric pressure. The container may be a truck, barge, or pipeline. On most land locations. sales may begin from a well as soon as the production equipment is installed. Also, since oil is contained and moved easily, the buyer of the crude oil may not always be constant. Natural gas must be kept in some type of container so it will not dissipate into the atmosphere. The high compressibility of the gas permits a smaller container to be used with increased confining pressures. Economics dictates that gas is to be transported through pipelines. The pipeline company must be assured that sufficient reserves are present to justify the expense of installing the pipeline. These capital expenditures often require long-term commitments from all the interested parties. Sufficient reserves must be proved to justify the expense of laying a pipeline. A number of wells may have to be drilled before any income is derived from the initial discovery. The operator must drill sufficient wells to ensure the quantities of gas required to be delivered over the contract period. The oil may be transported out by barge or tank truck if the reserves do not justify the expense of installing a pipeline in the case of oil production. Operating expenses are greater when oil is moved by tank truck or barge, but the capital investment is negligible when compared to pipeline installation. Development drilling in an oil field often may be conducted in a more growth-oriented manner than that in a gas field. Generally speaking, the capital investment required to develop a gas field is greater than for developing the same reserves in an oil field because a pipeline always is required to transport the gas. Non-capitalintensive barges or trucks may be used to transport oil.



The Best Depletion Technique There are fundamental differences between developing and depleting an oil reservoir and a gas reservoir. These differences are discussed next.



Oil Reservoirs. Every effort should be made to maintain reservoir pressure as high as possible during the depletion of an oil reservoir. A high reservoir pressure helps to preclude the installation of some type of artificial lift system or some method to aid in recovery. High reservoir pressures usually result from an active water drive or gas cap encroachment, both of which displace oil and help to push it toward the wellbore. These displacement mechanisms result in a reduced oil saturation at a relatively high abandonment pressure.



Gas Reservoirs. The compressibility of gas may be up to 1,000 times greater than relatively incompressible oils. These high compressibilities can allow the majority



DEVELOPMENT



PLAN FOR OIL & GAS RESERVOIRS



of the reserves in a gas reservoir to be depleted by simple gas expansion. In fact, ultimate recoveries of 80% of the original gas in place (OGIP) may be achieved by pressure depleting a gas reservoir, even though the remaining gas saturation may be quite high. Conversely, if a gas is trapped behind an advancing water front with a correspondingly lower residual saturation, the remaining gas left behind will be greater because the high compressibility of the gas allows a much greater quantity of gas to be trapped at these higher reservoir pressures. Example Problem l-Dry Gus Reservoir. The example given in Table 36.1 indicates the effect of the type of drive mechanism on ultimate recovery from a theoretical dry gas reservoir. Water is assumed to invade the reservoir uniformly in the water influx case. The assumption is not necessarily true in the operational context, but the illustration is made to show the necessity of abandoning gas reservoirs at low pressures. The effect of the gas FVF in the lower-pressure reaches of the reservoir allow the pressure depletion case to recover more gas. The previous discussion shows how the development of an oil reservoir may be conducted in a piecemeal and leisurely manner while development of a gas reservoir should be carried out with an eye toward maximizing the reservoir offtake rate to aid in the occurrence of pressure depletion conditions. To arrive at a development plan two basic steps need to be accomplished. These are (1) the characterization of the reservoir and (2) the prediction of the performance of the reservoir under various exploitation schemes and operating conditions.



Characterization of the Reservoir Geology Interpretation of Paleo-Environments.



The limits of a reservoir and the possible variation of the porosity and permeability within the reservoir may be inferred by studying the well logs and cores taken from wildcat and appraisal wells. The knowledge gained from these studies would be of great help in setting wellsite locations early in the life of a development drilling project. Usually the reservoir productive characteristics are known only after the field or reservoir is maturely developed. The nature of the reservoir rock often is reflected in the sedimentary record. The sedimentary section is penetrated during the drilling for oil and gas. The character of the sediments may be inferred by logs or by core analyses. For a number of years, geologists have been studying and relating currently occurring sedimentary processes to reservoir rock paleo-environments. Each sedimentary process has been shown to have a particular porosity and permeability distribution and to have a reasonably predictable area1 extent. The interpretation of the probable paleo-environment by log and core analysis of a sedimentary section could be of inestimable value early in the life of field development. The following discussion gives a brief overview of geological interpretive work. The literature contains an overabundance of work on the evolution of elastic sediments. The reservoir characteristics of a elastic sediment (mainly sandstones) often is related largely to its depositional history.



36-3



TABLE 36.1 -EXAMPLE OF EFFECTS OF DRIVE MECHANISM ON RECOVERY V,



= 6,400



acre-ft



Q = 22% s,



= 23%



s,



= 34%



G = 8.878 x log scf Cumulative Production (109 scf) Pressure (PW 3,150 2,500 2,000 500’



(set% ft) 188 150 120 28



Volumetric Reservoir 1 .a



3.2 7.6



Waterdrive Reservoir 5.8 6.4 -



‘LOW res?rvOlr press”res Will not be ObtaIned because Of me additlonal energy supplied by the encroaching water, therefore, Bg will be at a higher value at abandonment.



Therefore, a predictive interpretation may be some degree of certainty. Less is known of carbonates. The chemical the depositional processes to form carbonate and the usually extensive diagenetic history true nature of the reservoir character. A large data-i.e., a considerable number of wells-is before the nature of a carbonate reservoir discerned.



done with nature of reservoirs cloud the amount of required may be



Clastic Reservoirs. The depositional environment may be estimated by studying electric log sections that pass through the zone of interest and by analyzing core samples taken from the zone. l-3 The interpretation of these paleo-environments is derived from the study of modem depositional environments. The character of modem streams, deltas, and beaches has been well documented. 4-6 Bernard and LeBlanc’ divided the major depositional environments into continental, transitional, and deep marine classifications. Continental and deep marine deposits do not contain widespread oil or gas accumulations and are not discussed further. Transitional sediments may be divided into coastal interdeltaic and deltaic environments. The coastal interdeltaic area usually consists of linear, relatively narrow sand beaches, which extend seaward into a normal and then a deepwater environment. The sands composing the normal marine environment are usually very fine grained and are deposited in conjunction with a high percentage of clay. The generally low permeabilities displayed by normal marine sediments preclude a high incidence of commercial oil and gas deposits. 8 Deepwater marine sediments are composed mostly of shales and are on the whole nonproductive. The most common and important hydrocarbon-bearing sandstone reservoirs are of deltaic origin. These sediments usually are deposited in a high-energy, often fluctuating atmosphere. In deltaic environments encountered most often during oil and gas drilling operations, delta-bar and distributary channel sediments are the two most prevalent sedimentary environments found, while offshore bars may be found in the delta front areas.



PETROLEUM ENGINEERING



36-4



HANDBOOK



DELTAIC CHANNEL DEPOSITS GAMMA RN



a



PERML4Bltll-Y (MD1



b



DELTAIC BAR DEPOSITS GAMMA R&Y 01



PERMEABILITYIMDI a1 10 low



-3



r



Grain Fig. 1&l-idealized



Size



porosity and permeability profiles-bar



The delta-bar sequence is typified by an upward gradation from shallow, marine clays at the base through a section that shows an increasing grain size. The progressive upward coarsening of the sand-grain size is the result of the delta advancing over the marine clays. A high-energy regime is seen to increase in the vertical direction. A typical electric log section grades upward from a shale section (deeper water) to gradually increasing amounts of sand9 (see Fig. 36.1). The section contains crossbeds, ripple laminations, and modest amounts of quartz. Delta-bar sands grade downdip into pro-delta silts and clays and grade updip into the organic-rich, fresh- and brackish-water clays. Delta sands often are limited in areal extent, even though encompassing a thick sedimentary sequence. Vertical reservoir continuity may be restricted because of the large number of shale stringers present in the delta front sequence. Distributary (river) channels transport sediments to the delta front. Distributary channels cut through deltas in a variety of meandering ways. Even though they comprise only a small portion of sedimentary record, these sediments often transect deltaic or offshore bar sand reservoirs and incur reservoir discontinuities in an otherwise homogeneous system. Fig. 36.2 is an example of such a discontinuity in the South Pass 27 field located in the offshore waters of south Louisiana. lo The field is included in the sand/shale sequence generated by prior deposition of the Mississippi River. Notice how the channel cut through the previously deposited sediments and formed a reservoir separate from the original. Distributary channel sediments initially are deposited in a higher-energy atmosphere, and, hence, display a



and channel deposits.



coarser grain size toward the bottom of the section. The effect of grain size gradation may be seen in Fig. 36.1. These deposits are characterized by boxy log shapes with a very high sand content. The gradation of the sands is typified by an abrupt change from a shale to a very clean sand and then to a gradual increase in shale/sand ratio in the upward direction. Deposition is parallel to the source of the sediments. Shoreline or barrier-island sandstones are represented by a sequence of normal marine muds grading upward into laminated sandstones. The section may be overlain by aeolian dune sandstones, which are the emergent portion of the shoreline. Sand gradation is generally coarsening upward. The sand grains are well sorted, and the quartz content of the sand is quite high. Wave action has reduced the less resistant feldspars to clay-sized particles, which have been transported to lower-energy regimes. Deposition is normal to the source of the sediments. I1 The sand bodies contain very few shale laminations and they are characterized by excellent lateral continuity. I2 The lowermost layer of a barrier bar sand comprises interbedded sand, silts, and shales. The second layer is made up of a bioturbated thick sand sequence. The penultimate layer consists of laminated sands laid down on the beach or the upper shore face of the barrier bar. The uppermost layer usually consists of oxidized aeolian deposits. l3 Barrier bar reservoirs offer an excellent opportunity for hydrocarbon exploration. The reservoirs usually are overlain by lagoonal clays, which form an excellent trap. Barrier sands usually exhibit a high degree of internal continuity and are deposited parallel to the coastline.



DEVELOPMENT



36-5



PLAN FOR OIL & GAS RESERVOIRS



SOUTH



NORTH S.L. 1007 1 S.L. 1012 I NO.5.30 r



o R lttLo lM 2



no.54



llo.1n



MD.lM



‘a



9



-rit



DR ILL 1ED 2/ H



ORILLEDli74



ORILLEO



NO. 203



NO. 115



l



0 1Ull



DRILLED



ORILLED3,74



5hS



rA lOrr2



Fig. 36.2--Reservoir



discontinuity-channel



Carbonate Reservoirs. Carbonate reservoirs are completely different in nature from sandstone reservoirs. The composition of sandstone reservoirs is largely a product of the depositional environment; carbonate reservoirs are a product of not only the depositional environment but also mechanical processes that occur after deposition. l4 The heterogeneities caused by the variety of formative processes may form extremely complex fields such as the Means field shown in Fig. 36.3. Is Note the field heterogeneity. Carbonates may be deposited in both shallow- and deepwater marine environments. The fields may range from a few acres (pinnacle reefs) to regional in size (carbonate banks). Jardine16 has discussed how



Fig. 36.3-Means



carbonate



and fringe sands.



fields may be formed in a variety of settings.



Biohenn Reefs. Bioherm or pinnacle reefs usually are characterized by their relatively small size with a high degree of relief. The reefs contain a high percentage of skeletal material at the outermost portions of the accumulation. The interior of the reef is composed of finergrained material and has less porosity and permeability than the outer limits. Biostrome Reefs. Biostrome reefs were formed in less rapidly subsiding basins and may extend for hundreds of square miles. Like the biohenn reefs, the biostrome reefs contain a high percentage of skeletal material. Horizontal stratification is present.



field schematic.



IF



36-6



PETROLEUM ENGINEERING



PRIMARY POROSITY



HANDBOOK



SECONDARY POROSITY



CONFIGURATION



BI0HEP.M REEF



INCREASE K R



0



DOLOMIlIZAllON BANK



(SHELF)



DECREASE d



R K



DECREASE 0



& K



PORE SIZE AND K CEMENTATION



Fig. 36.4-Distribution



of porosity within various types of carbonate reservoirs.



Shelf Carbonates. Shelf carbonates are usually sheetlike or tabular bodies composed of a high percentage of skeletal material, enclosed by surrounding fine-grained material. Nearshore Deposits. Nearshore deposits are usually of a thin and restricted nature andare generally fine grained. This type of deposit is of minor significance in oil and gas production. the characteristics of the Fig. 36.4 I6 summarizes variety of carbonate deposits. Note the different types of porosity and the processes that affect the reservoir quality. The development of oil and gas fields in carbonate sediments requires the study of the fossil content, any postdepositional alterations, and characterization of the pore space. This type of reservoir often displays two dissimilar porosity-permeability systems.



Extent of Shale Stringers. The knowledge of the probable lateral composition of a sandstone body soon after



Fig. 36.5-Continuity



of shale intercalations.



discovery would be of considerable aid for planning of the future development drilling program. Weber” combined studies done principally by Zeito, I8 Verrien et al., I9 and Sneider et al. *’ to arrive at Fig. 36.5. The figure summarizes a number of efforts to estimate the effect of depositional environment on the extent of shale stringers on sandstone reservoirs. Note how the marine sands possess the most extensive shale barriers, while the more poorly sorted point bars and distributary channels possess the shale members of least extent. Of course, the more widely correlative a producing interval is, the easier it is to predict future productive patterns. Many channels and point bars have beenlaid down in such a widely fluctuating atmosphere that correlation between wells is often difficult if not impossible. The recognition of the possible extent of the shale intercalations early in the life of development in the field would be of tremendous aid in the spotting of well locations.



Engineering In&awe11 continuity of the producing zone is one of the main ingredients for successfully depleting an oil or gas reservoir of the majority of the potential reserves. Additional development drilling often is required in a field when sand stringers are found to be discontinuous between producing wells. The differential movement of fluids within a reservoir caused by rock heterogeneities was noted first in the engineering sense by Stiles.2’ Poor response to the installation of many of the waterflood projects installed in some of the west Texas carbonate reservoirs in the 1950’s and 1960’s produced a spate of studies investigating the often discontinuous nature of the reservoirs. Refs. 15 and 22 through 25 are good reviews of some of these investigations. The determination of the degree of noncommunication between adjacent wells may be quantified to a certain degree by geological and reservoir engineering studies. The better-known techniques for estimating the degree of reservoir continuity are discussed next.



DEVELOPMENT



36-7



PLAN FOR OIL & GAS RESERVOIRS



Net Pay/Net Connected Pay Ratio. Irregularities within sedimentary rocks often cause discontinuous productive horizons between wells. The degree of these discontinuities may be discerned by correlating the individual pay zones between adjacent wells. If a particular sand stringer is seen in one well but not in the other then it is called discontinuous. Sands are known to become more discontinuous with distance. A method to estimate the degree of producing-sand-interval intrawell communication is discussed in a paper by Stiles. 23 The continuity between wells is defined as the fraction of the total pay sand volume that is connected to another well. A productive stringer is defined as continuous when correlatable between two wells. The stringer is classed as discontinuous if it is not correlative. Well pairs are compared, and eventually a figure may be drawn that summarizes the decline in reservoir continuity with distance. Fig. 36.6 is the result of one of these studies.26 Notice the decline of continuity with distance between wells. The figure shows that the number of producing zone discontinuities was found to be much greater than expected when additional infill drilling was carried out in the Means field. A similar type of investigation by Stiles23 in the Fullerton-Cleat-fork Unit had indicated a degree of reservoir continuity of 0.72. The estimate compares quite favorably with a material balance of the field. \ A more recently published paper indicated that the material balance and the volumetric in-place estimate for a number of reservoirs in the Meren field compared very correlation of these same favorably. *’ A sand-by-sand reservoirs in the Met-en field indicated a degree of continuity approaching that of unity. One could gather from these studies that communication was uniform throughout the reservoirs and additional infield drilling in all likelihood would not discover many discontinuous sand members. However, infield drilling in the Fullerton-Clearfork Unit could prove fruitful because of the good probability of penetrating previously undrained sand members. Material Balance Studies. The results of volumetric reserves estimates may be compared to the material balance estimate. The material balance estimate is a function of production, which is derived from the movement of fluid through connected producing zones. Volumetric calculations are determined from net sand maps, which often do not take into account the effect of sand discontinuities on production. The difference between the results of the calculations gives an idea of the degree of discontinuity of a particular reservoir. Stiles23 used the idea when studying the Fullerton-Clearfork Unit. The material balance method indicated 738 million bbl OIP. A volumetric estimate showed 1.03 billion bbl OIP. The ratio of the material balance estimate to volumetric estimate is 0.72. The low degree of communication would be an indicator of the successful outcome of an infield drilling project.



Computer Simulation Methods. Reservoir



simulation studies are simply an extension of the material balance technique. However, the reservoir simulator allows one to take into account the producing and rock characteristics of individual areas within the reservoir.



Fig. 36.6-Continuous



pay-Means



field.



Details of reservoir simulation are given in Chap. 48. A study by Weber 28 is an excellent example of the use of core and log interpretation principles to aid in determining the paleo-environment. These interpretations then were used in a computer simulation program, which was able to typify the D 1.30 reservoir in the Obigbo field with a high degree of accuracy. Fig. 36.7 is the type log of the reservoir. Note the differentiation of the producing interval into four discrete depositional environments. Each of the environments is represented by an interval of differing productive characteristics. The variations of these environments were noted in the section of each well penetrating the D 1.30 sand interval. Core analyses indicated the range of permeabilities that each of the units would exhibit. A permeability distribution map was drawn for the reservoir as a whole from these machinations. Subsequent modeling of the drainage patterns within the reservoir could be carried out with a high degree of certainty since the pattern of deposition had been replicated.



Interference Testing. The analysis of reservoir pressures has been an age-old reservoir evaluation tool in the petroleum industry. The similarity of pressures within a group of wells usually helps prove or disprove the interwell communication. An abnormally different pressure from a particular well is often the first indication of reservoir separation. Further analysis may disclose a previously undetected fault separating the wells in question. Sometimes wells are seen to display similar static bottomhole pressures even though there is a known fault separation. The similarity of pressures is caused by the production from each well being sufficient to draw the reservoir pressure down to the same degree. A transient pressure test must be run between the well pairs to estimate the degree of interwell communication. The alteration in the producing or injection rate of a well will have an effect on the pressure in a connected observation well. The study of these effects is called “transient-pressure” or “interference” testing. Interference testing may be done by either a long-term production or injection change in a well (interference testing) or by very short-term rate alterations (pulse testing). Ref. 29 presents a detailed description of the two methods.



PETROLEUM ENGINEERING



36-8



HANDBOOK



D I.30 RESERVOIR LITHOLOGY



Fig. 36.7-Type



log-D



Interference tests comprise a relatively long-term rate alteration. The effect of the rate alteration will be noted in the observation well when there is interwell continuity. Of course, one would assume the presence of a discontinuity if the pressure fluctuation is not seen in the observation well. The field application of an interference test is well documented in Ref. 30. A fieldwide spacing rule of 40 acres per well had been instituted in the North Anderson Ranch field in Lea County, NM. The engineering effort



29



28



V&ELL “B”



WELL “C” 0



T



32



WELL



l



15 s



0 ‘A”



l



was designed to estimate the true drainage area with the field. Four wells were produced and the resulting pressure decline was noted in a central observation well. (See Fig. 36.8 for the plan of the well layout.) The production from the four offsetting wells declined 11 psi after 165 hours’ production. The diffusivity equation was used to calculate the expected pressure drop for similar conditions. The theoretically predicted pressure drop was 12 psi. The use of interference tests indicated a well drainage area greatly in excess of the initial 40-acre estimate. An go-acre drilling pattern would effect a similar recovery with a greatly reduced number of wells. Pulse testing is often more convenient than interference testing. 3* The use of very precise pressure gauges coupled with individual design characteristics often allows pulse tests to be carried out within 1 or 2 days. Minor variations in production or injection volumes are able to send a pulse to observation wells. The variation of rates provides a “footprint,” which may be noted by precision gauges placed in the observation wells. A pulse test is able to discern reservoir heterogeneities in a manner similar to the previously discussed interference test. However, the test may be carried out in a much shorter time because of the precision of the equipment. Ramey 32 discusses the use of the pulse testing technique to determine reservoir anisotropy.



3D Seismic Techniques.



T l



1.30 sand, Obigbo field.



Geophysics



l l



16 S



+



2



R 32 Fig. 36.8~Interference



AND ENVIRONMENT



W test plan.



The three-dimensional (3D) seismic technique is a system of seismic data collection and processing that permits the proper vertical images to be developed and displayed by solving three orthogonalwave equation migrations. The 3D method is a useful technique to map subsurface structures and to define the field configuration better previous to development. The detailed results allow the fault boundaries and



DEVELOPMENT



PLAN



FOR OIL & GAS RESERVOIRS



Fig. 36.9-Comparison



36-9



of 2- and 3D seismic surveys.



stratigraphic limits of a reservoir to be mapped accurately soon after discovery. The number of appraisal wells would be reduced, and a more reliable estimate of the reserves could be obtained early in the life of the prospect. The knowledge of these two important facts would materially affect the overall drilling program. The method is considerably more expensive than the more mundane seismic techniques, but it has been estimated that 100 sq miles of seismic covera e may be obtained for the cost of one appraisal well. 35 The 3D method provides greater structural definition than the better-known two-dimensional (2D) techniques for the following reasons. 34-38 1. The placement of the vertical and horizontal reflection images is more accurate. Additionally, both vertical and horizontal sections may be presented for any depth and for any direction. 2. Defraction events are eliminated. 3 The signal strength normally lost because of scattering problems is restored. 4. The increased control point density permits more accurate mapping. 5. The greater amount of data improves the statistical base for estimating near-surface corrections and velocities. A particularly interesting example of using the 3D seismic method to evaluate a prospect and to help plan the drilling program may be seen in a study conducted in the Gulf of Thailand. 39 Three wildcat wells had disclosed the presence of probably commercial quantities of



gas. However, the prospect appeared to be faulted and a number of appraisal wells would be required to evaluate the potential in this relatively unexplored region. A region of 120 km2 was subjected to a 3D seismic reconnaissance shot at 100-m intervals. The program afforded a greater definition of the megastructure, indicated faulting was much more prevalent than previously indicated, and also helped prove the viability of the prospect. Figs. 36.9a and 36.9b compare the structural interpretations obtained by conventional 2D results with those obtained by 3D vertical migration. Note the increase in the complexity of the structure. The clarity of the 3D subsurface structural interpretation results from the more sharply focused nature of the process. The 2D interpretations give a more blurred or distorted picture because of the coarser sampling, which results in a statistically poorer presentation. A survey conducted in offshore Trinidad4’ resulted in a change in the platform location and drilling plan of one prospect and the deletion of another prospect from development until additional exploration in other faultblocks was conducted.



Prediction of Reservoir Performance After the reservoir has been characterized adequately, as described previously, a development plan must be selected. Performance of the reservoir under various exploitation schemes needs to be determined before selecting the final development plan. The modem tools used



36-10



by the reservoir engineer to predict the performance of the reservoir are reservoir simulators or mathematical models (see Chap. 48). A general description of the simulation steps and the results from simulation follows.



S!mulation Steps Data Preparation. 1. Select the appropriate simulator to use in the study-i.e., black oil. compositional, 2D, 3D, etc. 2. Divide the reservoir into a number of cells-i.e.. establish a grid system for the reservoir. 3. Assign rock properties, geometry, initial fluid distribution, and fluid properties for each cell. The rock properties include permeability, porosity. relative permeability, capillary pressure, etc. The cell geometry includes depth, thickness, and location of wells. Fluid properties are specified by the usual PVT data and phase behavior if required. 4. Assign the production and/or injection schedule for wells and the well constraints that need to be maintained.



Performance Prediction. If no historical data are available, the next step is to make the necessary computer runs to obtain the performance of the wells and the reservoir as a function of time and various plans of development. If historical data are available, the first step is to match the historical performance. The reservoir performance is calculated and the results are compared with the fieldrecorded histories of the wells. If the agreement is not satisfactory, adjustments in the data (such as the relative permeability, the specific permeability. the porosity, the aquifer, etc.) are made until a satisfactory match is achieved. The model then is used to predict the performance for alternative plans of operating the reservoir. In summary, the reservoir engineer obtains from the simulators the reservoir performance for different including various displacement development plans, mechanisms (such as water or gas injection, miscible displacement, etc.), different number and location of wells, and effect of flow rates. The reservoir performance then is used in the appropriate economic analysis to decide on the optimal development plan.



References I. Krueger, W.C. Jr.: “Depositional Environments of Sandstones as Interpreted from Electrical Measurements-An Introduction.” Trans.. Gulf Coast Assoc. Geol. Sot. (1968) XVIII, 226-41. Selly, R.C.: “Subsurface Environmental Analysis of North Sea Sediments,” AAPG (Feb. 1976) 60, No. 2. 184-95. Berg, R.R.: “Point Bar Origin of Fall River Sandstone Reservoirs, Northeastern uiyormng.” AAPG (1968) 2116-22. Sedimenza~ Environmenrs and Fucies. H.G. Reading (ed.), Elsevier Press, New York City (1978). Remeck, H.E. and Singh, 1.B.: DeposittonaL Seduncniur$ Environments, second edition, Springer-Verlag Inc., New York City (1975). 6. Scholle, P.A. and Spearing, D.: “Sandstone Depositional Environments,” AAPG (1982) Memoir 3 1. I. Bernard, H.A. and LeBlanc, R.J.: Resume ofQuatemaq Geology ofrhe Northwestern GulfofMexico Province, Princeton U. Press, Pnnceton, N.J. (1965) 137-85. 8. Berg, R.A. : Studies of Reservoir Sun&ones. Prentice Hall, En&wood Cliffs, N.J. (1985).



PETROLEUM



ENGINEERING



HANDBOOK



9. Sneider. R.M.. Tinker. C.N.. and Meckel. L.D.: “Deltaic Environmental Reservoir Types and Their Characteristics,” .I. Per. Tech. (Nov. 1978) 1538-46. IO Hartman. J.A. and Paynter, D.D.: “Drainage Anomalies m Gull Coast Tertiary Sandstones,” J. Per. Tech. (Oct. 1979) 1313-22. II Pryor. W.A. and F&on, K.: “Geometry of Reservoir-Type Sandbodies in the Holocene Rio Cirande Delta and Comparison With Ancient River Analogs.” paper SPE 7045 prcsentcd at the 1978 SPEiDOE Enhanced Oil-Recovety Symposium. Tulsa. April 16-19. 12 Poston, S.W., Berry, P., and Molokowu. F.W.: “Meren Field-The Geology and Resewou Characteristics of a Nigenan Offshore Field,” /. Per. Tech. (Nov. 1983) 2095-2 104. 13 LeBlanc. R.J.: “Distnbutlon and Continuity of Sandstone Rehervain-Parts I and 2,” J. Per. Twh. (July j977) 776-804. 14 Harris, D.G. and Hewitt, C.H.: “Synergism in Reservoir Management-The Geologtc Perspectwe.” j. Per. Tech. (July 1977) 76 I-70. 15 Kunkel. G.C. and Bagley, J.W. Jr.: “Controlled Waterflooding. Means Queen Reservoir,” J. Pe/. Tech. (Dec. 1965) 1385-90. 16. lardine, D., er (I/.: “Distribution and Contmu~ty of Carbonate Reservoirs,” J. Per. Tech. (July 1977) 873-85. 17. Weber. K.J.: “Influence of Common Sedimentar): Structure\ on Fluid Flow in Reservoir Models,” J. Pet. Tech. (March 1982) 665-72. 18. Zeito, G.A.: “Interbedding of Shale Breaks and Reservoir Heterogeneities,” J. Pet. Tech. (Oct. 1965) 1223-28: Trcrns.. AIME, 234. 19. Verrien, J.P., Courand. G., and Montadert. L.: “Applications of Production Geology Methods to Reservoir Characteristics -Analysis From Outcrops Observations.” Proc . Seventh World Pet. Gong.. Mexlco City (1967) 425. 20. Sneider. R. M., er al. : “Predicting Reservoir Rock Geometry and Continuity in Pennsylvanian Reserwr. Elk City Field, Oklahoma,” J. Pet. Tech. (July 1977) X5 l-66. 21. Stiles. W.E.: “Use of Permeability Distributmn in Wateflood Calculations,” Trans., AIME (1949) 189. 9-14. 22. Driscoll, V.J. and Howell, R.G.: “Recovery Optimization Through Intill Drilling-Concepts, Analysis, and Field Results.” paper SPE 4977 presented at the 1974 SPE Annual Fall Meeting, Houston, Oct. 6-9. 23. Stiles, L.H.: “Optimizing Waterflood Recover), in a Mature Waterflood, The Fullerton Clearfork Unit,” paper SPE 6198 presented at the 1976 SPE Annual Fall Meeting, Houston, Oct. 3-6. 24. George, C.J and Stiles, L.H.: “Improved Techniques for Evaluating C; bonate Waterfloods in West Texac,” J. Pet. Tech. (Nov. 1978) 1547-54. 25. “Application for Waterflood Response Allowable for Wasson Denver Unit,” Shell Oil Co., testimony presented before Texas Railroad Commission, Austin (March 21. 1972) Docket 8-A-61677. 26. Barber, A.H. Jr. etal.: “Intill Drilling to Increase Reserves-ACtual Experience in Nine Fields in Texas, Oklahoma and Illinois.” J. Pet. Tech. (Aug. 1983) 1530-38. 27. Poston, S.W., Lubojacky. R.W. and Aruna. M.: “Meren Field-An Engineering Review.” J. Pet. Tech. (NW 1983) 2105-12. 28. Weber, K.J. er al.: “Simulation of Water InJection in a BanierBar-Type, Oil-Rim Reservoir in Nigeria.” J. Pet. Tech. (Nov. 1978) 1555-65. 29. Earlougher, R.C. Jr.: Adwnce.s in Well Tat Analysis. Monograph Series. SPE, Richardson (1977) 5. 264 30. Matthies, E.P.: “Practical Application of Interference Tests,” J. Per. Tech. (March 1964) 249-52. 31. Johnson, C.R., Greenkom, R.A., and Woods, E.G.: “PulscTesting: A New Method for Describing Reselvou Flow Properties Between Well,” J. Pet. Tech. (Dec. 1966) 1599-1602; Trans.. AIME, 237. 32. Ramey, H.J. Jr.: “Interference Analysis for Anisotropic Formatlons-A Case History,” J. Pet. Tech. (Sept. 1975) 1290-98. 33. Brown, A.R.: “Three-D Seismic Surveying for Field Development Comes of Age.” Oil & Gas J. (Nov. 17, 1980) 63-65. 34. Johnson, J.P. and Bone. M.P.: “Understanding Field Development History Utilizing 3D Seismic,” paper OTC 3849 presented at the 1980 Offshore Technology Conference, Houston, Mav 5-8. ot 35. Graebner. R.J., Steel. G., and Wuwn. C.B.: “Evolutwn Scivnic Technology I” the XO‘r.” APkA J (19801 20. I 10-X)



DEVELOPMENT



PLAN



FOR OIL & GAS RESERVOIRS



36. French. W.S.: “Two Dimenknal and Three Dimcns~onal M~eration of Model-Experiment Reflection Profiles,” Gwphrticx (April 1974) 39. No. 4. 265-77. 37. Hikerman. F.J.: “interpretation Lessons From ThreeGmphwics (May 1982) 47, No. 5. Dimensional Modeling.” 784-808. 38. McDonald. J.A., Gardner, G.H.F., and Kotcher. J.S : “Areal Seismic Methods For Determining the Extent of Acoustic Discon-



36-l 1



tinuitles,” Geo~hwi~~.c (Jan. 1981) 46. No I. 2-16. 39. Dahm. C.G. and Graebner. R.J.: “Field Development with Three Dimensional Seismic Methods-Gulf of Thailand-A Case History,” Geophysits (Feb. 1982) 47. No. 2. 149-76. 40. Galbraith, M. and Brown, R.B.: “Field Appraisal with ThreeDimensional Seismic Surveys-Offshore Trinidad.” Grophwicx (Feb. 1982) 47, No. 2, 177-95.



Chapter 37



Solution-Gas-Drive Reservoirs Roger



J. Steffensen,



Amoco Production Co.*



Introduction An oil reservoir



is a solution-gas-drive



dergoes primary



depletion



supplied



reservoir



if it un-



with the main reservoir



energy



by the release of gas from the oil and the expan-



sion of the in-place fluids as reservoir pressure drops. This excludes reservoirs affected significantly by fluid injection or water influx. Also, reservoirs that have vertical segregation



of the gas and oil by gravity



special analysis. (In combination duction practices, gravity drainage ery significantly.) are sometimes



Reservoirs included



internal-



with an initial



free-gas cap



of solution-gas-



(gas expansion)



drive also is called dispersed-gas



(as opposed



to injected)



merit



with appropriate procan increase oil recov-



in the category



drive reservoirs; the gas-cap drive plements the solution-gas drive. Solution-gas



drainage



gas drive



sup-



drive or



because the



gas comes out of solution throughout the portion of the oil zone that has a pressure below the bubblepoint. Initially,



pore space in a solution-gas-drive



tains interstitial



reservoir



con-



water plus oil that contains gas in solution



Abundant literature reservoir performance production-rate



is available on solution-gas-drive and prediction methods i--2 and on



computations



for



wells



in those



voirs. 23-3o Special methods have been developed dicting



the behavior



of volatile



oil reservoirs.



reserfor pre-



3’-3x



Definitions Bubblepoint



pressure is the saturation



pressure of the oil;



as pressure drops below bubblepoint, gas starts coming out of solution from the oil. Critical gas saturation is the minimum saturation at which gas starts to flow. Gravity drainage refers to vertical segregation of gas and oil by countercurrent flow because of gravity (i.e., density difference); gas moves up and oil moves down. In differential gas separation, moved



as pressure



the evolved is lowered,



gas is continuously



re-



so that the gas does not



remain in contact with the liquid. Flash gas separation occurs when the evolved gas remains in contact with the liquid as pressure is lowered.



because of pressure. No free gas is assumed to be present in the oil zone. As reservoir pressure drops below the



Typical Performance



bubblepoint because of production, the oil shrinks. Part of the pore space is filled by gas that comes out of solu-



Fig. 37.1 shows typical performance for a solution-gasdrive reservoir with an initial pressure above the bub-



tion. The water expansion, a much smaller effect, is often neglected. The drive mechanism (gas evolution and ex-



blepoint.



pansion)



is dispersed or scattered throughout



The evolved



gas (less any produced



space vacated by produced remaining oil. The amount the amount



the oil zone.



gas) fills the pore



oil and by shrinkage of the of oil recovered depends on



of pore space occupied



by gas (the gas satu-



ration .Sq) and the oil shrinkage (5, vs. pressure). Gas/oil relative-permeability characteristics and viscosities of oil and gas are important because they determine the flowing GOR at a given produced



along



with



S, (and thus the amount the oil).



of free gas



During



the early production,



pressure



is above



bubblepoint but is dropping rapidly. Gas saturation is zero. and the only gas produced was in solution in the produced oil at reservoir conditions (producing GOR, R=R,;). The rapid pressure decline is caused by the relatively



low com-



pressibility of the system. The only sources of pressure support are fluid and rock expansion. Once



the



reservoir



pressure



reaches



bubblepoint,



solution-gas drive begins, and pressure declines less rapidly. The additional pressure support is a result of the liberation of gas as pressure declines and the expansion of this gas as it undergoes further pressure reduction. As pressure drops below bubblepoint, the evolved gas is immobile until the gas saturation exceeds the critical



37-2



PETROLEUM ENGINEERING



HANDBOOK



by calculating the rate for an average or representative well and then multiplying by the number of active wells.



pi



Gridded



reservoir



a number



Pb



models



of gridblocks,



subdivide



sure, and saturations. Some blocks ded models enable consideration reservoir heterogeneity, individual characteristics, Tank-type



and fluid



Rsi



pres-



contain wells. Gridof such details as well locations and



migration



between certain



answering



into



its own PV,



models are adequate-in



preferable-for



Gas Saturation



the reservoir



each having



regions. cases even



some questions.



while



being



simpler and quicker to use than the gridded models. Understanding tank-type models aids the understanding of gridded simulators because both use basic continuity (material balance) principles. Even for reservoirs that ultimately



Cumulative Oil Production, N p Fig. 37.1-Typical



solution-gas-drive



reservoir performance.



may be studied



with a gridded



model.



the calcu-



lated tank-type primary performance can provide useful. quick information and can serve as a reference point for comparison. Also, a very important use of tank-type models is in interpretation of a reservoir’s pressure/production histo-



value, S,, . For this period, there is no free-gas production. and the produced GOR declines because the produced oil now contains less gas in solution (lower R,). begins, and Once S,s, is exceeded. free-gas production the total (free plus solution) produced GOR increases. This ratio rises to a peak much higher than the solution GOR (most of the gas produced at that time is free gas), then drops at low pressures. This drop is caused by insufficient additional gas evolution to sustain the high gas production.



Solution-gas-drive



characterized



by



(I)



reservoir



relatively



performance



rapid



pressure



is



decline



ry to determine the OIP and whether the reservoir is volumetric or has water influx. Havlena and Odeh I5 presented particularly useful techniques for doing this with the material-balance



duction



history



application



given



or



no



water



production;



and



(5) relatively low oil recovery-typically I5 to 20% of original oil in place (OOIP), but occasionally as low as 5% or as high as 30% OOIP. A notable



exception



is that reservoirs



benefiting



true if the oil production



lowler part of the oil column GOR



is taken



from



where the gas saturation



the and



are lower.



categories: models.



prediction tank-type



Tank-type



can consider propriately.



models



and



gridded



are simpler;



gridded



into two reservoir models



more details. Each is useful when used apTank-type models for solution-gas-drive



reservoirs are described in this chapter, models are discussed in Chap. 48.



and gridded



Before gridded models were made practical by the introduction of modern computers. the main methods available for tank-type



reservoir performance calculations were the models. These treat the reservoir as a single tank



or region average



that is described saturations



by the average



at a given



time.



production



but variations



with position



rate vb. time for tank-type



pressure



This is equivalent



assuming that the reservoir is at equilibrium form prcssurc and saturation). Variations considered,



Calculation first.



This



balances and



reservoirs.



Grid-



studies that are used to evaluate the range of tank-type models are also discussed.



methods



for ordinary



These are normally



B,, less than roughly chapter volatile



material



to solution-gas-drive



(nonvolatile)



adequate



2.0 RBISTB.



discusses performance oil reservoirs.



oils are



for oils having



The last part of the



prediction



methods



for



Basic Assumptions of Tank-Type Material Balance



2. The reservoir



methods can be divided models



with wells).



1. The reservoir PV is constant (except in some cases where nonzero rock compressibility is considered).



Types of Models Used Performance



is communicating



from



gravity drainage may have sustained production at a lower GOR and, consequently, a higher oil recovery. This is particularly



(i.e.,



This chapter focuses on tank-type their



I



(4) little



as the equation



plete communication.



ded simulator of applicability



2;



rearranged



may or may not agree with the volumetrically calculated OIP because of uncertainties in volumes and/or incom-



(faster than with fluid injection); (2) low initial producing GOR (equal to solution GOR) rising to a much higher GOR; (3) oil production rates declining because of both and



equation



of a straight line. They noted that OIP calculated by this equation is the oil that contributes to the preasureipro-



and to



(i.e.. has uniwith timc are



arc not. The field models is predicted



temperature



3. The reservoir relative-permeability



4. Equilibrium conditions voir at all times. Pressure throughout



is constant.



has uniform porosity characteristics.



the reservoir;



and uniform



exist throughout the reseris assumed to be uniform



consequently.



fluid



properties



at any time (i.e., any pressure) do not vary with position in the reservoir. The effects of pressure drawdown around wells are neglected.



The liquid



saturation



is assumed to



be uniform throughout the oil zone. Thus, at a particular time. the value of the gas/oil relative-permeability ratio (k,,/k,,,) is regarded as constant throughout This includes the assumption of no gravity For reservoirs assumption



having



an mltial



the oil zone. segregation.



gas cap. this mcludes the



of no gas coning at wells. Gas cap and oil zone



volumes are assumed not to change with time. Any gas leaving the cap because of gas expansion is assumed to be distributed uniformly throughout the oil zone.



SOLUTION-GAS-DRIVE



37-3



RESERVOIRS



5. The PVT properties



arc representative



of rcscrvoir



conditions. The fluid sample from which the PVT data are determined is assumed to be representative ofthe fluid in the reservoir, reservoir



and the gas liberation



mechanism



in the



is assumed the same as that used to determine



As the gas saturation mobility



increases



increases rapidly.



than the oil,



above



critical.



the gas



the gas becomes more mobile



and the gas moves



faster than the oil.



Be-



cause the evolved gas moves ahead of the oil, the process is closer to differential. Overall. the process in the



the PVT data. Usually. differential vaporization is assumed to be most representative of conditions in the



reservoir is approximated more closely by the laboratory differential PVT data than by the laboratory flash data.



reservoir.



This is particularly of the differential



fluid



With the possible exception



properties



are assumed



pressure-i.e.,



of volatile



to be functions



any effects of composition



oils. the of only



change are ne-



the pressure



true for high-solubility crudcs. Use PVT data is recommended. Even for



range just



below



bubblepoint.



PVT data are more appropriate.



glected. 6. The recovery is independent of rate. 7. Production is assumedsto result entirely ation of solution gas of any initial



gas and the expansion of the liberated gas cap and of oil as reservoir pressure



decreases. This includes assumptions injection;



from liber-



that water



is immobile



that there is no fluid



and there



is no water



where



the differential



tlash



data do



not cause significant errors because flash and differential data are almost identical in this pressure range. If laboratory sometimes



data are not available.



may be obtained



(see Chap. 22). Gas liberation



from



reasonable estimates



published



in the separators



correlations



is closer



to a flash



production and no water influx; and that reservoir water and rock compressibility can be neglected (note that this



vaporization much lower



assumption is used only below bubblepoint and that these effects should be considered above the bubblepoint).



ferential PVT data are used in the material-balance computations, the computed recoveries could be adjusted to



8. A relationship is assumed for specifying oil production rate as a function of reservoir pressure and saturation.



account different



9. Reservoir performance data, if used. are assumed to be reliable. This refers. for example, to average pres-



to stock-tank conditions (see Chap. 22 and Page 64 of Dakej’). For typical crudes. however. this adjustment



sure vs. cumulative



oil production



used to determine



OIP.



and producing GOR vs. pressure used to determine check the curve of k,.g/k,,, vs. saturation.



or



Basic Data Required



is often within



the range of other data and model limita-



tions and consequently



Because saturations



Two sources of OIP data are volumetric values determined from the reservoir’s



calculations and pressureiproduc-



tion history. Often, only the volumetric estimate able. When there is enough solution-gas-drive (reservoir



average



pressure



vs. oil produced),



is availhistory this volu-



metric value can be checked by a comparison with the history-derived OIP. A convenient method for determining the OIP from pressure/production history is given by and Odeh”



and will



be described



not warranted.



are assumed to be uniform,



value is used for initial



water saturation,



oil saturation is then Soi = I .O-S,,i. are initial fluid saturations obtained analysis



of representative



given



rule of thumb



is that 5 to 10% of



the fluid in place must be produced before the performance history is sufficient for calculation of OIP. For a solution-drive the ultimate the amount



reservoir,



this would



be a large fraction



recovery, which is typically of production is important,



reservoir



pressure



of



1.5to 20%. While good values for



at a sequence of times (based



on well pressure tests) are equally important. If you have a sequence of pressure points that were determined from field measurements, try Havlena and Odeh’s method: if several points form an essentially ably have enough



data to confirm



straight



a combination



Alternatively,



these



values can be based on logs or on other reservoirs same or similar formations.



in the



Relative-Permeability



A frequently



The preferred data from a laboratory



cores or from



of core analysis and well log analysis.



a single



S,,;. The initial



in a later



section.



average



for the different process (and particularly the temperature of gas separation) from bottomhole



Initial Fluid Saturations



OIP



Havlena



process and frequently is at a temperature than the reservoir temperature. Because dif-



line, you prob-



the OIP (even at less



Generally, laboratory-determined k,s/k,, and k,, data are averaged to obtain a single representative set for the reservoir that is consistent with the interstitial water saturation.



If laboratory



data are not available.



be based on other formations. For reservoirs



reservoirs



having



in the



sufficient



estimates



may



same or similar



solution-gas-drive



his-



tory, the calculated kg/k, values vs. saturation can be compared with the averaged laboratory or estimated k,y/k,, data. These values may be calculated” with Eqs. I and 2, and the laboratory data can be adjusted slightly to match more closely GOR



than 5% recovery).



Data



the observed



(R) vs. reservoir



pressure



history



of producing



if necessary.



PVT As reservoir pressure drops below the bubblepoint, the first gas liberation is by the flash vaporization process (the gas is not yet mobile and therefore stays in contact with the oil). Once the critical gas saturation is exceeded, some of the gas flows.



Thereafter.



the gas liberation



process



is somewhere between differential vaporization (gas is continuously removed from the oil) and flash vaporization.



s 0



=(N-Np)BoSoi NB,,i



~



.



.



67



37-4



PETROLEUM



]



;



,



I



/



ENGINEERING



NO- SEGREGATION ---COMPLETE-SEGREGATION 15,: i --j i



j



HANDBOOK



iL



i-10



GAS LAYER RESIDUAL OIL SATURATION)



(AT



------------I-------OIL LAYER (AT CRITICAL GAS SATURATION)



0



Fig.



37.2-Vertical gation.



saturation



distribution



for



complete



segre-



Fig.



where



k, = effective



permeability



(pL,, pLg,



37.3-Comparison segregation



1



,lL



08



IO



OF PORE



SPACE



of no-segregation relative-permeability



and data.



complete-



model, and the tank-type it. It is also possible to



kg/k, and k,



that has complete



gravity



data for the segregation,



The entire reservoir shown in Fig. 37.2 contains interstitial water saturation. Complete segregation means that



you need estimates reservoir



the upper



tion, enough



of the



pressure.



The at



Data vertical



gas and immo-



com-



above



S



moves



upward



rapidly



and leaves



that region, while 6 the upper region any oil above S,, drains downward and moves into the lower region. The flow



to wells



is assumed to be horizontal



of only gas in the upper region er region. tive kg/k,



that have enough



contains



S,,. Vertical communication is assumed to be high that, as gas evolves in the lower region, any gas



saturation



B,, and BR) are evaluated



Pseudo-Relative-Permeability for Complete Segregation



part of the reservoir



bile oil at residual oil saturation, S,, , while the lower part contains oil and immobile gas at the critical gas satura-



this pressure.



This refers to reservoirs



I



net pay).



OIP (N) and of the current properties



I



06



as shown in Fig. 37.2, and flow from the total net pay thickness (i.e., assuming wells are completed in the total



cp,



To use the above equations, fluid



I



case of a reservoir



;” I gas formation volume factor, RB/scf, B: = oil formation volume factor, RBLSTB, B,i = value of B, at initial pressure, RBISTB, s, = oil saturation, fraction PV, s,; = initial oil saturation, fraction PV, N, = cumulative oil production, STB, and N= initial OIP, STB.



initial



I



calculate I3 pseudo or effective



cp,



oil viscosity,



I



assumptions of the tank-type model is most suitable for



to gas, md,



to oil, md, ko = effective permeability R= producing GOR, scf/STB, R, = solution GOR, scf/STB,



cl8 = gas viscosity,



I



0.2 04 OIL SATURATION-FRACTION



and to consist



and of only oil in the low-



On the basis of these assumptions, and k,, are given by Eqs. 3 and



(SR-S,,.)(k,),r kg _ - (s, -S,,)(k,, )xc . ko



the effec4.



.



. (3)



munication for gravity segregation to occur, with evolved gas moving upward and oil draining downward. The literature on tank-type-model



predictions



of relative-permeability



includes description



modifications



to obtain



pseudo-



relative-permeability curves to account for complete gravity segregation within the reservoir. Consequently, the suitability of such pseudocurves in the tank-type materialbalance computations



should



given below, this approach should be avoided. The laboratory-measured ply to an unsegregated with height).



be discussed.



is potentially



For reasons



misleading



where = relative



(k,Lr



= relative



(b),,



relative-permeability



and



data ap-



(no change in saturation



This case is most consistent



with the basic



to gas at residual



permeability



to oil at critical



gas



saturation, S,



situation



permeability



oil saturation,



= gas saturation,



fraction



PV,



S,,



= critical



gas saturation,



fraction



PV.



S,,



= residual



oil saturation,



fraction



PV,



S,” = water



saturation,



fraction



PV.



and



SOLUTION-GAS-DRIVE



37-5



RESERVOIRS



Fig. 37.3 compares ordinary X-,/k, and kro curves for an unsegregated reservoir with the adjusted curves for the completely segregated assumption.



N(B,



-Boi)



(expansion



+N(R,yi -R,,)B,



(volume



Pseudo-relative-permeability data calculated with Eqs. 3 and 4 are consistent with the above assumptions. And one might be tempted to assume that results computed for no segregation



(unmodified



for



segregation



complete



permeability



(the



data) bracket



cases with



partial



approach?



The problem



thickness



relative-permeability above



data) and



What



is wrong



is that perforating



this



the entire pay



producing



GOR and maintaining



(gas-cap



NB,i(l



thereby



reducing



reservoir



S,c,(pj~



energy.



-PR)



NB,,;(l



cs (p ;R -pR)



under-



estimates the oil recovery compared to a good gravitydrainage project. Results of this model will lead to incor-



+



rect conclusions about the benefits of gravity about how to operate the field.



~ Gi



the gridded



model,



performance



you can study



and possible



sensitivity



the benefits



of oil recovery



tion rate and to the amount



of vertical



of



equation



G,,.B,



permeability.



keeps inventory



+



(production



minus



gas production)



Wf,B,, (water production)



- Wj B !,, (water -



injection)



W,B,,. (water



influx),



.



R,s = solution



GOR,



R,Yi = value



R,, at initial



injection



minus



of the in-place



ma-



terials. Van Everdingen et al. ‘” stated the material ance in reservoir volumes as follows:



bal-



of



scf/STB,



on all ma-



ing production and injection) of the oil, free gas, and water must equal zero. In other words, expansion equals voidmust be made up by expansion



gas production)



.



(5)



where



to a reservoir. it states that because reservoir volume is constant, the algebraic sum of volume changes (includ-



influx)



(gas-cap



solution



B, (gas injection)



scf/STB, pressure,



and



m = PV of gas capiPV



terial entering, leaving, and accumulating within a region. Sometimes called the Schilthuis ’ equation when applied



age; the net voidage



(liberated



to produc-



Material-Balance Equation The material-balance



-N,R,v)B,



predictions.



selective perforation low in the pay, possible benefits of producing mainly from downdip wells if the reservoir has dip.



expansion)



drainage and



If a reservoir has enough vertical communication to benefit from gravity drainage, consider use of a gridded With



(rock



Con-



+(G,,,



seriously



expansion)



1 -s,,.



the



model with pseudorelative



permeabilities



(water



+m)



+



=N,,B,, (oil production)



48) for primary



gas expansion)



+m)



+



sequently, the assumption of production from the entire pay thickness is inappropriate for this case. The tank-type



model (Chap.



gas)



1 -s,



Producing gas at high GOR from the upper part of the pay thickness reduces reservoir energy (pressure support). It is much better to produce such a reservoir only from part of the oil column,



(~)



by



solution



for



with



is not the best way to operate such a reservoir.



the lower



occupied



liberated



+mNB,j



oil)



pseudo-relative-



the results to be expected



segregation.



of initial



of oil zone,



dimensionless.



Solving Eq. 5 for N yields equation for initial OIP:



the general



material-balance



N=



N,,B,,+(G,,-N,R,)8,+fW,-W,-W,.)B,.-G,B,



~R,,-A,.,)t~R.,-R,)B.+mS.,,(~)+~Il+~~~tS,.i..+l-l~~:‘,~-Fr~ “(Cumulative



oil produced and its original



dissolved .



gas) + (Cumulative free gas produced) + (Cumulative water produced) - (Cumulative expansion of oil and dissolved tive expansion



gas originally



in reservoir)



of free gas originally



(Cumulative water entering reservoir), ”



original



where



- (Cumula-



in reservoir)



G, =G,,,



.



+G,,.



.



=cumulative



(61 gas production,



oil and water



the general equation can be simplified. discussed in the following sections. Water and tras iniection



could



also be considered



denoting



having



an initial



material balance expressed by Eq. 5.



in reservoir



with



Material



gas cap. with m



the ratio of gas-cap-volume/oil-zone-volume, volumes



These cases are



in the



material balince Gy replacing cumulative production cumulative production minus cumulative iniection. For an oii reservoir



in



standard cubic feet. By considering a case only above bubblepoint or only below bubblepoint, some terms are zero or negligible, and



=



the is given



Balance



For an undersaturated



Above Bubblepoint reservoir



(i.e., above bubblepoint),



no gas will be released from solution,



the produced



GOR



will remain constant at R,,;, and there would not be any gas cap. Thus (R,s, -R,)=O, m=O, and (G,-N,,R,)=O.



PETROLEUM



37-6



With



these simplifications



injection,



and the assumption



Eq. 6 reduces



Bo -Bc,i +p



Bo,



is often used in computations below bubblepoint.



. Because -pR),



for



(SL,.(.,,~+cf)(P;R



1 -SW,



of oil recovery



.....



-PR)



.. .



the single-phase



the material-balance



oil



Below



. . ..~...



B,, -II,,,



equation



Material (7)



= B,,, co ( p iR



above bubblepoint



Balance



Below



bubblepoint,



N,,B,, +CW,,- W, - W,.)B,,.



(gas evolution



compressibility



Bubblepoint



compared



because their



with gas evolution



be included case with



bubblepoint



are often neeffect



and expansion.



above bubblepoint. the following



.



c,,. =3X



cf=4



is small



they should



For example,



consider



a



data:



bubblepoint; production



Eq. 6 to Eq. I1



...



>



.



. (11)



pressure was above bubblepoint,



Eq.



N, and G, are the incremented oil and gas below bubblepoint; and the “initial” fluid



B,;



properties Another



and B,; are values at bubblcpoint. expression for N, often found in the literature to Eq.



11, is given



by Eq.



12.



vol/(vol-psi)



IO ph vol/(vol-psi)



x 10 -6



we simplify



I1 can be used to compute the performance below bubblepoint. In this case, the value used for N is the OIP at



and equivalent rr, = 15 X 10 +



oil shrinkage)



(Bo-Bo;) +(R.>ipR.5)B,y+mB,,,



(cLi) is



water and rock compressibility



below



of hydrocarbons



minus



N,J,, +(G, -NI,R,)B,s



N=



Even if the initial Although



recovery



is much greater than the expansion of rock and water. Consequently, the rock and water expansion terms can



(9)



glected



of the additional



plus gas expansion



(8)



’



B,ic,( prRppR) the effective



is straight-



the net expansion



and no net water production,



where



to bubblepoint



be omitted without serious error. By neglecting these terms and by assuming no water influx. no gas injection,



becomes



N=



HANDBOOK



forward, with bubblepoint pressure as the value of PR in Eq. 10. The remaining OIP is then N-N,,. This value



to



N,~B,,+(W,~-W;-W,,)B,,



N=



Calculation



of no gas



ENGINEERING



N=



N,,[B, +B,#,> -R.,;)l



(12)



vol/(PV-psi)



and



where



R,, = cumulative



produced



GOR,



scf/STB.



s,, =0.20 By use of Eq. 9,



R&



(0.20)(3x10--6)



..,...............,..........,.(13) P



+4xW6



r,=15x10-6+ l-O.2



l-O.2



B,



is the two-phase



reservoir



plus the gas that reservoir =20.75x



the water



and rock compressibility



contribute more than one-fourth of the total compressibility. Their omission would cause the OIP calculated by Eq. 8 to be too high by a factor of 20.75/15= I.383 (i.e., 38 %’too high). The error would be even greater for larger S,,



production, yields



of Oil Production undersaturated injection,



the following



Above Bubblepoint.



reservoir



or influx. expression



with negligible



For water



rearrangement



of Eq. 8



for cumulative



oil pro-



duction: NB,,~~,,(P,R N,,



=



hydrocarbon)



FVF-



was initially



dissolved



oil



in that oil



at



conditions.



B, =B,, +B,,,(R,,



B,,



-PR)



. .



. .



.. .



.



.(lO)



(14)



and Odeh I5 show



how



to use the material-



balance equation along with a reservoir’s pressure/production history to get information about whether the reservoir is volumetric or has water influx, plus the initial (N) and the ratio of gas-cap-volume/oil-zone-volume for a volumetric Chap. 38; only



.



-R,).



Material Balance as Equation of Straight Line for Determination of OIP and of Gas-Cap Size Havlena



an initially



total



by one barrel of stock-tank



IO p6 vol/(vol-psi)



For this example,



Calculation



(i.e.,



barrels occupied



here.



OIP (m)



reservoir. Water influx is discussed in the volumetric case will be considered



SOLUTION-GAS-DRIVE



RESERVOIRS



for reservoir withoul Fig. 37.4A-Straight-line material balance gas cap. Q, from Eq. 15 vs. A/3, =B, -B,,.



Havlena



and Odeh rearranged



of a straight



line,



grouping



Eq.



terms



12 as the equation



Fig. 37.4B-Straight-line



ing, and (4) gravity For a reservoir



Q,,=N,[B,+B,(R,-R,;)], AB,=B,-B,;,



.(15)



.(16)



.



for reservoir



with



gas



drainage



that is affecting



with an initial



reservoir



drive alone).



gas cap but no water in-



flux,



values of both N and m can be determined from the field performance data, as illustrated in Fig. 37.4B. By trial and error, the value of m yielding a straight line can be determined;



N is the slope of this line.



For cases with a gas cap, Havlena and Odeh recommended that a second method also be used as a check,



and AB,



=B,Y -B,;,



. .



(17)



even though powerful



where Qp = net fluid AB,



production,



= oil expansion RBISTB,



AB,



= expansion



RB,



on the horizontal



OIP.



and



(the gas cap)



free gas



free gas in place,



“is



a more must



(x) axis. If both sides of Eq. 18 are divid-



we can see that the plotted



points



should ap-



is equal to N. Conse-



the slope and the 4’intercept



of this plot enable



of these values calculation of both N and m. Comparison with those determined by the first method is a desirable



as



check,



.



Q, =NAB,+Ntn-ABR. B,;



A plot of Q,, vs. AB, +tr~(B,~lB,,)



.



(18)



AB,s should



result



absence of a gas cap, Qp =NAB,; a plot of Q, vs. AB, should be a straight line of slope N. going through the by Fig.



37.4A.



When field performance (Qp vs. AB,) is plotted, if it yields an approximately straight line, the slope indicates the value of the initial OIP (N). The data needed are fluid properties



vs. pressure and the reservoir



at several



times or pressures.



and also may aid selection



for use in the first



in a straight line going through the origin. The slope of this line represents N, the initial OIP. Similarly, in the



is illustrated



above)



specifies that the line



proximate a straight line with slope equal to NnzB,,/B,, Also, if this line is extrapolated so that it intercepts the quently,



B,;



This



(given



y axis, the y value at that intercept



RBiscf.



origin.



method



because “it



the origin.” The second method plots values on the vertical ( y) axis vs. values of AB,/AB,



ed by AB,,



of initial



12 can be rearranged



the first



method”



go through of Q,/AB,



per STB of initial



per scf of initial



Eq.



balance



(Rp lower than for solution-gas



performance



as follows:



material



cap.



performance



data



we are now ready to consider predictions of future performance by solution-gas drive. Techniques for this have been published three methods



by Muskat,” Tamer.’ and Tracy. ’ All yield essentially the same results when



small enough



intervals



cause Tracy’s



method



will



be described



of pressure



or time are used. Be-



is the most convenient



to use, it



first.



Material-Balance Calculations Using Tracy’s Method Prediction of solution-gas-drive performance involves the use of a material-balance equation such as Eq. 11, plus enough additional relationships (equation for producing



a straight



GOR,



flux



line,



possible



pressures



(see Chap.



and/or



38),



reasons fluid



include



properties,



(3) gas cap is present



data are



Having values of N and m that are based on reservoir performance and/or on other information or estimates,



N,, , G, , a,nd average reservoir pressure (for determination ot fluld properties). If the plot of Q,, vs. AB, is not average



The performance



of the best value of m



method.



(1) erroneous



(2) water inand expand-



and for relating



saturations



to N,,) to enable com-



putation of N,, and G, vs. pressure. The computations are performed for a sequence of pressure decrements. The



PETROLEUM



37-8



ENGINEERING



HANDBOOK



60-



2



30



a z



,/



90



r



25



20



I5 ,



IO



5



o-cl--



400



-50



6Oo



J



JO



;600



Go0



10 0 RESERVOIR



Fig.



37.5A-Oil



incremental production



pressure



function



oil production AC,



G0 vs.



AN,



Fig.



pressure.



and the incremental



for the pressure



to pn are determined cumulative production



reservoir



decrement



gas



from pn- 1



by an iterative method, and the values are then given by Eqs. 19



=(Np),r-,



.



+AN,



. .



.



.



.



. .



.(19)



=(G,),,-,



. .



+AG,.



. .



.(20)



Tracy



simplified



sure functions



the use of Eq. 11 by introducing 9,



pres-



and aO:



are infinite. lower With



.



. ..



. (21)



at bubblepoint



aW vs.



pres-



instead of



because R, is constant above are shown in pressure, the



in Eq. 21 is zero; consequently, This, however, 9 values



does not cause



9,



and a0



any difficulty



used are the finite



values



at



pressures. Tracy’s



9 functions,



Eq.



11 becomes



. . .



. ...



(23)



This form of the material-balance equation is particularly convenient because the + values are functions only of gas-cap size and of pressure. For each pressure level, the Cp values need to be calculated only once. Material-Balance



and



@9 and



also used R, at bubblepoint



N=N,,+,+G,9,.



..



functions



of 9, and Q, vs. pressure and 37.5B. At the initial



because the only Tracy



and used B,



R,,i, but these are equivalent bubblepoint.



denominator (G,),



water pressure pressure.



p.i(l



sure instead of the B,i used in Eq. 21. As discussed in the next section, use of B,i in Eq. 21 makes Tracy’s method also applicable above bubblepoint (in which case



Examples Figs. 37SA



and



and reservoir



started at bubblepoint



m is zero).



and 20. (N,),,



37.5B-Gas



PRESSURE,



decrement



from



Equation



23 is applied to the pressure



pn- 1 to p,, :



a(?=$-R,$ 9,. . . . . . . . . . . . . . . . . . . . . . (22) ( R > Actually, original



Eq.



equation.



21 is a slight Tracy



modification



gave an example



of Tracy’s problem



that



. .



. . .



.



(24)



SOLUTION-GAS-DRIVE



where



RESERVOIRS



the average



producing



37-9



GOR



is given



10. Compute the estimated OIP (N ) from Eq. 23 or 24. 11. To test GOR, check whether the new value of R



by



computed



at Step 7 is arbitrarily



timate



R for this same pressure decrement,



of



close to the previous



es-



denoted



Rold. An adequate test is Solving



Eq. 24 for ANp, 0.9991L51.001.



R old



ah, = N-O’,),,- I(@o)r,-(G/J,,-, (+) corresponding



(N,,), as in Step 3. 9. Compute (G,),



been considered



(G,),



=(G,),-1



+&N,.



Applicability of Tracy’s Above Bubblepoint Historically,



Method



two reasons were given



for not using Tra-



cy’s method above bubblepoint pressure: (1) use of Eq. 10 is simpler, and (2) according to the literature, Tracy’s method is not applicable above bubblepoint. The purpose of this section is to show how Tracy’s



method can be used



both above and below bubblepoint. Heretofore, the approach for calculating the total oil production N, for initially undersaturated reservoirs has been to calculate N,



PETROLEUM



37-10



to the bubblepoint



wjith Eq.



IO, to calculate



incremental



(Eq.



oil production below bubblepoint by Tracy’s method or another method, and to add these two produced volumes together native



to obtain



the total oil recovery.



is to use Tracy’s



method



The new alter-



for the entire



pressure



range. Existing computer programs that use Tracy’s method only below bubblepoint can be applied for the entire



pressure



range



if data are modified



as described



GOR,



28)



and determine



ENGINEERING



k,Y/k,,.



R,, (Eq. 27). Third.



HANDBOOK



Second.



calculate



calculate



the



the incremental



pas



production AC, (Eq. 25). Then the cumulative gas production is calculated by G,, =(G,, at previous pressure) +AG,, The correct



value



of N,,



is the value



at which



both



methods above yield identrcal values of G,, Tarner suggested plotting both sets of calculated G,, values vs. N,,



below.



The intersection



The literature contends that Tracy’s method cannot be used above bubblepoint pressure because the @ functions



G,, and N,. Tamer’s method works if the plotting is done accurately. It should yield the same results as Tracy’s method be-



are infinite



at bubblepoint.



This is true for Tracy’s



B,, at the bubblepoint



tions in Ref. 6 that used



equa-



(Tracy’s



initial condition) instead of B,,, as in Eq. 21. However, if B,,, is used in Eq. 2 1, Tracy’s method becomes more general. It can be used for all pressure intervals because the @ functions (Eqs. 21 and 22) are infinite only at the irtiriul pressure, which does not have to be the bubblepoint. Values of the @ functions at the initial pressure are not used in Tracy’s formulation; only the finite values at lower pressures



are used. Consequently,



B,,, is used in Eq.



if



2 I, Tracy’s method can predict performance for the entire pressure range from any initial pressure down to abandonment When used above bubblepoint, Tracy’s method does not require iteration because an accurate initial estimate can be made for R tR=R, ). When Tracy’s is used for the full pressure range of an initially



method under-



saturated oil, however, three considerations are pertinent: (1) the computed recovery will be a fraction of the initial OIP. not of the OIP at bubblepoint: sure should be one of the pressure sideration



that starts at bubblepoint;



and be



for realistic



computation



above bubblepoint.



A technique



point



the



by ad.justing



cause the same relationships



approach



is more straightforward



ly converges



within



of pressure



indirectly,



initial pressure. Eq. 29.



however, volume



for considering



and usual-



no initial



gas cap. It is intended



point pressure.



mainly



change in oil saturation, AS,,, during lated by use of the following depletion ential



for below-bubble-



For a sequence of pressure



steps. Ap. the



each step is calcuequation in differ-



form.



4



decline the third



are relative-



by use of pseudovalues



factor



to implement



a few iterations.



Muskat and Taylor’s method4 is applicable to the tanktype depletion performance of a volumetric reservoir with



B dR S(,A---l+S,sB,s-



B,, data is given below.



Because rock and water compressibilities



included



method



Material-Balance Calculations Using Muskat and Taylor’s Method



B,



=



at pressures



below



d(l’B,s)



d17R



+ Sox.,-,q~,, dB,, _______ B,,A-,-,,P,~



‘k’R



ly unimportant below bubblepoint, they were not included in the Tracy material-balance formulation. They can be the oil formation



are used. Tamer’s



is time-consuming because you have to calculate and plot the two curves of G,, vs. N,, and then determine their intersection. While this graphical interpolation approach can be implemented on digital computers. Tracy’s iterative



(2) bubblepoint preslevels for proper con-



(3) the effects of rock and water compressibility,must considered



of gas evolution



of the two curves then yields the correct



1



I



kyqvo



k m Ps



of the



......... ..



. . . . . . . (30)



Bz, are given by



These pseudovalues,



dpR



The stepwise



solution



of this depletion



equation



yields



the reservoir oil saturation, S,,, vs. reservoir pressure. PR. For each pressure at which S,, has been calculated. the cumulative



recovery



can be calculated These pseudovalues



include the additional



port of water and rock compressibilities balance computations.



and Tracy6



solved



referenced



Having



the same material-balance



and consequently



will



cumulative



the value of S,, ,



k&k,.,, can be determined



the plot of krh,/k,.(, vs. S,‘or vs. S,, +S,, , which quired data. The producing GOR is then



from is re-



be described.



For each pressure in the Tarner method. several estimates are made of the cumulative oil production, N,, For each N,, . the corresponding



OIP



pressure sup-



equation for a sequence of pressure decrements. Although Tracy’s method is more convenient, Tarner’s method is often



of the original



in the material-



Comparison of Tarner’s and Tracy’s Methods Tarner’



as a fraction



by use of Eq. 3 1.



R=R,+%



* k 10 ( hBs



_. _.



(32)



>



gas production,



G ,,, is calculated two ways: from Material Balance Equation I I, or on the basis of relative permeability. To calculate G, from relative permeability, first calculate S,,



Because throughout



this



method



assumes



the reservoir,



is appreciable



segregation



uniform



it is not applicable of gas and oil.



oil



saturation when there



SOLUTION-GAS-DRIVE



37-11



RESERVOIRS



Ratio of Original Gas Cap Volume to Reservoir



Eq. 30 can be solved tither explicitly or implicitly. Explicit means each term on the right side of Eq. 30 is evaluated on the basis of the pressure and saturation



at the start



of the pressure step. Each pressure step must be small so that these values are representative of conditions during the step. While this approach has the advantage of not requiring iteration, it is not self-checking. Significant cumulative



errors



may occur



unless the pressure



inter-



vals are sufficiently small. In the implicit (iterative) solution, the terms on the right side of Eq. 30 are evaluated on the basis of estimated the middle



conditions



(PR and S,,) at tither



or the end of the pressure



step. This requires



making an initial estimate of these conditions, computing the pressure step. checking agreement between esttmated



and



computed



recomputing



values.



the step with



and,



if



necessary,



the most recently



computed



values as the new estimates. This iterative solution involves more work but can handle larger pressure steps suitably.



Comparison



With Gridded



Simulator



Because Eq. 30 looks rather formidable



Equations and mysterious.



it may be helpful to show where the terms come from. This will also show the relationship of Eq. 30 to the equations used in gridded multiphase reservoir simulators; tank-type models and gridded models use similar continuity (material-balance) gridded



principles.



model omitting



gravity



For a two-phase and capillary



Oil Produced, Percent of Oil In Place



(gas/oil)



forces.



the



oil phase partial differential equation that combines Darcylaw flow and continuity is Eq. 33. This equation is in Dar-



Fig. 37.6~-Reservoir pressure vs. percent oil recovery for several values of m.



cy units.



0. (%vp) where



V denotes



and d(S,/B,)l&



S,,/B,



with



=d$($)



-4(>,,,



(33)



the gradient,



is the partial



derivative



of the quantity



respect to time.



The left side of Eq. 33 represents



Darcy-law



flow



of



Oil Produced, Percent of Oil In Place



oil in the reservoir (between blocks in a gridded model) and would be zero for a tank-type (one-block) model. The right-side terms represent oil accumulation and production. The corresponding equation for total (free+solution)



Fig.



37.7-Producing values



GOR of



vs. percent



oil recovery



for several



m.



gas is Eq. 34.



>VP 1 cl/,B,, C14B.c kk,,



V.



A+R,-



of each equation, multiplying by the bulk changing to oilfield units yields



kk i-o



v,,; (2,I>=y,, (34)



The corresponding



equations



obtained



that the left sides of Eqs. 33 and 34



arc zero for the tank-type



for a tank-type



model.



Deleting



model



and



(35)



and



v,,g(?+R,J$ by noting



volume,



,.. (36)



=ys.



are



the left side



This total gas rate q,q is the sum of the free-gas duction



rate and the solution-gas



production



rate



pro-



PETROLEUM



37-I 2



ENGINEERING



\‘I 12 13 Cumulative Recovery in Per Cent of Pore Space Fig. 37.8-Pressure



and



GOR



histories



of solution-gas-drive



reservoirs



14



15



I6



17



18



producing



oil of different



vis-



producing



oil of different



gas



cosities.



16



Cufrubtii



Fig. 37.9-Pressure solubilities



Recovery in Fw Cent of Pore Space



and GOR histories and oil viscosities.



of solution-gas-drive



reservoirs



HANDBOOK



SOLUTION-GAS-DRIVE



37-l3



RESERVOIRS



Rotios



5 Cumulative



Fig. 37.10-Reservoir cap



The producing



GOR



volume



6



7



Recavery



pressure and to oil-zone



(R), scf/STB,



in Per Cent of Pare Space



GOR histories of gas-drive reservoirs with volume (H = thickness of gas cap/thickness



is given



by



By equating



various ratios of gasof oil zone).



the two expressions



for



R given by Eqs.



37 and 39, using dS, = -dS,, and rearranging we obtain Eq. 30. Thus, the Muskat material balance for a tanktype reservoir



The producing from



Eq.



GOR can also be expressed



35 and qg from



by use of q.



method (e.g.,



Eq. 36:



(Eq. 30) can be derived



as a special case



of the equations for a gridded multiphase simulator. Because we use compatible equations, the results from a gridded simulator using special data to match the Muskat no flow between



those obtained



by Muskat’s



gridblocks)



method.



simulator with flow between blocks, showed results agreeing with Muskat’s formation



is given



Simulator



Studies.



in the section



should match



Even for a gridded Ridings method.



entitled



et al. I4 More in-



Insights



from



(38)



Sensitivity of Material-Balance Results Several authors have discussed the sensitivity From the chain



dx -=--



dxdp



dt



dp



rule for derivatives,



cap-volume/oil-reservoir-volume) of 0 (no cap), 0.1, 0.5, and 1.0. Oil recovery vs. pressure is shown in Fig. 37.6; Fig. 37.7 shows GOR vs. oil recovery. Tarner discussed applicability of assumptions about the gas initially in the



dt ’



gas cap: (1) the gas cap and the oil zone are each assumed to remain constant in size, and (2) all gas leaving the gas



Eq. 38 becomes



cap is assumed bypassing-such



R=



to pass through the oil zone (i.e., no as by gas coning at wells). Tarner stated



that such assumptions are obviously in error but they in part will compensate each other. The assumption of no



d(l/f&) s,dPR



of material-



balance results to data variations. Tarner2 showed the effect of gas-cap size on performance for values of m (gas-



B,



@R



B,,



d(llB,,)



s.. ~ ~” dp,q . . . . .



.



dPR



B,,



I



dS



Bo



dPR



+--’ .



dl)R



. . . . . . . . .



bypassing tends to overestimate oil recovery, while assumption of a constant oil-zone size (corresponding



dPR



low gravity . . . .



(39)



drainage)



tends to underestimate



the to



oil recovery.



Muskat and Taylor3 provided informative results about the sensitivity of oil recovery to oil property variations



PETROLEUM



37-14



RECOVERY, TOTAL



Fig.



37.1



LIQUID



SATURATION



l-Relative-permeability vs. liquid saturation.



IN PERCENT



ratio



(S,



for sands



as oil viscosity



higher producing



is increased.



GOR’s



and sandstones



bined effects of varying



Fig.



in oil



It also shows the



for cases with higher oil viscosi-



ty, Note the large variations than 8% to more than 17%.



in oil recovery, from less Fig. 37.9 shows the com-



oil viscosity



Fig. 37. IO shows performance



and solution



for several



GOR.



values of the



gas-cap-volume/oil-zone-volume ratio, which Muskat and Taylor denoted by H. It can be seen that calculated oil recovery and peak GOR both increase with increasing gascap size. Muskat and Taylor emphasized the assumptions that gas-cap size remains constant throughout the production history and that depletion of the cap takes place by gas moving from the cap into the oil zone where it is assumed



to be mixed or dispersed throughout the oil zone and produced along with the oil and gas originally in the oil zone. Arps and Roberts’ plotted several sets of sandstone permeability ratio vs. liquid-saturation data and determined the three curves designated maximum, average, and



minimum



in Fig.



37.11.



Maximum



means highest



oil



recovery (lowest k,/k, at a given liquid saturation), while minimum means lowest oil recovery (highest kg/k,,). For each k,/k,,



curve,



(acre-ft)(percent



they porosity)]



computed



oil



vs. pressure



HANDBOOK



STS/(acre-ft)(% porosity)



+S,)



and to gas-cap size. Fig. 37.8 shows the reduction recovery



ENGINEERING



recovery



[STBi



for several



sets



of oil fluid properties. Fig. 37.12 is for the minimum recovery (maximum k,Jk,,) case. Do not be confused by



37.12--Reservoir ft/percent minimum



pressure vs. recovery porosity for sandstone oil recovery.



kg/k,



the label of minimum the average



case, and Fig.



in this figure. 37.14



shows



factor, STElacrewith k,/k, giving



Fig. 37.13 results



is



for the



k,/k,) case. Again the maximum recovery (minimum label (maximum kg/k,) is misleadmg. Note the large variation in oil recovery, 12 for the minimum



STB/acre-ft/percent



and 9 to 26 for the maximum case. Arps also presented results with limestone k,/k, puted recovery



2 to case,



and Roberts’ curves. Com-



ranges were I to 7 for the minimum



3 to 16 for the average mum recovery Fig. 37.15



porosity:



case, 6 to 18 for the average



case,



case, and 13 to 32 for the maxi-



case.



is the comparison



by Sikora



‘s’of reservoir



performance for no segregation vs. complete segregation. The complete segregation case has a lower calculated oil recovery and a faster rise in producing GOR. This illustrates the adverse effects of assumed segregation



on per-



formance calculations in a tank-type model that, among other things, assumes production from the entire pay thickness. For a reservoir with high vertical communication, oil recovery



could be increased by selective



produc-



tion from perforations in the lower part of the oil zone.The tank-type prediction with production from the entire pay thickness would be inapplicable ed applicability of the tank-type



and misleading. The limitmodel to cases with segre-



gation was discussed previously. Performance predictions that consider the selective production would require a more



detailed



model,



such as a gridded



simulator.



SANDSTONE



RECOVERY,



Fig.



37.13--Reservoir ft/percent kg/k,.



pressure porosity



in gas-cap



water saturation, volume



factor



sensitivity



size (in),



permeability



(B,),



RECOVERY,



Fig.



vs. recovery factor, STWacrefor sandstone with average



I8 showed



Singh and Guerrero to variations



STB/(acre-ft)(% porosity)



ratio



solution



STB/(acre-ft)(% porosity)



pressure vs. recovery porosity for sandstone oil recovery.



factor, STBlacrewith kg/k, giving



of recovery



interstitial



(kg/k,),



37.14--Reservoir ftjpercent maximum



,HAXIYUM‘P/b



(connate) oil reservoir



GOR (R,Y), and initial



pres-



sure (P;R). Fluid properties are shown in Table 37.1 and Figs. 37.16 through 37.18. Singh and Guerrero used permeability-ratio average



data that approximated



permeability



ratio characteristics



and Roberts. 8 Interstitial



water saturation



the sandstone by



Arps



was 22%.



They



given



calculated



performance from bubblepoint pressure of 2,500 psi down to a loo-psi abandonment pressure using



200-psi



pressure



Fig. 37.19



decrements.



shows oil recovery



(below



bubblepoint)



vs.



pressure for three base cases with m values of 0,0.5, and 0.75. For each of the base cases, performance was computed for R,s or B,,



of calculated



performance



to these k3OW



changes in data values. These figures and Table 37.2 show that oil recovery percentage increased with reductions in B,,, pIR, or



k,/k,



and with increases



in



R, and Si,. Ta-



ble 37.2 shows that the changes in oil recovery were largest for cases with m=O (no gas cap). The presence of a



Fig.



II I



0 0



pC,/pn. The percentage change or error in oil recovery resulting from the 530% change in these data items is shown in Table 37.2. Figs. 37.20 through 37.24 show the sensitivity



I



400



f30% changes in each of the following: B,, PiR, interstitial water saturation, and k,qlk,, or



Ii I\



I



Y



I



: 2



37.15-Comparison segregation



4 PR&“CEDB(% OIL



OF



kAL)



of no-segregation reservoir performance.



12



and



\



\



4 0 14



complete-



37-16



PETROLEUMENGINEERING



HANDBOOK



TABLE 37.1-FLUID PROPERTY DATA FOR MATERIAL BALANCE PERFORMANCE SENSITIVITY STUDIES Pressure



Oil



Volume Factor



(psi4



(RBISTB)



Volume Factor (RBkcf)



1.315 1.325 1.311 1.296 1.281 1.266 1.250 1.233 1.215 1.195 1.172 1.143 1.108 1.057



0.000726 0.000796 0.000843 0.000907 0.001001 0.001136 0.001335 0.001616 0.001998 0.002626 0.003481 0.005141 0.009027 0.028520



3,000 2,500 2,300 2,100 1,900 1,700 1,500 1,300 1,100 900 700 500 300 100



Pressure,



Fig. 37.16-FVF’s



100



vs. pressure computations.



Gas



Solution GOR (scf/STB)



Viscosity of Oil (CP)



650 650 618 586 553 520 486 450 412 369 320 264



(CP)



1.200



194 94



0.02121



1.260



0.02046



1.320 1.386 1.455 1.530 1.615 1.714 1.626 1.954 2.103 2.281 2.539



0.01960 0.01869 0.01770 0.01670 0.01570 0.01472 0.01380 0.01298 0.01221 0.01165 0.01125



Pressure,



psia



used



Viscosity of Gas



in performance



sensitivity



Fig. 37.17--Solution



100



GOR vs. pressure computations



sitivity



psia



used



In performance



sen-



I Pool pcrformmccfor different gal topr



a



II Pressure,



Fig. 37.18-Gas ance



21



A I = 0.7s I m=OlO I l----l



n



3, 100



psia



and oil viscosities vs. pressure sensitivity computations.



used



in perform-



Cumulative



Fig. 37.19-Depletion-drive with



different



oil recovery, performance gas-cap sizes



% OIP for three



base



cases



SOLUTION-GAS-DRIVE



37-17



RESERVOIRS



TABLE 37.2-COMPUTED



CHANGE OR ERROR IN OIL RECOVERY CAUSED BY + 30% CHANGE IN DATA Percentaoe



Factor



m=O Factor BO



B, P,



and



SW kg/k,



and



“‘Factor”



gas cap moderated



R,



I&,



denotes



- 30.00



+ 30.00



- 30.00



+ 30.00



+ 11.0553 - 10.9920 +9.1756 - 9.8654 + 10.3020



-8.0781 +8.1900 - 7.8326 + 11.6368 - 7.2521



+3.6011 -2.7845 + 3.6844 -8.6772 + 8.3833



-2.1059 + 2.5720 -5.3114 + 10.3622 - 5.9272



+2.5361 - 2.0157 +2.6490 - 8.5560 + 7.9464



- 1.5338



sensitivity.



This does not reduces the For actual



reservoirs, there will be additional uncertainties. such as gas-cap size and applicability of the tank-type model (e.g., and no gas coning



at wells).



Production Rate and Time Calculations Rate and time were not considered



in the material-balance



computations



described



in the previous



performance



(recovery



vs. pressure)



sections would



are completed,



the incremental



tion for each pressure decrement time required for this production oil production



+ 1.0008 - 4.7911 + 10.0768 - 5.6907



All wells are assumed to have the same oil production rate at a given reservoir pressure (or equivalently an average well is considered). The production rate for the entire reservoir is calculated as the rate per well times the number of wells. Two different



approaches



have been



used



for calculat-



ing the oil production rate, 4,)) as a function of average reservoir pressure, p R, and well flowing BHP ( pIVf). The simpler shown



approach in Fig.



assumes



37.25



a straight-line



and given



relationship



by Eq. 40.



because



be indepen-



dent of rate and time for the assumed tank-type behavior with pressure equilibrium. Once the material-balance computations



0.75



+ 30.00



the type of data changed



performance



drainage



m=



- 30.00



mean that the presence of a gas cap always overall uncertainty about future performance.



no gravity



Varied’



m=0.50



oil produc-



has been calculated. The can be calculated if the



rate can be determined.



qO=J(PR-pnf).



. . . . . . . . . . . . . . . . . . . . . . . ..(40)



The other approach does not assume a straight-line



rela-



tionship. Curves that are called the well’s inflow performance relationship (IPR) aid in calculation of q,, Each approach



is discussed



34.



I



below.



9



I



Pool pcrformcmtcfor various initial prcrrurcr -8



Cumulative oil recovery, Fig. 37.20-Sensitivity change



of depletion-drive in interstitial water



% OIP performance



to 2 30%



saturation.



Cumulative



Fig. 37.21-Sensitivity change



oil recovery,



of depletion-drive in initial pressure.



0% OIP



performance



to f 30%



PETROLEUM



37-18



21



Cumulative



oil



recovery,



c



A m=O 0. R,



decreased 30%



6 ",=O



increased 30%



0. R,



37.22-Sensitivity change



of depletion-drive in 6,.



performance



to f 30%



Fig.



37.23-Sensitivity change



oil recovery,



\



Fig. 37.24-Sensitivity change



oil recovery,



of depletion-drive performance in permeability ratio k,/k,.



to f 30%



Fig.



performance



to + 30%



: ig pw,



I



OIL



% OIP



% OIP



of depletion-dnve in R,.



DR&wD~wN



Cumulative



HANDBOOK



% OIP Cumulative



Fig.



ENGINEERING



PRODUCING



37.25-Straight-line tionship.



inflow



RATE,



qO,



performance



BID



(q,,



vs.



pwf) rela-



SOLUTION-GAS-DRIVE



Rates Based on Productivity Well



production



proportional



Index



rates are often



to the pressure



tween reservoir



37-19



RESERVOIRS



(lower



pressures),



state flow



a tank-type



this equation



presented



The generalized



by Odeh”



and circular



4, =



drainage



form of



has a shape factor or



constant CA to enable characterization



of both noncircular



well



saturation



interval-because



index



for



is then



‘.-‘.(42)



is sometimes



some



variations



based on a differ-



it more consistent



to stay with the



assumption by using rates based on values? Although such questions are



note that the nonuniformity



tions tends to affect



mainly



in near-well



rates. The overall



saturamaterial-



than of near-well conditions. The IPR approach is also of interest for predicting oilwell productivity in other types of calculations for solution-gas-drive Vogel 24 used a computer program



reservoirs. to determine



oil pro-



duction rate (qo) vs. BHP, J.J,,~%for each of a sequence of declining reservoir pressures. This was done for a cir-



section



least



with rate calculations Isn’t



combine



that assumes



cular reservoir with a completely penetrating well at its center using Weller’s I6 approximation described in the



~L(,B,, ln(C,-z+r)



productivity



constant-at



for a single



Eqs. 40 and 41:



O.O0708k,,kh



J=L= P R -P wf



A well’s



index



by combining



uniform



why we would computation



balance results (oil recovery vs. average reservoir pressure) are more a function of average reservoir conditions



For a radial system, the shape factor is CA =relr,, , where rcl is the external radius and r,r is the wellbore The productivity



More



BHP.



material-balance



ent assumption.



logical.



.(41)



P,,B,, MC,4 - 4 +d



radius.



rate at a given



The reader may be wondering



uniform saturation productivity index



areas:



0.00708k ,,kh( p R -p ,,,f) 3 .



determined



not be uniform.



k,,,). This increased flow resistance reduces the oil pro-



(difference



duction



system.



will



be-



to be directly



40. The proportionality term is the productivity index, J, which is often based on the equation for pseudosteadyin a bounded



gas saturation



as shown by Eq.



drawdown



and wellbore



BHP),



gas will be evolved in the near-well region, causing higher gas saturations and more resistance to oil flow (lower



assumed



limited



treated



time



in pLo, B,,, and



as a



or pressure



k,,, are small.



entitled



Insights



from



Simulator



Studies.



Vogel simulated several circular reservoirs with different oil properties, relative-permeability characteristics, well spacings (i.e., sizes of the circular reservoir), and well skin conditions. in Fig.



37.26.



His results



for one case are shown



Each line shows q.



vs. p,!./ for a given



For performance predictions over larger pressure ranges, however. it is important to consider these variations.



cumulative oil recovery (or for a given reservoir pressure that is the pressure corresponding to zero qn). Note



The initial productivity index, Jj, can be determined two ways: (1) from well pressure and flow-rate tests (see



that, in contrast to the straight



Chap. 32). conditions.



or



(2) by Eq.



The expression



k,.,,= 1.0 at initial



with



\



.



(43)



No matter



at a later time (i.e.,



J is



the lines



This is a rem



and Muskat,



J who present-



ed theoretical calculations to show that plots of q. vs. pI,,f for two-phase flow result in curved lines rather than straight lines. Vogel



how J, is determined,



pressure),



curvature.



suit of the greater resistance to oil flow with increasing gas saturation. Vogel pointed out the compatibility of his



found that in plotting



as shown in Fig. 37.27,



a lower



line of Fig. 37.25,



have a downward



results with those of Evinger



for J, based on Eq. 42 is



0.00708kh , /



J, =



42



in Fig. 37.26



dimensionless



IPR curves,



the curves group closely.



He ap-



proximated this group of curves by a single average or reference curve shown in Fig. 37.28. This curve can be an approximation for all wells. An equation for this curve is



(



where k,,, is evaluated



..,



at the current



liquid



-=



saturation



(45)



and



p(, and B,, arc evaluated at the current reservoir pressure. Eq. 44 assumes pseudosteady-state flow conditions as the



where



average reservoir pressure declines [i.e., ai& (S,,/B,,) is the same at all points]. J from Eq. 44 is used in Eq. 40



Vogel did not provide a way to compute y,, given p,,~ and PR. His approach required knowledge of y. at some



to calculate y(,. Consequently, the well’s production rate is directly proportional to pressure drawdown (17~ -p ,,:,),



p,,f from a well test. Eq. 45 could then be used to calculate the y(, at any other value of put. In 1971,



but the proportionality saturation.



Standing’6



term (J ) varies wjith pressure and



(y,,),,,,



use Vogel’s Standing



Rates Based on Inflow



Performance



oil production



provided



the additional



results



in performance



The basic idea is that with increasing



drawdown



&=(I-$



insights



rate, STBID.



necessary



prediction



noted that Eq. 45 can be rearranged



Ratio (IPR)



The uniform saturation assumption of tank-type material balances is avoided in rate calculations using the IPR approach.



=maximum



(,+o.*~).



to



models. to



PETROLEUM



37-20



ENGINEERING



HANDBOOK



REFERENCE



CUM,,,



ATIYE



PERCENT



CURVE



RECOVER”.



OF ORl‘lNIL



OIL IN PLACE



RESERVOIR SAME AS



40



80



OIL



Fig.



37.26-Computed well



I20



PRODUCING



160



RATE,



q,,



200



”



210



BID



for



a



RATE



0.4



Standing



06



(~o/(qoh,),



37.28-Comparison curves.



Fig.



CONDITIONS 37.26



02



PRODUCING



inflow performance relationships in a solution-gas-drive reservoir



FIG.



of reference



curve



noted that the physical



Eq. 48 are that reservoir as reservoir



pressure,



0.8



FRACTION



OF



with



computed



conditions



with distance



IPR



inherent



gas and oil saturations,



vary



1.0 MAXIMUM



from



in



as well the well-



bore and that the well’s skin factor is zero. Standing also considered the situation in which fluid saturations are uniform within the reservoir. This would be the case for production with minimal drawdown. The well’s productivity under these conditions of essentially uniform saturations and pressure was denoted by J*. Note that J* is based on the same conditions assumed for the productivity index, J, in Eq. 42; J* is identical to the J of Eq. 42 and can be evaluated RESERVOIR SAME AS



FIG.



1



0.2



0



PRODUCING



Fig.



RATE



(qo



CONDITIONS 37.26



/



I



0.6



0.8



FRACTION



OF



where 1.0 MAXIMUM



37.27-Dimensionless Inflow performance relationships a well in a solution-gas-drive reservoir.



and that the productivity



J=-



index



of a well



90



PR-P,,l..



Substituting



?



/



I



/(so)mod,



0.00708 k,kh



J*=



04



is defined



for



the same way:



k,



is evaluated



the reservoir, age reservoir



\



,



at the average



.



fluid



saturations



(47)



tem the shape factor CA is simply t-h,,.. Standing used J to denote the true (or at least more ac-



small



drawdown



J*=



(i.e.,



J=



lim



as p,,f



approaches



1W1o)max PR



.



p R):



.



P,,/‘P R



(qdmaxbR,



(48)



index. The differof the inaccuracy



that occurs because J* is based on uniform conditions. Standing noted that J* is the limiting value of J for very



Combining



Eq. 47 into Eq. 46 yields



in



and p0 and B, are evaluated at the averpressure pR. Recall that for a radial sys-



curate) value of the well’s productivity ence between J and J* is an indication



by



. (49)



.



Eqs.



48



yielding



and



50



enables



elimination



(50) of



SOLUTION-GAS-DRIVE



37-2 1



RESERVOIRS



Eqs. 49 and 5 I enable calculation



of the well’s



J once



the average fluid saturations. p \,.,. and p R are known. By combining Eqs. 45 and 50, Standing eliminated (qo)m‘lx and obtained Eq. 52, which is a general relationship for IPR curves at various average reservoir pressures.



Insights from Simulator Studies Because reservoir



simulation



is the topic of Chap. 48, we



will not discuss it in detail here. For solution-gas-drive reservoirs, several comparisons have been made of gridded simulator results vs. simpler approaches, such as tanktype material



balances.



These comparisons



help to con-



firm the range of applicability of the simpler approaches. The key questions addressed by these studies are the same questions Vogel 24 considered in getting the computed results on which he based the IPR method for well rate cal-



. Thus,



Standing



(52)



has shown



how production



solution-gas-drive



performance



by use of Vogel’s



IPR information.



J* can be calculated



rate in a



model can be calculated Because a value of



with Eq. 49. all terms in Eq. 52 can



be evaluated. Later. Al-Saadoon”’ suggested that a different expression should be used for J. However, Rosbaco” clarified the situation by noting that although Standing?6 and AlSaadoon”



used different



formulas



for J and for



J/J*.



both yield the same results for q,, vs. ~,~f. Consequenly, it is workable and acceptable to use Standing’s equations. Standing’” discussed application of the IPR approach to damaged wells and Dias-Couto



and Golan “’ developed



a general IPR for wells in solution-gas-drive that is applicable to wells with any drainage any completion voir depletion.



flow efficiency,



and at any stage of reser-



These questions



saturation



distribution



does this influence



for Oil Production



oil recovery



vs. reservoir



pressure is known



are (I)



to what extent



is the



nonuniform,



and (2) how



much



performance.



The most informative



study



was by Ridings



Also.



they



used a gridded



radial



simulator



effect of rate and spacing on performance drive reservoirs. Their homogeneous, horizontal cluded



the following.



I. “Ultimate



recovery



essentially



2. “GOR



depends somewhat



high rates or close spacings,



of rate



predicted



on rate and spacing.



GOR’s



initially



For



are higher,



but later become lower than a Muskat prediction would indicate. At low rates or wide spacings. GOR behavior the Muskat



prediction.”



3. Computed depletion time agreed closely with conventional analysis (productivity index method) at low prcssure drawdowns, but differed more for high drawdowns.



productivity



by Vogel. ” 4. “Intermittent operation greatly GOR behavior, but the cumulative



(Eq. 42) or the IPR approach



is independent



and spacing, and agrees closely with recovery by the conventional Muskat method.”



This is in qualitative



index approach



to study the



of solution-gas-



conclusions concerning thin. solution-gas-drive reservoirs in-



from



the material-balance calculations. The oil producto a specified minition rate per well, q,, 1 corresponding mum P,,? can be calculated by use of either the



rt ul., ”



who compared laboratory vs. computed solution-gas-drive results for linear systems and obtained close agrcement.



approaches



Time Required At this point,



reservoirs area shape,



culations.



agreement



with the results obtained affects instantaneous GOR is not affected



(Eqs. 49 and 52). This y. is the calculated rate that the well is capable of producing. The well also may be sub-



significantly. Also, oil recovery apparently is not affected.” This refers to the cumulative oil recovery, not the



ject to a scheduling



amount



constraint,



such as an allowable



pro-



duction rate. Consequently, the well’s oil production rate q,, at pressure P,~ is the smaller of these two rates:



type models least



.



4,~ =(40)min, where (qo)min =minimum



. value of calculated



from



ij, =OS(q,,



p,i-



and sched-



rate q,, during



t to P,~ is given



the pressure



by Eq. 54.



.



+qn-,).



(54)



This average rate is used in Eq. 55 to calculate &,, required for the incremental oil production from



&



the time (AN,,),,



= (UP),, 4,T



. .



.



. . . .



. . . .



. . .



. (5%



II



The cumulative



time,



by Eq. 56. with



initial



+At,,.



for predictions



low



rates)



for



time



of recovery



period. the use of tankand of GOR (at



solution-gas-drive



reservoirs.



Although Muskat’s method is mentioned, other tank-type approaches, such as Tracy’s method, would be equally suitable. ” compared



one-dimensional



(1 D)



gridded simulator results vs. pressure and production data measured on a laboratory model produced by solutiongas drive. Computed and measured pressures vs. percent oil recovery



were



In 196 1, Levine of solution-gas-drive radial gridded



in close agreement. and Prats ” presented a comparison results for an “exact method” (a 1D



simulator)



vs. an “approximate



method.”



t,, , to reach pressure p,i is given time



t,,=O.



at any instant)-and “constant GOR,” which actually meant uniform GOR (i.e., the total GOR is the same at all points at any instant). Levine and Prats showed close agreement between results of the simulator and the approximate method. These results, for various stages of depletion,



t,,=t,,-,



in a given



1 and 2 support



The approximate method was based on assumptions of semisteady state-often called pseudosteady state (i.e.. the stock-tank-oil desaturation rate is the same at all locations



P,,- I to P,~.



II



for



Stone and Carder



uled oil rate, STBID. The average oil production decrement



(53)



of oil recovered



Note that Conclusions



(56)



were pressure and saturation



corresponding



values of producing



vs. radius and the



GOR and of percent



PETROLEUM



37-22



I-



Liquid



I



----



GOR(scf/slb)



oil



---?!



10



100



1,000



10,000



100,000



10,000



1.000



100



lOo,OoO 10 CGR(stb/MMscl



---------------Gas condensate



---



Fig.



HANDBOOK



Volatile Black



I



ENGINEERING



Dry gas



Gas



37.29--Solution GOR range from black oilto gases. Volatileoilstypically are in the range of 1,500 to 3,500 scf/STB.



oil recovery. Only limited information was given about the approximate method. This method would require derivation of additional equations and development of a computer program. Levine and Prats also discussed the extension of results to other sets of fluid and rock properties by use of dimensionless groups. Later, Weller I6 presented a different approach that retained the semisteady-state assumption but eliminated need for the “constant GOR” assumption. Weller showed that his method matched simulator results more closely than Levine and Prats’ constant-GOR method. Weller developed equations for the radial distribution of saturation and pressure based on the combination of a transient period before the effects of a change in producing rate are felt at the drainage boundary with semisteady state (same rate of tank-oil desaturation everywhere) thereafter. Because these equations serve mainly as an alternative to a gridded simulator, details will not be given here (see Ref. 16).



Volatile Oil Reservoir Performance Predictions Volatile oils are characterized by significant hydrocarbon liquid recovery from their produced reservoir gas. Also, volatile oils evolve gas and develop free-gas saturation in the reservoir more rapidly than normal black oils as pressure declines below the bubblepoint. This causes relatively high GOR’s at the wellhead. Thus, performance predictions differ from those discussed for black oils mainly because of the need to account for liquid recovery from the produced gas. Conventional material balances with standard laboratory PVT (black-oil) data underestimate oil recovery. The error increases for increasing oil volatility. A volatile oil can be defined as hydrocarbon that is liquid-phase oil at initial reservoir conditions but at pressures below bubblepoint evolves gas containing enough heavy components to yield appreciable condensate dropout at the separators. This is in contrast to black oils for which little error is introduced by the assumption that there is negligible hydrocarbon liquid recovery from produced gas. Cronquist 38 used Fig. 37.29 to show the position of volatile oils in the GOR range between black oils and gases. Compared to black oils, volatile oils have higher solution GOR (1,500 to 3,500 scf/STB), generally higher



oil gravities (greater than 40 or 45”API), and higher B, (above about 2.0 RB/STB). Volatile oils tend to shrink rapidly with pressure decline below bubblepoint. Cronquist used Fig. 37.30 to illustrate this behavior. The curves are made dimensionless (i.e., normalized to maximum values of unity) to facilitate comparisons. The ordinate bcjD is the dimensionless shrinkage: b,D =@ob -~oWo/,



-B,,,).



The abscissa PRD is a special form of dimensionless reservoir pressure:



where PRD = reservoir pressure, dimensionless, PR = reservoir pressure, psi, and pb = bubblepoint pressure, psi. The curve labeled BO in Fig. 37.30 represents the typical behavior of a black oil. Shrinkage is almost proportional to pressure reduction below bubblepoint. In contrast, Curves E, F, and G are for progressively more volatile oils and show much greater shrinkage as pressure drops below bubblepoint. This large shrinkage corresponds to substantial gas evolution (i.e., a large reduction in the solution GOR as pressure drops below bubblepoint). This is illustrated by Fig. 37.3 1, which shows dimensionless cumulative gas evolved, R @ =R,IR,J,, vs. dimensionless pressure. Rsb is the solution GOR at bubblepoint, and R, is the reduction in solution GOR below bubblepoint: R,, =R,h -R, . The trend line in Fig. 37.31 shows typical behavior for a black oil. Gas evolution is almost proportional to pressure reduction below the bubblepoint. Curves E, F, and G, which are for volatile oils, show much more gas evolution as pressure declines below bubblepoint. Consequently, depletion performance of volatile oil reservoirs below bubblepoint is strongly influenced by the rapid shrinkage of oil and by the large amounts of gas evolved. This results in relatively high gas saturations, high producing GORs, and low to moderate production of reservoir oil. The produced gas can yield a substantial



SOLUTION-GAS-DRIVE



37-23



RESERVOIRS



Dimensionless pressure, PRD = Pn/Pb



Dimensionless pressure,p AD = p R Ipb Fig. 37.30-Dimensionless shrinkage vs. dimensionless pressure.Curves E, F, and G are forprogressively more volatile oils.Curve 60 isfora black oil.Curve VO is for a volatile oil.



Fig. 37.31-Dimensionless evolved gas vs.dimenslonlesspressure.Curves E, F, and G are forprogreswely more volatile oils. The trendlinetypifies black-oil behavior



volume of hydrocarbon liquids in the processing equipment. This liquid recovery at the surface can equal or exceed the volume of stock-tank oil produced from the reservoir liquid phase. 31.33.34.38Depletion-drive recoveries are often between 15 and 25% of initial OIP. Improved recoverv through injection of gas or water is sometimes con;dered but is beyond the scope of this chapter. For volatile oil reservoir primary-performance prediction methods, the key requirements are correct handling of the oil shrinkage, gas evolution, gas and oil flow in the reservoir, and liquids recovery at the surface. For oil with a low volatility but a higher shrinkage than a typical black oil, simple corrections to differential shrinkage data are sometimes made. 33,3s.39 For volatile oils, however, it is essential to account for their special behavior more thoroughly. This includes determination of the composition of the gas evolved in the reservoir for a sequence of pressure steps below bubblepoint. Methods for predicting volatile-oil reservoir-depletion performance that assume tank-type behavior (i.e., ignore pressure gradients) have been published by Cook et al., ” Reudelhuber and Hinds, j3 and Jacoby and Berry. ” In Refs. 31 and 33, laboratory data determined fluid compositions, while in Ref. 34, fluid compositions were computed from data for equilibrium constants. Cronquist’s stated that there was no significant advantage of one method over the other two methods because “each method appears to yield acceptable results.” The multicomponent-flash method of Jacoby and Berry34 is particularly appealing because a comparison of predicted vs. actual reservoir performance is available. Sections to follow describe the prediction methodj” and discuss a comparison of predicted vs. field performance. 36 The description of the multicomponent-flash method is from Sikora. I3



Multicomponent-Flash of Jacoby and Berry



Method



Data required to predict volatile-oil reservoir performance by the multicomponent-flash method include (1) the state and composition of the reservoir fluid at initial pressure; (2) appropriate sets of equilibrium vaporization ratios (K values) for the reservoir pressure range at the reservoir temperature and covering the temperature and pressure of surface separation; (3) some experimental liquid-phase densities at reservoir conditions to check correlations for calculating the required liquid densities during the depletion process; (4) experimental oil-phase viscosity data at reservoir temperature; and (5) relativepermeability-ratio data. Calculation Procedure. Prediction of reservoir performance by the Multicomponent-Flash Method consists of the following steps, starting at pressure p 1. For convenience, the calculation is made for a unit of hydrocarbon PV. 1. Select a pressure p2 that is lower than p 1. 2. Flash the number of moles of the reservoir composite fluid in the unit pore space at p 1 to the next lower pressure p2. 3. Assume a gas saturation at p2 and calculate the average flowing bottomhole GOR with Eq. 57. l&F.



. 0



.



(57)



,q



4. Calculate the number of moles in each phase of the unit volume, the overall composition. and the number of moles of reservoir composite remaining in the unit volume at pl. 5. Determine the difference between the reservoir composite at p I and p2, which is the total amount and com-



37-24



PETROLEUM



TABLE



37.3-CALCULATED COMPOSITION OF THE WELLSTREAM



(MOLE



ENGINEERING



HANDBOOK



FRACTIONS)



Reservoir Pressure (psia) Component Nitrogen Methane Carbon dioxide Ethane Propane Butanes Pentanes Hexanes Heptanes plus



TABLE



4.836



4,768



4,556



4.300



3.750



2,750



1,750



750



0.0167* 0.6051' 0.0218* 0.0752* 0.0474' 0.0412' 0.0297' 0.0138' 0.1491'



0.0147 0.5718 0.0215 0.0764 0.0496 0.0442 0.0325 0.0154 0.1739



0.0170 0.6109 0.0218 0.0751 0.0470 0.0407 0.0292 0.0135 0.1448



0.0205 0.6711 0.0224 0.0737 0.0437 0.0359 0.0246 0.0108 0.0973



0.0235 0.7298 0.0236 0.0736 0.0411 0.0315 0.0200 0.0082 0.0487



0.0235 0.7582 0.0250 0.0775 0.0412 0.0296 0.0171 0.0064 0.0215



0.0215 0.7570 0.0267 0.0838 0.0451 0.0308 0.0161 0.0057 0.0133



0.0165 0.7001 0.0274 0.1004 0.0616 0.0466 0.0246 0.0076 0.0152



37.4-CALCULATED



RESERVOIR



FLUID



COMPOSITIONS



(MOLE



FRACTIONS)



Reservoir Pressure



(psia) Component



4,836



-4,700 -4,600 -4,500 Composite or Overall Mixture in the Reservoir



-4,400



-4,000



-3,500



3,000 __



~2,000



~1,000



Nitrogen Methane Carbon dioxide Ethane Propane Butanes Pentanes Hexanes Heptanes plus



0.0167* 0.6051* 0.0218' 0.0752' 0.0474' 0.0412' 0.0297' 0.0138' 0.1491'



0.0168 0.6060 0.0218 0.0752 0.0473 0.0411 0.0296 0.0138 0.1484



0.0168 0.6062 0.0218 0.0752 0.0473 0.0411 0.0296 0.0137 0.1483



0.0168 0.6062 0.0218 0.0752 0.0473 0.0411 0.0296 0.0137 0.1483



0.0167 0.6057 0.0218 0.0752 0.0474 0.0412 0.0296 0.0138 0.1486



0.0164 0.6001 0.0217 0.0753 0.0477 0.0416 0.0301 0.0140 0.1531



0.0160 0.5926 0.0216 0.0754 0.0480 0.0422 0.0307 0.0144 0.1592



0.0152 0.5766 0.0214 0.0754 0.0488 0.0434 0.0319 0.0151 0.1722



0.0128 0.5194 0.0201 0.0743 0.0510 0.0476 0.0367 0.0179 0.2203



0.0085 0.3937 0.0163 0.0674 0.0527 0.0559 0.0475 0.0244 0.3336



0.0142 0.5632 0.0214 0.0767 0.0502 0.0449 0.0332 0.0159 0.1803



0.0131 05447 0.0213 0.0772 0.0512 0.0464 0.0346 0.0166 0.1948



0.0123 0.5297 0.0212 0.0775 0.0520 0.0476 0.0358 0.0174 0.2065



0.0115 0.5146 0.0210 0.0776 0.0528 0.0487 0.0368 0.0180 0.2189



0.0087 0.4667 0.0202 0.0777 0.0549 0.0520 0.0404 0.0199 0.2595



0.0066 0.4205 0.0192 0.0776 0.0568 0.0555 0.0440 0.0221 0.2978



0.0047 0.3682 0.0177 0.0754 0.0587 0.0592 0.0485 0.0246 0.3430



0.0025 0.2662 0.0141 0.0681 0.0600 0.0663 0.0580 0.0303 0.4345



0.0010 0.1561 0.0090 0.0521 0.0542 0.0706 0.0679 0.0371 0.5520



0.0256 0.7546 0.0231 0.0698 0.0376 0.0279 0.0171 0.0065 0.0379



0.0256 0.7571 0.0230 0.0702 0.0379 0.0281 0.0173 0.0067 0.0341



0.0256 0.7575 0.0231 0.0705 0.0380 0.0283 0.0174 0.0066 0.0330



0.0257 0.7617 0.0231 0.0710 0.0380 0.0282 0.0173 0.0065 0.0285



0.0262 0.7700 0.0237 0.0722 0.0384 0.0283 0.0170 0.0065 0.0177



0.0262 0.7780 0.0243 0.0730 0.0386 0.0278 0.0163 0.0061 0.0098



0.0253 0.7770 0.0248 0.0754 0.0393 0.0282 0.0160 0.0059 0.0081



0.0230 0.7720 0.0261 0.0804 0.0420 0.0290 0.0155 0.0055 0.0066



0.0198 0.7492 0.0274 0.0902 0.0504 0.0339 0.0170 0.0056 0.0066



Reservoir Oil Phase Nitrogen Methane Carbon dioxide Ethane Propane Butanes Pentanes Hexanes Heptanes plus Reservoir Gas Phase Nitrogen Methane Carbon dioxide Ethane Propane Butanes Pentanes Hexanes Heptanes plus



position of the produced wellstream for this pressure decrement. 6. Calculate the bottomhole GOR by flashing the wellstream composition from p 1 to the average pressure (p 1 +p2)/2, for this pressure decrement. 7. If the difference between the GOR from Step 6 and the average GOR from Step 3 exceeds the desired tolerance. select a new gas saturation and repeat Steps 3 through 7 to continue iterations for the current pressure decrement. If this difference is within the tolerance, the final answer has been obtained for this pressure decrement. For the next decrement, set p , =p2 and select a



p2 that is lower than the previous p2. Repeat Steps 1 through 7. Example From Jacoby and Berry.34 Reservoir temperature, 246°F Initial pressure, 5,070 psia Bubblepoint pressure, 4,836 psia Initial GOR, 2 Mscf/STB Oil gravity, 5O”API Conventional B,, 4.7 RB/STB Original reservoir fluid composition, Table 37.3 (column 1)



SOLUTION-GAS-DRIVE



RESERVOIRS



Z60003



37-25



IOOOm LA $



-VOLATILE OIL ---- CONVENTIONAL -100



MATERIAL



BALANCE



2



0 3 -IO $ 0.1;



MATERIAL-



3



0’



CUMULATIVE OIL PRODUCTION BBL/BBL HC PORE SPACE



CUhWL ATIrE



Fig. 37.32-Comparison of oil and gas production for volatk011materialbalance (multicomponent flashmethod) vs. conventionalmaterialbalance.



Fig. 37.34-Main



STOCK



TANK O/L PROD. -THOUSAND



BBl



Reservoircumulative oilproductionvs. reser-



voir pressure.



160,OOC 4 8 k 2 I



CUMULATIVE



CONVENTIONAL MATERIAL



BALANCE



L9



120,000



E! 2



4



80,000



Q VOLATILE



2



OIL



MATERIAL BALANCE



QT 40,000 i? z F $



Fig. 37.33-Mam



Reservoir performance history



Solution. Results calculated by Jacoby and Berry” with the above method are given in Tables 37.3 and 37.4 and in Fig. 37.32. Table 37.3 shows the calculated wellstream compositions, and Table 37.4 shows the fluid compositions in the reservoir. The oil and gas production in Fig. 37.32 was obtained by separating the wellstream data in Table 37.3 at separator conditions of500 psia and 65°F and stock-tank conditions of 14.7 psia and 70°F. Fig. 37.32 also shows the comparison of oil and gas production



with



conventional



performance



predictions.



’



Comparison of Predicted vs. Actual Reservoir Performance Jacoby and Berry’s example was a performance prediction published in 1957 for a volatile-oil reservoir in north Louisiana that was discovered in 1953 and produced from



0



Fig. 37.35-Main



Reservoircumulative oilproductionvs. GOR.



the Smackover lime. 34 The reservoir was believed to be volumetric. The comparison vs. actual performance was published in 1965 by Cordell and Ebert. 36 They called this field the Main Reservoir. The field was completely developed with 11 wells on 160-acre spacing by 1956 and was 90% depleted by the time of their publication. Fig. 37.33 shows performance history for the Main Reservoir. j6 Figs. 37.34 and 37.35 compare actual performance (cumulative stock-tank-oil production vs. reservoir pressure) vs. erformance predicted by the volatile-oil material balance- P, and by conventional material balance.6 Cordell and Ebert stated that actual ultimate recovery would be 10% greater than predicted by the volatile-oil material balance and 175% greater than indicated by the conventional (black-oil) material-balance calculation.



37-26



Fig. 37.35 illustrates the large errors in applying a conventional black-oil material balance to volatile oils: oil recovery is underestimated, and producing GOR is overestimated. This emphasizes the importance of considcring the varying reservoir and wellstream compositions in volatile-oil reservoir-performance predictions by use of a volatileotl material-balance method.



Nomenclature h,, = oil shrinkage factor ~~dl = oil shrinkage factor, dimensionless B,Y = gas formation volume factor (gas FVF), RBiscf B,qj = initial gas formation volume factor, RBiscf B,, = oil formation volume factor, RBiSTB B 0 * = pseudovalues for formation volume factor, RBLSTB B 01, = B,, at atmospheric pressure and reservoir temperature. RBLSTB B oh = B,, at bubblepoint pressure. RBiSTB B,,, = initial oil formation volume factor. RBiSTB B, = two-phase FVF, RBISTB Bti = initial two-phase FVF, RBiSTB B,,. = water formation v,olume factor, RBiSTB AB, = expansion of initial free gas in place, RBlscf As, = expansion of initial OIP. RB/STB c (, = effective compressibility, voli(vo-psi) cf. = formation compressibility. vol/(vol-psi) CO = oil compressibility. vol/(vol-psi) c,,. = water compressibility. vol/(vol-psi) CA = shape factor or constant. dimensionless G, = cumulative gas injection. scf G,, = cumulative gas production. scf G,,, = cumulative production of gas that was initially in the gas cap, scf G,,, = cumulative production of gas that was initially solution gas, scf (G,,),, = cumulative gas production to pressure n, scf CC,,),, t = cumulative gas production to pressure II- I. scf AC,) = incremental gas production. scf H = thickness of gas cap/thickness of oil-zone (Fig. 37. IO) J = productivity index. STBIDipsi .I* = productivity index under conditions of uniform saturation and pressure, STB/D/psi J, = initial productivity index, STBlDipsi X = permeability, md k, = effective permeability to gas. md kh = formation flow capacity, md-ft k,, = effective pcrmcability to oil, md X,., = relative permeability to gas (k,., ) ,,, = relative permeability to gas at residual oil saturation



PETROLEUM



ENGINEERING



HANDBOOK



k r AIME (1955, 204. 767-70. Arp\. J.J. and Robert\. T.G.: “The Eltcct ol the Relative Permcublllty Ratlo. the Oil Cra~lty. and the Solutwn Gas-Oil Ratio on the Prmwy Recowry From a Dcpletwn Type Rewwr.” /‘wt.\. AlME (19.55, 204. 120~27. W;rhl. W.L.. MulInt\. L P.. and Eltrtnk. E.B.: “E\timation ofUItimate Recovery Irom Solution Ga+-Drlvc.” ‘r/wfr , AIME (1958, 213. 132-38 H:md>. L.L.: “A Lahoratorq Study 01 Oil Rccovq hy Sulutwn Gas DrI\c.“ Twc\., AIMt t 195X) 213. 310-1.5. Crali. B.C. and H,I\\ kins. M F : ,4/~/>/w/ P~~rrdrwr~rR~~~~~II. GI~,wo,-~)~q. Prcntlcc~H;~ll Inc.. En$cHood Chf\. NJ ( IYSY) Lwinu. 5.5. and Przat\. M.: “The Calculated Pcr(i)rmdncc\ ot Slrlutlon~Ga~~Drl\c Rcxrboirs.” S&. Per. Ejrq. J. iScpt 1961 ) 142~52: %.i/w.. AIME. 222 Slhor,l. V.J.: “Solution-Ga\~Drl\,e Oil Rexrvoir\.” f~~rrr~/cfw~/‘,?I~ ,) =mp



(5)



w,!, = cumulative water influx to end of interval, +c,,,har,,.’ ____._. ._. .(9) = 0.17811 “P for radial aquifers, = 0.17811 $r ,,.,hb 2 .(lO) MI] for infinite linear aquifers, AP(~~+I-~, = average pressure drop in interval n+l-j, W PD = dimensionless water-influx term, rw = field radius, ft, and c.,i = total aquifer compressibility, psi - ’. The solution of Eq. 8 requires the use of superposition, in a manner similar to that shown by the expansion of Eq. 6. A modification presented by Carter and Tracy3 permits calculations of W, that approximate the values



WATER



DRIVE OIL RESERVOIRS



38-3



obtained from Eq. 8 but does not require the use of superposition. This method is advantageous when the calculations are to be made manually. since fewer terms are required. Using Carter and Tracy’s method, Eq. I I, the cumulative water influx at time t,, is calculated directly from the previous value obtained at t,,-,



4\



FAULT



0



+ bpA~,,r~,, - W,,,, ,,P’D,,IVo,,-[I+,, ,, 1



Fig. 38.3~Infiniteaquiferbounded on one side by a fault.



PD,,-tDd”D,, ..



.... .....



........



(11)



where p,D



=pD,, -pD,,, ,>



I,



.



. . . . . . . ..I.....



(12)



ID,,-rD,,,-,,



and Ap,,=p,-pn,



. .... .....



.. .....



(13)



Reservoir Interference. Where two or more reservoirs2 are in a common aquifer, it is possible to calculate the change in pressure at Reservoir A, for example, caused by water influx into another reservoir, B, using Eq. 14 or 15. These are Eqs. 2 and 3 with modified subscripts. For unequal time intervals,



A~Pnwo,, =tnr



Ii [~doi‘,-,) J=I



.... .



-enB,,,JPD(A.R),~



.. ....



... ...



(14)



and for equal time intervals,



*P~(A,B),, =m,



A



e MB,,,+,mj ,APD(A,B),



>



.



.(I3



Hicks et al. 4 used the past pressure and production history in an analog computer to obtain influence-function curves for each pool in a multipool aquifer. The influence function F(r) can be defined as the product of m, and PO, F(r)=m,pD,



.. ..



.



.(l7)



and can be substituted in Eqs. 59 and 60 to calculate the future performance. Nonsymmetrical Aquifers. By use of the images method,2 the procedure for calculating reservoir interference can be extended to the case where one boundary of an infinite aquifer is a fault. For example, Fig. 38.3 shows Reservoir A located in this type of aquifer. To calculate the pressure performance at Reservoir A, first locate the mirror-image Reservoir A’ across the fault. The water-influx history for the mirror-image Reservoir A’ will be taken to be the same as Reservoir A. Next, assume that the fault does not exist so that there are two identical reservoirs in a single infinite aquifer, with Rexrvoir A’ causing interference at Reservoir A. The pressure drop at Reservoir A now can be calculated by use of Eq. I9 (for equal time intervals).



j=l APIA,,



where



=mr



2



[~NzA~,,+,~, , APO,



1



J=t



pressure term for PD(A,B) = dimensionless Reservoir B with respect to Reservoir A, AP,~(~,J) = pressure drop at Reservoir A caused by Reservoir B, and e,,,B = Water inflUX rate at Reservoir B.



Because e ,,,A=e Lr,A, ,



n The total pressure drop at Reservoir A at any given time is the sum of the pressure drops caused by all reservoirs in the common aquifer, or



APoA,,



=m,



c



j=l



e)+,A ,,!+,-, j [APO,



.., . . . . . . . . . APIA,, =AP~(A,A I,, +AP~(A.B),,



+AP~(A,cJ,,



. . . . . . . . . . . . . . . . . . . . . . .



+. . . .



.



(16)



Since dimensionless pressure differences are available only for homogeneous infinite radial aquifers, pressureinterference calculations are limited at the present time to aquifers that can be approximated by a uniform, infinite, radial system.



-APD(A.AY,



.... ....



1. . (1%



If other reservoirs in the aquifer also are causing reservoir interference at Reservoir A, each mirror image will cause reservoir interference at Reservoir A. The total pressure drop at Reservoir A, therefore, will be the sum of the pressure drops caused by each reservoir and each mirror image (see Fig. 38.4). Nonsymmetrical aquifers will be discussed further under Methods of Analysis, Method 2.



PETROLEUM



38-4



ENGINEERING



.



Ap~=-$Aro.



HANDBOOK



..



. . (25)



ID



and pD=tD+o.33333,



..... ...



.



. . .(26)



where to = dtmensionless time, rD = dimensionless radius =T,/T,, ru = aquifer radius, ft, rw = field radius, ft, and d = a geometry term obtained from Table 38.1. Methods of Analysis Fig. 38.4-Dimensionless pressure drop forinfinite aquifersystem for constant flow rate.,8



pn and W,~Values. Values ofpn, PD(A,B),and W,D are functions of dimensionless time rg (Eq. ZO), aquifer geometry, and aquifer size (to for radial aquifers). Table 38.1 gives the substitution for d in Eq. 20 to calculate tD and the table, graph, or equation to obtain po, P&A-B), or W,D for various types of aquifers. The following equations are used in conjunction with Table 38.1. 0.006328kr tD = ~C~,~?ftL,d2,



po=l.l284JtD, pD=o.5(h pD=h



. .



.(21) .(22)



tD+0.80!?07), .................



.(23)



,-D, ............................. .....



TABLE



_..



.



. ..(24)



3&l--REFERENCE



Aquifer Type Infinite radial Smaller t, Larger t, Finiteoutcropping radial Smaller t, Larger t, Finiteclosed radial Smaller lo Larger t, Infinite linear Finiteclosed linear Larger to Interference(infinite radial) Larger to



where AZ, =Zi -Zj- r . Method 2 is not limited to homogeneous linear or radial aquifers because the final Z is obtained by adjusting previous approximations to Z. Techniques for applying Method 2 to the case where reservoir interference exists are not available at this time, except for unusual circumstances. ‘Personal



TABLE



FOR



Value of d in Eq. 20 * rw rw rw r, rw rw rw r, rw b” Lf L r(A.B15 ‘W)



*r* = radus of pwl bang analyzed, f, “b +P*D



(27)



e, fn+, , ,AZj,



Apwj,, = 2 j=l



(20)



.



........................



WeD=0.5(rD’-I),



Reservoir Volume Known. Rigorous Methods. There are two methods for obtaining the coefficient m, and APO in Eq. 6 from the past pressures and the waterinflux rates from a material balance on the reservoir. Method l* is used whenever the aquifer can be approximated by a uniform linear or radial system; therefore, published values of pD are used. If the aquifer can be approximated by a homogeneous, infinite, radial system, the method can be extended to handle reservoir interference. In Method 2,5 the product of m, and pD is replaced by Z (the resistance function).



= width Of aquifer. ft = We,



1 = length of aqwfei, ft §r ,A,Bj =distance between centers of Reservoirs A and 8. ft



communication



OBTAINING



from Allant~c Refining



WeD AND



PD Table 38.3 Eq. 21 Eq. 22 Table 38.7 Table 38.7 Eq. 23 Table 38.6 Table 38.3 Eq. 25 Eq. 21 Table 38.8 Eq. 26 Fig. 38.4 pDcA,E) Table 38.3, Eq. 22



p.



WC?0 Table 38.3 Eq. 21 Table 38.5



Table 38.6 Table 38.3 Eq. 24 Eq. 21+



Co



WATERDRIVEOIL RESERVOIRS



TABLE



36-5



38.2-COMPARISON



QuaXer or Interval No



OF RESULTS



MZtLal Balance (B/D) 500 1.100



APf” Field (Psi)



OF METHODS



PO 210



AI,



rD=m



1 AND



2 FOR SAMPLE



Z” 4PW” Method 1 fi (psi/B/D) (Psi)



CALCULATION



Mzi%d 2 (psi)



478 581



1.651 1.960 2.147 2.282 2.389



1.000 1.414 2.732 2.000 2.236



55 136 318 478 581



55 135 317 477 584



2.476 2.550 2.615 2.672 2.723



2.449 2.646 2.828 3.000 3.162



663 616 599 652 733



672 630 614 664 739



2.770 2.812 2.851 2.887 2.921



3.317 3.464 3.606 3.742 3.873



761 803 858 928 949



761 607 860 934 946



55 136



318



8 9 IO



3,100 3,600



663 616 599 652 733



11 12 13 14 15



3,500 3,600 3,800 4,100 3,900



761 803 858 928 949



The procedure for both methods can be illustrated best by an application to a single-pool aquifer. Assume that a reservoir has produced for 15 quarters and that Cols. 2 and 3 in Table 38.2 are, respectively, the pressures at the end of each quarter and the average water-influx rates obtained by material balance for each quarter.



If the AZD selected is the correct value, m, as a function of n will be constant. Variations from a constant can result from (1) incorrect AtD, (2) production and pressure errors, (3) incorrect aquifer size or shape, or (4) aquifer inhomogeneities. An examination of the m, plot will aid in the analysis of the cause.



Example Problem 1. Method 1. From the following assumed best set of aquifer properties, check Table 38.1 for the substitution of d in Eq. 20.



Value of m,



Possible Remedy



increase decrease constant, constant,



decrease with At, increase AtD finite-closed aquifer finite-outcropping aquifer



c,,, = /.i,,, = h = 01 = k = q5 = r,, =



5.5X10-’ psi-‘, 0.6 cp, 50 ft , 27~ radians, 76 md, 0.16, 3,270 ft,



For a finite-closed aquifer or finite-outcropping fer, Eq. 29 or 30 is used to find rD. rD=2.3(NilAtD)0.518 for N;,At,



and the aquifer geometry is infinite radial. Calculate a convenient value (to minimize interpolation) of dimensionless time interval (AZ,) for the quarterly interval (Ar=91.25 days) by varying the permeability (if necessary) in Eq. 20. In this case, AID = 10, corresponding to k=91 md, was selected. A check of Table 38.1 shows that pi is to be obtained from Table 38.3 (also tabulated in Table 38.2, Col. 4).



m



APS,, ?I=



),



with II with n then increasing then decreasing



.



(28)



where Ape is the known field pressure drop at original woe. Calculate ApD as a function of interval number. Then calculate m, as a function of interval number using Eq. 28 and plot m, as a function of n (Curve 1, Fig. 38.5). Fig. 38.6 shows an example of the calculation procedure for n=5 using equal time intervals.



~3.4,



... .



aqui-



. . (29)



and



r~=3(A’i,Af~)“.30’



.



.



.



.(30)



for NirAtD 63.4, where N;, is the time interval number where m, vs. n increases from a constant value. In this example, m,. increased with n (Fig. 38.5. AtD = 10). Therefore, AtD was decreased from 10 to 1 (large changes are recommended) and m, for At, = 1 was calculated (Curve 2). Now m, is constant until about Interval 9 and then increases, indicating the possibility of a finite-closed aquifer. Using Ni, =9 and AtD = I in Eq. 29 gives a first approximation of 7 (rounded from 7.2) for rD. The m,. calculated for AtD = 1 and rD =7 is rem duced after Interval 9 (Curve 3) but is still too high and therefore indicates that the aquifer is still too large. An rg of 6 is taken for the next approximation, and this results in a constant value of m, (Curve 4). This shows that the past field behavior (Col. 3, Table 38.2) can be duplicated by assuming a finite-closed aquifer where AtD = 1 and rD=6 (Col. 6, Table 38.2). Because these aquifer properties gave the best match to the past field performance, they should be taken as the best set for predicting the future performance.



38-6



TABLE



PETROLEUM



38.3-DIMENSIONLESS



WATER



INFLUX



AND



DIMENSIONLESS



t,



W c?D



0.112 0.278 0.404 0.520 0.606



PO ~___ 0.112 0.229 0.315 0.376 0.424



1.5x103 2.0 x 103 2.5x IO3 3.0 x 103 4.0 x IO3



4.136x10' 5.315x10" 6.466x IO2 7.590x10' 9.757x10'



2.5x 10 -' 3.0x10-' 4.0x10-' 50x106.0x 10 -'



0.689 0.758 0.898 1.020 1.140



0.469 0.503 0.564 0.616 0.659



5.0 x103 6.0 x IO3 7.0 x lo3 8.0 x103 9.0 x lo3



11.88 13.95 15.99 18.00 19.99



7.0x10-' 8.0x10-' 9.0x10-' 1.0 1.5



1.251 1.359 1.469 1.570 2.032



0.702 0.735 0.772 0.802 0.927



1.0x 1.5x 2.0 x 2.5 x 3.0 x



2.0 2.5 3.0 4.0 5.0



2.442 2.838 3.209 3.897 4.541



1.020 1.101 1.169 1.275 1.362



6.0 7.0 8.0 9.0 1.0x10'



5.148 5.749 6314 6.661 7417



1.5x10' 2.0x10' 2.5x10' 3.0x IO' 4.0x10' 5.0x10' 6.0x IO' 7.0x10' 8.0x10' 9.0x10'



w eD



PRESSURES tD



FOR W eD



ENGINEERINGHANDBOOK



INFINITE RADIAL



to



AQUIFERS



W



1.5~10~ 2.0x107 2.5x10' 3.0x107 4.0x107



1.828~10~ 2.398x106 2.961~10~ 3.517x106 4.610~10"



1.5x 2.0x 2.5x 3.0x 4.0x



5.0x107 6.0~10~ 7.0x107 8.0~10~ 9.0x107



5689x10' 6.758~10~ 7.816~10~ 8.866x10e 9.911xlO~



5.0x10" 6.0x IO" 7.0x IO" 8.0x IO" 9.0x IO"



3.75xlO'O 4.47x 10" 5.19x IO'O 5.89x 10'0 6.58~10'~



21.96 x102 3.146~10~ 4.679x103 4.991 x103 5.891 x IO3



1.0~10~ 1.5~10' 2.0~10~ 2.5~10' 3.0x 10'



10.95 x 106 1.604x 10' 2.108x 10' 2.607~10' 3.100x10'



1.0~10'~ 1.5x10" 2.0~10'~



7.28x IO" 1.08x10" 1.42~10"



4.0x10" 5.0x104 6.0 x lo4 7.0~10~ 8.0x lo4



7.634~10~ 9.342x103 11.03 x104 12.69 x104 14.33 x104



4.0x10* 50x108 6.0~10" 7.0~10' 8.0~10'



4.071x10' 5.032~10~ 5.984x10' 6.928x10' 7.865~10'



1.436 1.500 1.556 1.604 1.651



9.0 x IO4 l.OxlO~ 1.5~10~ 2.0~10~ 25~10~



15.95 x104 17.56 x104 2.538~10~ 3.308x104 4.066x IO4



9.0x10* 1.0~10~ 1.5~10' 2.0x10" 2.5~10"



8.797x10' 9.725x10' 1.429x10n



9.965 1.229x10' 1.455x10' 1.681~10' 2.088~10'



1.829 1.960 2.067 2.147 2.282



3.0x105 40~10~ 5.0~10~ 6.0~10~ 7.0~10"



4.817~10~ 6.267~10~ 7.699x IO4 9.113x104 10.51 x105



3.0~10" 40x10' 5.0~10~ 60x10' 7.0~10~



2.771~10' 3.645~10' 4.510x108 5.368~10' 6.220~10'



2.482~10' 2.860x10' 3.228~10' 3599x10' 3.942x 10'



2.388 8.0~10~ 2.476 9.0x10" 2.550 1.0~10" 2.615 1.5~10" 2.672 2.0 x lo6



11.89 x105 13.26 x105 14.62 x105 2.126~10~ 2.781x lo5



8.0~10' 9.0x10" 1.0~10'~ 1.5~10'~ 2.0~10'~



7.066~10' 7.909x 108 8.747x10B 1.288~10" 1.697x10"



2.723 2.5 x IO6 2.921 3.0 x106 3.064 4.0 x lo6 3.173 5.0x lo6 3.263 6.0 x IO"



3.427x lo5 4.064x lo5 5.313x105 6.544~10~ 7.761 x IO5



2.5x 10" 3.0x IO" 4.0x10'" 5.0~10'" 6.0~10'"



2.103~10~ 2.505~10~ 3.299x10" 4.087~10" 4.868~10~



7.0~10" 8.0~10'" 9.Ox1O'o 1.0~10"



5.643~10" 6.414~10~ 7.183~10~ 7.948x10'



1.0x10* 1.5x10* 2.0x 102 2.5x102 3.0x10'



4.301x10' 5.980x10' 7.586~10' 9.120x10' 10.58 x10'



4.0x10* 5.0x10* 6.0x 10' 7.0x 102 80x102 9.0x10' l.OxlOJ



13.48 16.24 18.97 21.60 24.23 26.77 29.31



x10' 3.406 x10' 3.516 x10' 3.608 x.10' 3.684 x10' 3.750 x10' 3.809 x10' 3.860



lo4 lo4 lo4 lo4 IO4



7.0 x106 8.0~10" 9.0x106 1.0x10'



x103 x103 x103 x103 x 103



8.965x10' 10.16 x106 11.34 x106 12.52 x106



If an infinite aquifer had been indicated, it may be desirable in some cases to predict the future performance assuming first an infinite aquifer and then a finite-closed aquifer having a calculated rg based on the best estimate of AtD and setting N;, equal to the last interval number in Eq. 20 or 30. Note that, in general. the plot of m,. will not be a smooth plot because of errors in basic data. The first few values are particularly sensitive to errors and generally may be ignored. If it is possible to obtain a relatively constant value of v?,., check the production and pressure data for errors. If the production and pressure data are correct, try Method



10" IO" 10" 10" 10"



1.17xs100'" 1.55x 1o'O 1.92x10'" 229x1o'o 3.02~10"



2. If it appears that the production and/or pressure data may be in error, refer to the following discussion of Errors in Basic Data. Example Problem 2. Method 2. This method is based on the following principles: (I) the slope of Z (m, times J>I)) as a function of time is always positive and never increases; (2) a constant slope of Z vs. time indicates a finite aquifer (see Eqs. 25 and 26) and therefore the extrapolated slope is constant; and (3) a constant slope of Z vs. log time indicates an infinite radial aquifer (Eq. 22). Extrapolation of this constant slope continues to simulate an infinite aquifer.



WATER



DRIVE



OIL RESERVOIRS



38-7



e l-l “15



0.18



e, e 0.14



ew



t %+I-,



e



*p,



5



0.1 6



4



-3



AP



e



i



= 467.5



1 Apo I



AP *2



0.10



= 1050.6



D2



ApD



e



E 3.12



=6 108.7



D4



II



*P



%I



=



148.5



=



53.5



0.08 u 0.06



3



5 7 9 II 13 TIME INTERVAL YUMBER



Fig. 38.5-Estimation of m,, N,, and roP fordata inTable 38.2 (Method 1).



As in the first procedure, time is divided into equal intervals. The first approximation to 2 can be obtained as in Method 1 or by arbitrarily using the square root of the interval number (Col. 5, Table 38.2, and Trial 1, Fig. 38.7). A fitting factor m is calculated as a function of time for Trial 1 in exactly the same manner used to calculate M r in Method 1.



APf,, mn=



(31)



n



c



e,,,,+,m,,AZ,



“.“““““.‘.



j=l



However, instead of m being plotted, m is used to calculate the next approximation of Z by use of Eq. 32. New Z, =m,(old



Z,,).



.



n=5



I5



m



581 =--0074



r5 7828.8



Fig. 38.6-Sample



. pressure-drop calculation



Fig. 38.7 shows that three trials were needed to obtain a constant value of 1 for m. Col. 7, Table 38.2, shows that the final Z’s will duplicate the past pressure performance and therefore may be used to predict the future performance. Because Z becomes a straight line as a function of n, a finite-closed aquifer is indicated (Principle 2). Therefore, Z can be extrapolated as a straight line to calculate the future performance. Errors in Basic Data. Good results were obtained for both methods, since accurate water influx and pressure data were used. In many cases a solution for m, and Ape in Method 1 or Z in Method 2 is impossible because of errors in basic data. In these cases the errors may be eliminated by smoothing the basic data or may be adjusted somewhat by using Eqs. 33 and 34.5



. .(32) 6Apf,, = -0. l-



The new values of Z are plotted as a function of n (Trial 2, Fig. 38.7), and a smooth curve is drawn through the points, making certain the slope is positive and never increases (Principle 1). This procedure is repeated with values of 2 from this smoothed curve until the fitting factors are relatively constant and equal to 1 (Trial 3, Fig. 38.7). The final 2 curve then is extrapolated to calculate the future performance as follows. 1. If the final slope of Z as a function of time is constant, extrapolate Z at a constant slope (Principle 2). 2. If the final slope is not constant as a function of time but is constant as a function of log time, first assume that the aquifer is an infinite radial system and will continue to behave as such (Principle 3) and extrapolate Z as a straight line as a function of log time; then assume that the aquifer is immediately bounded and extrapolate Z as a straight line on a linear plot of time using the last known slope (Principle 2). 3. If the final slope is not constant for either time or log time, extrapolate Z as a straight line using half the last known slope.



.087 I= 7828.8



m, -m



Apf,,



. (33)



m,



“0



2



4



6



8



IO



12



14 ”



n Fig. 38.7-Estimation of Z for data in Table 38.2 (Method 2).



PETROLEUM



38-0



TABLE



38.4-WATER



ENGINEERING



DRIVE



BEHAVIOR



Type Aquifer



0.06



EQUATIONS



Basis



Infinite radial Infinite hear Finiteoutcropping Finiteclosed



0.1



HANDBOOK



lo ; Li L t



t



Eq. Eq. Eq. Eq. 25



22 21 23 or 26



0.04 EL 0.02 0.0 I 0.006



TIME



( QUARTERS



1



Fig. 38.8-Estimation of mF and F function for approximate water drive analysisof data in Table 38.2.



and



--!---&e n,i,i+,-, , AZ,, AZ



..



.



.(34)



I j=2



where @f” = correction to Apf,, , 6e% = correction to eM? n , and ti = average value of m. In applying Eqs. 33 and 34 to Method 1, replace m by m, and AZ by ApD. Note that, since Eqs. 33 and 34 imply that the last values of Z (or APO) are reasonably correct, some judgment must be exercised when making these adjustments. Approximate Methods. If the water influx rate is constant for a sufficiently long period of time, the following equations can be used to estimate water drive behavior roughly. A P w,,, =mFervr,,F



..



.



(35)



and W e,,,m,l,=-



1



‘2 4M.r



s mF, I



...............



-



F



The equations for the infinite-radial and finiteoutcropping aquifers are commonly referred to in the literature as the “simplified Hurst” and “Schilthuis”6 water drive equations. The procedure consists of calculating mF for the past history using Eq. 35 or 36, plotting mF as a function of time, and extrapolating m,V to predict the future water drive performance. Since the method assumes a constant water influx rate, the use of these equations should be limited to short-term rough approximations of future water drive behavior. Large errors may be obtained if the method is used to predict the behavior for large changes in reservoir withdrawal rates. Fig. 38.8 shows a comparison of mF as a function of time for various values of F and the data in Table 38.2. These curves seem indicative of either an infinite linear or radial aquifer (the curves for these assumptions more nearly approach a constant value), whereas the more rigorous analyses indicated a finite aquifer. The selection of the best curve to use in predicting the future performance is difficult because of the fluctuations in the curves caused by variations in water influx rates. Note that this difficulty would be compounded if there were errors in the production and pressure data. Fetkovitch’ presented a simplified approach that is based on the concept of a “stabilized” or pseudosteadystate aquifer productivity index and an aquifer material balance relating average aquifer pressure to cumulative water influx. This method is best suited for smaller aquifers, which may approach a pseudosteady condition quickly and in which the aquifer geometry and physical properties are known. In a manner similar to single-well performance, the rate of water influx is expressed by Eq. 37. ew,=Ja(Pa



where e wp= J, = p, = P W’=



-p,),



..



... . ..



.



. (37)



water influx rate, B/D, aquifer productivity index, B/D-psi, average aquifer pressure, psi, and pressure at the original WOC, psi.



Combining Eq. 37 with a material-balance equation for the aquifer, the increment of influx over a time interval t,, -t,- 1 is given by Eq. 38.



(36)



’



where F is an approximation to pD and a function of the type of aquifer and m,G is a proportionality factor. See Table 38.4 for function and aquifer type.



Aw



= wet[Pa(n-j)



e



-p wn [l -,(-J,*‘,)‘((,,V,,)] Pd



. . . . . . . . ..~......_...._.___



(38)



WATERDRIVEOIL RESERVOIRS



38-9



where WC,, = ~C..,P,,, total aquifer expansion capacity, bbl, IJ’,~,;= initial water volume in the aquifer, bbl, PO1 = initial aquifer pressure, psi, and c ,I’, = total aquifer compressibility, psi -1 .



~~~~,~,,=p~j[l-~],



7.08x



Jo = ~,,,(ln



.t...,



10 -’ kh rD-0,75)



.



(39)



RESERVES IN) .



.



(40)



Fig. 38.9-Estimation of reservoir volume and water drive (Brownscombe-Collins method).



for a closed radial system, and



Jo =



3(1.127x



IO-‘)kbh (41)



tiplied by the factor X calculated by Eq. 43 gives the best estimate of OOIP for the selected permeability. Eq. 44 gives the minimum variance for this permeability.



PJ



-*of, for a closed linear system.



x=“-



n



Original Oil in Place (OOIP) Occasionally. it may be necessary to estimate the OOIP and to make a water drive analysis simultaneously. In general. the methods available are very sensitive to errors in basic data so that it is necessary to have a large amount of accurate data. Also, since the expansion of the reservoir above the bubblepoint is relatively small, generally only the data obtained after the reservoir has passed through the bubblepoint will be significant in defining the OOIP. In the three methods to be discussed, the aquifer will be assumed to be infinite and radial. Brownscombe-Collins Method. This method’ assumes that the OOIP and the aquifer permeability are unknown and that the reservoir and aquifer properties other than permeability are known. The pressure performance and the variance are calculated using Eqs. 7 and 42 for a given assumed aquifer permeability and various estimates. The minimum variance from a plot of variance vs. OOIP (Fig. 38.9) will be the best estimate of OOIP for the selected permeability.



c2=i



-$



(AP.~, -a~,,.).



(42)



/ This procedure is repeated for various estimates of permeability until it is possible to obtain a minimum of the minimums. The permeability and the OOIP associated with this minimum should be the best estimates for the assumptions made. It is possible to calculate the best estimate of OOIP for each selected permeability by the following procedure. Using the best available estimate of OOIP. calculate the reservoir voidage and expansion rates as a function of time. Select an aquifer permeability and use these rates in place of the water influx rates in Eq. 6 to calculate pressure drops Ap, ,, and APE,, The estimated OOIP mul-



WPE,



c j=l



.



.



(43)



(APE,)~



and .d



.. .



i W~+P~,-XA~~,)~, n j=1



.



where A~,z = total pressure drop at original WOC (field data), psi, Ap, = total pressure drop at WOC (calculated using reservoir voidage rates), psi, and ApE = total pressure drop at WOC (calculated using reservoir expansion rates). psi. van Everdingen, Timmerman, and McMahon Method. This method9 assumes that the OOIP, aquifer conductivity k/m/p, and diffusivity kI(@pc) are unknown. Combination of the material-balance equation and Eq. 8 and solving for the OOIP yields Eq. 4.5. N=A +m/,F(t),



.



. .



.



.



(45)



where 1 A=



V’V- 1P,;



U’,JvB,



+N,,(R,,



-R,)&



+ w,,l.



. . . . . . . . . . . . . . . . . . . . . . .._... II



1 F(t) = CFVmllBoi



F”=Ph-P -+I, PY



[



C j=I



*PC,,+ 1-j) Wa/,



1 ,



(46)



(47)



. . . . . . . . . . . . . . . . . . . . . . . . ..(48)



PETROLEUM



38-l 0



TABLE To =I.5 t,



36.5-DIMENSIONLESS



WATER



FOR



rD =2.5



70 =2.0 to



INFLUX



W eD



t,



FINITE OUTCROPPING fD =3.0



W eD



ENGINEERING



RADIAL



rD =3.5



HANDBOOK



AQUIFERS rD =4.0



--~~



rD =4.5



w,D



t,



weD



5.0x 10 -? 6.0x10-* 7.0x10-2 8.0x10-' 9.0x10m2



0.276 0.304 0.330 0.354 0.375



5.0~10~' 7.5x10-" 1.0x10-' 1.25x10-' 1.50x10-'



0.278 0.345 0.404 0.458 0507



1.0x10-' 1.5x10-' 2.0x10-' 2.5x10-' 3.0x10-'



0.408 0.509 0.599 0.681 0.758



3.0x10m 4.0x105.0x10 6.0x10 7.0x10 -



0.755 0.895 1.023 1.143 1.256



1.00 1.20 1.40 1.60 1.60



1.571 1.761 1.940 2.111 2.273



2.00 2.20 2.40 2.60 2.80



2.442 2.598 2.748 2.893 3.034



2.5 3.0 3.5 4.0 4.5



2.835 3.196 3.537 3.859 4.165



1.0x10-' 11x10~' 1.2x10-' 1.3x10-' 1.4x10-'



0.395 0.414 0.431 0.446 0.461



1.75x10-' 2.00x10-' 2.25x10-l 2.50~10 -' 2.75x10-l



0.553 0597 0.638 0.678 0.715



3.5x10-' 4.0x 0-l 4.5x 10-l 5.0x 0-l 5.5x 0-l



0.829 0.897 0.962 1.024 1.083



8.0~10~ 9.0x10 -' 1.00 1.25 1.50



1.363 1.465 1.563 1.791 1.997



2.00 2.20 2.40 2.60 2.80



2.427 2.574 2.715 2 649 2.976



3.00 3.25 3.50 3.75 4.00



3.170 3.334 3.493 3.645 3.792



5.0 5.5 6.0 6.5 7.0



4.454 4.727 4.986 5.231 5.464



1.5x10m' 1.6x10-' 1.7x10m1 1.8~10~' 1.9x10-'



0.474 3.00x 10 -' 0.486 3.25x10-l 0.497 3.50x10-' 0.507 3.75x10-1 0.517 4.00x10 -'



0.751 0.785 0.817 0.848 0.677



6.0x 0-l 6.5x 0-l 7.0x 0-l 7.5x 0-l 8.0x10 -'



1.140 1.195 1.248 1.229 1.348



1.75 2.00 2.25 2.50 2.75



2.184 2.353 2.507 2.646 2.772



3.00 3.25 3.50 3.75 4.00



3.098 3.242 3.379 3.507 3.628



4.25 4.50 4.75 5.00 5.50



3.932 4.068 4.198 4.323 4.560



7.5 8.0 8.5 9.0 9.5



5.684 5.892 6.089 6.276 6.453



2.0x 10 -' 2.1x10-' 2.2x10-l 2.3~10~' 2.4x10-l



0.525 0.533 0.541 0.548 0.554



4.25 x 10 -' 4.50 x IO -' 4.75 x IO -' 5.00 x 10 -' 5.50x10-'



0.905 0.932 0.958 0.982 1.028



8.5x10-' 9.0x10 -' 9.5x10m' 1.0 1.1



1.395 1.440 1.484 1.526 1.605



3.00 3.25 3.50 3.75 4.00



2.886 2.990 3.084 3.170 3.247



4.25 4.50 4.75 5.00 5.50



3.742 3.850 3.951 4.047 4.222



6.00 6.50 7.00 7.50 8.00



4.779 4.982 5.169 5.343 5.504



10 11 12 13 14



6.621 6.930 7.200 7.457 7.680



2.5~10.' 2.6x10 -' 2.8x 10 -' 3.0x 10 -' 3.2x 10 -'



0.559 6.00x10 -' 0.565 6.50x IO-' 0.574 7.00x10m' 0.582 7.50x10-' 0.588 8.00x10 -'



1.070 1.108 1.143 1.174 1.203



1.2 1.3 1.4 1.5 1.6



1.679 1.747 1.811 1.870 1.924



4.25 4.50 4.75 5.00 5.50



3.317 3.381 3.439 3.491 3.581



6.00 6.50 7.00 7.50 8.00



4.378 4.516 4.639 4.749 4.846



8.50 9.00 9.50 10 11



5.653 5.790 5.917 6.035 6.246



15 16 18 20 22



7.880 8.060 8.365 8.611 8.809



3.4x10-' 3.6~10~' 3.8x10-' 4.0x10m' 4.5x10-' 5.0x10m' 6.0x10-' 7.0x10 -' 8.0x10-'



0.594 9.00x 10-l 0.599 1.00 0.603 1.1 0.606 1.2 0.613 1.3 0.617 1.4 0.621 1.6 0.623 1.7 0.624 1.8 2.0 2.5 3.0 4.0 5.0



1.253 1.295 1.330 1.358 1.382 1.402 1.432 1.444 1.453 1.468 1.487 1.495 1499 1.500



1.7 1.8 2.0 2.2 2.4 2.6 2.8 3.0 3.4 3.8 4.2 4.6 5.0 6.0 7.0 8.0 9.0 10.0



1.975 2.022 2.106 2.178 2.241 2.294 2.340 2.380 2.444 2.491 2.525 2.551 2.570 2.599 2.613 2.619 2.622 2.624



6.00 6.50 7.00 7.50 8.00 9.00 10.00 11.00 12.00 14.00 16.00 18.00 20.00 22.00 24.00



3.656 3.717 3.767 3.809 3.843 3.894 3.928 3.951 3.967 3.985 3.993 3.997 3.999 3.999 4.000



8.50 9.00 9.50 10.00 11 12 13 14 15 16 17 18 20 25 30 35 40



4.932 5.009 5.078 5.138 5.241 5.321 5.385 5.435 5.476 5.506 5531 5.551 5579 5.611 5621 5.624 5.625



12 13 14 15 16 17 18 20 22 24 26 30 34 38 42 46 50



6.425 6.580 6.712 6.825 6.922 7.004 7.076 7.189 7.272 7.332 7.377 7.434 7.464 7.481 7.490 7.494 7.497



24 26 28 30 34 38 42 46 50 60 70 80 90 100



8.968 9.097 9.200 9.283 9.404 9.481 9.532 9.565 9.586 9.612 9.621 9.623 9.624 9.625



weD



and y=



ph-p P(FV-,).



~.~..............,._.._,,,,



FV = ratio of volume of oil and its dissolved



N = N,, = W,] = R,, = B,, = B,q = p/1 =



original gas at a given pressure to its volume at initial pressure, OOIP. STB, cumulative oil produced, STB, cumulative water produced. bbl, cumulative produced GOR, scf/STB. oil FVF, bbl/STB, gas FVF. bbhscf, and bubblepoint pressure. psia.



tD



weD



tD



weD



tD



Generally, Y is calculated with laboratory-determined values of FV - 1. Because Y vs. p is generally a straight line, smoothed values of Ycan be calculated with Eq. 50: Y=b+m,



.



.



.



(50)



where h= intercept and m =slope. The equations for obtaining the least-squares tit to Eqs. 46 and 47 for a given dimensionless time interval, At,. and n data points are



II



nN=



c j=l



A,-m,



i J=I



F(t),



.(51)



WATER



DRIVE



OIL RESERVOIRS



38-11



TABLE 38.5-DIMENSIONLESS



WATER INFLUX FOR FINITE OUTCROPPING rD = 6.0



r, = 5.0



rD = 7.0



tD



to



W ell



tD



W eD



3.0 3.5 4.0 4.5 5.0



3.195 3.542 3.875 4.193 4.499



6.0 6.5 7.0 7.5 8.0



5.148 5.440 5.724 6.002 6.273



9.00 9.50 IO 11 12



5.5 6.0 6.5 7.0 7.5



4.792 5.074 5.345 5.605 5.854



8.5 9.0 9.5 10.0 10.5



6.537 6.795 7.047 7.293 7.533



8.0 a.5 9.0 9.5 10



6.094 6.325 6.547 6.760 6.965



11 12 13 14 15



11 12 13 14 15



7.350 7.706 8.035 8.339 8.620



16 18 20 22 24



___~



rD =8.0



weD



tD



7.389 7.902 6.397



11



7.920



12 13



a.431 8.930



13 14 15 16 17



a.876 9.341 9.791 10.23 10.65



14 15 16 17 18



9.418 9.895 10.361 10.82 11.26



26 28 30 32 34



7.767 8.220 8.651 9.063 9.456



18 19 20 22 24



11.06 11.46 11.85 12.58 13.27



19 20 22 24 26



11.70 12.13



16 17 18 19 20



9.829 10.19 10.53 10.85 11.16



26 28 30 35 40



13.92 14.53 15.11 16.39 1749



8.879 9.338 9.731 10.07 10.35



22 24 25 31 35



il.74 12.16 12.50 13.74 14.40



45 50 60 70 80



26 28 30 34 38



10.59 10.80 10.89 11.26 il.46



39 51 60 70 80



14.93 16.05 16.56 16.91 17.14



42 46 50 60 70



11.61 11.71 11.79 11.91 11.96



90 100 110 120 130



17.27 17.36 17.41 17.45 17.46



80



il.98 11.99 12.00 12.0



140 150 160 180 200



17.48 17.49 17.49 17.50 17.50



220



17.50



j=l



weD



rD



13.98



9.965 12.32 13.22 14.09 14.95



14.79 15.59 16.35 17.10 17.82



28 30 32 34 36



15.78 16.59 17.38 18.16 18.91



13.74 14.50



36 38 40 42 44



18.52 19.19 19.85 20.48 21.09



38 40 42 44 46



19.65 20.37 21.07 21.76 22.42



28 30 34 38 40



15.23 15.92 17.22 18.41 18.97



46 48 50 52 54



21.69 22.26 22.82 23.36 23.89



48 50 52 54 56



23.07 23.71 24.33 24.94 25.53



18.43 19.24 20.51 21 45 22.13



45 50 55 60 70



20.26 21.42 22.46 23.40 24.98



56 58 60 65 70



24.39 24.88 25.36 26.48 27.52



58 60 65 70 75



26.11 26.67 28.02 29.29 30.49



90 100 120 140 160



22.63 23.00 2347 23.71 23.85



80 90 100 120 140



26.26 27.28 28.11 29.31 30.08



75 80 a5 90 95



28.48 29.36 30.18 30.93 31.63



80 85 90 95 100



31.61 32.67 33.66 34.60 35.48



180 200 500



23.92 23.96 24.00



160 180 200 240 280



30.58 30.91 31.12 31.34 31.43



100 120 140 160 180



32.27 34.39 35.92 37.04 37.85



120 140 160 la0 200



38.51 40.89 42.75 44.21 45.36



320 360 400 500



31.47 31.49 31.50 31.50



200 240 280 320 360



38.44 39.17 39.56 39.77 39.88



240 280 320 360 400



46.95 47.94 48.54 48.91 49.14



400 440 480



39.94 39.97 39.98



440 480



49.28 49.36



J=f



The variance of this fit from field data can be calculated by Eq. 53.



12.95



7.417 9.945 12.26 13.13



I u Id



I



02=1 i {A,,-N+m,[F(r)],}? n /=I



weD



15 20 22 24 26



I



6.861 7.398



rD _-_



10 15 20 22 24



and



J=i



weD



9 10



100 120



r, =lO.O



rD =9.0



6.861 7.127



90



RADIAL AQUIFERS (continued)



(53)



The minimum in a plot of variance vs. various assumed values of At, will be the best estimate of At, and can be used in Eqs. 51 and 52 to solve for the best estimate of N and m,, (see Fig. 38. IO).



BEST ESTIMATE OF At,



Ato Fig. 38.10-Estimation of reservoirvolumeand waterdrive(van Everdingen-Timmerman-McMahon



method).



PETROLEUM



36-12



TABLE 38.6-DIMENSIONLESS



ID=1.5



tLl



HANDBOOK



PRESSURES FOR FINITE CLOSED RADIAL AQUIFERS r,=25



rD =2.0



PO



to



ENGINEERING



PO



tD



rD = 3.0



rD = 3.5



rn =4.5



r. =4.0



PO tD PO __- tD -~-



tD



PO



--



PO



t,



PD



6.0x10-' 8.0x10-' 1.0x10-' 1.2x10-' 1.4x10-l



0.251 0.288 0.322 0.355 0.387



2.2x10-' 2.4x10-l 2.6x10-l 2.8x10-l 3.0x10-'



0.443 0.459 0.476 0.492 0.507



4.0x 10-l 4.2x10-l 4.4x 10-l 4.6x 10-l 4.8% lo-'



0.565 0.576 0.587 0.598 0.608



5.2x10 5.4x 10 5.6x10 6.0x10 6.5x10



0.627 0.636 0.645 0.662 0.683



1.0 1.1 1.2 1.3 1.4



0.802 0.830 0.857 0.882 0.906



1.5 1.6 1.7 1.8 1.9



0.927 0.948 0.968 0.988 1.007



2.0 2.1 2.2 2.3 2.4



1.023 1.040 1.056 1.072 1.087



1.6x10-' 1.8x10-' 2.0x10-' 2.2x10-l 2.4x10 -'



0.420 0.452 0.484 0.516 0.548



3.2x10-l 3.4x10-l 3.6x10-l 3.8x10-l 4.0x10 -'



0.522 0.536 0.551 0.565 0.579



5.0x lo-' 5.2x 10-l 5.4x 10 -' 56x10-' 5.9x10-'



0.618 0.682 0.638 0.647 0.657



7.0x10 75x10 8.0x 10 8.5x10 9.0x IO



0.703 0.721 0.740 0.758 0.776



1.5 1.6 1.7 1.8 1.9



0.929 0.951 0.973 0.994 1.014



2.0 2.2 2.4 2.6 2.8



1.025 1.059 1.092 1.123 1.154



2.5 2.6 2.7 2.8 2.9



1.102 1.116 1.130 1.144 1.158



2.6x10-l 2.8x10 -' 3.0x10 -' 3.5x10 -' 4.0x 10 -'



0.580 0.612 0.644 0.724 0.804



4.2x10-l 4.4x10-' 4.6x10-l 4.8x IO-' 5.0x10-'



0.593 0.607 0.621 0.634 0.648



6.0x 10-l 6.5x 10-l 7.0x 10-l 7.5x10-' 8.0x IO-'



0.666 9.5x10 0.688 1.0 0.710 1.2 0.731 1.4 0.752 1.6



0.791 2.0 0.806 2.25 0.865 2.50 0.920 2.75 0.973 3.0



1.034 1.083 1.130 1.176 1.221



3.0 3.5 4.0 4.5 5.0



1.184 1.255 1.324 1.392 1.460



3.0 3.2 3.1 3.6 3.8



1.171 1.197 1.222 1.246 1.269



4.5x10m' 5.0x 10 -' 5.5x10m' 6.0x10-'



0.884 0.964 1.044 1.124



6.0x IO -' 7.0x 10-l 8.0x10-' 9.0x10-'



8.5x10-' 9.0x10-' 9.5x 10-l 1.0 2.0



0.772 0.792 0.812 0.832 1.215



1.076 4.0 1.328 5.0 1.578 6.0 1.828



1.401 1.579 1.757



1.o



0.715 0.782 0.849 0.915 0.982



5.5 6.0 6.5 7.0 8.0



1.527 1.594 1.660 1.727 1.861



4.0 4.5 5.0 5.5 6.0



1.292 1.349 1.403 1.457 1.510



2.0 3.0 5.0



1.649 3.0 2.316 4.0 3.649 5.0



9.0 1.994 10.0 2.127



7.0 8.0 9.0 10.0 11.0



1.615 1.719 1.823 1.927 2.031



12.0 13.0 14.0 15.0



2.135 2.239 2.343 2.447



1.596 1.977 2.358



Havlena-Odeh Method. In this method, lo the materialbalance equation is written as tire equation of a straight line containing two unknown constants, N and m,, Combination of the material-balance equation and Eq. 8 yields Eq. 54. (See Fig. 38.10.) Nfm, vR,,



EN,,



c



j=i



*PW I -;) WA, .



. . .



(54)



EN,,



where



E,tr =B,-B, I/



VR,,= EN = B, =



W,, = Wi = G, = B,, =



+p Bf, (cf+Sw~w)(P; ’ I-S,,.



2.0 3.0 4.0 5.0



-P,,)



cumulative voidage at the end of interval II, RB. cumulative expansion per stock-tank barrel OOIP. RB, two-phase FVF, bbl/STB. cumulative water produced, STB, cumulative water injected. STB. cumulative gas injected. scf. water FVF, bbl/STB,



cf = formation compressibility, psi t , Cl, = formation water compressibility, psi t , s,,. = formation water saturation, fraction, and m = fitting factor. Eq. 54 is the equation of a straight line with a slope of mP and a y intercept of N. Estimates of TD and Are are made and the appropriate values of W,D are obtained from Table 38.3 or 38.5, according to system geometry. The summation terms in Eq. 54 then may be calculated and a graph plotted, as shown in Fig. 38.11. If a straight line results, the values of mp and N are obtained from the slope and intercept of the resulting graph. An increasing slope indicates that the summation terms are too small, while a decreasing slope indicates that the summation terms are too large. The procedure is repeated, using different estimates of TD and/or Ato until a straight-line plot is obtained. It should be noted that more than one combination of i-o and AND may yield a reasonable straight line-i.e., a straight-line result does not necessarily determine a unique solution for N and mp. Future Performance The future field performance must be obtained from a simultaneous solution of the material-balance and water drive equations. If the reservoir is above saturation pressure, a direct solution is possible; however, if the reservoir is below saturation pressure, a trial-and-error procedure is necessary.



WATER



38-13



DRIVE OIL RESERVOIRS



PRESSURES FOR FINITE CLOSED



TABLE 3&G-DIMENSIONLESS



rD =6.0



rD = 5.0



t,



PO



‘0



PO



rD =7.0



tD



PD



10.0 10.5 11.0 11.5 12.0



1.651 1.673 1.693 1.713 1.732



t, __~ 12.0 12.5 13.0 13.5 14.0



PO



1.556 1.582 1.607 1.631 1.653



8.5 9.0 9.5 10.0 11.0



1.586 1.613 1.638 1.663 1.711



1.675 1.697 1.717 1.737 1.757



12.5 13.0 13.5 14.0 14.5



1.750 1.768 1.786 1.803 1.819



14.5 15.0 15.5 16.0 17.0



1.817 1.832 1.847 1.862 1.890



12.0 13.0 14.0 15.0 16.0



1.757 13.0 1.776 1.801 13.5 1.795 1.845 14.0 1.813 1.888 14.5 1.831 1.931 15.0 1.849



15.0 15.5 16.0 17.0 18.0



1.835 18.0 1.917 1.851 19.0 1.943 1.867 20.0 1.968 1.897 22.0 2.017 1.926 24.0 2.063



4.0 4.5 5.0 5.5 6.0



1.275 1.322 1.364 1.404 1.441



6.0 6.5 7.0 7.5 8.0



3.5 3.6 3.7 3.8 3.9



1.227 1.238 1.249 1.259 1.270



6.5 7.0 7.5 8.0 8.5



1.477 1.511 1.544 1.576 1.607



4.0 4.2 4.4 4.6 4.8



1.281 9.0 1.638 9.5 1.668 1.301 1.321 10.0 1.698 1.340 11.0 1.757 1.380 12.0 1.815



1.598 1.641 1.725 1.808 1.892



12.0 13.0 14.0 15.0



1.975 2.059 2.142 2.225



18.0 19.0 20.0 25.0 30.0



rD = 10.0



PO



1.167 1.180 1.192 1.204 1.215



7.5 8.0 9.0 10.0 11.0



rD = 9.0



1.436 8.0 1.470 8.5 1.501 9.0 1.531 9.5 1.559 10.0



t,



3.0 3.1 3.2 3.3 3.4



5.0 1.378 13.0 5.5 1.424 14.0 6.0 1.469 15.0 6.5 1.513 16.0 7.0 1.556 17.0



rD =8.0



RADIAL AQUIFERS (continued)



PO



t,



10.5 11.0 11.5 12.0 12.5



1.732 1.750 1.768 1.784 1.801



1.873 170 1.974 17.0 1.919 19.0 1.931 18.0 2.016 19.0 1.986 20.0 1.988 19 0 2.058 21.0 2051 22.0 2.045 20.0 2.100 23.0 2.116 24.0 2.103 22.0 2.184 25.0 2.180 26.0



1.955 1.983 2.037 2.090 2.142



26.0 28.0 30.0 32.0 34.0



2.108 2.151 2.194 2.236 2.278



2.160 2.217 2.274 2.560 2.846



2.193 2.244 2.345 2.446 2.496



36.0 38.0 40.0 50.0 60.0



2.319 2.360 2.401 2.604 2.806



24.0 26.0 28.0 30.0



2.267 30.0 2.340 2.351 35.0 2.499 2.434 40.0 2.658 2.517 45.0 2.817



28.0 30.0 34.0 38.0 40.0



45.0 2.621 70.0 3.008 50.0 2.746



There are several methods of solution because there are several possible combinations of the various materialbalance and water drive equations. However, only one combination will be used to illustrate the general application to (1) a reservoir above the bubblepoint pressure, and (2) a reservoir below the bubblepoint pressure. In either case, it will be necessary to know (1) the saturations behind the front from laboratory core data or other sources, (2) the water production as a function of frontal advance, and (3) the pressure gradient in the flooded portion of the reservoir. ,’ Pressure Gradient Between New and Original Front Positions. Eq. 55 shows that the difference between the average reservoir pressure and the pressure at the original WOC is a function of water-influx rate, aquifer fluid and formation properties, and aquifer geometry.



00 0



1 AP%,



e



EN



where FG is the reservoir geometry factor. The linear frontal advance is given by FG=



L.f .,_...,.....,..........I



0.001127hb



(56)



and the radial frontal advance is given by 27r In@, irf)



.____.____............



FG= 0.00708ha



:



(-57)



Fig. 38.11-Estimation of OOIP and mp.



PETROLEUM



38-14



TABLE



38.7- DIMENSIONLESS



r,=1.5



rD =2.0



PRESSURES



FOR



r. =2.5



rD =3.0



PD



PD t,



FINITE OUTCROPPING



RADIAL



rD =3.5



HANDBOOK



AQUIFERS



rD = 4.0



rD =6.0



to



PD to



~___



PO



5.0x10-' 5.5x10-2 6.0x10-' 7.0x10 -2 8.0x10-'



0.230 2.0~10~' 0.240 2.2x10-l 0.249 2.4~10~' 0.266 2.6x10-l 0.282 2.8x10-'



0.424 0.441 0.457 0.472 0.485



3.0x10-' 3.5x10-' 4.0~10~' 4.5~10~' 5.0x10-'



0.502 0.535 0.564 0.591 0.616



5.0~10~' 5.5~10~' 6.0~10~' 7.0x10m' 8.0x10-'



0.617 0.640 0.662 0.702 0.738



5.0x 10 -' 6.0x10-' 7.0x10-' 8.0x10 -' 9.0x10-'



0.620 0.665 0.705 0.741 0.774



1.0 1.2 1.4 1.6 1.8



0.802 0.857 0.905 0.947 0.986



4.0 4.5 5.0 5.5 6.0



1.275 1.320 1.361 1.398 1.432



9.0x10-' 1.0x 10-l 1.2x10-' 1.4x10-' 1.6x10-'



0.292 0307 0.328 0.344 0.356



3.0~10~' 3.5~10~' 4.0x IO-' 4.5x10-l 5.0~10~'



0.498 0.527 0.552 0.573 0.591



5.5x10-l 6.0~10~' 7.0x 10-l 8.0x10-' 9.0x10-'



0.638 0.659 0.696 0.728 0.755



9.0x10m' 1.0 1.2 1.4 1.6



0.770 0.799 0.850 0.892 0.927



1.0 1.2 1.4 1.6 1.8



0.804 0.858 0.904 0.945 0.981



2.0 2.2 2.4 2.6 2.8



1.020 1.052 1.080 1.106 1.130



6.5 7.0 7.5 8.0 8.5



1.462 1.490 1.516 1.539 1.561



1.8x10-' 2.0x10m' 2.2x10-l 2.4x10-' 2.6~10~'



0367 0.375 0381 0.386 0390



5.5x10-l 6.0x10-' 6.5~10~' 7.0~10~' 7.5x10-'



0.606 0.619 0.630 0.639 0.647



1.0 1.2 1.4 1.6 1.8



0778 0.815 0.842 0.861 0.876



1.8 2.0 2.2 2.4 2.6



0.955 0.980 1.000 1.016 1.030



2.0 2.2 2.4 2.6 2.8



1.013 1.041 1.065 1.087 1.106



3.0 3.4 3.8 4.5 5.0



1.152 1.190 1.222 1.266 1.290



9.0 10.0 12.0 14.0 16.0



1580 1.615 1.667 1.704 1730



2.8~10~' 3.0x10-' 3.5x10m' 4.0x10-' 4.5x10-'



0.393 0.396 0.400 0.402 0.404



8.0x10-' 8.5x 10-l 9.0x IO-' 9.5x10-' 1.0



0.654 0.660 0.665 0.669 0.673



2.0 2.2 2.4 2.6 2.6



0.887 0.895 0.900 0.905 0.908



2.6 3.0 3.5 4.0 4.5



1.042 1.051 1.069 1.080 1.087



3.0 3.5 4.0 5.0 6.0



1.123 1.158 1.183 1.215 1.232



5.5 6.0 7.0 8.0 9.0



1.309 1.325 1.347 1.361 1.370



18.0 20.0 22.0 24.0 26.0



1.749 1.762 1.771 1.777 1.781



5.0x10 -' 6.0x IO-' 7.0x lo8.0x10-'



0.405 0.405 0.405 0.405



1.2 1.4 1.6 1.8 2.0



0.682 0.688 0.690 0.692 0.692



3.0 3.5 4.0 4.5 5.0



0.910 0.913 0.915 0.916 0.916



5.0 5.5 6.0 6.5 7.0



1.091 1.094 1.096 1.097 1.097



7.0 8.0 9.0 10.0 12.0



1.242 1.247 1.240 1.251 1.252



10.0 12.0 14.0 16.0 18.0



1.376 1.382 1.385 1.386 1.386



28.0 30.0 35.0 40.0 50.0



1.784 1.787 1.789 1.791 1.792



2.5 3.0



0.693 5.5 0.693 6.0



0.916 0.916



8.0 10.0



1.098 1.099



14.0 16.0



1.253 1.253



where Lf = linear penetration of water front into reservoir, ft, rf = radius to water front after penetration. and (Y = angle subtended by reservoir, radians.



‘D



ENGINEERING



PO



tD



+Apo,,,- ,/,



. .



where *P,,,, = total reservoir pressure drop from initial pressure at end of interval n, = total production rate, RB/D, q,,, V,, = total reservoir PV, bbl, and c 0, = total reservoir compressibility, psi - ’,



tD



e w,, = *P (,,,, ,) +(*tqr,r/V,+-,,,)-mr



Reservoir Above Bubblepoint Pressure. Above the bubblepoint pressure the total compressibility can be assumed to be constant; so the material-balance equation



vl7co,



PO



ft,



%*PD,



(qr,, -e,,,8 W



tD



can be combined with Eqs. 6 and 5.5 and solved for the water-influx rate:



Note that FG is a function of distance traveled by the front so that, if the pressure gradients between the reservoir and the original reservoir boundary are known for the past history, F, may be evaluated as a function of frontal advance. Future values of FG then can be obtained by extrapolating FG as a function of frontal advance on some convenient plot (linear, semilog, etc.)



APO,, =



PO



(58)



2 oil ,,,,, ,,*PD, .,= 2



+(*tlv,,~,,,)+(ll.,,.F,B/~I, . .. . .



. .. .



.. . . .



1 . . .



(59)



The calculated water-influx rate now can be used in Eq. 58 to calculate Ap(,,, and the whole procedure is repeated for the next time interval. If Eq. 27 is used instead of Eq. 6, mr= 1 and ApD is replaced by AZ in Eq. 59. Reservoir Below Bubblepoint Pressure. To simplify the calculation procedure, it was assumed that (1) uniform saturations exist ahead of and behind the front, (2) the saturations do not change as any portion of the reservoir is bypassed, and (3) the changes in pressure are selected small enough that the changes in oil FVF’s are very small. Fig. 38.12 shows the saturation changes as the front advances into the unflooded reservoir volume I/,- 1 during time interval n. The following equations will be used in this method. Water influx rate: II .I



-



(60) m,ApD,



-(p,,,.FGlk,,.)



”



WATER



DRIVE



TABLE



38.7-



DIMENSIONLESS



r,=lO



ID =8.0 to



38-15



OIL RESERVOIRS



FOR



PO



PO



tD



FINITE OUTCROPPING



r,=20



,,=I5



tD



PO



PRESSURES



RADIAL



PO



tD



(continued)



r,=30



r,=25



PO



tD



AQUIFERS



r,=40



to



PO



to



PO



7.0 7.5 8.0 8.5 9.0



1.499 1.527 1.554 1.580 1.604



10.0 12.0 14.0 16.0 16.0



1.651 1.730 1.798 1.856 1.907



20.0 22.0 24.0 26.0 28.0



1.960 2.003 2.043 2.080 2.114



300 35.0 40.0 45.0 50.0



2.148 50.0 2.219 55.0 2.282 60.0 2.338 65.0 2.388 70.0



2.389 70.0 2.434 80.0 3.476 90.0 2.514 10.0x10 2.550 12.0x 10



2.551 2.615 2.672 2.723 2.812



12.0x IO 14.0~10 16.0~10 18.0~10 20.0x10



2.813 2.888 2.953 3.011 3.063



9.5 10.0 12.0 14.0 16.0



1.627 1.648 1.724 1.786 1.837



20.0 25.0 30.0 35.0 40.0



1.952 2.043 2.1I1 2.160 2.197



30.0 35.0 40.0 45.0 50.0



2.146 2.218 2.279 2.332 2.379



60.0 70.0 80.0 90.0 10.0x10



2.475 2.547 2.609 2.658 2.707



75.0 80.0 85.0 90.0 95.0



2.583 2.614 2.643 2.671 2.697



2.886 2.950 2.965 2.979 2.992



22.0x 24.0x 26.0x 28.0x 30.0x



10 10 10 10 10



3.109 3.152 3.191 3.226 3.259



18.0 20.0 22.0 24.0 26.0



1.879 1.914 1.943 1.967 1.986



45.0 50.0 55.0 60.0 65.0



2.224 2.245 2.260 2.271 2.279



60.0 700 800 90.0 10.0x10



2.455 2.513 2.558 2.592 2.619



10.5x10 11.0x10 11.5x10 12.0~10 12.5x10



2.728 2.747 2.764 2.781 2.796



10.0x10 12.0x10 14.0~10 16.0x10 18.0~10



2.721 18.0x10 2.807 20.0x10 2.878 25.0~10 2.936 30.0x10 2.984 35.0x10



3.006 3.054 3.150 3.219 3.269



35.0x 10 40.0x 10 45.0x10 50.0x10 55.0x10



3.331 3.391 3.440 3.482 3.516



28.0 30.0 35.0 40.0 45.0



2.002 2.016 2.040 2.055 2.064



70.0 75.0 80.0 90.0 10.0~10



2.285 2.290 2.293 2.297 2.300



12.0~10 14.0x10 160x10 18.0~10 200x10



2.655 2.677 2.689 2.697 2.701



13.0x10 13.5x10 14.0~10 14.5x10 15.0~10



2.810 20.0x10 2.823 22.0x10 2.835 24.0~10 2.846 26.0x10 2.857 28.0~10



3.024 3.057 3.085 3.107 3.126



3.306 60.0x 10 3.332 65.0x 10 3.351 70.0x10 3.375 80.0x 10 3.387 90.0x10



3.545 3.568 3.588 3.619 3.640



50.0 60.0 70.0 80.0



2.070 2 076 2.078 2 079



11.0x10 12.0x 10 13.0x10 14.0x10 16.0x 10



2.301 2.302 2.302 2.302 2.303



22.0x10 24.0x10 26.0~10 28.0x10 30.0x10



2.704 2.706 2.707 2.707 2.708



16.0~10 180x10 200x10 240x10 28.0x10



2.876 2.906 2.929 2.958 2.975



30.0x10 35.0~10 40.0x10 45.0~10 50.0x10



3.142 80.0~10 3.171 90.0x10 3.189 10.0x10* 3.200 12.0~10' 3.207 14.0~10'



3.394 3.397 3.399 3.401 3.401



10.0x10' 12.0x10' 14.0x10~ 16.0x10* 18.0x10*



3.655 3.672 3.681 3.685 3.687



30 0x10 40.0~10 50.0x10



2.980 2.992 2.995



60.0x 10 70.0x10 80.0x 10 90.0 x10



3.214 3.217 3.218 3.219



20.0 x 10' 25.0x 10'



3.688 3.689



14.0x10 16.0x 10 16.5x 10 17.0x 10 17.5x10



40.0x10 45.0~10 50.0x10 60.0x10 70.0x10



Flooded and unflooded volumes: (e I\.,, - 4 it ,, W,,



Al’,, =



f~(I-sj,,.-sor-s~,)



,,-,



S



(61)



“.‘..“’



%



and



On-l n-l



Siw



V,,=V,,-,



.



-AL’,.



.



s



Oil saturation in V,:



On-l



S T' r



+



~RAV,



[So,,vm,, -S,,,,



I



-q,,,At,



B C’,,



.



.



(63)



Gas production:



L:



gn-I



S



S



On



Orn



s4'"



S



Siw



wn



% Sii



(b)



vrz[s,,,t ,,-s,,? 1 aGPft = B



Fig. 38.12-Saturation change with frontaladvance.



h’w,,



+ fRAv&,,,



I, -‘ 1 +q B KI,



II,,



At jj



!I .’8,



(64)



PETROLEUM



38-16



TABLE



38.7-DIMENSIONLESS



r,=50



PRESSURES



t,



PO



to



20.0x 22.0x 10 24.0 x10 26.0x10 28.0 x10



3.064 3.111 3.154 3.193 3.229



3.0 x 10' 4.0x10' 5.0x IO2 6.0 x IO* 7.0 x10*



3.257 3.401 3.512 3.602 3.676



5.0x10" 6.0 x 10’ 7.0x10' 8.0~10' 9.0x 10'



30.0 x10 35.0x10 40.0x10 45.0 xl0 50.0x 10



3.263 3.339 3.405 3.461 3.512



8.0 x lo* 9.0 x 102 10.0xlo2 12.0~10~ 14.0~10~



3.739 3.792 3.832 3.908 3.959



55.0 x10 60.0x 10 65.0x 10 700x10 75.0 x 10



3.556 16.0x IO2 3.595 18.0~10~ 3.630 20.0x10* 3.661 25.0x IO2 3.668 30.0 x IO2



80.0x10 85.0 x10 90.0x10 95.0x10 10.0x 102



3.713 3.735 3.754 3.771 3.787



12.0x10' 14.0x 102 16.0~10~ 18.0~10~ 20.0 x102



3.833 3.662 3.881 3.892 3.900



22.0x 10' 24.0~10' 26.0~10~ 28.0~10'



3.904 3.907 3.909 3.910



10



35.0x 102 40.0x 10" 450x10 50.0x102 55.0~10'



3.512



For these fR = S, = S, = S,,. = Sj,,. =



r,=lOO



r,=90 PO



t,



(continued)



PO



t,



PD



3.680 3.746 3.803



3.603 3.680 3.747 3.805 3.857



8.0 x10* 9.0x10' 1.0~10~ 1.2x 103 1.3 x IO3



3.747 3.806 3.858 3.949 3.988



1.0x 1.2x 1.4x 1.6x 1.8x



10" 103 lo3 IO* IO3



3.859 3.949 4.026 4.092 4.150



10.0~10' 12.0x 102 14.0x 10' 16.0x 10' 18.0~10~



3.854 3.937 4.003 4.054 4.095



12.0x 14.0x 15.0x 16.0x 18.0x



IO" 102 lo2 10' IO'



3.946 4.019 4.051 4.080 4.130



1.4~10~ 1.5x IO3 18~10~ 2.0 x103 2.5 x103



4.025 4.058 4.144 4.192 4.285



2.0x 2.5x 3.0x 3.5x 4.0x



IO3 IO3 IO3 103 lo3



4.200 4.303 4.379 4.434 4.478



3.996 4.023 4,043 4.071 4.084



20.0~10~ 25.0~10' 30.0~10~ 35.0~10' 40.0~10'



4.127 4.181 4.211 4.228 4.237



20.0x 10' 25.0x 10' 30.0~10~ 35.0x 10' 40.0~10~



4.171 4.248 4.297 4.328 4.347



3.0 x 103 3.5 x102 4.0 x lo3 4.5 x103 5.0 x103



4.349 4.394 4.426 4.446 4.464



4.5x 5.0x 5.5x 6.0x 6.5x



103 IO3 IO3 IO3 lo3



4.510 4.534 4.552 4.565 4.579



4.090 4.092 4.093 4.094 4.094



45.0~10' 50.0~10~ 55.0~10' 60.0~10' 65.0~10~



4.242 4.245 4.247 4.247 4.248



45.0x 10' 50.0x IO2 60.0~10~ 70.0~10" 80.0~10~



4.360 4.368 4.376 4.380 4.381



6.0 x lo3 7.0 x103 8.0~10~ 9.0 x lo3 10.0~10~



4.482 4.491 4.496 4.498 4.499



7.0x lo3 7.5x IO3 8.0x IO3 9.0x IO3 10.0~10~



4.583 4.588 4.593 4.598 4.601



70.0x102 75.0x102 80.0~10'



4.248 4.248 4.248



90.0x102 10.0~10~ 11.0~10~



4.382 4.382 4.382



11.0x103 12.0~10~ 14.0~10~



4.499 4.500 4.500



12.5~10~ 15.0x IO3



4.604 4.605



3.603



GOR (production): .



tD



AQUIFERS



HANDBOOK



6.0x IO* 7.0x 10" 8.0x10' 9.0x10' 10.0x10'



(65)



AGn



RADIAL



r,=80 PO



GOR (relative permeability):



R,=----qo,, At,



FINITE OUTCROPPING



r,=70



rD =60 PD



tD



FOR



ENGINEERING



.



..



(66)



equations, fraction of reservoir swept, oil saturation, fraction, gas saturation, fraction, water saturation, fraction, and interstitial water saturation, fraction.



One method for solutions using equal time intervals is as follows. 1. Estimate the pressure drop during the next time interval. 2. Calculate the water-influx rate with Eq. 60. 3. Calculate AL’, and V, with Eqs. 61 and 62. 4. Calculate the oil saturation in V, for the predicted oil production during Interval n with Eq. 63. 5. Calculate gas production with Eq. 64.



6. Calculate the GOR with Eq. 65. 7. Calculate the GOR with Eq. 66 for average values of pressure and saturation. 8. Compare the GOR’s obtained in Steps 6 and 7 and, if they agree, proceed to the next interval. If they do not agree, estimate a new pressure drop and repeat Steps 2 through 8. If the water drive equation for unequal time intervals is used, the need for re-evaluating the pressure functions for each trial in a given interval can be eliminated. This procedure calls for selecting a given pressure drop and estimating the length of the next time interval in Steps 1 and 8 and this program. The remaining steps are unchanged. Reservoir Simulation Models. The capability of mathematical simulation models to calculate pressure and fluid flow in nonhomogeneous and nonsymmetrical reservoir/ aquifer systems has been thoroughly described in the literature since the early 1960’s. Widespread availability of computers and models throughout the industry has helped to remove many of the idealizations and restrictions regarding geometry and/or homogeneity that are a practical requirement for analysis by traditional methods. These models have the capability to analyze performance for virtually any desired description of the physical system, including multipool aquifers. See Chap. 48 for more information.



38-17



WATER DRIVE OIL RESERVOIRS



PRESSURES FOR FINITE OUTCROPPING



TABLE 38.7-DIMENSIONLESS rD =200



rD =400



fD =300



fD = 500



RADIAL



AQUlFERS(contlnued)



r,=600



rD = 700



to



PO



t,



PO



t,



PO



to



PO



t,



PO



t,



PO



1.5~10~ 2.0x103 2.5x lo3 3.0x 103 3.5x 103



4.061 4.205 4.317 4.408 4.485



6.0 x lo3 8.0~10~ 10.0~10~ 12.0~10~ 14.0~10~



4.754 4.896 5.010 5.101 5.177



1.5x104 2.0~10~ 3.0~10~ 4.0x104 5.0~10~



5.212 5.356 5.556 5.689 5.781



2.0x104 2.5~10~ 3.0 x lo4 3.5x104 4.0 x lo4



5.356 5.468 5.559 5.636 5.702



4.0~10~ 4.5~10~ 5.0~10~ 6.0~10~ 7.0~10~



5.703 5.762 5.814 5.904 5.979



5.0~10~ 6.0~10~ 7.0~10~ 8.0~10~ 9.0~10~



5.814 5.905 5.982 6.048 6.105



16.0~10~ 18.0~10~ 20.0~10~ 24.0~10" 28.0~10"



5.242 5.299 5.348 5.429 5.491



6.0~10" 7.0~10~ 8.0~10~ 9.0x104 10.0~10~



5.845 5.889 5.920 5.942 5.957



4.5x IO4



5.759



6.0~10~ 7.0x104 8.0x10"



5.894 5.960 6.013



8.0x10" 9.0x104 10.0~10~ 12.0~10~ 14.0~10~



6.041 6.094 6.139 6.210 6.262



10.0~10~ 12.0~10~ 14.0~10~ 16.0~10~ 18.0~10~



6.156 6.239 6.305 6.357 6.398



9.0~10~ 10.0x103 12.0x103 14.0x103 16.0~10"



4.949 30.0~10~ 4.996 40.0~10" 5.072 50.0~10~ 5.129 60.0~10~ 5.171 70.0~10"



5.517 5.606 5.652 5.676 5.690



11.0~10~ 12.0~10~ 12.5~10~ 13.0~10~ 14.0~10~



5.967 5.975 5.977 5.980 5.983



9.0x104 10.0x10' 12.0~10" 14.0~10~ 16.0x10"



6.055 16.0~10~ 6.088 18.0~10~ 6.135 20.0~10~ 6.164 25.0~10~ 6.183 30.0~10~



6.299 6.326 6.345 6.374 6.387



20.0~10~ 25.0~10~ 30.0~10~ 35.0~10~ 40.0~10~



6.430 6.484 6.514 6.530 6.540



18.0~10~ 20.0x 25.0~10~ 30.0x103 35.0x 103



5.203



5.264 5.282 5.290



80.0~10~ 90.0x103 10.0~10~ 12.0~10~ 140~10~



5.696 16.0~10~ 5.700 18.0~10~ 5.702 200x10" 5.703 24.0~10~ 5.704 26.0~10~



5.988 5.990 5.991 5.991 5.991



18.0~10~ 20.0~10~ 25.0~10~ 30.0x104 35.0~10~



6.195 6.202 6.211 6.213 6.214



6.392 45.0~10~ 6.395 50.0~10~ 6.397 60.0~10~ 6.397 70.0~10~ 80.0~10~



6.545 6.548 6.550 6.551 6.551



1035.294



15.0x10"



5.704



40.0~10"



6.214



1034.552



4.0x 5.0x10" 6.0~10~ 7.0x103 8.0~10~



4.663 4.754 4.829 4.894



1035.227



40.0x



5.0x104 5.810



35.0~10~ 40.0~10~ 50.0~10~ 60.0~10~



Nomenclature A



= constant described by Eq. 46



b = intercept B,



B, B,



B,,.



cf (,, c,~



C



cwt d



= = = = = = = = =



gas FVF, bbl/STB oil FVF, bbl/STB two-phase FVF, bbl/STB water FVF, bbl/STB formation compressibility, psi -I total reservoir compressibility, psi-’ formation water compressibility, psi -I total aquifer compressibility, psi - ’ geometry term obtained from Table 38.1



e,,. = water influx rate, B/D e WB = water influx rate at Reservoir B, B/D e I,,111,,I = water-influx rate at interval n+ 1 -j, BID c 1v1 ,, = total water influx rate at interval n, B/D E,li = cumulative expansion per stock-tank barrel OOIP, bbl f~ = fraction of reservoir swept F = approximation to po and a function of type of aquifer FG = reservoir geometry factor F(r) = influence function FV = ratio of volume of oil and its dissolved original gas at a given pressure to its volume at initial pressure G, = cumulative gas injected, scf !I = aquifer thickness, ft j = summation of time period 1 fo,,



J, = k = L = Lf = m =



mF = mrJ = m,.



n N N,, y,,



=



aquifer productivity index, B/D-psi permeability, md aquifer length, ft linear penetration of water front into reservoir, ft fitting factor (see Page 38-7); ratio of initial reservoir free-gas volume to initial reservoir oil volume; slope proportionality factor influx constant, bbl/psi (see Eqs. 9 and IO) rate constant, psiibbl-D (see Eqs. 3 through 5) interval OOIP, STB time interval number cumulative oil produced, STB average aquifer pressure, psi initial aquifer pressure, psi



= = = = P ‘I = PN, = ph = bubblepoint pressure, psi pi = dimensionless pressure term PD(A,B) = dimensionless pressure term for Reservoir B with respect to Reservoir A P II’= pressure at original WOC, psi P II’,, = cumulative pressure drop at the end of interval n, psi Ape = known dimensionless field pressure drop at original WOC = dimensionless pressure drop to time APO, period i



38-18



PETROLEUM



TABLE 38.7rD = 800 to



PO



7.0x10" 8.0~10~ 9.0x lo4 100x10~ 12.0x104



DIMENSIONLESS rD = 900 tLJ



PO



PRESSURES FOR FINITE OUTCROPPING rD =I,000 tL7



t,



PO



tD



PO



6.507



6.785 6.849



2.5~10~ 3.0x 105 3.5x105 4.0x105 5.0x lo5



6.619 6.710 6.787 6.853 6.962



7.0x10" 8.0x10" 9ox105 10.0x10~ 120x105



7.013 7.038 7.056 7.067 7.080



5.0x105 6.0~10~ 7.0x IO5 8.0x lo5 9.0x105



6.950 7.026 7.082 7.123 7.154



6.0~10~ 7.0x 105 8.0~10~ 9.0 x 105 10.0x lo5



7.046 7.114 7.167 7.210 7.244



6.813 6.837 6.854 6.868 6.885



14.0x105 16.0 x lo5 18.0~10" 19.0x105 20.0 x 105



7.085 7.088 7.089 7.089 7.090



10.0x 105 15.0x IO5 20.0~10~ 25.0~10~ 30.0~10~



7.177 7.229 7.241 7.243 7.244



15.OxlO~ 20.0x IO5 25.0~10~ 30.0~10~ 35.0~10~



7.334 7.364 7.373 7.376 7.377



6.895 6.901 6.904 6.907 6.907 6.908



21.0x105 22.0x105 23.0 x10' 24.0 x lo5



7.090 7.090 7.090 7.090



31.0~10~ 32.0~10~ 33.0x 10'



7.244 40.0~10~ 7.244 42.0~10~ 7.24 44.0x IO5



7.378 7.378 7.378



6.049 6.108 6160 6.249



1.0x IO5 1.2~10~ 1.4~10~ 1.6~10" 1.8~10~



6.161 6.252 6.329 6.395 6.452



140x104 16.0~10~ 180x104 20.0x104 250x10"



6322 6.382 6432 6.474 6551



160~10~ 18.0x lo4 20.0 x lo4 25.0 x lo4 300x10"



6.392 6.447 6.494 6.587 6652



2.0~10~ 2.5~10~ 3.0x105 3.5~10~ 4.0~10"



6.503 6.605 6.681 6.738 6.781



30.0x104 35.0x104 40.0x lo4 45.0 x lo4 50.0x10"



6.599 6.630 6.650 6.663 6.671



40.0 x104 45.0x10" 50.0x10" 55.0x10" 60.0~10"



6.729 6.751 6.766 6.777 6.785



4.5x lo5 5.0~10~ 5.5~10~ 6.0~10~ 7.0~10~



550x104 60.0x lo4 70.0x10" 80.0 x lo4 100.0x10"



6.676 6.679 6.682 6.684 6.684



70.0 x104 80.0x IO4 90.0 x IO4 10.0 x IO5



5.794 6.798 6.800 6.801



8.0~10~ 9.0x lo5 10.0~10~ 12.0~10~ 14.0~10~ 16.0~10~



=



APL



=



Apy



=



*PO,+I-.;)



=



AP,,,A,B) =



APIA,,



=



A,-.],. = Yo,, =



r,,, = J/,



=



R .’3, =



St, = fD =



AIn = VP = VR =



rD =1,600



2.0x lo5 2.5~10~ 3.0x IO5 3.5x 105 4.0x105



6.049 6.106 6.161 6251 6.327



APO,



RADIAL AQUIFERS (continued) fD =1.400



rD =I,200



HANDBOOK



6.507 6.704 6.833 6.918 6.975



PO



8.0x 10' 9.0 x104 10.0x lo4 120~10~ 14.0x lo4



5.983



ENGINEERING



dimensionless pressure drop to time period j total pressure drop at WOC (calculated using reservoir expansion rates). psi total pressure drop at original WOC (field data), psi average pressure drop in interval, psi pressure drop at Reservoir A caused by Reservoir B, psi total pressure drop at Reservoir A at end of interval H. psi total pressure drop at WOC (calculated using reservoir voidage rates), psi total oil production rate at end of interval n. BID total production rate. B/D aquifer radius, ft dimensionless radius=r,,/r,,. radius to water front after penetration, ft field radius, ft cumulative produced GOR, scf/STB average solution GOR at end of interval n, scf/STB gas saturation, fraction interstitial water saturation, fraction oil saturation, fraction residual oil saturation at end of interval n. fraction formation water saturation, fraction dimensionless time dimensionless time interval total reservoir PV. bbl cumulative voidage, bbl



t,



PO



2.0 x105 3.0x 105 4.0 x lo5 5.0 x 105 6.0~10~



6.619 6.709



initial water volume in the aquifer, bbl aquifer width, ft W rD = dimensionless water-influx term we,, = cumulative water influx at end of interval n, bbl w,, = W,.,,,p,i, total aquifer expansion capacity, bbl w; = cumulative water injected, bbl w,, = cumulative water produced, bbl Y= constant described by Eqs. 49 and 50 z= resistance function z,, = new values of Z CY= angle subtended by reservoir, radians 6e ,,,,, = correction to e,,.,, @?f,, = correction to A pi,, Pl!, = water viscosity, cp 02 = variance porosity, fraction dJ= v



= M, w =



TABLE 38.8-DIMENSIONLESS PRESSURES FOR FINITE-CLOSED LINEAR AQUIFERS to



PO



o.005 0.01 0.02 0.03 0.04



0.07979 0.11296 0.15958 0.19544 0.22567



-!k0.18 0.20 0.22 0.24 0.26



PO 0.47900 0.50516 0.53021 0.55436 0.57776



0.05 0.06 0.07 0.08 0.09



0.25231 0.27639 0.29854 0.31915 0.33851



0.28 0.30 0.4 0.5 0.6



0.60055 0.62284 0.72942 0.83187 0.93279



0.10 0.12 0.14 0.16



0.35682 0.39088 0.42224 0.45147



0.7 0.8 0.9 1.0



1.03313 1.13326 1.23330 1.33332



WATER



DRIVE



OIL RESERVOIRS



38-19



TABLE 38.7-DIMENSIONLESS rD =2,000



r,=1,800 PO



tD



PRESSURES FOR FINITE OUTCROPPING RADIAL AQUIFERS (continued) rD =2,200



rD =2,400



rD =2,800



rD = 2,600



rD = 3,000



PO



t,



PD



tD



PO



7.057 7.0~10~ 7.134 8.0~10~ 7.200 9.0x105 7.259 10.0~10~ 7.310 12.0~10~



7.134 7.201 7.259 7.312 7.401



8.0x lo5 9.0x lo5 10.0x IO5 12.0x105 16.0~10~



7.201 7.260 7.312 7.403 7.542



1.0~10~ 1.2x106 1.4~10~ 1.6~10" 1.8~10~



7.312 7.403 7.480 7.545 7.602



7.167 12.0x lo5 7.199 16.0~10~ 7.229 20.0~10~ 7.256 24.0x IO5 7.307 28.0~10~



7.398 7.526 7.611 7.668 7.706



14.0~10~ 16.0~10~ 18.0~10~ 20.0x lo5 24.0~10~



7.475 7.536 7.588 7.631 7.699



20.0x lo5 24.0~10~ 28.0x105 30.0x 105 35.0x lo5



7.644 7.719 7.775 7.797 7.840



2.0 x 10" 2.4 x IO6 2.8 x 106 3.0 x106 3.5~10~



7.651 7.732 7.794 7.820 7.871



to



PO



t,



PO



tD



6.966 7.013 7.057 7.097 7.133



6.0~10~ 7.0~10" 8.0~10~ 9.0x105 10.0~10~



PO



tD



3.0~10~ 4.0~10~ 5.0x IO5 6.0~10~ 7.0x 105



6.710 6.854 6.965 7.054 7.120



4.0x105 5.0x105 6.0x105 7.0x 105 8.0~10~



6.854 6.966 7.056 7.132 7.196



5.0~10~ 5.5~10~ 6.0~10~ 6.5~10~ 7.0~10~



8.0~10~ 9.0x IO5 10.0x lo5 15.0x 105 20.0x 105



7.188 7.238 7.280 7.407 7.459



9.0 x lo5 lO.Ox10~ 12.0x105 14.0x105 16.0~10"



7.251 7.298 7.374 7.431 7.474



7.5x105 8.0~10" 8.5~10~ 9.0x105 10.0~10~



30.0 x lo5 40.0x105 50.0x lo5 51.0x105 52.0x i05



7.489 7.495 7.495 7.495 7.495



18.0~10~ 20.0 x lo5 25.0~10" 30.0x10" 35.0x105



7.506 12.0~10~ 7.530 16.0~10~ 7.566 20.0~10~ 7.584 25.0~10~ 7.593 30.0~10~



7.390 30.0~10" 7.507 35.0~10' 7.579 40.0~10" 7.631 50.0~10" 7.661 60.0~10"



7.720 7.745 7.760 7.775 7.780



28.0~10~ 30.0~10~ 35.0~10~ 40.0~10~ 50.0~10~



7.746 7.765 7.799 7.621 7.845



40.0x 50.0x 60.0x 70.0x 80.0x



lo5 105 lo5 IO5 i05



7.870 7.905 7.922 7.930 7.934



4.0 x IO6 4.5x106 5.0x106 6.0x lo6 7.0x106



7.908 7.935 7.955 7.979 7.992



53.0x 105 54.0x lo5 56.0x IO5



7.495 40.0x10" 7.495 50.0x10" 7.495 60.0~10" 64.0x IO5



7.597 35.0~10" 7.600 40.0~10" 7.601 50.0x IO5 7.601 60.0~10" 70.0 x105 80.0~10"



7.677 70.0~10~ 7.686 80.0~10" 7.693 90.0x10" 7.695 95.0x10" 7.696 7.696



7.782 7.783 7.783 7.783



60.0~10~ 70.0~10~ 80.0~10~ 90.0x105 1O.OXlO~



8.656 7.860 7.862 7.863 7.863



90.0x 10.0x 12.0x 13.0x



lo5 10" 10" IO6



7.936 7.937 7.937 7.937



S.OXlO~ 9.0x106 10.0~10~ 12.0 x 106 150x10~



7.999 8.002 8.004 8.006 8.006



Key Equations With SI Units The equations in this chapter may be used directly with practical SI units without conversion factors, except for certain equations containing numerical constants. These equations are repeated here with appropriate constants for SI units. P 112



r



II



.



=



8.527~10-~



kha’



.“““’



(3)



P ,I’ mr=



8,527x10-”



kh’



(4)



...“.“’



J,=



3(8.527 x 10 -5)kbh tLM.L



Lf FG= 8,527x,o-5



8.527x10-”



m,,=(l)&



khb’



(5)



“‘....’



,,‘bar,,?, ,



m,,=(1)r#x,,.,hb2,



(9)



.



.



(10)



8.527 x 10 -s kt tD =



(#)(‘b,,,p,,p



’



5.36x 1O-1 kh Jo = p,,,,(ln rD -0.75) -



.



(20)



(40)



. .(41)



..



...



and 2a In(r,/rf) FG= 5,36x1o-4 ha,



where !J ,J



t?lr=



hb,



,



k is in md, h is in m, b is in m, L is in m,



rD is dimensionless, r,,. is in m. p,,. is in mPa*s, c,,., is in kPa - ’, J, is in mj/d*kPa, ~1,. is in kPa/m3 *d, tnp is in m3/kPa, FG is in m-‘, and 01 is in radians.



.............. .....



38-20



References 1. Van Everdmgen. A.F. and Hut-Q. W.: “The Appltcatton of the Laplace Transformation to Flow Problems in Reservoirs.” Twns., AIME (1949) 186. 305-24. 2. Mottada, M.: “A Practical Method for Treating Oillield Interference in Water-Drive Reservoirs,” J. Per. Twh. (Dec. 1955) 217-26; Trurts.. AIME. 204. 3. Carter, R.D. and Tracy, F.W.: “An Improved Method for Calculatmg Water Influx,” J. Pet. Tech. (Dec. 1960) 58-60; Trms., AIME. 219. 4. Hicks. A.L. ( Weber, A.G., and Ledbetter, R.L.: “Computing Techmques for Water-Drive Reservoirs,” J. PH. Twh. (June 1959) 65-67; Trum.. AIME. 216. 5. Hutchwon. T.S. and Sikora. V.J.: “A Generaltzed Water-Drive Analysis.“J. Prt. T&r. (July 1959) 169-78; Trclns.. AIME, 216. 6. Schilthuis. R.J.: “Active Oil and Reservoir Energy.” 7rctn.s.. AIME 11036) 118. 33-52. 7. Fetkovich. M.J.: “A Simplified Approach to Water lntlux Calculations-Finite Aquifer Systems.” J. Pc~t. T&I. (July 1971) 814m28. 8. Brownscombc. E.R. and Collins. F.A.: “Estimation of Reserves and Water Drive from Pressure and Production Hratory,” Trtrnv., AIME (194Y) 186, 92-99. 9. Van Everdingen. A.F.. Timmerman. E.H., and McMahon, J.J.: “Application of the Material Balance Equation to a Partial WaterDrive Reservoir.” J. Prr. Tech. (Feb. 1953) 51-60; Trm\., AIME. 198. IO. Havlena. D. and Odrh. A.S.. “The Material Balance as an Equation of a Straight Line.” J. &f. Twh. (Aug. 1963) 896-900: Trwrc.. AIME. 228.



General References Chatas, A.T.: “A Practical Treatment of Nonstcady-State Flow Problems in Rew-voir System-I.” Per. Enx. (May 1953) B42Chatas, A.T.: “A Practical Treatment of Nonsteady-State Flow Prob PH. Enq. (June 1953) B3Xlems in Reservoir System-II,” Chatas. A.T.: “A Practical Treatment of Nonsteady-State Flow Problems in Reservoir Systems-III.” Per. Eng. (Aug. 1953) B46-



PETROLEUM



ENGINEERING



HANDBOOK



Closman. P.J.: “An Aquifer Model for Fissured Reservoirs,” Eng. J. (Oct. 1975) 385-98.



Sue. Pet.



Henaon. W.L., Beardon, P.L., and Rtce, J.D.: “A Numertcal Solutton to the Unsteady~State PartiallWater-Drive Reservoir Performance Problem,” .Soc. Per. Eng. J. (Sept. 1961) 184-94; Trans., AIME. 222. Howard, D.S. Jr. andRachford, H.H. Jr.: “Comparison of Pressure Distributions During Depletion of Tilted and Horizontal Aquifers,” J. Per. Tech. (April 1956) 92-98; Trans., AIME. 207. Hurst, W.: “Water Influx Into a Reservoir and Its Application to the Equation of Volumetric Balance.” Trans., AIME (1943) 151, 57-72. Hutchinson. T.S. and Kemp, C.E.: “An Extended Analysis of BottomWater-Drive Reservoir Performance,” J. Pet. Tech. (Nov. 1956) 256-61; Trum., AIME, 207. Lowe. R.M.: “Performance Predictions of the Marg Tex Oil Reservoir Using Unsteady-State Calculations,” J. Per. Tech. (May 1967) 595-600. Mortada, M.: “Oiltield Interference in Aquifers of Non-Uniform Propc&s.” J. Pej. Tech. (Dec. 1960) 55-57: Trms AIME, 219. Mueller, T.D. and Witherspoon, P.A : “Pressure Interference Effects Within Reservoirs and Aquifers.” J. Per. Tech. (April 1956)471-74; Trum., AIME, 234. Nabor. G.W. and Barham, R.H.: “Linear Aquifer Tdr. (May 1964) 561-63: Truns., AIME. 231. Odeh. A.S.: “Reservoir 1969) 13X3-88.



Simulation-What



Behawor.”



J. Per.



Is It’?” J. Prr. Twh. (Nov.



Stewart, F.M.. Callaway. F.H., and Gladfelter. R.E.: “Comparisons ot Methods for Analyzing a Water Drive Field. Torchlight Tensleep Reservoir. Wyommg.” J. Per. Tech. (Sept. 1954) 105-10; Trms.. AIME, 201. Wooddy, L.D. Jr. and Moore, W.D.: “Performance Calculations for Reservoirs with Natural or Artificial Water Drtves.” J. PH. Twh. (Aug. 1957) 245-5 I; Trans., AIME, 210



Chapter 39



Gas-Condensate Reservoirs Phillip L. Moses, Core Laboratories ~nc.* Charles W. Donohoe. Core Laboratories I~C



Introduction The importance of gas-condensate reservoirs has grown continuously since the late 1930’s. Development and operation of these reservoirs for maximum recovery require engineering and operating methods significantly different from crude-oil or dry-gas reservoirs. The single most striking factor about gas-condensate systems (fluids) is that they exist either wholly or preponderantly as vapor phase in the reservoir at the time of discovery (the critical temperature of the system is lower than the reservoir temperature). This key fact nearly always governs the development and operating programs for recovery of hydrocarbons from such reservoirs; the properties of the fluids determine the best program in each case. A thorough understanding of fluid properties together with a good understanding of the special economics involved is therefore required for optimum engineering of gascondensate reservoirs. Other important aspects include geologic conditions. rock properties, well deliverability, well costs and spacing, well-pattern geometry, and plant costs. Engineers have a wealth of literature on gas-condensate reservoirs available for reference. From this mass of material, Refs. 1 through 5 are especially recommended for fundamental background, and Refs. 6 through 8 are recommended for information on properties of pure compounds and their simple mixtures related to gas-condensate systems. For information regarding reservoir engineering processes and data, Refs. 5 and 9 through 16 are recommended. The best single bibliography on gas-condensate reservoirs is that of Katz and Rzasa “; however, later pertinent literature listings will be found in Refs. 6 through 14. The collection of references in Refs. 11 and 12 is particularly recommended for case histories of various gascondensate operations. Petroleum production papers pub-



lished by SPE (AIME) ‘s and API ” have been indexed separately through the years 1985 and 19.53, respectively. The practicing field engineers should have the following minimum library on gas-condensate systems available for their use: either Ref. 1, 2, or 3; Refs. 5, 9, 13, and 15; and selected volumes of Refs. 11 and 12.



Properties and Behavior of Gas-Condensate Fluids Sloan*’ described the general occurrence of petroleum in the earth: “. think of all the hydrocarbons, beginning with the lightest, methane, to the heaviest asphaltic substances as a series of compounds of the same family, consisting of carbon and hydrogen in a limitless number of proportions. A hydrocarbon reservoir then. is a porous section of the sedimentary crust of the earth containing a group of hydrocarbons, which is probably unique and whose overall properties such as reservoir phase, gas/oil ratio, gasoline content, viscosity. etc., is the direct result of this composition, together with the temperature and pressure that happen to exist in this particular spot in the porous sediment. “It is now easy to conceive of any possible combination of these hydrocarbons in a given reservoir, and it is also easy to visualize a reservoir fluid whose physical state may range from a completely dry gas in the reservoir, shading gradually through the wet gas, the condensate, the critical mixture, the highly compressible volatile liquid, the more stable light crude oil whose color is beginning to darken, the heavier crudes with decreasing solution gas, and ending with the semisolid asphalts and waxes with no measurable solution gas. “The condensate reservoir that is the topic under discussion is therefore first a hydrocarbon reservoir. Due to the composition and proportion of the individual hydrocarbons in the mixtures, the content is gas phase at the temperature and pressure of the reservoir.”



PETROLEUM ENGINEERING



39-2



TABLE 39.1~-HYDROCARBON



ANALYSES



HANDBOOK



AND PROPERTIES OF EXAMPLE CRUDE OILS AND GAS CONDENSATES Mole Fraction Condensate 1143”



0.4404 0 0432 0.0405 0.0284 0.0174 0.0290 0.4011 287 0.9071



0.5345 0.0636 0.0466 0.0379 0.0274 0.0341 0.2559 247 0.8811



0.00794 0.01375 0.76432 0.07923 0.04301 0.03060 0.01718 0.01405 0.02992 120 0.7397



0.00130 0.00075 0.89498 0.04555 0.01909 0.00958 0.00475 0.00365 0.02015 144 0.7884



0.00695 0.01480 0.89045 0.04691 0.01393 0.00795 0.00424 0.00379 0.01098 143 0.7593



loo+



42 34.5 1,078



73 18,000+



53.2 43,000 f



61.1 69,000 a



A’



-



Carbon dioxide Nitrogen Methane Ethane Propane Butanes Pentanes Hexanes Heptanes and heavier Molecular weight C, plus Specific gravity C, plus, 60’/6O”F Viscosity C, plus, Saybolt universal seconds at lOOoF Tank-oil gravity, OAPI at 60°/600F Producing gas/oil ratio, cu ft/bbl



27.4 525



approwmal~ng



Composition Ranges of Gas-Condensate



Condensate 944”



Crude Oil



Component



-see Ref 12, D 327 “See Ref. 2, Vbl I, Table 8 8, pp. 402-W ‘Viscosity 01 residual 011 left in apparatus,



8’



Condensate 843”



Crude Oil



Ihe hexanes-plus



Systems



Approximate composition indices for gas-condensate systems are the gas/liquid ratio of produced fluids (sometimes called the GOR) or its reciprocal, the liquid/gas ratio, and the gravity of the tank liquid separated out under various surface conditions. These two indices vary widely; they do not necessarily prove whether a hydrocarbon system is in the vapor phase in the reservoir. Eilerts et al. ’ (Vol. 1, Chaps. 1 and 8) show in a survey that the liquid/gas ratios of gas-condensate systems can vary from more than 500 (very “rich”) to less than 10 bbl/MMscf; tank condensate produced from the wells varied from less than 30 to more than 80”API, and more than 85% was within the range of 45 to 65”API. Eilerts et al.’ (Vol. 1) also quote a rule of thumb that a gascondensate system exists when the gas/liquid ratio exceeds 5,000 cu ftibbl (200 bbl/MMscf and less) and the liquid is lighter than 5O”API. This appears to be on the conservative side because there is evidence that systems exist as single-phase vapor in the reservoir when the surface gas/liquid ratio is less than 4,000 cu ft/bbl (more than 250 bbl/MMscf) and the API gravity of the liquid in the stock tanks is lower than 40”API. A more accurate representation of the composition of gas-condensate fluids is provided by fractional analyses of the well streams coming from the reservoirs. The contrast of the fluid composition with the total stream coming from crude-oil reservoirs is fairly large for the relative amounts of the lighter vs. heavier ends of the paraffinhydrocarbon series. For example. Eilerts et ul. ’ (Vol. 1, Table 8.8) report a methane content from about 75 to 90 mol% for several gas-condensate systems, whereas Dodson and Standing” report 44 and 53 mol%, respectively, for two crude-oil systems (see Table 39.1). The table, however, shows much lower heptanes-and-heavier content for the gas-condensate systems than for the crude oil. These are the two outstanding composition features of gas-condensate systems.



material



Pressure and Temperature Ranges of Gas-Condensate Reservoirs Gas-condensate reservoirs may occur at pressures below 2.000 psi and temperatures below l00”F20 and probably can occur at any higher fluid pressures and temperatures within reach of the drill. Most known retrograde gas-condensate reservoirs are in the range of 3,000 to 8,000 psi and 200 to 400°F. These pressure and temperature ranges, together with wide composition ranges, provide a great variety of conditions for the physical behavior of gas-condensate deposits. This emphasizes the need for very meticulous engineering studies of each gascondensate reservoir to arrive at the best mode of development and operation.



Phase and Equilibrium



Behavior



An understanding of the behavior of pure paraffin hydrocarbons and simple two-component or threecomponent systems (involving such compounds as methane, pentane, and decane) is of considerable benefit to the engineer working with gas-condensate reservoir problems. Excellent coverage is given this subject by Sage and Lacey ’ and a more condensed discussion by Burcik.’ Occasional review of such material will assist the engineer concerned with more complex hydrocarbon mixtures. Chap. 23 describes the phase and equilibrium behavior of complex (multicomponent) hydrocarbon mixtures (see Fig. 23.14 and the accompanying discussion). Note that the critical state (critical point) is that state or condition at which the composition and all other intensive properties of the gas phase and the liquid phase become identical-i.e., the phases are indistinguishable. In gascondensate reservoirs, the portion of the phase diagram to the left of and above the critical point will not be involved.



GAS-CONDENSATE



RESERVOIRS



39-3



i 1T O50



Fig. 39.1-Phase



I00



150 200 TEMPERATURE.‘F



250



300



diagram of Eilerts’ Fluid 843.



(discussed in The term “retrograde condensation” Chap. 23) is used more loosely than implied by its rigorous definition, ’ In field practice, the term may imply any process where the amount of condensing liquid phase passes through a maximum, whether the process is isothermal or not. While Fig. 23.14 provides a simplified picture of the phase diagram, reservoir engineers will find that very few quantitative phase diagrams on naturally occurring gascondensate mixtures have been published. Figs. 39.1 through 39.3 come from extensive work’ and represent quantitative measurements on the flow streams from wells in the Chapel Hill, Carthage, and Seeligson fields in Texas. The critical points are not shown because they are at temperatures below those of interest to field operations. This emphasizes that the compositions of gas-condensate systems vary widely and strongly affect the form of the phase diagrams encountered in actual gas-condensate reservoirs. These three phase diagrams represent a reasonable spread in the properties of gas-condensate systems. from a gas/liquid ratio of about 18,000 to 69,000 cu ftibbl (56 to 14.5 bbl/MMscf). This does not mean, however, that all other gas-condensate systems would fall inside the limits of the properties suggested by these three phase diagrams. The three cases in Figs. 39.1 through 39.3 imply that the dewpoint boundary approaches zero pressure at a relatively high temperature. Other condensate systems are believed to approximate the qualitative picture shown in Fig. 23.14 more closely. Note that all three systems exhibit both cricondentherm and cricondenbar points (maximum temperature and pressure, respectively, beyond which there is no liquid present in the vapor); the critical temperatures all fall to the left of each diagram at lower temperatures and pressures than the maxima for the dewpoint boundaries. Liquid-content lines on phase diagrams can be represented by a number of different units. Figs. 39.1 through 39.3 use gallons per thousand cubic feet of separator gas.



Fig. 39.2-Phase



diagram of Eilerts’ Fluid



1143.



The approximate behavior of condensate fluids while being produced from the reservoir into surface vessels can be represented advantageously on phase diagrams. In Fig. 39.2, for example, Line FT shows a flow path for fluids that starts at formation conditions (outside the dewpoint boundary, indicating that the formation fluids were all in vapor phase); proceeds to sandface pressure, Point S i , at the well; declines as the fluid rises from the bottom of the hole to the wellhead, Point WH; passes through the choke to separator conditions, Point S2 ; and reaches Point T, representing tank conditions. The phase diagram is thus helpful to the engineer in visualizing what happens to gas-condensate fluids as they flow from the formation to the wellbore and from there to surface equipment.



4,5OOf7777777



TEMPERATURE,



Fig. 39.3-Phase



*F



diagram of Eilerts’ Fluid 944



39-4



Methods have been proposed by Organick” and Eilerts et al. * for predicting the critical temperatures and pressures of hydrocarbon mixtures and for computing the phase diagrams (including dewpoint curves) of gascondensate fluids. The dependability of these methods for a wide range of gas-condensate compositions has not yet been established. For reservoir engineering work, direct laboratory measurements of phase diagrams or of pressure-depletion behavior are necessary because of the large recoveries at stake. Laboratory work may not be required for other problems.



PETROLEUM ENGINEERING



HANDBOOK



imate method that may be used when there are no intermediate separator stages and stock tanks for individual well measurements and when the atmospheric temperature and pressure do not vary appreciably from stock-tank conditions. Gas/liquid ratios usually are reported in cubic feet per barrel of liquid (or thousands of cubic feet per barrel) and liquid contents or liquid/gas ratios in barrels of liquid per million standard cubic feet of gas. The separator streams used in the ratio must be specified. Properties of Separated Phases



Gas/Liquid Ratios and Liquid Contents of Gas-Condensate Systems As discussed earlier, it is difficult to specify whether a hydrocarbon system is in the vapor phase in the reservoir from measurements of field gas/liquid ratio and tankoil gravity. Fluid production with tank-oil gravities as low as 30”API and gas/liquid ratios as low as 3,000 cu ft/bbl may be from true gas-condensate systems; this possibility should always be checked by laboratory measurements of phase behavior for these and intermediate values. “Liquid content” and “gas/liquid ratio” can be direct reciprocals, depending on the type of problem considered. The terms must be carefully defined in each case because gas-condensate systems in the field frequently undergo different types of separating procedures that involve several stages before the final liquid phase (“liquid” means hydrocarbon liquid unless otherwise specified) reaches the tanks at atmospheric pressure. To study the properties of gas-condensate fluids at reservoir conditions, it is convenient to define gas/liquid ratios and liquid contents on the basis of the gas and liquid outputs of the first-stage separator through which the fluids pass. These two output streams then represent the total composition of the gas-condensate fluid in the reservoir if sampling, producing, and measuring conditions have been properly set and maintained. Other gas/liquid ratios may be reported, however, including the total gas output of all stages of separation divided by the tank-liquid volumes corresponding to the gas output: note that the total gas output would include a measurement of tank vapors as well as separator gas to represent the full composition of the wellstream. The gas/liquid ratio at stock-tank conditions may be roughly approximated when field facilities are not available for measurements. The gas and liquid flow rates from the high-stage separator are observed and a liquid sample collected from the separator in a stainless-steel cylinder of known volume. If all the cylinder contents are bled into a calibrated graduate at atmospheric pressure and the volume of the resultant liquid phase is compared with the original liquid volume, an approximate value of the liquidphase shrinkage may be determined. From this, the highstage gas/liquid ratio may be converted to stock-tank conditions. This procedure ignores the volume of gas liberated between high-stage separator and stock-tank conditions. This volume can be approximated by using a calibrated glass separator with gas meter attached in place of the graduate. Ignoring this gas volume adds further errors to those resulting from not simulating the existing field stage separation conditions. The higher the first-stage separation pressure, the greater the error in total gas volume of the gas/liquid ratio. This is only an approx-



The properties of both liquid and gas phases separated from gas-condensate streams can vary considerably. One of the dominant properties of the gas is high methane content. Eilerts et al. 2 (Vol. I, Chap. 8) list the compositions of the gas and liquid phases of eight gas-condensate systems. Methane contents of the gas phases (simulated from field separators) varied from about 0.83 to 0.92 mole fraction; the hexanes and heavier (“hexanes plus”) varied from 0.004 to about 0.008 mole fraction. The liquid phases varied from about 0.1 to nearly 0.3 mole fraction methane; hexanes plus varied from about 0.4 to 0.7 mole fraction. In the absence of measured data, properties of the separated phases of gas-condensate systems (including volumetric and density behavior) can be approximated by methods described in Chaps. 20 through 23, especially Chaps. 20 and 22 (see also Refs. 9 and 14). Viscosities of Gas-Condensate



Systems



The viscosity of a gas-condensate system is of interest in various reservoir calculations, particularly with respect to cycling operations and the representation of such reservoirs in computer models. Whenever possible, viscosity of the vapor phase at reservoir conditions should be measured directly. Carr et al. 23 presented a method to estimate the viscosities of gas systems from a knowledge of compositions or specific gravities (see also Chap. 20 and Ref. 14). Viscosities of separate gas and liquid phases at the surface conditions usually encountered can be obtained by direct measurement or by the use of the correlations for gas previously mentioned and the correlation of Chew and Connally24 for liquid (see also Chap. 22). Viscosity information on separated materials is needed mainly for separator or plant residue gases to be injected during cycling and for some types of reservoir calculations.



Gas-Condensate Well Tests and Sampling Proper testing of gas-condensate wells is essential to ascertain the state of the hydrocarbon system at reservoir conditions and to plan the best production and recovery program for the reservoir. Without proper well tests and samples, it would be impossible to determine the phase conditions of the reservoir contents at reservoir temperature and pressure accurately and to estimate the amount of hydrocarbon materials in place accurately. Tests are made on gas-condensate wells for a number of specific purposes: to obtain representative samples for laboratory analysis to identify the composition and properties of the reservoir fluids; to make field determinations on gas and liquid properties; and to determine formation



,



GAS-CONDENSATE



RESERVOIRS



and well characteristics, including productivity. producibility, and injectivity. The first consideration for selccting wells for gas-condensate fluid samples is that they be far enough from the “black-oil ring” (if present) to minimize any chance that the liquid oil phase will enter the well during the test period. A second and highly important consideration is the selection of wells with as high productivities as possible so that minimum pressure drawdown will be suffered when the reservoir fluid samples are acquired. Well Conditioning Proper well conditioning is essential to obtain representative samples from the reservoir. The best production rates before and during the sampling procedure have to be considered individually for each reservoir and for each well. Usually the best procedure is to use the lowest rate that results in smooth well operation and the most dependable measurements of surface products. Minimum drawdown of bottomhole pressure during the conditioning period is desirable and the produced gas/liquid ratio should remain constant (within about 2%) for several days; the less-permeable reservoirs require longer periods. The farther the well deviates from constant produced gas/liquid ratio. the greater the likelihood that the samples will not be representative. Recombined separator samples from gas-condensate wells are considered more representative of the original reservoir fluid than subsurface samples. Accurate measurements of hydrocarbon gas and liquid production rates during the well-conditioning and wellsampling tests are necessary because the laboratory tests will later be based on fluid compositions recombined in the same ratios as the hydrocarbon streams measured in the field. The original reservoir fluid cannot be simulated in the laboratory unless accurate field measurements of all the separator streams are taken. (Gas/liquid ratios may be reported and used in several different forms, as discussed previously.) If the produced gas/condensate (gas/liquid) ratio from field measurements is in error by as little as 5 %, the dewpoint pressure determined in the laboratory may be in error by as much as 100 psi. Water production rates should be measured separately and produced water excluded as much as possible from hydrocarbon samples sent to the laboratory. Separator pressure and temperatures should remain as constant as possible during the well-conditioning period; this will help maintain constancy of the stream rates and thus of the observed hydrocarbon gas/liquid ratio. If the well is being prepared during a period when atmospheric temperatures vary considerably from night to day. reasonably consistent average temperatures and pressures on the several vessels during the conditioning period should be adequate. Field Sampling and Test Procedures After the conditioning period has proceeded long enough to show that producing conditions are steady. exacting measurement methods must be used to obtain representative samples. The mechanics of well sampling is partially covered in Chaps. 12 through 14, 16, and 17. The help of experienced laboratory personnel is advisable in



39-5



acquiring gas and condensate-liquid samples. Certain minimum items of information in addition to all stream rates are essential, including regular readings of the pressures and temperatures of all vessels sampled, and of tubing heads and casing heads where available, and a recorded history of the well conditions before and during sampling. along with the actual mechanics of the sampling steps. Other information acquired during the sampling period that would help to explain reservoir and well conditions should also be recorded because it is useful in interpreting the results of the tests. Care must be taken that the compositions of gas and liquid samples obtained are representative and are properly preserved for laboratory analyses. API RP 44?’ outlines appropriate sampling methods. For cases when the liquid-phase sample is obtained at a low temperature (from low-temperature separation equipment), triethylene-glycol/water mixtures are convenient for collecting the samples. Ten percent or more of the cylinder volume for liquid-phase samples should be gas to prevent excessive pressure that could result from temperature rise during subsequent shipment. This 10% “gas cap” can be effected by closing the cylinder sampleinlet valve when 90% of the glycoliwater mixture has been displaced and then carefully withdrawing nearly all the remaining mixture from the bottom of the cylinder without losing the oil phase. The volumes of fluids requested for laboratory testing should be acquired during the sampling period. plus a reasonable amount (25 % or more) of extra sample materials in separate containers for emergency use should some of the main samples be lost by leakage or other adversity between the field site and the laboratory. At the end of actual sampling mechanics in the field, the well should remain on stream for a reasonable period of time, and its producing rate, gas/liquid ratio, and various pressures and temperatures should be observed to confirm that they are consistent with the information developed before and during the sampling period. Any radical changes should be analyzed carefully to decide whether resampling may be necessary to ensure accuracy of the samples and well statistics obtained during the sampling period. Equipment is available for making some determinations of gas-condensate properties in the field. ’ Among these properties are the gas/liquid ratios of several vessels simulating various separation conditions (numbers of stages, pressures and temperatures of the stages, and other conditions) and the “gasoline content” of the overhead gas at each stage. If hydrogen sulfide and carbon dioxide are present in the production streams, special sampling procedures should be used and the samples should be taken in stainless-steel cylinders. These corrosive gases could react with the sample cylinders during shipment. Field determinations of the hydrocarbon compositions of streams from gas-condensate wells can be made with appropriate fractionation equipment in mobile laboratories. Eilerts rt al. ’ described such equipment and the test procedures for determining the effect of individual hydrocarbons on liquid/gas ratios at different separation pressures and temperatures. These tests can assist in determining optimum field separation conditions for given production objectives. They require special equipment and experienced personnel.



39-6



PETROLEUM



Measurements of gas-condensate well productivity, producibility, and injectivity are of considerable importance for planning overall field operations and size of plants for either gasoline recovery or condensate-liquids recovery and cycling, as bases for contracts for deliverability from a reservoir for pipeline purposes, and for various other needs. This topic is discussed more fully later; test procedures are described in Chap. 33 and in several published standards and regulations. 26-29



Sample Collection and Evaluation In taking samples for recombination to evaluate a gascondensate reservoir, the samples of gas and samples of liquid usually are taken from the first stage of separation. A representative portion of all the hydrocarbons produced from the well will be contained in these two samples. The first step in the laboratory study is to evaluate the samples taken. The first test is to measure the bubblepoint of the separator liquid. The bubblepoint should correspond to the separator pressure at separator temperature at the time the samples were taken. The hydrocarbon composition of the separator samples should then be determined by chromatography or lowtemperature fractional distillation or a combination of both. An example of the composition of typical separator products are shown in Table 39.2. These compositions may be evaluated by calculation of the equilibrium ratio for each component (see Chap. 23). The equilibrium ratio for a component is the mole percent of that component in the vapor phase divided by the mole percent of the same component in the liquid phase. As an example, the equilibrium ratio for methane in Table 39.2 is calculated by the equation K, =yl/x,



=83.01/10.76=7.71,



TABLE 39.2-HYDROCARBON



The experimental equilibrium ratio for methane is 7.71 for the temperature and pressure existing in the field separator at the time of sampling. The equilibrium ratios for each of the hydrocarbons methane through hexane are calculated in a similar manner. These data can then be compared with equilibrium ratios, such as those published in Ref. 16. If the equilibrium ratios compare favorably, then the samples are in equilibrium and the study should continue. If they do not compare well, then new samples should be obtained before proceeding. Recombination



(mol %)



Hydrogen sulfide C&bon dioxide Nitrogen Methane Ethane Propane iso-Butane n-Butane iso-Pentane n-Pentane Hexanes Heptanes plus



Total Properties of heptanes



of Separator Samples



The samples are now ready to be recombined in the same ratio that they were produced. Because we have samples of first-stage separator gas and first-stage separator liquid, we must have the produced gas/liquid ratio in the same form. If the producing gas/liquid ratio was measured in the field in this form, then we can proceed directly with the recombination. If the ratio was measured in the field in the form of primary-separator gas per barrel of second-stage separator liquid or per barrel of stocktank liquid, then a laboratory shrinkage test must be run to simulate field separation conditions. The shrinkage obtained can then be used to convert the field-measured ratio to the form necessary for the recombination. Once the separator products have been recombined, the composition can be measured and compared with the calculated composition. This will check the accuracy of the physical recombination.



PRODUCTS AND CALCULATED



Separator Gas



SeDarator Liauid



mol %



Well Stream mol % 0.00 0.01 0.11 68.93 8.63 5.34 1.15 2.33 0.93 0.85 1.73 9.99



2.295 1.461 0.374 0.730 0.338 0.306 0.702 6.006



100.00



12.212



0.00 0.01 0.13 83.01 9.23 4.50 0.74 1.20 0.31 0.25 0.21 0.41



2.454 1.231 0.241 0.376 0.113 0.090 0.085 0.185



100.00



100.00



4.775



-



separator



gal/l,000



cf gas



plus



0.827 158



103



Calculated separator gas gravity (air = 1.000) Calculated gross heating value for separator gas per cubic foot of dry gas at 14.65 psia and 60°F, Btu Primary-separator-gas/separator-liquid ratio at 60°F, scf/bbl* Primary-separator-liquid/stock-tank-liquid ratio at 60°F, bbl Primary-separator-gas/well-stream ratio, MscWMMscf Stock-tank-liquid/well-stream ratio, bbl/MMscf *Primary



WELL STREAM



gal/l ,000 cf gas



0.00 0.00 0.01 10.76 6.17 8.81 2.85 7.02 3.47 3.31 8.03 49.57



API gravity at 6O“F 39.0 Density, g/cm3 at 60aR).8293 Molecular weight 160



gas and primary



separator



liquid collected



HANDBOOK



where K, = the equilibrium ratio for methane, y1 = methane in the vapor phase, mol%, and Xl = methane in the liquid phase, mol%.



ANALYSES OF SEPARATOR



Component



ENGINEERING



at 440 psig and 87’F.



0.699 1,230 3,944 1.191 805.19 171.4



GAS-CONDENSATE



Dewpoint



RESERVOIRS



and Pressure/Volume



39-7



Relations



The laboratory personnel will next measure the pressure/volume relations of the reservoir fluid at reservoir temperature with a visual cell. This is a constant-composition expansion and furnishes the dewpoint of the reservoir fluid at reservoir temperature and the total volume of the reservoir fluid as a function of pressure. The volume of liquid at pressures below the dewpoint as a percent of the total volume may also be measured. Phase diagrams can be developed dy measuring the liquid volumes at several other temperatures. Table 39.3 is an example of the dewpoint determination and pressure/volume relations of a gas-condensate reservoir fluid.



TABLE 39.3-PRESSURE/VOLUME RELATIONS OF RESERVOIR FLUID AT 256OF (Constant-Composition Expansion)



Pressure (PSW 7,500 7,000 * 6,500 6,300 6,200 6,100 6,010+ 5.950 5,900 5,800 5,600 5,300 5,000 4,500 4,000 3,500 3,000 2,500 2,100 1,860 1,683 1,460 1,290 1,160 1,050



Simulated Pressure Depletion Pressure depletion of gas-condensate reservoirs may be simulated in the laboratory by use of high-pressure visual cells. In these depletion studies made in the laboratory, the assumption is that the retrograde liquid that condenses in the reservoir rock will not achieve a sufficiently high saturation to become mobile. This assumption appears to be valid except for very rich gas-condensate reservoirs. For very rich gas-condensate reservoirs where the retrograde liquid may achieve a high enough saturation to migrate to producing wells, the gas/liquid relative permeability data should be measured for the reservoir rock system. These data can then be used to ad,just the predicted recovery from the reservoir. Table 39.4 is an example of a depletion study on a gascondensate reservoir fluid. Note from Table 39.4 that the dewpoint pressure of this reservoir fluid is 6,010 psig. The composition listed in the 6,010-psig-pressure column in Table 39.4 is the composition of the reservoir fluid at the dewpoint and exists in the reservoir in the gaseous state



Relative Volume



Deviation Factor, z



0.9341 0.9523 0.9727 0.9834 0.9891 0.9942



1.328 1.264’ * 1.19s 1.175 1.163 1 150 1.140f



1.oooo 1.0034 1.0076 1.0138 1.0267 1.0481 1.0749 1.1268 1.2024 1.3096 1.4689 1.7169 2.0191 2.2747 2.5150 2.9087 3.3173 3.7153 4.1342



‘Reservoir preSS”re ‘;Gas



ev~ans~on



factor = 1 545 Mscllbbl



‘Gas



expansion



factor = 1 47,



oewpolnlpressure



TABLE 39.4--DEPLETION



Mscfibb,



STUDY AT 256°F Reservoir Pressure, psig



6,010



700 2,100 4,000 3,000 1,200 5,000 Hydrocarbon Analysis of Produced Well Stream, mol %



700*



Component Carbon dioxide Nitrogen Methane Ethane Propane iso-Butane n-Butane iso-Pentane n-Pentane Hexanes Heptanes plus



Molecular weight of heptanes plus Density of heptanes plus



0.01 0.11 68.93 8.63 5.34 1.I5 2.33 0.93 0.85 1.73 9.99



0.01 0.12 70.69 8.67 5.26 1.10 2.21 0.86 0.76 1.48 8.84



0.01 0.12 73.60 8.72 5.20 1.05 2.09 0.78 0.70 1.25 6.48



0.01 0.13 76.60 8.82 5.16 1.01 1.99 0.73 0.65 1.08 3.82



0.01 0.13 77.77 8.96 5.16 1.Ol 1.98 0.72 0.63 1.Ol 2.62



0.01 0.12 77.04 9.37 5.44 1.10 2.15 0.77 0.68 1.07 2.25



0.01 0.11 75.13 9.82 5.90 1.26 2.45 0.87 0.78 1.25 2.42



Trace 0.01 11.95 4.10 4.80 1.57 3.75 2.15 2.15 6.50 63.02



100.00



100.00



100.00



100.00



100.00



100.00



100.00



100.00



158 0.827



146 0.817



134 0.805



123 0.794



115 0.784



110 0.779



109 0.778



1.140 1.140



1.015 1.016



0.897 0.921



0.853 0.851



0.865 0.799



0.902 0.722



0.938 0.612



0.000



6.624



17.478



32.927



49.901



68.146



77.902



Deviation factor, z Equilibrium gas Two-phase Well stream produced, cumulative % of initial



174 0.837



39-8



PETROLEUM ENGINEERING



1.6



50



15



45 45



1.4



40



I 3



35



12



30



i I



25



10



20



09



15



08



10



07



5



HANDBOOK



0



0.6 0



1000



2000



3000



4000



Pressure.



5000



6000



7000



8000



0



1000



2000



Fig. 39.4-Deviation factor, z, of well stream during depletion at 256OF.



The depletion study is performed by expanding the reservoir fluid in the cell by withdrawing mercury from the cell until the first depletion pressure is reached; this is 5,000 psig in the example. The fluid in the cell is brought to equilibrium and the volume of retrograde liquid is measured. The mercury is then reinjected into the cell and, at the same time, gas is removed from the top of the cell so that a constant pressure is maintained. Mercury is injected into the cell until the hydrocarbon or reservoir volume of the cell is the same as the volume when the test was begun at the dewpoint pressure. The gas volume removed from the cell is measured at the depletion pressure and reservoir temperature. The gas removed is charged to analytical equipment where its composition is determined and its volume is measured at atmospheric pressure and temperature. The composition determined is that listed in Table 39.4 under the heading 5,000 psig. The volume of gas produced in this manner is then divided by the standard volume of gas in the cell at the dewpoint pressure. The produced volume is presented in Table 39.4 as cumulative well stream produced. As mentioned earlier, as the gas is removed from the top of the cell, its volume is measured at the depletion pressure and reservoir temperature. From this volume, the “ideal volume” of this displaced volume may be calculated with the ideal-gas law. When the ideal volume is divided by the actual volume of the gas produced at standard conditions, we get the deviation factor, z, for the produced gas. This is listed in Table 39.4 under



3000



4000



Pressure.



osi



Fig. 39.5--Retrograde



5000



6000



7000



8000



psi



condensation



during depletion.



“Deviation Factor z, equilibrium gas” and plotted in Fig. 39.4. The actual volume of gas remaining in the cell at this point is the gas originally in the cell at the dewpoint pressure minus the gas produced at the first depletion level. If we divide the actual volume remaining in the cell into the calculated ideal volume remaining in the cell at this first depletion pressure, we obtain the two-phase deviation factor shown in Table 39.4. We call this value a two-phase deviation factor because the material remaining in the cell after the first depletion level is actually gas and retrograde liquid and the actual gas volume we calculated above is the gas volume plus the vapor equivalent of the retrograde liquid. The two-phase z factor is significant in that it is the z factor of all the hydrocarbon material remaining in the reservoir. It is the two-phase z factor that should be used when a plz-vs.-cumulativeproduction plot is made in evaluating gas-condensate production. This series of expansions and constant-pressure displacements is repeated at each depletion pressure until an arbitrary abandonment pressure is reached. The abandonment pressure is considered arbitrary because no engineering or economic calculations have been made to determine this pressure for the purpose of the reservoirfluid study. In addition to the composition of the produced well stream at the final depletion pressure, the composition of the retrograde liquid was also measured. These data are included as a control composition in the event the study is used for compositional material-balance purposes.



GAS-CONDENSATE



RESERVOIRS



39-9



TABLE 39.5--RETROGRADE CONDENSATION DURING GAS DEPLETION AT 256’F



The volume of retrograde liquid measured during the course of the depletion study is shown in Fig. 39.5 and Table 39.5. The data are shown as a percent of hydrocarbon pore space. These are the data that should be used in conjunction with relative permeability data and water saturation data to determine the extent of retrograde liquid mobility. As mentioned earlier, this is a significant factor only with extremely rich gas-condensate reservoirs. Also obtained from the reservoir fluid study is Table 39.6. This table was calculated with the results of the laboratory depletion study described previously applied to a unit-volume reservoir. The unit volume chosen was 1,000 Mscf in place at the dewpoint pressure (note the 1,000 Mscf in Table 39.6 in the first column of numbers). Equilibrium ratios were then used to calculate the amount of stock-tank liquid, primary-separator gas, second-stage gas, and stock-tank gas contained in the unit-volume reservoir. The equilibrium ratios used were for the separator conditions listed at the bottom of Table 39.6. The separator conditions used for these calculations should be the conditions in use in the field or those conditions anticipated for the field. The relative amounts of gas and liquid produced will be a function of the surface separation conditions, among other things. These calculations may be made at a variety of conditions to determine optimum separator pressures and temperatures. For the purpose of this table, production was begun at the dewpoint pressure. The amount of total well effluent (well stream) produced from this unit-volume reservoir as a function of pressure is listed in the table. The amount of stock-tank liquid produced as a function of pressure is also listed. The primaryseparator gas, second-stage gas, and stock-tank gas are presented in a similar manner. Various other factors associated with the production of the gas and condensate from this reservoir are also presented in the table.



TABLE 39.6-CALCULATED



CUMULATIVE



RECOVERY



Initial in Place Well stream, Mscf Normal temperature separation’ Stock-tank liquid, bbl Primary separator gas, Mscf Second-stage gas, Mscf Stock-tank gas, Mscf Total plant products in primary separator gas, gal Ethane Propane Butanes (total) Pentanes plus Total plant products in second-stage gas, gal Ethane Propane Butanes (total) Pentanes plus Total plant products in well stream, gal Ethane Propane Butanes (total) Pentanes plus ‘Primary



separator



at 450 psig and ,!YF,



second-stage



Retrograde Liquid Volume (% hydrocarbon pore space)



Pressure W9) 6,010’ 5,950



0.0



Trace



5,900



0.1 0.2 0.5 2.0 7.8 21.3 25.0 24.4 22.5 21.0 17.6



5,800 5,600 5,300 5,000’ * 4,000 3,000 2,100 1,200 700 0 ‘Dewpmt pressure “First depletion level.



Table 39.6 shows the initial stock-tank liquid in place to be 181.74 bbl for this unit-volume reservoir. After production to 700 psig, 51.91 bbl had been produced. The difference between these two numbers (18 1.74 - 5 1.9 1), 129.83 bbl, is the amount of retrograde loss or liquid still unproduced at 700 psig expressed in terms of stock-tank barrels. The value of 181.74 bbl may be considered the recovery by pressure maintenance, assuming 100% conformance and 100% displacement efficiency. Table 39.7 furnishes the gravity of the stock-tank liquid that may be expected to be produced as a function of reservoir pressure. Also reported are the instantaneous gas/liquid ratios as a function of reservoir pressure.



DURINGDEPLETIONPER MMscf OF ORIGINALFLUID



6.010



1.OOo



0



181.74 777.15 38.52 38.45



Reservoir Pressure (wig) 4,000 3,000 2,100



1,200



700



66.24



174.78



329.27



499.01



681.46



779.02



10.08 53.18 2.26 2.29



21.83 145.16 5.17 5.38



31 .a9 283.78 8.03 8.73



39.76 440.02 10.51 11.85



47.36 608.25 13.21 15.51



51.91 696.75 14.99 18.05



1,474 749 374 177



1,709 873 441 206



5.000



1,841 835 368 179



0 0 0 0



126 58 26 12



344 163 73 35



674 331 155 73



1,050



204 121 53 23



0 0 0 0



12



27 17 8 3



42 27 13 5



55 36 17



70 47 23 10



80 54 27 11



2,295 1,461 1,104 7,352



0 0 0 0



153 95 70 408



404 250 178 890



767 468 325 1,322



1,171 707 486 1,680



1,626 979 674 2,037



1,880 1,137 789 2,249



separatora, 100 ps,gand75OF,



3



stock tank a, 75DF



526 256 122



PETROLEUM ENGINEERING



39-10



These data may be calculated without the benefit of rock propertles or interstitial water values. The assumption is that the retrograde liquid does not achieve significant mobility. Because only one phase is flowing, water and hydrocarbon liquid saturations do not enter into the calculations. The assumption that the retrograde liquid does not flow in the reservoir except in the drawdown area immediately around the wellbore appears to be good. Only with very rich reservoirs does movement of retrograde liquid add significantly to well production. It was mentioned earlier that the most popular form of material balance on a gas-condensate reservoir is the p/zvs.-cumulative-production curve. It was stated that the z factor to be used must be the two-phase : factor. The cumulative production must be the total production from the well. This includes. in most instances, the first-stage separator gas, second-stage separator gas, tank vapors. and the vapor equivalent of the stock-tank liquid. The most accurate production figures from a gas-condensate field are usually the sales-gas volumes. This usually includes the first- and second-stage separator gas. To make the p/zvs.-cumulative plot, the tank vapors and the vapor equivalent of the stock liquid must be accounted for. Without the benefit of laboratory data, the tank vapors must be estimated and the vapor equivalent of the stock-tank liquid calculated with an average or estimated number. Table 39.7 furnishes the data to make these calculations. If sales gas is the primary- and second-stage gas, and the average reservoir pressure is 5.000 psig, then the total well-stream volume can be calculated by dividing the sales volume by 0.83704. This factor accounts for the tank vapors and the vapor equivalent of the tank liquid. If the sales gas is only the first-stage gas, then the appropriate factor would be 0.80285.



Operation by Pressure Depletion



curacy) on the basis of the composition of the gas-condensate system. Whenever possible, the predictions should be made with actual laboratory data because the better accuracy obtained at the reservoir conditions is justified by the large gas and liquid reserves involved in reservoir calculations. Predictions With Laboratory-Derived and Hydrocarbon Analysis



Data



With the assumption that the liquid condensate in the reservoir during a pressure-depletion operation stays in place (does not build up sufficiently to provide liquid-phase permeability for flow), reservoir behavior can be predicted from the laboratory constant-composition depletion study discussed previously. Pertinent information is shown in Tables 39.3 through 39.6 and Figs. 39.4 and 39.5. Liquid-phase change in the reservoir is shown in Fig. 39.5 derived from Table 39.5. Note that the amount of liquid remaining in the reservoir passes through a maximum but does not return to zero, indicating that pressuredepletion operations leave some liquid hydrocarbons behind at abandonment pressure. Economic analyses of pressure-depletion operations are necessary for estimating the magnitude of this loss and its effect on development and operating policy for the reservoir. The ultimate recoveries by pressure depletion of wet gas. condensate, and plant products can be calculated for the reservoir described in Table 39.8 by use of the data given in Table 39.6. Gas in place ut original pressure: (500x 106)(1.545)(178. l)= 137,582 MMscf. Gas in place at dewpoint pressure: (500x106)(1.471)(178.1)=130,992



Pressure-depletion gas-condensate reservoir behavior can be predicted from the laboratory data described previously, or if necessary, by various correlation and computation procedures that provide similar information (with less ac-



TABLE 39.7-CALCULATED



HANDBOOK



MMscf.



Wet gas produced to dewpoint pressure: 137,582-



INSTANTANEOUS



130,992=6,590



MMscf.



RECOVERY DURING DEPLETION



Reservoir Pressure (asiai 6,010 Normal temperature separation’ Stock-tank liquid gravity at 6OOF. OAPl Separator-qaslwell-stream ratio, Mscf/MMscf primary-separator gas only primary and second-stage separator gases Separator-gas/stock-tank-liquid ratio, scf/STB primary-separator gas only pnmary and second-stage separator gases Recovery from smooth well stream compositions, gal/min Ethane plus Propane plus Butanes plus Pentanes plus ‘Primary



separator



at 450 ps~g and 75T



second-stage



separator



5,000



4.000



3,000



2,100



1,200



700



49.3



51.7



55.4



60.4



64 6



67.5



68.6



777.15 815.67



802.85 837.04



847.45 874.26



897.28 915.77



920.44 935.04



922.04 936.84



907.14 925.38



4,276 4,488



5,277 5.502



7,828 8.076



13,774 14,058



19,863 20.178



22,121 22.476



19,475 19.867



12.212 9.917 8.456 7.352



10.953 8.648 7.209 6.158



9.175 6.856 5.434 4.437



7.509 5.164 3.752 2.800



6.851 4.469 3.057 2.108



6.970 4.479 2.990 1.959



7.574 4.963 3.349 2.171



at 100 pslg and 75OF. stock tank at 75OF



GAS-CONDENSATE



RESERVOIRS



39-11



Wet gas produced Sfom dewpoint pressure to abandonment: (130,992)(0.77902)=



102,045 MMscf.



Total wet gas produced: 6,590+ 102,045 = 108,635 MMscf. Condensate produced to dewpoint pressure: (6,590)(181.74)=1,197,667 Condensate producedfiom donment:



Original reservoir pressure, psig Dewpoint pressure, psig Assumed abandonment pressure, psig Average reservoir temperature, OF Hydrocarbon pore space (by volumelrics), cu ft Gas expansion factor (8,) of produced fluid at original pressure, Mscflbbl Gas expansion factor (B,) of produced fluid at dewpoint, Mscf/bbl



7,000 6,010 700 256 500x 10” 1.545 1.471



bbl. dewpoint pressure to aban-



(130,992)(51.91)=6,799,795 Total condensate produced: 1,197,667+6,799,795=7,997,462 Percent recoveries by pressure depletion from dewpoint pressure to abandonment: 102,045 Wet gas= ~ x 100=77.9%; 130,992 Condensate =



TABLE 39.8-FORMATION AND FLUID DATA FOR A GAS-CONDENSATE RESERVOIR



6,799,795



x 100=28.6%.



181.74x 130,992 The total plant products can be calculated in a similar manner, depending on the flow streams to be processed and the recovery efficiencies anticipated.



Predictions With Vapor/Liquid Calculation and Correlations



Equilibrium



In the absence of direct laboratory data on a specific gascondensate system, pressure-depletion behavior can be estimated with vapor/liquid equilibrium ratios (i.e., equilibrium constants, equilibrium factors or K values) to compute the phase behavior when the composition of the total gas-condensate system is known. Correlations for estimating phase volumes must also be available. When multicomponent hydrocarbon gases and liquids exist together under pressure, part of the lighter hydrocarbons (light ends) are dissolved in the liquid phase, and part of the heavier hydrocarbons (heavy ends) are vaporized in the gas phase. A convenient concept to describe the behavior of specific components quantitatively is the equilibrium ratio. The ratios vary considerably with the pressure, temperature, and composition of the system involved The equilibrium ratio is defined as the mole fraction of a given constituent in the vapor phase divided by the mole fraction of the same constituent in the liquid phase, the two phases existing in equilibrium with each other. The equilibrium ratio is designated as K. The basis for this definition is discussed in Chap. 23 and by Standing. 9 Fig. 23.21 illustrates the behavior of equilibrium ratios for a particular system and shows the rather wide variation possible for a given constituent at different pressures. The



figure shows a tendency of the equilibrium ratios to converge isothermally to a value of K= 1 at a specific pressure. The pressure is roperly called the “apparent convergence pressure. ” g The selection of equilibriumratio values for calculations usually is based on the system’s apparent convergence pressure, which can change in a pressure-depletion process because of changing system composition with pressure decline. Large inaccuracies can occur in pressure-depletion calculations with equilibrium ratios when the heavier hydrocarbons (e.g., heptanes and heavier) are not adequately described. To obtain satisfactory results in calculating pressure-depletion behavior of a gas-condensate system, an extended analysis of the CT+ fraction should be made. A determination of the the molar distribution of CT+ through at least C!z=,is recommended. As can be observed in Table 39.4, the CT+ component of the subject gas-condensate fluid exhibited a change in molecular weight from 158 at a pressure of 6,010 psig to 109 at a pressure of 700 psig. The change in density of the C 7 + component was from 0.827 to 0.778 over the same pressure range. Table 39.4 also shows that at 700 psig, the molecular weight of the CT+ in the liquid phase is 174, compared to 109 in the gas phase, and the density is 0.837 in the liquid phase, compared to 0.778 in the gas phase. This change in composition of the C7+ fraction with pressure reduction leads to large errors in the vapor/ liquid split of the CT+ fraction when equilibrium ratios are used and in the resultant molecular weight and density of the calculated gas and liquid volumes. Should such an extended analysis of the CT+ component not be available, then a statistical split should be made that maintains the integrity of the average molecular weight and density of the CT+ component. Once the CT+ component has been divided into multiple pseudocomponents, the physical properties required to make reservoir flash calculations must be developed. Wbitson30 presents a method for determining the molar distribution of single-carbon-number (SCN) groups that are defined by their boiling points as a function of each group’s molecular weight. To make the distribution, a three-parameter gamma probability function is used. Whitson also presents equations for calculating the required physical properties with the Watson3’ characterization factor. This method can be easily programmed for a personal computer and permits rapid development of molar distribution and physical properties. A statistical expansion of the C7+ component of the gas-condensate fluid presented in Table 39.2 has been made with the teehnique Whitson described. The results of this expansion



PETROLEUM ENGINEERING



39-12



are presented in Table 39.9. The ability to calculate accurately the pressure-depletion performance of a gascondensate reservoir depends on proper characterization of the vapor/liquid equilibrium ratios (K values) of the hydrocarbon system. Equilibrium ratios for nonhydrocarbon components and hydrocarbons C, throu h C 10 can be found in the Engineering Data Book. 15 Hoffman et al. 32 and Cook et al. 33 have presented methods for developing K values for the pseudocomponents. Hoffman et al. ‘s procedure can be programmed easily for a personal computer for rapid development of equilibrium ratios. An alternative method is to plot the methane and normal pentane K values as a function of their boiling points on a semilog graph for each depletion pressure to be calculated. An equation can be determined for a straight line connecting these two points. The K value for each of the other components and pseudocomponents can then be calculated for each pressure point with their individual boiling points. This method of obtaining K values was used in the earlier example calculation. There are some limitations on the accuracy of the data derived by these methods unless some measured data on similar hydrocarbon systems are available. However, the data should be usable for the quick, rough approximations often needed in the preliminary reservoir evaluation stage. The C t through Cc composition of the gas-condensate fluid presented in Table 39.2 was used to develop a K-value relationship for the extended C7+ compositions. The resultant relationship is presented in Fig. 39.6. Chap. 23 describes the general techniques of the use of vapor/liquid equilibrium ratios to compute the phase compositions and magnitudes of hydrocarbon gas/liquid mixtures. Standing’ also has an excellent presentation of this usage, including a discussion of the serious errors that can result in calculating the phase behavior of gascondensate systems. When these methods are used to estimate the pressure-depletion behavior of a gas-condensate reservoir, the following procedure is used. 1. Assume that the original (known) composition flashes from original pressure (and volume) to a lower pressure, at which the compositions and amounts (in moles) of the liquid and gas phases are computed with the best K values available. 2. Estimate the volume of each phase with the methods discussed below. 3. Assume that enough vapor-phase volume is removed (produced) at constant pressure to cause the remaining gas plus all the liquid to conform to the reservoir’s original constant volume.



TABLE 39.9-STATISTICAL



HANDBOOK



BOILING POINT CONDENSATE NO7 FLUID



000,



0



COUPONENT



BOILING POINT OR



CO2 N



275



w



MO



E: I% NC4



,"d, 462 482



CT.



869



200



HO 217



400



Kc BOILING



Fig. 39.6-K-value



803



1000



12M)



1403



POINT,'RANKlNE



correlation for Condensate 7 depletion.



4. Subtract the number of moles of each component in the vapor represented by this gas removal from the original system composition. 5. With the new total composition from Step 4, consider the system flashed to the next lower pressure step and repeat the procedure. Removal of vapor phase alone is required by the assumption that fluid flowing into the wells will not be accompanied by any liquid phase at any step of the process. As indicated previously, the calculations require knowledge of the volume occupied by each phase at each pressure step. Methods to estimate these volumes are described in Chaps. 20 and 22 and also by Standing. 9 To estimate phase volumes, smoothed values should be used from curves drawn through the points computed from properties of the phase at each known composition.



EXPANSION OF C,,



COMPONENT, CONDENSATE 7



C 7+ Mole fraction 0.0999 Molecular weight 158.0 Density, g/cm 0.827 Component



Mole Fraction



Mole Weight



Density (g/cm3)



C7 2



0.01685 0.01535 0.01235



100.9 113.6 126.9



0.7486 0.7648 0.7813



Boiling Point (W 658 702



40 C ,I+



0.00941 0.04594



139.5 205.1



0.7960 0.8641



791 748 1,020



GAS-CONDENSATE



39-13



RESERVOIRS



These calculations are intended to approximate the experimental procedure used in the PVT cell during a laboratory pressure-depletion study. The number of pressure steps used in making such calculations is arbitrary but probably should conform to about SOO-psi intervals, with points usually closer together at the start and at the end of the calculations. The calculated depletion performance of Condensate 7 is presented in Table 39.10. The dewpoint pressure of 5,277 psig was calculated with an empirical relationship Nemeth and Kennedy j4 presented. The best method to determine the dewpoint pressure is by direct measurement, as in the laboratory PVT analysis. If these data are not available, then one must resort to estimation by empirical methods. such as that used in this example, or by gas/liquid production performance. In the latter choice, one must deplete the reservoir to a pressure below the dewpoint. In Table 39.10 a comparison of wet gas and condensate recoveries is made between the laboratory-measured and calculated depletion performance. As can be seen from the comparison, large errors are possible in the calculated data resulting from estimation of the dewpoint pressure and the physical properties of the reservoir fluid. Hydrocarbon/Liquid Condensation; Gas-Condensate Behavior



Effect on



For some gas-condensate systems, large amounts of liquid can be condensed during pressure depletion, resulting in high liquid saturations in the formation pores. When this probability is indicated by either laboratory tests or calculations, the possibility of hydrocarbon/liquid flow through and out of the reservoir must be examined. Relative permeability information (usually curves showing k,/k, vs. liquid saturation in the formation) should be combined with viscosity data (pO/pR) to estimate the volumetric proportion of liquid in the flowing stream (thus removed from the reservoir), thereby affecting the remaining reservoir phase compositions at each of the depletion steps. The best k,gpu,/k,p., data to use are those determined in the laboratory with actual samples of the reservoir rock and hydrocarbon system in question. In the absence of such information, k,/k, can be estimated by the methods explained in Chap. 28; viscosity approximations may be made by the methods described by Carr et al. 23 After the amount of gas and liquid removed at each step has been estimated, the calculation procedures can be adjusted to obtain the desired behavior predictions. Pressure Drawdown at Wells; Effect on Well Productivity and Recovery The previous discussion has taken liquid condensation in the formation into account as though it occurred uniformly throughout the reservoir (uniform pressure at any instant of time). In low-permeability formations, however, there can be appreciable pressure drawdown at the producing wells because the pressures near the wellbores are much lower than in the main part of the reservoir. This tends to increase the early condensation of liquids around the wells considerably, thus decreasing the gas permeability and affecting the phase behavior of the system near the wells. This is important from at least two standpoints: (1) composition history of fluids produced from the reservoir may diverge from that predicted by assuming uniform pressure in the reservoir at any instant of time and



(2) adverse effects on the ability of the wells to produce may occur, potentially affecting the optimum well spacing and the rate of gas-condensate recovery from the zone as pressures decline. The effects of well-pressure drawdown on the composition history (and ultimate liquid recoveries) of gascondensate reservoir production have had little discussion in the literature. The general expectation would be that in lower-pressure areas around the wells, liquid hydrocarbons are precipitated earlier and in greater amounts than in the main volume of the reservoir. The main factors involved in this phenomenon are the richness of the gas condensate, the retrograde characteristics of the reservoir fluid, and the permeability of the reservoir rock. Normally, the area around the wellbore that is affected will be small and the condition will stabilize. Normal operating practices to restrict the pressure drawdown to reasonable values will alleviate the problem. In those reservoirs that exhibit extremely low permeability and contain fluids exhibiting condensable liquids of more than 200 bbl/MMscf, the problem can be severe. When separator samples are taken for the laboratory, the analysis procedure discussed previously should be followed to minimize the drawdown effect on the gas and liquid compositions. The effects on well productivity of precipitated liquid in the vicinity of the wellbore theoretically can be appreciable. Normally, estimates of future well productivity ignore the drawdown effects of production on liquidphase distribution in the reservoir. The greater liquid accumulations and lower gas permeabilities near the wells thus are ignored in theoretical predictions of well productivity (or extrapolations from early tests); these predictions then tend to show minimum decline rates. The operating engineer should be alert to this possibility whenever calculated well or reservoir rates approach undesirably close to the minimum necessary for the operating objectives of the project. Well productivity is discussed later. Relative Merits of Measured vs. Calculated Pressure-Depletion Behavior This chapter has emphasized that for purposes of reservoir analysis and prediction, measured properties and observed behavior of gas-condensate systems are much superior to the use of correlations or approximations. This applies in particular to the use of equilibrium ratios for simulating or predicting the pressure-depletion behavior of a reservoir. The problem is discussed and illustrated by Standing 9 in his Vapor Liquid Equilibria and GasCondensate Systems chapters. In particular, Standing’s Fig. 36 shows that serious errors (in excess of 40%) can be incurred in the computation of the liquid volume of a gas-condensate system from errors of less than 10% in the equilibrium ratios for heptanes-plus and methane. The literature contains reports on the use of equilibrium ratios for calculating the reservoir behavior of gascondensate systems. Allen and Roe3” computed the pressure-depletion behavior of a gas-condensate reservoir and observed certain discrepancies with the actual behavior. These authors did not report laboratory-measured equilibrium ratios for the specific fluids involved, however; consequently, there were no means to compare computed fluid behavior with actual fluid behavior. All the observed discrepancies were assigned arbitrarily by Allen



PETROLEUM ENGINEERING



39-14



TABLE 39.10-CALCULATED



HANDBOOK



COMPOSITION OF PRODUCED STREAM, mol% Reservoir pressure (psig)



5,277



5,000



4,000



3.000



2,100



1,200



700



0.01 Cl.11 68.93 8.63 5.34 1.15 2.33 0.93 0.85 1.73 1.685 1.535 1.235 0.941 4.594



0.01 0.11 70.74 8.67 5.28 1.12 2.26 0.89 0.81 1.62 1.55 1.38 1.09 0.81 3.66



0.01 0.13 74.77 a.77 5.13 1.06 2.10 0.79 0.71 1.35 1.21 1 .Ol 0.73 0.49 1.74



0.01 0.13 77.09 8.88 5.05 1.Ol 1.99 0.73 0.64 1.15 0.97 0.75 0.49 0.30 0.81



0.01 0.13 78.05 9.04 5.10 1.01 1.96 0.69 0.61 1.03 0.82 0.59 0.35 0.19 0.42



0.01 0.12 77.55 9.37 5.41 1.08 2.09 0.73 0.64 1.04 0.78 0.52 0.28 0.14 0.24



0.01 0.12 75.53 9.76 5.95 1.22 2.41 0.86 0.75 1.23 0.90 0.59 0.31 0.15 0.21



Trace 0.01 12.29 4.22 5.02 1.62 3.80 2.14 2.16 5.97 7.33 7.92 7.34 6.14 34.04



100.000



100.00



100.00



100.00



100.00



100.00



100.00



100.00



9.990 156 0.825



8.49 155 0.822



5.18 146 0.812



3.32 137 0.802



2.37 129 0.793



1.96 124 0.784



2.16 121 0.780



62.77 166 0.832



Deviation factor, z equilibrium gas two-phase Gas FVF, Mscf/scf



1.021 1.021 0.2561



0.987 1.009 0.2511



0.901 0.922 0.2201



0.861 0.845 0.1730



0.863 0.782 0.1211



0.899 0.695 0.0668



0.930 0.595 0.0380



Retrograde liquid volume, % hydrocarbon pore space



0.000



Carbon dioxide Nitrogen Methane Ethane Propane Iso-butane n-butane Iso-pentane n-pentane Hexanes Fraction C, Fraction C, Fraction C, Fraction C ,0 Fraction C , , + Heptanes-plus mol% molecular weight density



15.3



26.96



27.89



26.43



21.95



23.85



Cumulative recovery per MMScf of original flurd



Well stream, Mscf Normal temperature separation’ * Stock-tank liquid, bbl



Reservoir pressure (psig)



Initial in place



5.277



5,000



4,000



3,000



2,100



1,200



700



1.ooo



0.00



40.73



160.03



311.34



478.33



662.91



768.03



183.13



0.00



6.91



21.98



34.00



42.98



50.71



55.05



Primary separator gas, Mscf Second-stage gas, Mscf Stock-tank gas, Mscf



776.98 37.01 38.31



0.00 0.00 0.00



32.46 1.42 1.50



138.96 4.76 5.26



280.26 7.74 8.92



437.60 10.21 12.19



610.03 12.58 15.60



707.57 14.08 17.93



Total separator gas, Mscf



852.30



0.00



35.38



148.98



296.92



460.00



638.21



739.58



Comparison of Recovery Calculations



Gas in place at original pressure, MMscf Gas in place at dewpoint pressure, MMscf Wet gas produced to dewpornt pressure, MMscf Wet gas produced from dewpoint to abandonment,



MMscf



Total wet gas produced, MMscf Condensate produced to dewpoint pressure, bbl Condensate produced from dewpoint to abandonment, Total condensate produced, bbl



bbl



Laboratory Depletion



Calculated Depletion



137,582 130,992



137,582 128,050



6,590 102,045



9,532 98,346



108,635



107,878



1,197,667 5,297,156



1,745,595 5,413,947



6,494,823



7,159,542



GAS-CONDENSATE



RESERVOIRS



and Roe to factors other than the possible inaccuracies of equilibrium ratios from correlations compared with actual measured ratios for the particular system composition and reservoir conditions involved. Some of these discrepancies were probably attributable to the equilibrium ratios used. Berrymanj6 compared calculated gas-condensate fluid performance with that actually obtained in the laboratory; however, he made observations on actual vapor/liquid equrlibrium in the laboratory cell and adjusted the literature equilibrium ratios to conform to this actual behavior. With the adjusted vapor/liquid equilibrium ratios, the calculated performance was found to match actual reservoir performance during early life satisfactorily. Rodgers ef ul. j7 provided detailed laboratory data, vapor/liquid equilibrium calculations, and actual reservoir performance for a small gas-condensate reservoir in Utah. The pressure range involved was moderate compared with most cases. Even at these moderate pressures, however, the literature-derived equilibrium ratios for heptanes-plus did not agree favorably with measured values for the system. The authors commented that the “appearance of the data. clearly shows the need for improved techniques in establishing proper equilibrium data.” On the basis of this experience and for the reasons Standing stated, it would appear desirable to use measured values of phase and volumetric behavior for a gas-condensate system in predicting the pressure-depletion behavior of a gas-condensate reservoir. As more data are obtained and better correlating methods developed, it is possible that equilibrium ratios may achieve suitable accuracy for reservoir-type calculations in the future. Numerous equation-of-state (EOS) calculation techniques have been developed that produce phase equilibrium data that can be used to perform depletion calculations for gascondensate reservoirs. Many are discussed in Refs. 38 through 40. The use of EOS methods, while more flexible and in many cases more accurate, requires sophisticated computer programs that may or may not be available or warranted. Continued improvement in techniques using EOS’s may enhance the accuracy of calculated pressuredepletion performance.



Operation by Pressure Maintenance or Cycling Pressure maintenance of a gas-condensate reservoir can exist by virtue of (1) an active water drive after moderate reduction of pressure from early production, (2) pressure maintenance through water injection operations, (3) injection of gas, or (4) combinations of all of these. From time to time, certain reservoirs may be encountered that have fluids near their critical points and that thereby may be candidates for special recovery methods, such as the injection of specially tailored gas compositions to provide miscibility and phase-change processes that could improve recovery efficiency. These usually are not regarded as gascondensate cases. Water Drive and Water Injection Pressure Maintenance Very few cases of gas-condensate reservoirs operated under natural water drive have been reported in the litera-



39-15



ture. To be attractive economically. a water drive would have to be sufficiently strong to maintain pressure high enough to minimize condensed hydrocarbon losses in the formation. Under these conditions, expenditures for cycling or other pressure-maintenance operations might not be justified: a careful engineering and economic analysis should be made if this possibility seems imminent. The analysis should include a geologic review of conditions surrounding the reservoir to estimate whether any indicated early water drive is apt to last for the life of the operation. There are also other considerations to be studied carefully. including the expenses of dewatering or working over invaded producing wells, the displacement efficiency of water moving gas. and the potential bypassing and loss of condensate fluids when wells become watered-out prematurely through permeable stringers [invasion efficiency (see Pages 39- 17 and 39- 18) of the natural flood]. Should this last possibility exist, use of a natural water drive would be of doubtful value if the amount of hydrocarbons in place is large. In any case, predictions of recovery by natural water drive should take into account the factors for water injection discussed below. The injection of water into a gas-condensate reservoir to maintain pressure is sometimes considered. A number of factors must be weighed carefully before a decision is reached. The mobility ratio (mobility of driving fluid over mobility of the driven fluid, water/gas) in this case is favorably low because of the very high mobility of the gas, thus tending to provide high areal sweep and pattern (@S-weighted) efficiencies. There is strong evidence, however. that displacement efficiency by the water is not high. While Buckley et al. 4’ indicated that the displacement efficiency of water driving out gas can be as high as 80 to 85%, experiments and field observations by Geffen et al. ” indicate that it may be as low as 50%. This is offset to some extent by the improved area1 sweep efficiency enjoyed at a low mobility ratio. All things considered, the recovery of gas condensate in the vapor phase by water injection is likely to be appreciably lower than by cycling, and any consideration of water injection for gas-condensate recovery should be accompanied by detailed experimental work on cores from the specific reservoir involved. This will help to determine whether the water can, in fact, accomplish a high enough displacement efficiency to justify its use. Should water injection be decided on, gas and liquid recovery predictions for the reservoir can be made by combining the pattern (h&Gweighted). invasion, and displacement efficiencies with a knowledge of the condensable-liquids content of the gas-condensate system at the pressure chosen for pressure maintenance. As an example, an area1 sweep efficiency of 90% (based on an extremely low mobility ratio for water displacing gas) might be applied to the case cited on Page 39-24. Taking into account the thickness variations of the reservoir, this might provide a pattern (h&S-weighted) efficiency of about 95 % With an assumed invasion efficiency of 65 % within the invaded volume, water injection for this case would have swept out about 55% (product of the above three efhcienties) of the vapor phase in place at the start of injection. This compares with the actual recovery of more than 86% of the wet vapor by cycling operations. as discussed on Page 39-22.



39-I 6



These estimates of possible gas recoveries by either a natural water drive or water injection can be affected materially by the permeability distribution in the reservoir. The presence of large differences in permeability will result in premature water breakthrough. Flowing gas wells tend to “load up” when producing water and, depending on the vertical flow velocity and bottomhole flowing pressure, may cease to flow. This inability to flow results from sufficient water dropping out in the tubing to form a hydrostatic water column that exerts a pressure equal to the bottomhole pressure. It is difficult to obtain economical flow rates by artificial lift. This loss of productivity may result in premature abandonment of the project. The problems would be particularly serious for deeper reservoirs where the cost of removing water would be a significant factor. Yuster4’ discusses possible remedial methods for drowned gas wells. Bennett and AuvenshineM discuss dewatering gas wells. Dunning and Eakin4” describe an inexpensive method to remove water from drowned gas wells with foaming agents. Generally, the use of water injection for maintaining pressure in a gas-condensate reservoir is unlikely to be attractive where a wide range of permeabilities exists in a layered reservoir and selective breakthrough of water into producing wells might be expected before an appreciable fraction of the gas condensate in place could be recovered. Reservoir Cycling, Gas Injection Dry-Gas Injection. Comparative economics determines whether a gas-condensate reservoir should be produced by pressure depletion or by pressure maintenance. The objective of using dry-gas injection in gascondensate reservoirs is to maintain the reservoir pressure high enough (usually above or near the dewpoint) to minimize the amount of retrograde liquid condensation. Dry field gases are miscible with nearly all reservoir gascondensate systems: methane normally is the primary constituent of dry field gas. Dry-gas cycling of gas-condensate reservoirs is a special case of miscible-phase displacement of hydrocarbon fluids for improving recovery. Experimentation has shown that the displacement of one fluid by another that is miscible with it is highly efficient on a microscopic scale; usually the efficiency is considered 100% or very nearly so. This is one of the factors that explain the effectiveness and attractiveness of cycling. Another advantage of cycling is that it provides a means to obtain liquid recoveries from reservoirs at economical rates while at the same time avoiding waste of the produced gas when a market for that gas is not available; the operation provides at its termination a reservoir of dry gas with a potentially greater economic value. Inert-Gas Injection. The demand for dry gas as a marketable commodity varies, and the economic aspects of retaining dry cycled gas in reservoirs for future use have a changing significance. Most conservation laws in the U.S. still provide for minimizing waste of condensable liquids that would result if gas-condensate reservoirs were depleted through the retrograde range in a manner that left large liquid volumes unrecoverable. The use of inert gas to replace voidage during cycling of gas-condensate reservoirs can be an economical altemative to dry natural gas. One of the first successful inert-



PETROLEUM ENGINEERING



HANDBOOK



gas injection projects was in 1949 at Elk Basin, WY,46 where stack gas from steam boilers was used for injection. In 1959, the first successful use of internal combustion engine exhaust was seen in a Louisiana oil field.47 The first use of pure cryogenic produced nitrogen to prevent the retrograde loss of liquids from a gas-condensate fluid was in the Wilcox 5 sand in the Fordoche field located in Pointe Coupee Parish, LA.48 In the Fordoche field, the nitrogen was used as makeup gas. The nitrogen amounted to about 30% of the natural-gas/nitrogen mixture injected. Moses and Wilson’s49 studies confirmed that the mixing of nitrogen with a gas-condensate fluid elevated the dewpoint pressure. Moses and Wilson also presented data to show that the mixing of a lean gas with a rich-gas condensate would also result in a fluid with a higher dewpoint pressure. The increase in dewpoint pressure was greater with nitrogen than with the lean gas. In the same study, results are presented from slim-tube displacement tests of the same gas-condensate fluid both by pure nitrogen and by a lean gas. In both displacements, more than 98% recovery of reservoir liquid was achieved. These test results were also observed by Peterson, 5o who used gascap gas material from the Painter field located in southwest Wyoming. The authors concluded that the observed results were obtained because of multiple-contact miscibility. Cryogenic-produced nitrogen possesses many desirable physical properties. 5’ Those that make nitrogen most useful for a cycling fluid are that it is totally inert (noncorrosive) and that it has a higher compressibility factor than lean gas (requires less volume). The latter advantage is partially offset by increased compression requirements when compared with lean gas. Until the mid 1970’s, most inert-gas injection consisted of injection of combustion or boiler gas into oil zones. The need for an alternative source of gas for gas-condensate-cycling projects emerged because of the high cost of hydrocarbon gas needed to replace reservoir voidage. The combustion and boiler gas that had been used to displace oil miscibly contains byproducts (CO. 02, HzO, and NO, +) that are highly corrosive5* and decrease cost effectiveness. Economic parameters used to evaluate any process are by their nature representative only under the general economic conditions during which they are prepared. Therefore, there will be no attempt here to present representative economic data. However, one should be cognizant of and take into account those variables peculiar to a particular process when applying current economic parameters to compare different processes. Many factors affect the economics of a gas-cycling project. The major factors are product prices, makeup gas costs, liquid content of reservoir gas, and degree of reservoir heterogeneity. When inert-gas injection is considered, some important additional factors should also be considered. Donohoe and Buchanan” and Wilson”’ have discussed these factors. The use of inert gas as a cycling fluid offers both advantages and disadvantages. The major advantages are that it permits early sale of residue gas and liquids, resulting in greater discounted net income and that a higher recovery of total hydrocarbons is achieved because the reservoir contains large volumes of nitrogen rather than hydrocarbon gas at abandonment.



GAS-CONDENSATE



RESERVOIRS



Offsetting these advantages are some disadvantages: production problems and increased operating costs caused by corrosion if combustion or flue gas is used as cycling fluid; possible additional capital investments to remove the inert gas from the sales gas, to pretreat before compression, and/or to fund reinjection facilities; and early breakthrough of inert gas caused by high degrees of heterogeneity in the reservoir, resulting in excessive operating costs to obtain marketable sales gas. All these factors should be evaluated properly when the depletion method is selected. Calculation of Cycling Performance. Methods of calculating reservoir performance under gas-cycling operations generally fall into one of two categories: feasibility and/or sensitivity analysis or detailed design and evaluation. The calculation method selected usually is determined after consideration of the quality and quantity of data available and the ultimate use of the engineering study. When the potential of a gas-condensate reservoir for cycling is first considered, it is generally desirable to make calculations that require the use of some reasonably simplifying assumptions. In this manner, relatively rapid and inexpensive results can be obtained that define the approximate cycling rate, cycling life, ultimate recovery, and profitability. If, at the conclusion of these studies, it appears that gas cycling is feasible, more detailed and exacting studies can be made with mathematical simulators to evaluate the earlier results and to design the most advantageous distribution of injection and producing wells. Efficiency and Effectiveness of Cycling. The principal factors determining reservoir cycling efficiency have been used with interchangeable labels and definitions in the literature. It is therefore necessary to define the various efficiencies clearly. The engineer should define and explain terms carefully when reporting estimates on gascondensate reservoir behavior. Reservoir Cycling Efficiency. ER is defined as the reservoir wet hydrocarbons recovered during cycling divided by the reservoir wet hydrocarbons in place in the productive volume of the reservoir at the start of cycling. Both figures must be computed at the same pressure and temperature; e.g., at reservoir conditions or at standard conditions. The reservoir cycling efficiency can be visualized as the product of three other efficiencies: pattern (h@S-weighted), invasion, and displacement. A fourth efficiency factor, area1 sweep, can be evaluated for various injection patterns using analog or mathematical models. All efficiency terms used (except “displacement efficiency”) must be identified as to time-i.e., time of dry-gas breakthrough into first producing well, time of breakthrough into last well, end of cycling, or other suitable designation. Area1 Sweep Efficiency. EA is the area enclosed by the leading edge of the dry-gas front (outer limit of injected gas) divided by the total area of reservoir that was productive at the start of cycling. (Black oil, if present, is usually excluded from these areas.) Area of sweep can be estimated closely from analog or mathematical model studies (discussed later) or by observing the locations of wells developing dry-gas content during actual operations. The area1 sweep efficiency depends primarily on the injection and production well patterns and rates and the lateral



39-17



homogeneity of the formations from a permeability and porosity standpoint. Lesser factors affecting areal sweep efficiency include variations in water content of the pores; time of operation of the compression plant in relation to the input capacities of the wells and their locations in the reservoir; the activity, if any, of a natural water drive; and the presence and handling of black-oil wells if an oil ring exists in the reservoir. Mathematical model techniques (Chap. 48) provide a useful means for predicting the areal sweep efficiencies of gas-condensate reservoirs and, simultaneously, the rate of frontal advance of the injected dry gas. For such studies, a reasonable amount of subsurface data is needed on sand characteristics, reservoir fluid properties, properties of injected fluid, and geologic description. Pattern (hcpS- Weighted) Efficiency. E,, is the hydrocarbon pore space enclosed by the projection (through full reservoir thickness) of the leading edge of the dry-gas front divided by the total productive hydrocarbon pore space of the reservoir at start of cycling. (Black oil, if present, is usually excluded from these volumes.) The hydrocarbon volume contained within the dry-gasfront projection can be determined by outlining the farthest-advanced position of the front (from model studies or field observations) on a hydrocarbon isovol map (isovol maps are developed from data on sand thickness, porosity, and interstitial water content), determining the hydrocarbon volume enclosed by this line, and comparing the volume with total reservoir productive hydrocarbon pore space. Note that the definition specifies “projection of the leading edge” and avoids stating whether either the entire gross or entire microscopic PV’s are invaded or displaced by the injected gas. For the special cases in which productive thickness, porosity, interstitial water content, and effective permeability are each uniform, the pattern (h&S-weighted) and areal sweep efficiencies are the same. The pattern (&S-weighted) efficiency in general depends on the same factors discussed for areal sweep efficiency. Expected pattern (&S-weighted) efficiencies of nearly 95 % have been predicted under favorable conditions. ” Invasion Efficiency. El is the hydrocarbon pore space invaded (contacted or affected) by the injected gas divided by the hydrocarbon pore space enclosed by the projection (through full reservoir thickness) of the leading edge of the dry-gas front. (Sometimes volumetric sweep efficiency, E,, =E, X El, is used.) The definition says nothing about the effectiveness of the invading fluid in forcing original fluid out of the pores contacted. The term “vertical sweep efficiency” has sometimes been used in the sense of invasion efficiency. This is misleading in that it uses a one-dimensional term (“vertical”) when dealing with a three-dimensional problem. Invasion efficiencies can be as high as 90% under favorable conditions.” However, invasion is affected significantly by large variations in reservoir flow properties, These might be strictly lateral variations in horizontal permeability (and to a lesser extent in porosity and interstitial water content) of a singlebed reservoir that does not have any variations vertically at any location; strictly layering effects by which the reservoir may comprise several strata, each relatively uniform in properties but differing appreciably in permeability from all the others; or combinations of these extreme cases. Performance of cycling operations can vary ap-



PETROLEUM ENGINEERING



39-18



TABLE 39.11 -EFFICIENCY Areai Sweep Efhoency Area enclosed by leadtng edge 01 ~n,ected-gas (dryugas, lronl dlwded by total area of re*erYoll r,rod”ctlve at 51111 01



sweep efllciency’ IReI 5 pages 657 77, and 777 Ret 51 Pages 246 and 247 and Rel 13 Pages 308-09)’



Pattern



TERMS USED IN RESERVOIR CYCLING OPERATIONS



IhoS-weIghted) Eil~ciency



lnvas~on Elflcency



Hydrocarbon pore space enclosed by Ihe pro,ectlo” jrhrough full resewxr Ihlckness) of leadmg edge of drygas front diwded bv total



sweep elficlency’ iReI 5. Pages 755 763 and 770 and Ret 13 pages 40s09)‘.



Displacement



Hydrocarbon pore space invaded by (contacted Or affected by) dry gas dlwded by hydrocarbon pore S!XXX’ enclosed by Itw pro,ecmn (Ihrouqh full leservolr



Elf,c~ency caused by permeab!My stral!flcatlon IRet 13. pages 408-09)



Conformance Sweeping (&I 57)



laclot’ elhclency



Pattern elilclency’ (M 60 pages 63 64. 98 and 99 and Rel 54 Page 77)



lRel



56



Pages



Conformance 571



Reservar



Efflclency



volume Wet hydrocarbon swepl out of lndlvldual pores or Small groups 01 pores dwded bv



Displacement etficlency (Rel 56. Pages 130 and 136 and Ret 13 Pages 408-09)



Flood efiumcy (Rel Pages 358 and 374)



Flood coverage’ IRel 59 pages 358 and 374,’



HANDBOOK



Cycling



Efflcencv



Reservmr we, hydrocarbons recovered d”rl”g Cycling dwded by resewo~r we, hvdrocarbons I” place at starI ai cycl,ng (calculated at same temperature and pressure)



sweep pages



ehxncy IRet 5 612 771. and 7881



59



130 and 136) factor’



(Ref



D6placement’ Page 110)



l&f



61



‘Flushing elf,cencv’ (Ref 4 1, Pages 246 and 247)



preciably according to what combination of the two extremes may exist for a given reservoir. Mathematical models can handle reservoir heterogeneities, both horizontally and vertically, if the data are available. Maximum use of core analysis data, pressure buildup and drawdown analysis, and detailed analysis of downhole logs is required to ensure an accurate evaluation of a reservoir’s potential as a cycling project. Displacement Efficiency. ED is the volume of wet hydrocarbons swept out of individual pores or small groups of pores divided by the volume of hydrocarbons in the same pores at the start of cycling; note that both volumes must be calculated at the same conditions of pressure and temperature. This term is used here because it has received wide acceptance in the literature (on immiscible as well as miscible processes) for the microscopic displacement of fluids. Displacement efficiency is controlled mainly by the miscibility of the driving and driven fluids and their mobilities. For a cycling operation in which the pressure is being maintained at or above the dewpoint, the displacement efficiency resulting from action of the dry gas against the wet-gas phase in the individual pores will be virtually 100% because of nearcomplete miscibility and the near-identical mobility ratios of the two fluids. If the pressure is well below the dewpoint, the displacement efficiency will be less than 100% because of the immobility of the condensed liquid and incompleteness of revaporization of the dry gas. Evaluation



of a case of this type requires trial calculations of vapor/liquid equilibrium to estimate the extent to which dry gas coming into contact with the condensed liquid would revaporize some of the components and carry them toward the producing wells. Thus the reservoir cycling efficiency is the product of the pattern (&S-weighted), invasion, and displacement efficiencies, as summarized in Table 39.11, along with the previous discussion, and usage of terms appearing in some of the literature. Permeability Distribution. Permeability variation, both laterally and vertically, can have a strong influence on recoveries by cycling. Vertical stratification of horizontal permeability is probably the primary factor controlling invasion efficiency. In reservoirs containing layers or regions of contrasting permeabilities, the leading edge of the dry-gas front (used in calculating invasion efficiency) is at a different position for each layer. Field observations usually establish the front on the basis of breakthrough in the most-permeable layer, whereas mathematical model studies may have been based on an average permeability of layers or a discrete number of layers. thus predicting later breakthrough. This possibility should be understood when model predictions of breakthrough time are compared with field observations. Detailed reservoir analysis is required in developing a mathematical model to ensure that the model used adequately reflects the properties of the reservoir.



GAS-CONDENSATE



39-19



RESERVOIRS



TABLE 39.12-CALCULATIONS ILLUSTRATING THE DILUTION CAUSED BY WEIGHTED-AVERAGE PERMEABILITY PROFILE-BASED ON 16 WELLS (COTTON VALLEY BODCAW GAS-CONDENSATE RESERVOIR)



1866 ,860 1855 1825 I8 10



14 64 37 20 74 20 36 50 18100



77 4 78 9 80 4 84 3 86 2 88 2 90 4 1000 105 7



There can be several sources of comparative permeability information for reservoir layers, including direct measurements of permeabilities on cores removed from wells, formation tests during drilling and completion, comparative transmissibilities from carefully run injection profiles, and flow, drawdown, and buildup tests on wells completed in different layers. If different kinds of information are to be used together, they should all be adjusted to the same units for calculating the effects of permeability variation on gas-condensate reservoir performance. Much discussion has been published regarding the effects of permeability variation on the recoveries of hydrocarbons from reservoirs. Discussions with particular reference to as-condensate reservoirs have been provided by Muskat, B+I Standing et al., 65 Miller and Lents, 66 and others. 67-70 Generally, the proposals to account for the effect of permeability variations on gas-condensate reservoir performance use two different methods of wellto-well averaging of horizontal permeabilities. The first method averages all high permeabilities from all wells together (irrespective of vertical positions of the highpermeability samples in the section) and all low permeabilities from all wells in another group, with intermediate permeabilities classified into one or more subgroups. Each of the average permeabilities is regarded as a single stratum continuous throughout the reservoir. This type of averaging would appear to give maximum probability of computed early breakthroughs of dry gas to producing wells. In the second method, permeabilities are averaged from well to well according to vertical position in the sec-



3 14 6 16 1195 14 78 25 83 28 53 33 82



15 39 19 20 22 96 57 35 33 37 38 45 45 52 106 67 150 2173 239 0



89 30 95 20 10280 10740 11090 1,430 118 10 122 20 126 70 131 60 139 60 148 30 159 20 275 30 503 50



36 43 48 50



1 23 1 36 1 19 76 3 77 4



40 76 55 88



55 40 57 41 66 84 70 49 77 54 79 24 84 15 87 29 90 31 91 76 94 40 98 17 99 30 99 74 10000



22 46 20 76 15 85 12 71 9 69 8 24 5 60 1 88 0 70 0 26 0 00



0 0 0 1 0



89 38 27 52 62



78 4 80 9 81 9 83 0 84 0 88 3 90 8 91 9 92 9 93 7 94 5 96 2 96 8 96 3 97 2 97 6 97 8 99 4 1000



tion. For example, permeabilities in the top 10% of each well’s productive section might all be averaged together, the next 10% together, and so on to the bottom. This procedure maintains layers in their relative vertical positions in the reservoir, and thus, by averaging laterally, the effects of any individual high-permeability samples tend to be damped out unless high-permeability streaks are actually persistent in one or more layers of the section. Either of these methods can be used in solutions presented by Muskat, 5XA who used the “stratification ratio” to develop mathematical means of evaluating the effects of vertical variation of permeability on cycling. “Stratification ratio” is the ratio of the permeability of the mostpermeable recognizable layer in the section to that of the least-permeable layer in the same section (these permeabilities are the layer average in each case, determined by whatever means, rather than individual high or low permeabilities from single plugs or cores from the layer). The Muskat development also assumes simple parallel superposition of layers of different horizontal permeabilities with no crossflow between. The resultant correlations are presented graphically in the references. Miller and Lents66 used the second type of lateral permeability averaging in their analysis of the Cotton Valley Bodcaw reservoir. Their work should be reviewed for an understanding of the detailed procedure used. The table of permeabilities they developed (rearranged in descending order of magnitude) for illustrating the calculation of dilution behavior of the subject reservoir with time is shown here as Table 39.12. The calculation assumes no



PETROLEUM ENGINEERING



39-20



HANDBOOK



sweep is sufficiently great in length. Few reservoirs conform to a parallel deposition of lens, each of different uniform permeability, unless one wishes to subscribe to the worst possible consequences for cycling, which can condemn the application of such a program in a rich gas-condensate field. Such unpublished information as has come to our attention tends to substantiate the belief that most reservoirs are not composed of continuous layers of contrasting pcrmeabilities (with no crossflow) that would tend to produce quick breakthrough during injection operations. Hurst’s viewpoint should therefore be considered seriously by the engineer predicting the behavior of cycling projects, because overemphasis on the permeability variation within a reservoir could produce too pessimistic a view of possible recoveries and thereby condemn cycling in gascondensate reservoirs that might, in fact, yield profitable cycling performance. The second method for lateral averaging of permeabilities is recommended, whether the Miller and Lents66 analysis or other techniques are applied to the handling of permeability variation in gas-condensate reservoirs. Proper consideration for pattern (&S-weighted) efficiency must be given in each case.



Fig. 39.7~-Boundary of invaded area predicted by early potentiometric model studies.



crossflow, and the reservoir is treated as though it were composed of alternating layers of variable porosity and permeability. It is also assumed that parallel flow occurs simultaneously in the various layers with the same potential distribution throughout the layers. The injection wells are treated as a line source, and the producing wells as a “line sink.” Hence, the calculations in the table predict the percentage of original reservoir hydrocarbon volume at constant pressure produced at the instant each layer has been displaced and the percentage of dry gas (and wet gas) in the producing stream as more and more layers are displaced (breakthrough). The recovery to any stage of dilution in the produced gas can then be predicted; the recovery Miller and Lents calculated (supported by later production history, as shown by Brinkley’ss5 Fig. 7) is in good agreement with predictions from Muskat’s correlations. Very little has been published comparing the actual behavior and final recoveries of gas-condensate reservoirs with those predicted with the different methods of accounting for permeability variation. Stelzer63 reports on the performance of the Paluxy gas-condensate reservoir of the Chapel Hill field, TX, the cycling behavior of which had been predicted earlier by Marshall and Oliver. 58 This analysis is discussed further later. In a discussion of Stelzer’s paper, Hurst takes the position that permeability variation or stratification in a reservoir can be of minor significance in controlling the ultimate recovery by cycling: The lithological nature of a reservoir is such that with the interspersion of shale throughout, it can virtually reproduce the configuration of a uniform sand if the



Prediction of Cycling Operations with Model StudiesAnalog Techniques. The steady-state flow of fluids through porous media, when governed by Darcy’s law, is analogous to the flow of current through an electrical conductor governed by Ohm’s law. Thus steady-state electrical-model studies have been used quite successfully in the prediction of gas-condensate cycling operations. The fundamental analogy between an electrical model of a gas-condensate reservoir and the flow system of the reservoir depends on the equivalence of electrical charge to reservoir fluid, current flow to fluid flow, specific conductivity to fluid mobility, and potential (voltage) distribution in the model to a function ap, (not to pressure distribution in the reservoir, as in an oil/water system) defined by Muskat as



where pg = gas density, px = gas viscosity, and p = pressure. This analogy holds, provided the sources, sinks, and boundary conditions are made equivalent in shape and distribution. Steady-state models can be divided into two general classes: electronic and electrolytic. The former depends on the movement of electrons through resistive solids, such as metal sheets, carbon paper, and graphiteimpregnated cloth or rubber sheeting. Electrons are introduced at one boundary and move into the model to displace free electrons throughout the entire body of the model. The electrons moving out of the model at the other boundary produce a current that causes a potential drop in the solid resistive medium in accordance with Ohm’s



GAS-CONDENSATE



RESERVOIRS



law. As a result, the movement of the equivalent fluid interface can be traced. In the case of a graphiteimpregnated cloth model, the reservoir is represented by layers of cloth, the number of layers of which are some function of the permeability/net-thickness product (kh) of the producing strata. The shape of each layer of cloth conforms to the shape of the kh range it represents. Copper electrodes are fixed in the cloth model at positions corresponding to the wells in the reservoir and direct currents are passed through these electrodes in proportion to the well flow rates. The electrodes are not usually scaled to the actual well diameters. Electrolytic models depend on the mobility of the ions in the medium. Because the velocity of an ion in an electrolyte system is proportional to the potential gradient, just as the velocity of a liquid particle in a porous medium is proportional to the pressure gradient, an electrolytic model can be set up that provides a good analogy to singlephase flow in a porous system. The ions are moved into the model across one or more boundaries and displace ions throughout the entire medium, causing ions to leave through other boundaries. The flowing current and potential drop are established in exactly the same way as in the electronic models. Electrolytic models can be divided into three major types: gel, blotter, and liquid. Although the first two types can be used to determine the area1 sweep patterns in twodimensional uniform media, the potentiometric model that uses a liquid electrolyte is the most flexible and accurate. In this type, the fluid conductivity of the porous medium is usually represented by an open container that has its bottom shaped to produce electrolyte depths proportional to the kh of the producing strata and its sides shaped to conform to the productive limits of the strata. This construction implies that there is no vertical variation in permeability and no bedding at any location in the reservoir, as represented by the model. Copper electrodes (not scaled to well diameter) are fixed in the model at positions corresponding to the locations of the wells in the reservoir, and alternating currents of proper phase are passed through these electrodes. The magnitudes of these currents are made proportional to the production and injection rates to be used in the reservoir. The direction of current flow at every point in the model is considered analogous to the direction taken by the flowing fluid in the reservoir. The general assumptions applicable to steady-state analog techniques are that (1) a vertical and discrete interface exists between the displacing and the displaced phases; (2) because the history of advance of only one front can be traced at any one time, if two interfaces or fronts are present (such as gas/gas and gas/water), one is considered a stationary boundary; (3) average reservoir pressure is constant regardless of the injection or production schedule (this avoids compressibility effects in the model study); and (4) gravitational effects are neglected. In addition, if the mobility ratio of the system is not (near) unity or infinity, the necessary procedures become tedious and costly. An example case history by Marshall and Oliver5* reported results of a potentiometric model study of the Paluxy sand reservoir of the Chapel Hill field. Smith County, TX. This gas-condensate reservoir is bounded on the north by a gas/water contact, on the west by a fault, and on the south and east by a pinchout. It was assumed



A B C D-W E F G HI -C.



I, WALTON #I (INJ) I. WALTON #Z (INJ) W. WALTON “8” #I WALTON #I S. WALTON # I H CAMPBELL #I B MOSLEY #I M WARREN #l-A G



------PHASE



FINCH



#I PHASE PHASF



I II III



Fig. 39.8-Boundaries of invaded areas predicted by later potenliometric model studies.



that the gas/water contact was a fixed impermeable boundary; that the permeability, porosity, and interstitial water content were each uniform throughout the producing zone; that the reservoir volume rate of dry-gas injection was equal to the corresponding rate of gas-condensate production; and that gravity effects were negligible. Fig. 39.7 shows the final dry-gas/wet-gas interface position at time of breakthrough into Well 1 (determined after several trials of well arrangement and production- and injectionrate schedules) that yielded an optimum pattern (h&Tweighted) efficiency prediction of 83 %. Injection was into Wells 1 and A with production from Wells 2 through 4 and B as indicated in Fig. 39.7. This program provided a sustained capacity of 35 MMscf/D for the life of the operation. Stelzer63 reported a comparison of model study predictions with actual performance for this reservoir. Actual gas injection was begun in accordance with the north/ south sweep indicated by the model study. During the initial period (first 15 months after cycling began) the production- and injection-rate program predicted by the initial model study was followed quite closely. New structural data revealed in the drilling of additional wells, however, required some changes in the isopach map of the Paluxy sand. The results from a second model study, which incorporated these changes plus injection into only Wells A and B, are shown in Fig. 39.8. Three interface boundaries (dry-gas fronts) are shown for three



PETROLEUM ENGINEERING



39-22



ND OF PHASE -



IO



SAME



PHASES 0 0



1 IO



l (ADUSTED



INJECTION



RATES



TO AS,



I



.



II AND III1



I 20



30



40



50



60



70



ACTUAL RESERVOIR OPERATING TIMEMONTHS AFTER START OF CYCLING



Fig. 39.9-Comparison of predicted with actual times of first drygas breakthrough, Paluxy gas-condensate reservoir, Chapel Hill field. TX.



production- and injection-rate schedules. The first schedule was maintained for the first 15 months of cycling; the second was continued until breakthrough of dry gas into Well E; the final schedule was maintained until first breakthrough at Well 1. There was close agreement between the model rates used and actual reservoir rates. The second model study indicated a pattern (h&G weighted) efficiency of 88 % , a 5 % increase over that obtained by the initial study. Stelzer estimated the amount of reservoir gas in place at start of cycling to be 78.4 Bscf. The new model study thus implies an additional 4 Bscf of predicted recoverable gas as a result of better reservoir definition and better operating schedules. The data in Fig. 39.9 compare model (predicted) breakthrough times with the actual times to dry-gas appearance in corresponding field wells. (Phases 1, 2, and 3 of actual behavior correspond to Schedules 1, 2, and 3 of the model study.) Field data on breakthrough were taken from breaks in content curves of isobutanes-plus; the dashed line shows the cumulative well-by-well breakthrough behavior of the dry-gas flood. Because predicted and actual injection and production rates were nearly equal and constant during the period shown (except for Phase 1, which was adjusted to the same average rates), times on the plot are directly proportional to cumulative reservoir volumes of gas. Therefore, the lower light line represents a hypothetical invasion efficiency of 100% that would prevail if actual breakthrough times coincided with those predicted by the model [and the area1 and pattern @@S-weighted) sweeps were identical with model predictions]. The upper light line represents an arbitrary invasion efficiency of 80% [assuming that predicted and actual pattern (h&-weighted) efficiencies are identical]. The straight heavy line from the origin through the last well to experience breakthrough



HANDBOOK



indicates an invasion efficiency a little greater than 90% and implies that more complete invasion of lowpermeability regions behind the dry-gas front was accomplished during the later stages of cycling. The agreement of predicted breakthrough times within 10% of actual breakthrough times illustrates the great utility of potentiometric models in planning cycling operations. Small further improvement in the pattern (k&S-weighted) and invasion efficiencies was to be expected before abandonment of the reservoir in this case. Stelzer’s63 figures (at the start of cycling) of 78.4 Bscf of gas in place and 74 bbl of condensable liquids in the vapor phase of the reservoir per 1 MMscf of gas indicate that 5,800,OOO bbl of condensable liquids is in the reservoir vapor phase at the start of cycling. Using the modelderived pattern (/#-weighted) efficiency of 88% (end of Schedule 3), 5,100,OOO bbl of liquids was subject to removal by dry-gas invasion. Stelzer’s Fig. 5 shows that about 4,640,OOO bbl of liquid products were recovered between the start of cycling and the breakthrough of gas at Well 1. This provides an invasion efficiency of 91% at that time, based on 100% displacement efficiency. Thus the product of the pattern (k&Y-weighted) and invasion efficiencies represents a reservoir cycling efficiency of 80% at the time of breakthrough into Well 1. In addition, later operations increased the cumulative recovery during cycling to more than 5 million bbl of condensable liquids, thus bringing final reservoir cycling efficiency to more than 86 % This is considered very good. Prediction of Cycling Operations With Mathematical Reservoir Simulators. The use of mathematical reservoir simulators to calculate reservoir performance during gascycling operations yields results superior to those obtained by the more simplified calculation procedures. Use of these simulators removes the necessity of making the assumptions required in an analog model. Some assumptions are required, however, which should be understood to perform a reservoir simulation study properly. The theory of reservoir simulation is presented in Chap. 48. Coats7’ presents a good discussion of reservoir simulation studies of gas-condensate reservoirs. One must keep in mind that the results from a mathematical reservoir simulator depend on the quality of the data used to prepare the reservoir model. If good data are not available, one should consider whether the expense and time required to perform a mathematical reservoir simulation are justified. Data Requirements for Gas-Condensate Cycling Study. To evaluate properly the potential of cycling a gas-condensate reservoir, the following data are required. 1. Geologic data-maps and cross sections showing net effective sand thickness, structural contours on the top and base of the productive formation, location of gas/ water interface originally and at the date the model study begins, and location of dry-gas/wet-gas interface at the start of study-and general information on lithology and lenticularity of the productive strata, such as extent of fissures, fractures, caverns, and other special conditions. If a black-oil ring is present, its size and extent should be shown. 2. Physical properties of the reservoir rock-isoporosity map (or average porosity), effective or specific isopermeability map (or average values), and interstitial water content.



GAS-CONDENSATE



RESERVOIRS



39-23



3. Fluid characteristics (produced, and injected where applicable)-fluid composition. retrograde dewpoint pressure of reservoir fluids, gas FVF or specific volume vs. pressure, deviation factor, condensate content of reservoir fluid. viscosity, and densities of liquid and gas phases, all from original reservoir pressure through the range of interest (usually to abandonment conditions). 4. Amount of original fluids in place (derivable from data in Points I through 3). 5. Reservoir pressure history (volumetrically weighted) from discovery to present. If this is not available, isobaric contour maps at the various pressure survey dates should be supplied. 6. Condensate. gas, and water production data, from the date of discovery. 7. Proposed future production rates. 8. Gas- and/or water-injection data, past and future projections. 9. Productivity, injectivity, and backpressure test data on wells. Ultimate Recovery of Gas and Condensate Liquids by Cycling. The same reservoir for which pressure-depletion calculations were made previously can be used to illustrate the effectiveness of a cycling operation. Table 39.8 lists the basic data for predicting the ultimate recoveries of wet gas, condensate, and plant products during cycling at original reservoir pressure (to avoid serious drawdown effects) followed by pressure depletion to abandonment pressure. Productive thickness, porosity. and interstitial water content are each assumed uniform. Consequently, the 79.0% areal sweep efficiency obtained by a potentiometric model study is also the pattern (@S-weighted) efficiency. The invasion efficiency is assumed to be 90% because permeability variations are moderate. Because a dry-gas/wet-gas cycling operation is a miscible flood, the displacement efticiency is essentially 100%. Therefore, the reservoir cycling efficiency would be 7 1.1%. To simplify the example. it is assumed that after cycling, the unswept pore space both inside and outside the dry-gas front will pressure deplete in the same manner as predicted previously for the noncycling case: it will also be assumed economical to recover the butanes-plus from the gas produced.



Reservoir Mvt gas produced during cycling period (original reservoir comnposition): 130.992x0.711=93.135



MMscf.



Reser\vir wet gas produced by pressure depletim ufter cycling (changing cornposition, as shown in pressuredepletion example): 102,045x(1,000-0.711)=29,491



MMscf



Resertjoir tvet gas produced at ahundomnentpressure, 700 p.sig. 93.135+29,491=



122.626 MMscf.



Total separator gas produced (see Table 39.6): During cycling, 777.15+38.52+38.45



x93,135



l,ooO =0.85412x93,135=79,548



MMscf.



During depletion, 696.75+ 14.99+18.05 1,000 =0,72979x29,491



x29,491



=21,522



MMscf.



Total : 79,548+21,522=



101,070 MMscf.



Total condensate produced: During cycling, 181.74x93,135=16,926,355



bbl.



During depletion, 51.91 x29,491 = 1,530,878 bbl Total: 16,926,355 + 1,530,878 = 18,457,233 bbl These figures represent a significant improvement over the recoveries previously estimated for pressure-depletion alone. Noninjection-Gas Requirements in Cycling Operations. The noninjection-gas requirements for cycling can affect the amount of gas available for injection. The amount of gas to be cycled is determined by the optimum pressure level to be maintained and the efficiency of reservoir fluid recovery to be achieved; the amount of gas readily available, including sources and costs; and the design and operating programs for surface facilities. The amount of gas that is economical to cycle through a gas-condensate reservoir varies with many factors, including richness of the vapor at reservoir cycling pressure, size and cost of the plant, and the price of the field products and of dry gas. Miller and Lents% expected to cycle the equivalent of about 115 % of the gas in place to recover some 85 % of the wet-gas reserve of the Cotton Valley Bodcaw reservoir. While Brinkley 55 indicated cycling-gas volumes of as much as 130% of original wet gas in place for various reservoirs, no general correlation has been prebented on the amount of gas that is economically sound to cycle; this should be the subject of a detailed engineering analysis in each case. The makeup gas needed for constant-pressure cycling is mainly the volume required to replace shrinkage by liquid recovery and the amount consumed



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39-24



for various fuel needs. For some composition, temperature, and pressure ranges, the removal of high-molecularweight constituents from the produced wet gas may result in a higher compressibility factor for the injected dry gas; hence, the greater volume per mole injected may require little or no makeup gas for constant-pressure cycling. The amount of gas not available for injection because of consumption for operating needs should be taken into account in determining makeup gas requirements if pressure is to be maintained. The amount of fuel for compression and treatment plants depends mainly on the total amount of gas to be returned to the reservoir and the discharge pressure for the plant. Discharge pressure, in turn, depends on the total rate of injection demanded and the number of injection wells and their intake capacities throughout the life of the operation. Other factors affecting the amount of gas required for overall operations are type of plant, type of liquid-recovery system used, and auxiliary field requirements (such as for drilling. completion, and well testing; camp fuel and power for maintenance shops, general service facilities, employee housing; and other factors that vary from one case to another). Moores4 reports that gas fuel consumption for the compression plant alone varies from 7 to 12 cu ft/bhp-hr; this is probably for gases with heat values of about 1,000 Btu/cu ft. Horsepower requirements per million standard cubic feet of gas compressed per day are correlated in Ref. 16 (Compressor section). An example based on Refs. 16 and 52 shows that, with 8 cu ft/bhp-hr, a compression ratio of 15.0 (compressing from, say, 461 to 7,000 psia) requiring three stages of compression with a ratio per stage of 2.47, and a specificheat ratio of 1.25, the cubic feet of compressor fuel used per million cubic feet of gas compressed can be calculated as follows. For a gas of 0.65 specific gravity and a stage compression ratio of 2.5. the chart in Ref. 16 reads 22 bhp. The allowance factor for interstage pressure drop (three compression stages) is 1.1. Fuel used per million cubic feet of gas compressed = bhp x cu ft of fuel/bhp-hr x ratio/stage x number of stages x allowance factor. Or compressor fuel consumption is m,.=22x8x24x2.47x3x1.1=34.4



MscfiMMscf.



This compares favorably with the factor presented in Moore’s54 Fig. 8. For an example reservoir originally containing 130,992 MMscf of wet gas, which might be cycled the equivalent of 1 l/4times, the approximate compressor fuel consumption would be 130.992x 1.25x34.4=5.633



MMscf.



This is approximately 3 % of the gas handled through the plant. Treatment plant fuel and other plant needs added to compressor fuel bring the range of consumption inside the plant fence to 3 to 7 % of the gas handled by a cycling plant. In addition to these needs and others mentioned earlier, possible gas losses can occur in a cycling operation: gas used in “blowing down” wells, should this be necessary for cleaning or treating purposes; small gas leaks at compressor plants and in field lines; and gas leaks



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resulting from imperfect seals or corrosion in well tubings, casings, and cement jobs. Remedial workover operations should be planned immediately when there is evidence of appreciable loss of gas between the compression plant and the reservoir sandface or between the outflow-well sandface and the plant intake. Combination



Recovery Procedures



Partial water drive-conditions of natural water influx at rates too low to maintain pressure completely at the desired production rates-can exist for gas-condensate reservoirs. In such cases, operation may be by partial water drive and depletion, supplemental water injection, or partial water drive and cycling. Prediction of reservoir behavior and recovery under these conditions requires knowledge or assumptions about the aquifer and the water drive it supplies. This information can be deduced from a study of geologic conditions and early producing history of the reservoir; sometimes the deductions are accurate, sometimes not. Projections of water drive magnitude into the future at selected reservoir pressure levels can be made by methods developed in Refs. 72 and 73. If sufficient early producing history of a reservoir is available, it can usually be matched (simulated) by a mathematical reservoir simulation study. The future behavior of the reservoir can then be predicted under the following producing methods: (1) producing history and ultimate recovery of gas and liquids under partial water drive and pressure depletion at the selected production rate; (2) amount of supplemental water injection required to maintain reservoir pressure fully at the selected pressure level and production rate; and (3) size of cycling plant required to maintain pressure at the selected pressure level and production rate.



General Operating Problems: Well Characteristics and Requirements As with any complex operation, gas-condensate recovery projects have many operating problems. Those pertaining to the plant, lines, and other surface facilities are best left to experienced plant and maintenance personnel, except as they affect reservoir operation (e.g., compressor-oil or corrosion-products carry-over into wells). Operating difficulties occurring at and below the wellhead are often concerns of the reservoir engineer and have an important bearing on the effectiveness of reservoir operation, whether by pressure depletion or by pressure maintenance. Among these are the maintenance of injection and production wells in good mechanical condition, the protection of wells against excessive corrosion, the general maintenance of well injectivity and well productivity (which are often interrelated), and the formation of hydrates that can interfere with the general injection and/or production operation. Well Productivity and Testing It is essential to maintain the producing capacities of gascondensate wells above minimum levels for good economic performance. Much has been written about the productivities of gas and gas-condensate wells, their general producing characteristics, and the optimum methods for testing and reporting their productivities. Loss of productivity of gas-condensate wells can occur from reservoir



GAS-CONDENSATE



39-25



RESERVOIRS



pressure decline (including possible effects from condensation of liquids in the reservoir and consequent reduction of effective gas permeability), from the invasion of water into producing wells, from solid precipitates in the pore space, from formation damage during well killing or workover operations, and from mechanical failure of downhole equipment. The engineer must have indices at his disposal that show the productivity histories of wells and whether productivity decline is excessive for prevailing producing conditions. Productivity Testing. In making productivity tests on wells, orderly well-conditioning and overall test procedures should be used. as suggested in Chap. 33 or in standards recommended by Texas, 26 New Mexico,” Kansas, 28 and the Interstate Oil Compact Commission. 2y It is common to use wellhead pressures in determining well productivity (or injectivity) characteristics with arbitrary correction procedures for estimating BHP’s from the observed surface pressures. No fully satisfactory methods have been devised for making accurate estimates of gas-condensate well BHP’s, either static or flowing. Calculated static pressures can have serious uncertainties because of unknown amounts of liquid hydrocarbons or water in the wellbore and tubing and unknown temperature distribution. Calculated flowing pressures can have uncertainties because of inaccuracies in the detailed temperature distribution and the particular friction factor assumed for each specific case. Lesem er ~1.‘~ provide helpful charts for approximating the temperature distribution in flowing gas wells. Errors and uncertainties of the above nature become worse as well depths increase. Consequently, for best results, downhole pressure measurements with accurate gauges should be used. Where this is not feasible, BHP’s may be estimated from surface pressure readings for gas-condensate wells with better accuracy than is usually true for oil wells. Chaps. 33 and 34 discuss methods for making such estimates. For these methods, measured fluid properties (e.g., density) should be used whenever available in preference to calculated or correlation values. For gas and gas-condensate wells, a plot of static and producing BHP’s vs. producing rates (in millions of standard cubic feet per day) is not a straight line. Smooth curves with closer approximations to straight lines can be obtained by plotting squares of the static and producing well BHP’s (absolute) vs. producing rate. A rough analogy to oilwell behavior is then obtained by plotting the differences in squares of the static and producing pressures vs. the corresponding producing rates (usually on log-log paper). If several pressures are obtained on a well at different rates, these procedures do not always yield straight-line relationships (see Chap. 33 and Ref. 75); however, they provide reasonable indices for limited extrapolation to future well behavior and for comparison of current with past well behavior. Estimation of future well productivity can be made by modifying initial well productivity to account for the changes in reservoir pressure and gas permeability as pressure declines and liquid is deposited in the pores. For no loss of gas permeability, a new productivity line can be drawn on the plot of pressure squared vs. rate, parallel to the original productivity line and through the square of the new static pressure selected: this yields an estimate of flowing rate for any



flowing pressure selected. If the original curve for rate vs. difference in squares of static and flowing pressures is used, rates can be estimated for any future flowing pressure by using the proper (future) static pressure; lowpermeability wells would require special adjustment of earlier isochronal test data obtained (see Chap. 33 and Ref. 7.5). These methods yield approximations of future productivity as affected by pressure decline in the absence of fluid-phase and viscosity changes in the reservoir. If gas permeability, k,, is likely to be seriously affected by condensation of liqutds in the pores (and gas viscosity by pressure decline), then the change in gas mobility k,/p,, must be approximated and radial-flow calculations made (see Chap. 35) to estimate the new productivity curve corresponding to the static pressure selected for prediction. Normally, the two aforementioned types of productivity estimates ignore the drawdown effects of production on liquid-phase distribution in the reservoir and any consequent additional reduction of gas permeability near the producing wells; minimum calculated reduction of productivity should, therefore, result from these two estimating methods. Large deviations from such estimates, based on a well’s early characteristics, would indicate that the well should be analyzed for productivity troubles. Excessive Productivity Loss. If the capacity of a producing well declines abnormally compared with that predicted from its original productivity (in the absence of excessive water production), and if appreciable liquid condensation around the wellbore within the formation is suspected, efforts to improve well productivity should be made. These could include the short-term injection of dry gas into the well (several days to several weeks) to evaporate part of the liquid, followed by immediate production to remove some of the vaporized liquid block. Loss of well productivity caused by excessive water production has been discussed briefly. In some cases, well workover operations would be justified to reduce or to shut off water entry. Other factors that can influence well productivity are deposits on the sandface or in the pores near the wellbore, perhaps caused by salts precipitated from reservoir water: any mechanical damage resulting from killing the well for pulling equipment or workover: mechanical failure of downhole equipment; and possible hydrates (see Chap. 33). In case of well productivity injury for mechanical reasons, conventional methods of well repair should be undertaken on the basis of the particular difficulty involved. Various means are available for stimulating lowproductivity wells; see Chaps. 54 and 55 and discussions by Clinkenbeard et al. 76 Well Injectivity Maintenance of well injectivity is essential for the economic operation of cycling programs. Injectivity decline can be caused by sandface plugging or by buildup of reservoir pressure. Lnjectivity Testing. The characterization of gas-injection wells is similar to that for gas-producing wells. In either case, analysis is made on the basis of plots of rates vs. the squares of BHP’s or rates vs. differences of squares



39-26



of pressures. Consequently, after suitable well conditioning. as previously described, injectivity testing should consist of a series of injection rates at different pressures to establish the early injectivity performance of the well when well conditions are known to be good and the sandface is clean. If facilities are not available for obtaining a range of injection rates and pressures, it is sometimes acceptable to obtain production rates and pressures for the injection well through a reasonable range and use the pressure-squared relationship for extrapolating across the zero-rate axis into higher injectionpressure ranges to approximate well characteristics. Plots of production rate vs. difference in squares of pressure can also be adapted to estimate later well-injectivity behavior. As in the case of producing wells, if injectivity declines with time, analysis of well conditions is required to decide whether corrective procedures should be used. If a gascondensate reservoir is being operated essentially at constant pressure, then the obvious index of injectivity decline is whether the rate for each injection well remains constant at the injection-well pressure. Injection-rate decline at constant well pressure or injection-pressure rise at constant irrjection rate shows that injectivity is declining. Injection-Well Plugging. Plugging of the sandface can occur in injection wells. This may result from liquid carryover from the compressors (probably lubricating oil components) or from corrosion products from surface lines or well equipment. Carry-over of lubricating oils from compressors can be serious. Usually. the remedy is to install high-efficiency aftercoolers, scrubbers, and/or mist extractors on the discharge side of the compressors. A particularly el’fective combination for this is the use of “drips” or collectors, followed by plate or screen impaction-type mist elitninators. followed by combination fibrous and wire-mesh filter elements. When liquid-blocking of the sand around an injection wellbore cannot be relieved by backflowing (as mentioned later), consideration can be given to “slugging” the well with suitable volatile solvents. The solvent used should preferably be miscible with both the normal injection gas and the liquid that is suspected to be blocking the pores. While propane is a good solvent for many hydrocarbon liquids, some lubricating oils have constituents not soluble or miscible with propane. In these cases, other solvents (possibly nonhydrocarbons) should be used. Sometimes solvent injection is followed immediately by resumption of dry-gas injection. If successful, this dissolves part or all of the liquid block and spreads out the materials in the reservoir sufficiently to relieve the problem. In other cases, the solvent is injected into the formation for short periods and then produced back out to provide a type of washing intended to remove the liquid accumulation from the formation. Corrosion products from steel lines between compressor discharge and the sandface can also provide serious well plugging. All well piping and casing and all surface lines should be cleaned thoroughly before they are installed to avert as much as possible the transportation of fine corrosion products to the sandface when injection starts. For continued protection during the life of injection equipment, liquid carry-over and mist-elimination measures should be combined with adequate control of corrosive



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agents in the field gas. Sometimes the use of internally coated or lined pipe is justified. These and other corrosioncontrol procedures are best carried out with the help of a competent corrosion engineer. Corrosion products that plug the sandface are sometimes removed by backflowing the injection well to blow the material off the sand and out of the well. Where this is feasible, such complete removal of the plugging agents from the borehole is believed to be the best for the well. Other remedies may include treating the well with inhibited hydrochloric acid to dissolve the corrosion products. Sometimes the acid is pushed back into the formation and injection is started immediately without backwashing or backflowing of the well. If repeated periodically, this procedure is questionable because it is possible to develop plugging farther away from the well face that could ultimately hinder injection and be difficult to correct. Number of Wells Required The number of wells used in exploiting gas-condensate reservoirs has varied from the equivalent of less than I60 acres/well to more than 640 acres/well. Bennett” discussed the general problem and pointed out that the first wells are “drilled to determine the upper and lower limits of condensate production; to determine the extent of the pool, the net pay, thickness, porosity. etc.; and to provide suitable production or injection wells to fit a final pattern,” which will not necessarily have a regular geometrical design. The number of wells to be drilled for gas-condensate operations must be analyzed for each specific case. Important factors to be considered are (I) contract commitments to deliver gas and products, (2) capacity of plant to be served, (3) productivities and injectivities of the wells, (4) maximum practical pattern (&S-weighted) efficiencies, controlled by number and location of wells (reservoir geometry is an important consideration), (5) amount of recoverable hydrocarbons and their value, and (6) project costs, including well-development costs. Items 3 through 5 must be balanced against Items 1. 2, and 6 to ensure that the economic objectives and contract commitments of the project are met. If wells are low in capacity, extra wells may be needed to meet production requirements during periods of well repair or workover.



Economics of Gas-Condensate Reservoir Operation Arthur” and Boatright and Dixon79 published discussions on the economics of cycling gas-condensate reservoirs. Arthur concluded that the most profitable method of operation depends on many factors. and the answer cannot be generalized. The following factors adapted from Arthur’s list are considered important. 1. Reservoir formation and fluid characteristics, including occurrence or absence of black oil, size of reserves of products, properties and composition of reservoir hydrocarbons, productivities and injectivities of wells, permeability variation (controls the degree of bypassing of injected gas), and degree of natural water drive existing. 2. Reservoir development and operating costs. 3. Plant installation and operating costs. 4. Market demand for gas and liquid petroleum proaucts.



GAS-CONDENSATE



39-27



RESERVOIRS



5. Future relative value of the products. 6. Existence or absence of competitive producing conditions between operators in the same reservoir. 7. Severance, ad valorem, and income taxes. 8. Special hazards or risks (limited concession or lease life, political climate, and others). 9. Overall economic analysis. In choosing between pressure depletion and pressure maintenance as operating methods for a gas-condensate reservoir, detailed analyses must be made for predicting optimum economics. Cycling and gas processing procedures require sizable plant expenditures. Possible processing methods, whether reservoir fluids are cycled or not, include stabilization. compression, absorption, and fractionation. The last two recover appreciably more condensables from wet gas than do the first two. If the removal of ethane from a gas stream is desirable for economic or other reasons, fractionation should be used. When reservoir characteristics appear favorable for recovery of condensable hydrocarbons, it must be considered whether cycling would be economical. The primary comparison is between value of the estimated additional recovery of liquid products by cycling and the actual cycling costs, taking into account deferment of gas income and other factors. Economic analyses of cycling and noncycling are required and must be carried out in detail for maximum dependability with information factors and assumptions pertinent to each particular case. General information on valuation of oil and gas properties is given in Chap. 41. Economic comparisons are of no value unless reasonably accurate predictions of physical reservoir behavior can be made. Consequently. in the gas-condensate reservoir case. the information given previously would have to be expanded to include schedules of annual production and injection volumes derived from the physical characteristics of the reservoir and from the external factors that would affect production rates. Schedules of investment, anticipated prices of products. operating costs, and taxes would also be required to complete the detailed information needed to make comparative economic analyses.



Nomenclature 8, = gas expansion factor (gas FVF) E, = area1 sweep efficiency ED = displacement efficiency E, = invasion efficiency E,, = pattern (h4.5weighted) efficiency ER = reservoir cycling efficiency El/ = volumetric sweep efficiency h= net pay thickness, ft k= permeability. md k,, = relative permeability to gas, fraction k,, = relative permeability to oil, fraction K= equilibrium ratio P= pressure, psi S= hydrocarbon fluid saturation of the pore space, %’ layer number deviation factor (compressibility factor) gas viscosity, cp oil viscosity, cp



0s = gas density. g/cm3 4 = porosity, X +,s = flow potential, psi



References



13. I?. IS. 16. D~,qirzerrirlg Daicr Book. ninth edItion. Gas Processors Suppliers Astn. and Natural Gas Assn. of America. Tulsa. OK (1981). 17. Katz. D .L. and Rzaaa. M .J. : B&liogrtrphJ .fiw Phwicd Ba/w\?or of Hw/mcarbons Under Prr.wurr md Rrlorrrl Phr~romv~~r. J W. Edwards Publisher Inc.. Ann Arbor (1946). 18. Genercrl Inde.~ to Pmolrum Puhlrcrrtiom of SPE-AIME. SPE, RIchardson, TX (1921-85) 1-5. 19. fnrfe.r c~Di~~isiori of’Producrion Prqwrs. 192 7- 1953. API. New York City (1954). 20. Sloan. J.P.: “Phase Behavior of Natural Gas and Condenaate Systems.” Pet. Eq. (Feb. 1950) 22. No. 2. B-54-8-64. 21. Dodson, C.R. and Standing. M.B.: “Prediction of Volumetric and Phase Behavior of Naturally Occurrq Hydrocarbon Systems.” Drill. trnd Pmcl. Pruc~.. API (194 I) 326-40. 22. Organick. E.L.: “Prediction of Critical Temperatures and Critical Pressures of Complex Hydrocarbon Mixtures.” Clirw. &q. Prqq. (1953) 49, No. 6, 81-97. 23. Carr. N.L.. Kobayashi, R.. and Burrows. D.B.: “Vlscnsity of Hydrocarbon Gases Under Pressure.” J. Per. Tdz. (Oct. 1954) 47-55: Tiww.. AIME. 201. 24. Chew, J.N. and Connally, C.A. Jr.: “A Viscosity Correlatmn for Gas-Saturated Crude Oils.” Trcrn.s.. AIME (19.59) 216. 23-25. 25. “API Recommended Practice for Sampling Petroleum Reservoir Fluids.” API RP 44. first edttion. Dallas (Jan. 1966) 26. Bock-Prc~ssurz Tesr for IV&~& Co.\ l+‘c,l/\ Texas Railroad Commission. Austin (1985). 27. Mutzual for Back Pwssuw Tcsr,fi~r Nuruml Grrs We//.\, New Mexico Oil Conservation Commission. Santa Fe (1966). 28. Munuul ofBud Pressurc~ Tcdna rf Gav W~,i/s. Kansas State Corp. Commission, Topeka (1959). 29 A Su,qp~fed Mumud for Standurtl Buck- Pw wuc Tc.viqq Mm’ml.~ Interstate Oil Compact Commission. Oklahoma City (1986). 30. Whitson. C.H.: “Characterizing Hydrocarbon Plus Fraction\.” paper EUR I83 presented at the 1980 SPE European Offshore Pctroleum Conference and Exhibition. London. Oct. 2 I-24. 71. Watson, K.M.. Nelson. E.F.. and Murphy. G.B.: “Characterization nf Petroleum Fractions.” fncl. ,%K. Chrw. 11935) 27. 1460-64.



39-28



32. Hoffman, A.E.. Grump, J.S.. and Hocott. CR.: “Equilibrnnn Constants for a Gas-Condensate System,” Trans., AIME (1953) 198. l-10. 33. Cook, A.B., Walker, C.J., and Spencer, G.B.: “Realistic K Values Hydrocarbons for Calculating Oil Vaporization During Gas OfC,, 1969) 9Oi-15; Cycling at High Pressures,” .I t’e;r. Tech:(July Tram.. AIME. 246. 34. Nemeth. L.K. and Kennedy. H.T.: “A Correlation of Dcwpoint Pressure With Fluid Composition and Temperature.” Sot. Per. Enx. I. (June 1967) 99-104. 35 Allen. F.H. and Roe, R.P.: “Performance Characteristics of a Volumetric Condensate Reservoir,” Trans., AIME (1950) 189, 83-90. 36. Berryman, J.E.: “Predicted Performance of a Gas-Condensate System. Washington Field, Louisiana,” J. Per. Tech. (April 1957) 102-07: 7-runs.. AIME, 210. 37. Rodgers, J.K., Harrison. N.H.. and Regier, S.: “Comparison Be~ween the Predicted and Actual Production History of a Condensate Reservoir.“J. Per. Tech. (June 1958) 127-31: Trans., AIME. 213. 38. Redlich, 0. and Kwong, J.N.S.: “On the Thermodynamics of Solutions V. an Equation of State Fuaacities of Gaseous Solutions.” Chem. Review (1949) 44. 233. 39. Peng. D.Y. and Robinson, D.B.: “A New Two-Constant Equation of State.” Ind. Eng. Chum Fundamentals (1976) 15. 15-59. 40. Martin, J.J.: “Cubic Equations of State-Which?” Ind. GI,~. Chrm. Fundamentals (May 1979) 18, 81, 41. Petroleum Conservafion, S.E. Buckley ef al. (eds.), AIME, New York City (1951). 42. Geffen. T.M. @al.: “Efticiency of Gas Displacement From Porous Media by Liquid Flooding.” Trans., AIME (1952) 195, 29-38. 43. Yuster, S.T.: “The Rehabilitation of Drowned Gas Wells,” Drill. and Prod. Prac.. API ( 1946) 209- 16. 44. Bennett. E.N. and Auvcnshine. W.L.: “Dewatering of Gas Wells,” Drill. and Prod. Prac.. API (1956) 224-30. 45. Dunning, H.N. and Eakin, J.L.: “Foaming Agents arc Low-Cost Treatment for Tired Gassers.” Oil and GasJ. (Feb. 2. 1959) 57. No. 6, 108-10. 46. Bates, G.O., Kilmer, J.W., and Shirley, H.T.: “Eight Years of Experience with Inert Gas Equipment.” paper 57-PET-34 presented at the 1957 ASME Petroleum Mechanical Engineermg Conference. Sept. 47. Barstow, W.F.: “Fourteen Years of Progress tn Catalytic Treating of Exhaust Gas.” paper SPE 457 presented at the 1973 SPE Annual Meeting. Las Vegas, Sept. 30-Oct. 3. 48. Eckies. W.W. and Holden, W.W.: “Unique Enhanced Oil and Gas Recovery Project for Very High Pressure Wilcox Sands Uses Cryogenic Nitrogen and Methane Mixture,” paper SPE 9415 presented at the 1980 SPE Annual Technical Conference and Exhibition, Dallas. Sept. 21-24. 49. Moses, P.L. and Wilson, K.: “Phase Equilibrium Considerations in Utilizing Nitrogen for Improved Recovery From Retrograde Condensate Reservoirs,” paper SPE 7493 presented at the 1978 SPE Annual Technical Conference and Exhibition, Houston. Oct. l-4. 50. Peterson, A.V.: “Optimal Recovery Experiments with Nz and co, \.’ Pet. Enx. Inrl. (Nov. 1978) 40-50. 51. “Physical Prop&es of Nitrogen for Use in Petroleum Reservoirs,” Eu[/. 1 Air Products and Chemical Inc.. Allentown. PA (1977). 52. Wilson, K.: “Enhanced-Recovery Inert Gas Processes Compared,” 011 and Gas J. (July 31, 1978) 162-72. 53. Donohoe. C.W. and Buchanan, R.D.: “Economic Evaluation of Cycling Gas-Condensate Reservotrs With Nitrogen.” paper SPE 7494 presented at the 1978 SPE Annual Technical Conference and Exhibition, Houston, Oct. 1-4. 54. Proc.. Ninth Oil Recovery Conference, Symposium on Natural Gas m Texas. College Station, TX (1956).



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s.5. BrinkIcy, T.W.. “Calculation of Rate and Ultimate Recovery from Gas Condensate Reservoirs.” paper 1028-G presented at the 1958 SPE Petroleum Conference on Production and Reservoir Engineering.” Tulsa, OK, March 20-2 I. 56. Patton, C.E. Jr.: “Evaluation of Pressure Matntenance by Internal Gas Injection in Volumetrically Controlled Rcscrvoirs.” Trrr,r.s.. AIME (1947) 170, 112-55. 57. API Standing Subcommittee on Secondary Recovery Methods, Circ. D-294. API (March 1949) Appendix B 58. Marshall, D.L. and Oliver, L.R.: “Some Uses and Limitations 01 Model Studies in Cycling,” Truns., AIME (1948) 174. 67-87. 59. Calhoun, J.C. Jr.: Fitndarnenrals r$Rrsrrwir EnKineerrnX, U. of Oklahoma Press. Norman (1953) 358, 374. 60. Hock, R.L.: “Determination of Cycling Efficiencies in Cotton Valley Field Gas Reservoir,” Oil alrd Gus J. (Nov. 4, 1948) 47. No. 27, 63-99. 61. Calhoun. J.C. Jr.: “A Resume of the Factors Governing Interpretation of Waterflood Performance,” paper presented at the 1956 SPE-AIME North Texas Section Secondary Recovery Symposium, Wichita Falls, Nov. 19-20. 62. Pirson, S.J.: Oil Reservoir Emginrering, McGraw-Hill Book Co. Inc.. New York City (1958) 406. 63. Stelzer, R.B.: “Model Study vs. Field Performance Cycling the Paluxy Condensate Reservoir,” Drill. und Prod. Prur., API (1956) 336-42. 64. Muskat, M.: “Effect of Permeability Stratification in Cycling Operations,” Trans., AIME (1949) 179. 3 13-28. 65. Standing. M.B., Linblad. E.N.. and Parsons. R.L.: “Calculated Recoveries by Cycling from a Retrograde Reservoir of Variable Permeability,” Trans., AIME (1948) 174, 165-90. 66. Miller, M.G. and Lents, M.R.: “Performance of Bodcaw Reservoir. Cotton Valley Field Cycling Project, New Methods of Predicting Gas-Condensate Reservoir Performance Under Cycling Operations Compared to Field Data.” Drill. and Prod. Prac.. API (I 946) 128-49. 67. Law, J.: “A Statistical Approach to the Interatttial Heterogeneity of Sand Reservoirs,” Trans.. AIME (1945) 155, 202-22. 68. Hurst, W. and van Everdingen. A.F.: “Performance of Distillate Reservoirs in Gas Cycling,” Trans., AIME (1946) 16.5, 36-51. 69. Cardwell. W.T. Jr. and Parsons, R.L.: “Average Permcabthttes of Heterogeneous Oil Sands,” Trcr,~s., AIME (1945) 160, 34-42 70. Sheldon, W.C.: ‘*Calculating Recovery by Cycltng a Retrograde Condensate Reservoir,” .I. Pel. Tech. (Jan 19.59) 29-34. 71. Coats, K.H.: “Simulation of Gas Condensate Reservoir Performance.” paper SPE 10512 presented at the 1982 SPE Reservoir Simulation Symposium. New Orleans. Jan. 3 I-Feb. 3. 72. Hurst, W.: “Water Influx into a Reservoir and Its Application to the Equation of Volumetric Balance,” Trcrris.. AIME (1943) 151. 57-72. 73. van Everdingen, A.F. and Hurst, W.: “Application of Laplace Transformation to Flow Patterns in Reservoirs.” Tram\. . AIME (1949) 186, 305-24. 74. Lesem, L.B. et ai. : “A Method of Calculating the Distribution of Temperature in Flowing Gas Wells,” J. Per. Tech. (June 1957) 169-76; Trans., AIME, 210. 75. Tek. M.R. 1Grove, M.L., and Pocttmann. F.H.: “Method for Predicting the Back-Pressure Behavior of Low Permeability Natural Gas Wells,” J. Pet. Tech. (Nov. 1957) 302-09: Truns.. AIME, 210. 76. Clinkenbeard. P., Bozeman, J.F., and Davidson. R.D.: “Gas Well Stimulation Increases Production and Profits,” J. Per. Tech. (Nov. 1958) 21-24. 77. Bennett. E.O.: “Factors Influencing Spacing in Condensate Fields.” Pet. Eq. (1944) 15, No. IO. 158-62. 78. Arthur, M.G.: “Economics of Cycling,” Drill. and Prod. Pram., API (1948) 144-59. 79. Boatright. B.B. and Dixon, P.C.: “Practical Economics of Cyclmg.” Drill. and Prod. Pram., API (1941) 221-27.



Chapter 40



Estimation of Oil and Gas Reserves Forrest A. Garb, SPE, Gerry L. Smith ,** H.J.



H.J. Grt~y & Assocs. Gruy



6i Asaoca.



Inc.*



Inc.



Estimating Reserves General Discussion Management’s decisions are dictated by the anticipated results from an investment. In the case of oil and gas, the petroleum engineer compares the estimated costs in terms of dollars for some investment opportunity vs. the cash flow resulting from production of barrels of oil or cubic feet of gas. This analysis may be used in formulating policies for (1) exploring and developing oil and gas properties; (2) designing and constructing plants, gathering systems, and other surface facilities; (3) determining the division of ownership in unitized projects; (4) determining the fair market value of a property to be bought or sold: (5) determining the collateral value of producing properties for loans; (6) establishing sales contracts, rates, and prices; and (7) obtaining Security and Exchange Commission (SEC) or other regulatory body approvals. Reserve estimates are just what they are calledestimates. As with any estimate, they can be no better than the available data on which they are based and are subject to the experience of the estimator. Unfortunately, reliable reserve figures are most needed during the early stages of a project, when only a minimum amount of information is available. Because the information base is cumulative during the life of a property, the reservoir engineer has an increasing amount of data to work with as a project matures, and this increase in data not only changes the procedures for estimating reserves but, correspondingly, improves the confidence in the estimates. Reserves are frequently estimated (1) before drilling or any subsurface development, (2) during the development drilling of the field, (3) after some performance data are available, and (4) after performance trends are well established. Fig. 40.1 demonstrates (I) the various periods in the life of an imaginary oil property, (2) the sequence



of appropriate recovery estimating methods, (3) the impact on the range of recovery estimates that usually results as a property ages and more data become available, (4) a hypothetical production profile, and (5) the relative risk in using the recovery estimates. Time is shown on the horizontal axis. No particular units are used in this chart, and it is not drawn to any specific scale. Note that while the ultimate recovery estimates may become accurate at some point in the late life of a reservoir, the reserve estimate at that time may still have significant risk. During the last week of production. if one projects a reserve of 1 bbl and 2 bbl are produced, the reserve estimate was 100% in error. Reserve estimating methods are usually categorized into three families: analogy, volumetric, and performance techniques. The performance-technique methods usually are subdivided into simulation studies, material-balance calculations, and decline-trend analyses. The relative periods of application for these techniques are shown in Fig. 40.1. ‘.2 During Period AB, before any wells are drilled on the property, any recovery estimates will be of a very general nature based on experience from similar pools or wells in the same area. Thus, reserve estimates during this period are established by analogy to other production and usually are expressed in barrels per acre. The second period, Period BC, follows after one or more wells are drilled and found productive. The well logs provide subsurface information, which allows an acreage and thickness assignment or a geologic interpretation of the reservoir. The acre-foot volume considered to hold hydrocarbons, the calculated oil or gas in place per acre-foot, and a recovery factor allow closer limits for the recovery estimates than were possible by analogy alone. Data included in a volumetric analysis may include well logs, core analysis data, bottomhole sample information, and subsurface mapping. Interpretation of these



PETROLEUM



40-2



Fig. 4&l-Range in estimates of reservoir.



of ultimate recovery during life



data. along with observed pressure behavior during early production periods, may also indicate the type of producing mechanism to be expected for the reservoir. The third period, Period CD, represents the period after delineation of the reservoir. At this time, performance data usually are adequate to allow derivation of reserve estimates by use of numerical simulation model studies. Model studies can yield very useful reserve estimates for a spectrum of operating options if sufficient information is available to describe the geometry of the reservoir, any spatial distribution of the rock and fluid characteristics, and the reservoir producing mechanism. Because numerical simulators depend on matching history for calibration to ensure that the model is representative of the actual reservoir, numerical simulation models performed in the early life of a reservoir may not be considered to have high confidence. During Period DE, as performance data mature, the material-balance method may be implemented to check the previous estimates of hydrocarbons initially in place. The pressure behavior studied through the materialbalance calculations may also offer valuable clues regarding the type of production mechanism existent in the reservoir. Confidence in the material-balance calculations



ENGINEERING



HANDBOOK



depends on the precision of the reservoir pressures recorded for the reservoir and the engineer’s ability to determine the true average pressure at the dates of study. Frequent pressure surveys taken with precision instruments have enabled good calculations after no more than 5 or 6 % of the hydrocarbons in place have been produced. Reserve estimates based on extrapolation of established performance trends, such as during Period DEF, are considered the estimates of highest confidence. In reviewing the histories of reserve estimates over an extended period of time in many different fields, it seems to be a common experience that the very prolific fields (such as East Texas, Oklahoma City, Yates, or Redwater) have been generally underestimated during the early “barrels-per-acre-foot” period compared with their later performance, while the poorer ones (such as West Edmond and Spraberry) usually are overestimated during their early stages. It should be emphasized that, as in all estimates, the accuracy of the results cannot be expected to exceed the limitations imposed by inaccuracies in the available basic data. The better and more complete the available data, the more reliable will be the end result. In cases where property values are involved, additional investment in acquiring good basic data during the early stages pays off later. With good basic data available, the engineer making the estimate naturally feels more sure of his results and will be less inclined to the cautious conservatism that often creeps in when many of the basic parameters are based on guesswork only. Generally, all possible approaches should be explored in making reserve estimates and all applicable methods used. In doing this, the experience and judgment of the evaluator are an intangible quality, which is of great importance. The probable error in the total reserves estimated by experienced engineers for a number of properties diminishes rapidly as the number of individual properties increases. Whereas substantial differences between independent estimates made by different estimators for a single property are not uncommon, chances are that the total of such estimates for a large group of properties or an entire company will be surprisingly close.



Petroleum Reserves-Definitions and Nomenclature3 Definitions for three generally recognized reserve categories, “proved,” “probable,” and “possible,” which are used to reflect degrees of uncertainty in the reserve estimates, are listed as follows. The proved reserve definition was developed by a joint committee of the SPE, American Assn. of Petroleum Geologists (AAPG), and American Petroleum Inst. (API) members and is consistent with current DOE and SEC definitions. The joint committee’s proved reserve definitions, supporting discussion, and glossary of terms, are quoted as follows. The probable and possible reserve definitions enjoy no such official sanction at the present time but are believed to reflect current industry usage correctly. Proved Reserves Definitions3 The following is reprinted from the Journal of PetroleUM Technology (Nov. 1981, Pages 2113-14) proved reserve definitions, discussion, and glossary of terms.



ESTIMATION



OF OIL AND GAS RESERVES



40-3



Proved Reserves. Proved reserves of crude oil, natural gas, or natural gas liquids are estimated quantities that geological and engineering data demonstrate with reasonable certainty to be recoverable in the future from known reservoirs under existing economic conditions.* Discussion. Reservoirs are considered proved if economic producibility is supported by actual production or formation tests or if core analysis and/or log interpretation demonstrates economic producibility with reasonable certainty. The area of a reservoir considered proved includes (1) that portion delineated by drilling and defined by fluid contacts, if any, and (2) the adjoining portions not yet drilled that can be reasonably judged as economically productive on the basis of available geological and engineering data. In the absence of data on fluid contacts, the lowest known structural occurrence of hydrocarbons controls the lower proved limit of the reservoir. Proved reserves are estimates of hydrocarbons to be recovered from a given date forward. They are expected to be revised as hydrocarbons are produced and additional data become available. Proved natural gas reserves comprise nonassociated gas and associated/dissolved gas. An appropriate reduction in gas reserves is required for the expected removal of natural gas liquids and the exclusion of nonhydrocarbon gases if they occur in significant quantities. Reserves that can be produced economically through the application of established improved recovery techniques-are included in the proved classification when these qualifications are met: (1) successful testing by a pilot project or the operation of an installed program in that reservoir or one with similar rock and fluid properties provides support for the engineering analysis on which the project or program was based, and (2) it is reasonably certain the project will proceed. Reserves to be recovered by improved recovery techniques that have yet to be established through repeated economically successful applications will be included in the proved category only after successful testing by a pilot project or after-the operation of an installed-p&g&~ in the reservoir provides support for the engineering analysis on which the project or program was based. Estimates of proved reserves do not include crude oil, natural gas, or natural gas liquids being held in underground storage. Proved Developed Reserves. Proved developed reserves are a subcategory of proved reserves. They are those reserves that can be expected to be recovered through existing wells (including reserves behind pipe) with proved equipment and operating methods. Improved recovery reserves can be considered developed only after an improved recovery project has been installed. Proved Undeveloped Reserves. Proved undeveloped reserves are a subcategory of proved reserves. They are those additional proved reserves that are expected to be recovered from (I) future drilling of wells, (2) deepening of existing wells to a different reservoir, or (3) the installation of an improved recovery project. ‘Most reserve,, engmeers



add the expression



“considering



current technology.”



Glossary of Terms Crude Oil Crude oil is defined technically as a mixture of hydrocarbons that existed in the liquid phase in natural underground reservoirs and remains liquid at atmospheric pressure after passing through surface separating facilities. For statistical purposes, volumes reported as crude oil include: (1) liquids technically defined as crude oil; (2) small amounts of hydrocarbons that existed in the gaseous phase in natural underground reservoirs but are liquid at atmospheric pressure after being recovered from oilwell (casinghead) gas in lease separators*; and (3) small amounts of nonhydrocarbons produced with the oil. Natural Gas Natural gas is a mixture of hydrocarbons quantities of nonhydrocarbons that exists gaseous phase or in solution with crude underground reservoirs. Natural gas may fied as follows.



and varying either in the oil in natural be subclassi-



Associated Gas. Natural gas, commonly known as gascap gas, that overlies and is in contact with crude oil in the reservoir. ** Dissolved Gas. Natural gas that is in solution with crude oil in the reservoir. Nonassociated Gas. Natural gas in reservoirs that do not contain significant quantities of crude oil. Dissolved gas and associated gas may be produced concurrently from the same wellbore. In such situations, it is not feasible to measure the production of dissolved gas and associated gas separately; therefore, production is reported under the heading of associated/dissolved or casinghead gas. Reserves and productive capacity estimates for associated and dissolved gas also are reported as totals for associated/dissolved gas combined. Natural Gas Liquids Natural gas liquids (NGL’s) are those portions of reservoir gas that are liquefied at the surface in lease separators, field facilities, or gas processing plants. NGL’s include but are not limited to ethane, propane, butanes, pentanes, natural gasoline, and condensate. Reservoir A reservoir is a porous and permeable underground formation containing an individual and separate natural accumulation of producible hydrocarbons (oil and/or gas) that is confined by impermeable rock and/or water barriers and is characterized by a single natural pressure system.



‘From a technical standpoint, these hqulds are termed condensate”, however, they are commmgled wth Ihe crude stream and it IS impractical to meawe and report their volumes separately All other condensate IS reported as either “lease condensate” or “plant condensate” and Included I” natural gas l,q”,ds . ‘Where resewar cond,,,ons are such lhat the production of associated gas does not substantlallv affect the recwerv of crude 011 I” the reser~oll. such aas rnav be reclassitled’as nonassoclated gis by a regulatory agency In this w&t, res&es and producbon are reported I” accordance wth the classlficatw used by the regulatory agency



PETROLEUM



40-4



ENGINEERING



HANDBOOK



Probable Reserves



OIL-WATER CONTACT -7450



Probable reserves of crude oil, natural gas, or natural gas liquids are estimated quantities that geological and engineering data indicate are reasonably probable to be recovered in the future from known reservoirs under existing economic conditions. Probable reserves have a higher degree of uncertainty with regard to extent, recoverability, or economic viability than do proved reserves.



0



Possible Reserves



Fig. 40.2-Geological



map on



Possible reserves of crude oil, natural gas, or natural gas liquids are estimated quantities that geological and engineering data indicate are reasonably possible to be recovered in the future from known reservoirs under existing economic conditions. Possible reserves have a higher degree of uncertainty than do proved or probable reserves.



top (-) and base (-7) of reservoir.



Computation of Reservoir Volume4



In most situations, reservoirs are classified as oil reservoirs or as gas reservoirs by a regulatory agency. In the absence of a regulatory authority, the classification is based on the natural occurrence of the hydrocarbon in the reservoir as determined by the operator.



When sufficient subsurface control is available, the oilor gas-bearing net pay volume of a reservoir may be computed in several different ways. 1. From the subsurface data a geological map (Fig. 40.2) is prepared, contoured on the subsea depth of the top of the sand (solid lines), and on the subsea depth of the base of the sand (dashed lines). The total area enclosed by each contour is then planimetered and plotted as abscissa on an acre-feet diagram (Fig. 40.3) vs. the corresponding subsea depth as the ordinate. Gas/oil contacts (GOC’s) and water/oil contacts (WOC’s) as determined from core, log, or test data are shown as horizontal lines.* After the observed points are connected, the combined gross volume of oil- and gas-bearing sand may be determined by the following methods.



Improved Recovery Improved recovery includes all methods for supplementing natural reservoir forces and energy, or otherwise increasing ultimate recovery from a reservoir. Such recovery techniques include (1) pressure maintenance, (2) cycling, and (3) secondary recovery in its original sense (i.e., fluid injection applied relatively late in the productive history of a reservoir for the purpose of stimulating production after recovery by primary methods of flow or artificial lift has approached an economic limit). Improved recovery also includes the enhanced recovery methods of thermal, chemical flooding, and the use of miscible and immiscible displacement fluids.



GROSS



‘lf working I” Sl umls, the depths WIII be expressed in meters and the planlmetered areas enclosed by each contour w,ll be expressed I” hectares The resultant hectaremeter plot can be treated exactly llke the following acre-foot example to yield reserw~ ~oI!mes m cubic meters. (1 ha, m = 10,000 m3 )



GAS BEARING SAND VOLUME:



[(0+8&42lt4(24)]



GAS-OIL



~2367



ACRE FEET



CONTACT



GROSS OIL BEARING y



[W-42+



SAND VOLUME:



378 -242)+4(209-1061]=m



OIL-WATER



100



200 AREA



300



ENCLOSED



400



500



BY CONTOUR



Fig. 40.3-Acre-feet



diagram



CONTACT



600



ACRE FEET



ESTIMATION



40-5



OF OIL AND GAS RESERVES



Fig. 40.4-lsopachous



map-gas



a. Planimetered from the acre-feet diagram. b. If the number of contour intervals is even, computed by Simpson’s rule:



So/3[(0+136)+4(24+103)+2(46)]=



12,267 acre-ft.



(The separate calculations of the volume of gross gasbearing sand and gross oil-bearing sand by means of Simpson’s rule are shown in the diagram of Fig. 40.3.) r. With somewhat less accuracy, computed by the trapezoidal rule:



SO[%(O+ 136)+(24+46+



103)] = 12,050 acre-ft.



d. Computed by means of the somewhat more complicated pyramidal rule:



ss[(O+136)+2(24+46+



103)+J24x88



sand



available on many wells, it is sometimes justified to prepare an isopachous map of the number of porosity feet (porosity fraction times net pay in feet) and compute the total available void space in the net-pay section from such an isopachous map by the methods discussed under Item la, b, or c.



Computation of Oil or Gas in Place Volumetric Method If the size of the reservoir, its lithologic characteristics, and the properties of the reservoir fluids are known, the amount of oil or gas initially in place may be calculated with the following formulas: Free Gas in Gas Reservoir or Gas Cap (no residual oil present). For standard cubic feet of free gas, GFj =



= 11,963 acre-ft.



43,5601/,@(1 -Siw) *, ,



.



(1)



where



e. If the sand is ofuniform thickness, it will oftentimes suffice to multiply the average gross pay thickness h I by



the area enclosed by the contour 1/2Zfi above the WOC. J If the area within the top contour is circular (area A, height Z), then the top volume is QrZ+ %AZ if treated as a segment of a sphere, and %AZ if treated as a cone. From a study of the individual well logs or core data, it is then determined what fraction of the gross sand section is expected to carry and to produce hydrocarbons. Multiplication of this net-pay fraction by the gross sand volume yields the net-pay volume. If, for example, in the case illustrated with Figs. 40.2 and 40.3, it is found that 15% of the gross section consisted of evenly distributed shale or dense impervious streaks, the net gas- and oilbearing pay volumes may be computed as, respectively, net acre-ft of gas pay



and 0.85x9,900=8,415



map-oil



+m



+d5icEm-m-J]



0.85 x2,367=2,012



Fig. 40.5-lsopachous



sand



net acre-ft of oil pay.



2. From individual well-log data, separate isopachous maps may be prepared for the net gas pay (Fig. 40.4) or for the net pay (Fig. 40.5) and the total net acre-feet of oil- or gas-bearing pay computed as under It&m la, b, or c. 3. If the nature of the porosity varies substantially from well to well, and if good log and core-analysis data are



V, = net pay volume of the free-gas-bearing



4 S;, B, 43,560



= = = =



portion of a reservoir, acre-ft, effective porosity, fraction, interstitial water saturation, fraction, gas FVF, dimensionless, and number of cubic feet per acre-foot.



Values for the gas FVF or the reciprocal gas FVF, l/B,, may be estimated for various combinations of pressure, temperature, and gas gravity (see section on gas FVF).” Oil in Reservoir (no free gas present in oil-saturated portion). For stock-tank barrels of oil, N= 7,758V,4(1 -S,,) B,



,



....



. . . .



. (2)



where N = reservoir oil initially in place, STB, V, = net pay volume of the oil-bearing portion of a reservoir, acre-ft, B, = oil FVF, dimensionless, and 7,758 = number of barrels per acre-foot. ‘Refer ,oChaps. 20 through 25 for delaled properties. and correlalions.



coverage of 011.gas, condensate and watel



40-6



PETROLEUM



TABLE



40.1--BARRELS



OF STOCK-TANK



OIL IN PLACE



ENGINEERING



HANDBOOK



PER ACRE-FT



Porositv. d B



0



1.0



1.5



2.0



3.0



iwS 0.10 0.20 0.30 0.40 0.50 0.10 0.20 0 30 0.40 0.50 0.10 0 20 0.30 0.40 0.50 0.10 0.20 0.30 0.40 0.50



0.05



0.10



0.15



0.20



0.25



0.30



0.35



349 310 272 233 194 233 206 182 155 128 175 155 136 116 97 116 105 89 78 66



698 621 543 465 388 465 411 365 310 256 349 310 272 233 194 233 209 178 155 132



1,047 931 615 698 582 698 617 547 465 384 524 465 407 349 291 349 314 268 233 198



1,396 1,241 1,066 931 776 931 822 729 621 512 698 621 543 465 388 465 419 357 310 264



1,746 1,552 1.358 1.164 970 1,164 1,028 912 776 640 873 776 679 582 485 582 524 446 388 330



2,095 1,862 1,629 1,396 1,164 1,396 1,234 1,094 931 768 1,047 931 815 698 582 698 628 535 465 396



2,444 2,172 1,901 1,629 1,358 1,629 1,439 1,276 1,086 896 1,222 1,086 950 815 679 815 733 625 543 462



Table 40.1 shows the number of barrels of stock-tank oil per acre-foot for different values of porosity, 4, interstitial water saturation, S,,,., and the oil FVF, B,, Solution Gas in Oil Reservoir (no free gas present). For standard cubic feet of solution gas,



Gs =



7,7581/,@(1 -s,,.)R., Bo



.



.



(3)



to small changes in the two-phase FVF, B,, an adjustment procedure, called the Y method, may be used for the pressure range immediately below the bubblepoint. The method consists of plotting values of y= (Ph-PRPoi



pR(B,-B,,i)



Method5-8



In the absence of reliable volumetric data or as an independent check on volumetric estimates, the amount of oil or gas in place in a reservoir may sometimes be computed by the material-balance method.5 This method is based on the premise that the PV of a reservoir remains constant or changes in a predictable manner with the reservoir pressure when oil, gas, and/or water are produced. This makes it possible to equate the expansion of the reservoir fluids upon pressure drop to the reservoir voidage caused by the withdrawal of oil, gas, and water minus the water influx. Successful application of this method requires an accurate history of the average pressure of the reservoir, as well as reliable oil-, gas-, and waterproduction data and PVT data on the reservoir fluids. Generally, from 5 to 10% of the oil or gas originally in place must be withdrawn before significant results can be expected. Without very accurate performance and PVT data the results from such a computation may be quite erratic, 6 especially when there are unknowns other than the amount of oil in place, such as the size of a free-gas cap, or when a water drive is present. When the number of available equations exceeds the number of such unknowns, the solution should preferably be by means of the “method of least squares. “’ Because of the sensitivity of the material-balance equation



..........



.



. . . .(4)



where ph = pR = B, = Boi =



where G, is the solution gas in place, in standard cubic feet, and R,T is the solution GOR, in standard cubic feet per stock-tank barrel. Material-Balance



,



bubblepoint pressure, psia, reservoir pressure, psia, two-phase FVF for oil, dimensionless, initial oil FVF, dimensionless.



and



vs. reservoir pressure, PR, and bringing a straight line through the plotted points, with particular weight given to the more accurate values away from the bubblepoint. This straight-line relationship is then used to correct the previous values for Y, from which the adjusted values for B, are computed. Values of B, computed with this method for pressures substantially below the bubblepoint should not be used if differential liberation is assumed to represent reservoir producing conditions. When an active water drive is present, the cumulative water influx, W,, should be expressed in terms of the known pressure/time history and a water drive constant,’ thus reducing this term to one unknown. A completely worked-out example of the use of material balance that uses this conversion and in which the amount of oil in place is determined for a partial water drive reservoir where 36 pressure points and equations were available at a time when about 9 % of the oil in place had been produced is given in Ref. 7. The material-balance equation in its most general form reads N=



N,,[B,+O.l7XIB,(R,~-R,,)I-(W,,-~,,) B,q B, B,,, rnB + B- -(m+ ,q, 0,



I) I -



&RR(‘.,+S,,,“,!) 1 -s,,,



. . . . . . . . . . . . . . . . . . . . . . . ..~....



II (5)



’



ESTIMATION



40-7



OF OIL AND GAS RESERVES



TABLE Reservoir



40.2-CLASSIFICATION



OF MATERIAL-BALANCE Material-balance



Type



Oil reservoir with gas cap and active water drive



Np]B, +0.1781B,(R,



EQUATIONS



Equation’



-R,,)]-(W,



Unknowns



Equation



- WP)



N=



N, W,,



m



6



mB,,



Oil reservoir with gas cap; no active water drive (W, = 0)



Np[B, +0.1781B,(Rp



-I?,,)]+



w,



N=



N. m



7



N, W,



8



N, W,



9



N



10



N



11



ma,,



Initially undersaturated oil reservoir with active water drive (m =0): 1. Above bubblepoint



N,U N=



APl(C, +c, -S,&,



2. Below bubblepoint



N=



-S,,) 1(1



we-WP +APpRco) - ~ B,,



-c,)l



Npl~,+0.f781B,(R,-R,,)1-(W,-W,) 8, -60,



lnltially undersaturated oil reservoir: no active water drive (m = 0),(W, = 0): 1. Above bubblepoint



N,(l



+W&J-



F



1



(1 -St,)



01



N=



QJDR[c,+c,-S,,(c,-c,)l



NJ!3, 2. Below bubblemint



+ O.l781B,(R,



-R,,)]+



W,



N= 6, -go,



G=



G,B,



-5.615(W,



- WP)



Gas reservoir with active water drive



G



W,



12



B, --By



Gas reservoir; no active water drive we



=O)



where N,, R,, R.,, w,, w,, Aj?R



= = = = = =



B,pi = III = “f = c,, =



G,B,



+5.615W, G



G= 6,



13



-B,,



cumulative oil produced, STB, cumulative GOR, scf/STB. initial solution GOR. scf/STB, cumulative water influx, bbl, cumulative water produced, bbl, change in reservoir pressure, psi, initial gas FVF. res cu ftiscf, ratio of initial reservoir free-gas volume and initial reservoir-oil volume, compressibility of reservoir rock, change in PV per unit PV per psi, and compressibility of interstitial water, psi -’



When a free-gas cap is present, this equation may be simplified to Eq. 6 of Table 40.2 by neglecting the reservoir formation compressibility cf and the interstitial water compressibility c,,..



When such a reservoir has no active water drive Eq. 7 results. For initially undersaturated reservoirs (m = 0) below the bubblepoint, Eqs. 6 and 7 reduce to Eqs. 9 and I I, depending on whether an active water drive is present. For initially undersaturated reservoirs (m=O) above the bubblepoint, no free gas is present (R,) -R,yi =O). while B, =Bo;+A~~c, (where c, is the compressibility of reservoir oil, volume per psi), so that general Eq. 5 reduces to Eqs. 8 and 10, depending on whether an active water drive is present. For gas reservoirs the material-balance equation takes the form of Eq. 12 or 13, depending on whether an active water drive is present. The numerator on the right side in each case represents the net reservoir voidage by production minus water influx, while the denominator is the gas-expansion factor (BR -B,;) for the reservoir. (W,,=O),



PETROLEUM



40-8



ENGINEERING



HANDBOOK



TABLE 40.3-CONDITIONS FOR UNIT-RECOVERY EQUATION. DEPLETION-TYPE RESERVOIR Initial Conditions’ Reservoir Interstitial Free gas, Reservoir



pressure water, @SW, bbllacre-ft &S,, bbllacre-ft oil, bbllacre-ft



$58 0



7pp58 7,758 7,758$~(l - S,, -S,,)



7,758$41 -s,,) 1-S



Stock-tank



oil, bbl/acre-ft



7,758



7,7584



d2 BO,



'SubstIMe



10 000 for the 7.758 constanf



11c"b,c melers per hectare.mefer



Saturated Depletion-Type Oil Reservoirs-Volumetric Methods General Discussion Pools without an active water drive that produce solely as the result of expansion of natural gas liberated from solution in the oil are said to produce under a depletion mechanism, also called an internal- or solution-gas drive. When a free-gas cap is present, this mechanism may be supplemented by an external or gas-cap drive (Page 40-13). When the reservoir permeability is sufficiently high and the oil viscosity low, and when the pay zone has sufficient dip or a high vertical permeability, the depletion mechanism may be followed or accompanied by gravity segregation (Page 40-14). When a depletion-type reservoir is first opened to production, its pores contain interstitial water and oil with gas in solution under pressure. No free gas is assumed to be present in the oil zone. The interstitial water is usually not produced, and its shrinkage upon pressure reduction is negligible compared with some of the other factors governing the depletion-type recovery. When this reservoir reaches the end of its primary producing life, and disregarding the possibility of gas-cap drive or gravity segregation, it will contain the same interstitial water as before, together with residual oil under low pressure. The void space vacated by the oil produced and by the shrinkage of the remaining oil is now filled with gas liberated from the oil. During the depletion process this gas space has increased gradually to a maximum value at abandonment time. The amount of gas space thus created is the key to the estimated ultimate recovery under a depletion mechanism. It is reached when the produced free GOR in the reservoir, which changes according to the relative permeability ratio relationship and the viscosities of oil and gas involved, causes exhaustion of the available supply of gas in solution. Unit-Recovery Equation The unit-recovery factor is the theoretically possible ultimate recovery in stock-tank barrels from a homogeneous unit volume of 1 acre-t? of pay produced by a given mechanism under ideal conditions. The unit-recovery equation for a saturated depletiontype reservoir is equal to the stock-tank oil initially in place in barrels per acre-foot at initial pressure pi minus the residual stock-tank oil under abandonment pressure pi,, as shown in Table 40.3.



Ultimate Conditions’



1 -s&v -s,, B w



IS used.



By difference, the unit recovery by depletion or solution-gas drive is, in stock-tank barrels per acre-foot, 1 - S,M - s,, B o(I



'



.'."



(14)



where S,, is the residual free-gas saturation under reservoir conditions at abandonment time, fraction, and B,, is the oil FVF at abandonment, dimensionless. The key to the computation of unit recovery by means of this equation is an estimate of the residual free-gas saturation S,, at the ultimate time. If a sufficiently large number of accurate determinations of the oil and water saturation on freshly recovered core samples is available, an approximation of S,, may be obtained by deducting the average total saturation of oil plus water from unity. This method is based on the assumption that the depletion process taking place within the core on reduction of pressure by bringing it to the surface is somewhat similar to the actual depletion process in the reservoir. Possible loss of liquids from the core before analysis may cause such a value for S,, to be too high. On the other hand, the smaller amount of gas in solution in the residual oil left after flushing by mud filtrate has a tendency to reduce the residual free-gas saturation. Those using this method hope that these two effects somewhat compensate for each other. A typical S,, value for average consolidated sand, a medium solution GOR of 400 to 500 cu ftibbl, and a crude-oil gravity of 30 to 4O”API is 0.25. Either a high degree of cementation, a high shale content of the sand, or a 50% reduction in solution GOR may cut this typical S,, value by about 0.05, while a complete lack of cementation or shaliness such as in clean, loose unconsolidated sands or a doubling of the solution GOR may increase the S,, value by as much as 0.10. At the same time, the crude-oil gravity generally increases or decreases the S,, value by about 0.01 for every 3”API gravity. Example Problem 1. A cemented sandstone reservoir has an interstitial water content a porosity $=0.13, S,,,.=O.35, a solution GOR at bubblepoint conditions, /?,I, =300 cu ftibbl, an initial oil FVF B,,; = 1.20, an oil FVF at abandonment B,, = I .07, and a stock-tank oil gravity of 40”API. Based on the above considerations, the higher-than-average oil gravity would just about offset the effect of the somewhat lower-than-average GOR. and the residual free-gas saturation S,, after a 0.05 reduction for the cementation can therefore be estimated at 0.20.



ESTIMATION



40-9



OF OIL AND GAS RESERVES



Solution. The unit recovery by depletion according to Eq. 14 would be



N,, =(7.758)(0.13)



l-0.35



l-O.35



-0.20



1.07 = 122 STBiacre-ft



This stepwise solution of the depletion equation yields the reservoir oil saturation S,, as a function of reservoir pressure pR. The results may be converted into cumulative recovery per acre-foot. In stock-tank barrels per acre-foot,



> (16)



[I57 m3/ha.mj.



where N,, is the unit recovery by depletion or solutiongas drive, STB. Muskat’s Method. 9 If the actual relationships between pressure and oil-FVF B,, gas-FVF B,, gas-solubility in oil (solution GOR) R, , oil viscosity p,), and gas viscosity ps are available from a PVT analysis of the reservoir fluids, and if the relationship between relative permeaand the total liquid saturation, S,, is bility ratio k,/k, known for the reservoir rock under consideration, the unit recovery by depletion can be arrived at by a stepwise computation of the desaturation history directly from the following depletion equation in differential form: As,, -1



The results may be converted into cumulative recovery as a fraction of the original oil in place (OOIP) by



L+L)



(?c), .,....,.......



N



(17)



while the GOR history, in standard cubic feet gasistocktank barrel, may be computed by



(18) where R is the instantaneous producing GOR, in standard cubic feet per stock-tank barrel, and the relative production rate in barrels per day by



APR



B, dR, d(liB,s) S,,‘+(I -s,, -s,,, )B,L!-+s,,--dl’R B,, ‘k’R



. ..t...



.I..........,.........



PL,,k,.,



dB,,



ko



I-‘,?k,,, Bdr’R



(15)



where S, = oil or condensate saturation under reservoir conditions, fraction, PLO= reservoir oil viscosity, cp, PLK= reservoir gas viscosity, cp, k, = relative permeability to gas as a fraction of absolute permeability, and k, = relative permeability to oil as a fraction of absolute permeability. The individual computations are greatly facilitated by computing and preparing in advance in graphical form the following groups of terms, which are a function of pressure only,



and the relative permeability ratio k,ik,,, which is a function of total liquid saturation S, only. The accuracy of this type of calculation on a desk calculator falls off rapidly if the pressure decrements chosen are too large, particularly during the final stages when the GOR is increasingly rapidly. With modern electronic computers, however, it is possible to use pressure decrements of IO psi or smaller, which makes a satisfactory accuracy possible.



Poi



PR .



where 90 kc, km Poi 40;



= = = = =



.



.



(19)



oil-production rate, B/D, effective permeability to oil. md, initial effective permeability to oil. md, initial reservoir oil viscosity, cp, and initial oil-production rate, B/D.



It should be stressed that this method is based on the assumption of uniform oil saturation in the whole reservoir and that the solution will therefore break down when there is appreciable gas segregation in the formation. It is therefore applicable only when permeabilities are relatively low. Another limitation of this method as well as of the Tarner method, discussed hereafter, is that no condensation of liquids from the produced gas is assumed to take place in the tubing or in the surface extraction equipment. It should therefore not be applied to the high-temperature, high-GOR, and high-FVF “volatile” oil reservoirs to be discussed later. Tarner’s Method. Babson ‘” and Tarner ” have advanced trial-and-error-type computation methods for the desaturation process that require a much smaller number of pressure increments and can therefore be more readily handled by a desk calculator. Both methods are based on a simultaneous solution of the material-balance equation (Eq. 11) and the instantaneous GOR (Eq. 18). Tarner’s method is the more straightforward of the two. The procedure for the stepwise calculation of the cumulative oil produced (N,,)I and the cumulative gas produced (Gp)* for a given pressure drop from p I to p, is as follows.



PETROLEUM



40-l 0



TABLE



40.4-COMPUTED



DEPLETION



RECOVERY



IN STBIACRE-FTIPERCENT



POROSITY



ENGINEERING



FOR TYPICAL



HANDBOOK



FORMATIONS



Solution GOR (cu ftlbbl)



Oil Gravity, (OAPI)



cRsb)



-70



Unconsolidated



Consolidated



Highly Cemented



Vugular



Fractured



;z 50 15 30 50 15 30 50 30 50 50



7.2 12.0 19.2 7.0 11.6 19.4 7.6 10.5 15.0 12.3 12.0 10.6



4.9 8.5 13.9 4.6 7.9 13.7 4.8 6.5 9.7 7.6 7.2 6.4



1.4 4.9 9.5 1.8 4.4 9.2 2.5 3.6 5.8 4.5 4.1 4.0



2.6 6.3 11.8 2.6 5.8 11.4 3.3



0.4 18 5.1 0.5 1.5 4.4 0.9



60 200



600



1,000 2,000



Sand or Sandstone (S,, = 0.25)



1. Assume that during the pressure drop from p , to pl the cumulative oil production increases from (N,) , to (N,,)* N, should be set equal to zero at bubblepoint. 2. Compute the cumulative gas produced (G,,)z at pressure p2 by means of the material-balance equation assuming (Eq. 111, which for this purpose-and Wp =0-is rewritten in the following form:



(G,,h =(N,h(R,,):!=N



(R.7,-R,\)-5.615



3. Compute the fractional total liquid saturation @,)I at pressure p2 by means of (s’);=S;~+(l-s;,,J~[l-~].



.., . ..(21)



4. Determine the k,lk,, ratio corresponding to the total liquid saturation (S,), and compute the instantaneous GOR at p2 by means of ....



R* =R,$ +ui15$+. RPK



..



(22)



ro



5. Compute the cumulative gas produced at pressure p2 by means of (G,)2=(Gp)1+



RI +R, ---[VP)2 2



-VP) 11,



. (23)



in which RI represents the instantaneous GOR computed previously at pressure p, . Usually three judicious guesses are made for the value (N,) 2 and the corresponding values of (G,,) 2 computed by both Steps 2 and 5. When the values thus obtained for (G,) 2 are plotted vs. the assumed values for (N,) 2 , the intersection of the curve representing the results of Step 2 and the one representing Step 5 then indicates the cumulative gas and oil production that will satisfy both equations. In actual application, the method is usually simplified further by equating the incremental gas production (Gp)z -(G,) I) rather than (G,)Z itself. This



Limestone, Dolomite or Chert (S,, =0.15)



4.7 7.2 5.4 4.8 (4.3)



(1.2) (2.1) (1.6) (1.2) (1.5)



equality signifies that at each pressure step the cumulative gas, as determined by the volumetric balance, is the same as the quantity of gas produced from the reservoir, as controlled by the relative permeability ratio of the rock, which in turn depends on the total liquid saturation. Although the Tamer method was originally designed for graphical interpolation, it also lends itself well to automatic digital computers. The machine then calculates the quantity of gas produced for increasing oil withdrawals by both equations and subtracts the results of one from the other. When the difference becomes negative, the machine stops and the answer lies between the last and next to last oil withdrawals. Tarner’s method has been used occasionally to compute recoveries of reservoirs with a free-gas cap or to evaluate the possible results from injection of all or part of the produced gas. When a free-gas cap is present, or when produced gas is being reinjected, breakthrough of free gas into the oil-producing section of the reservoir is likely to occur sooner or later, thus invalidating the assumption of uniform oil saturation throughout the producing portion of the reservoir, on which the method is based. Since such a breakthrough of free gas causes the instantaneous GOR (Eq. 18) as well as the entire computation method to break down, the use of Tamer’s method in its original form for this type of work is not recommended. It should also be used with caution when appreciable gas segregation in an otherwise uniform reservoir is expected. Computed Depletion-Recovery Factors. Several investigators9, 12-14 have used the Muskat and Tarner methods to determine the effects of different variables on the ultimate recovery under a depletion mechanism. In one such attempt I2 the k,lk, relationships for five different types of reservoir rock representing a range of conditions for sands and sandstones and for limestones, dolomites, and cherts were developed. These five types of reservoir rock were assumed to be saturated under reservoir conditions with 25 % interstitial water for sands and sandstones and 15 % for the limestone group and with 12 synthetic crudeoil/gas mixtures representing a range of crude-oil gravities from 15 to 5O”API and gas solubilities from 60 to 2,000 cu ft/STB. Their production performance and recovery factors to an abandonment pressure equal to 10% of the bubblepoint pressure were then computed by means of depletion (Eq. 15).



ESTIMATION



OF OIL AND GAS RESERVES



10.0 z 2



1.0



e = P



0.1 0.01 5 TOT PER



Notes: interstitial water is assumed to be 30% of pore space and deadoil viscosity at reservoir temperature to be 2 cp. Equilibrium gas saturation is assumed to be 5% of pore space. As here used “ultimate oil recovery” is realized when the reservoir pressure has declined from the bubblepoint pressure to atmospheric pressure. FVF units are reservoir barrels per barrel of residual oil. Solution GOR units are standard cubic feet per barrel of residual oil. Example 1: Required: Ultimate recovery from a system -having a bubblepoint pressure = 2,250 psia, FVF = 1.6, and a solution GOR. Procedure: Starting at the left side of the chart, proceed horizontally along the 2,250-psi line to FVF = 1.6. Now rise vertically 10 the 1,300-scflbbl line. Then go horizontally and read an ultimate recovery of 23.8%. Example 2: F)eqoired:Convert the recovery figure determined in Example 1 to tank oil recovered. Data requirements: Differential liberation data given in Example 1. Flash liberation data: bubblepoint pressure = 2,250 psia, FVF = 1.485, FVF at atmospheric pressure = 1.080 for both flash and differential liberation.



FORMATIONVOLUME FACTOR Procedure: Calculate the oil saturation at atmospheric pressure by substituting differential liberation data in the equation as follows:



Oil saturation at atmospheric pressure = 0.360. Next, substitute the calculated value of oil saturation and the flash liberation data into the previous equation and calculate the ultimate oil recovery as a percentage of tank oil originally in place.



N,, (ultimate place.



oil recovery)=29.3%



of tank oil originally



in



Fig. 40.6-Chart for estimating ultimate recovery from solution gas-drive reservoirs.



These theoretical depletion-recovery factors, expressed as barrels of stock-tank oil per percent porosity, will be found in Table 40.4 for the different types of reservoir rocks, oil gravities, and solution GOR’s assumed. In cases where no detailed data are available concerning the physical characteristics of the reservoir rock and its fluid content, Table 40.4 has been found helpful in estimating the possible range of depletion-recovery factors. It may be noted that the k,lk, relationship of the reservoir rock is apparently the most important single factor governing the recovery factor. Unconsolidated intergranular material seems to be the most favorable, while increased cementation or consolidation tends to affect recoveries unfavorably. Next in importance is crude-oil gravity with viscosity as its corollary. Higher oil gravi-



ties and lower viscosities appear to improve the recovery. The effect of GOR on recovery is less pronounced and shows no consistent pattern. Apparently the beneficial effects of lower viscosity and more effective gas sweep with higher GOR is in most cases offset by the higher oil FVF’s. In general, these data seem to indicate a recovery range from the poorest combinations of 1 to 2 bbl/acre-fi for each percent porosity to the best combinations of 19 to 20 bbllacre-Mpercent porosity. An overall average seems to be around 10 bbliacre-ftlpercent porosity. It is also of interest to note that when the reservoir is about two-thirds depleted, the pressure has usually dropped to about one-half the value at bubblepoint.



PETROLEUM



40-12



In another attempt ” nine nomographs were developed, each for a given combination of the k, lk ,.(,curve, “deadoil” viscosity, and interstitial water content. The nomograph for an average k,lk, relationship, an interstitial water content of 0.30. and a dead-oil viscosity of 2 cp is reproduced as Fig. 40.6. Instructions for its use are shown opposite the figure. The authors ” also introduced an interesting empirical relationship between the relative permeability ratio k,/k,, the equilibrium gas saturation S,,., the interstitial water saturation S,,., and the oil saturation S,: k ri: = i(O.0435 +0.4556E), k t-0



. (24)



where t;=(l -S,,.-S,, -S,)/(S, -0.25). A similar correlation I5 for sandstones that show a linear relationship between lip,’ (where p,.=critical pressure) and saturation is k rg -=



(1 -S*)I[



1 -@*)I]



(s*)4



k ro



,



.



(25)



where effective saturation S*=S,I(l -Si,). This tion represents a useful expression for calculating tive permeability ratios in sandstone reservoirs for an average water saturation has been obtained by electrical log or core analysis.



equarelawhich either



In a statistical study of the actual performance of 80 solution gas-drive reservoirs, the API Subcommittee on Recovery Efficiency I6 developed the following equation for unit recovery (N,,) below the bubblepoint for solution gas-drive reservoirs, in stock-tank barrels per acre-foot*: N,, =3,*44 [ 44;,y



1.‘6” x (2-J



0.1741



x(s,



,)O.3722x !k IM ( >



.



With progressively deeper drilling, a number of oil reservoirs have been encountered that, while lacking an active water drive, are in undersaturated condition. Because of the expansion of the reservoir fluids and the compaction of the reservoir rock upon pressure reduction, substantial recoveries may sometimes be obtained before the bubblepoint pressure pb is reached and normal depletion sets in. Such recoveries may be computed as follows. The oil initially in place in stock-tank barrels per acrefoot at pressure pi is according to Eq. 2, .. ‘.



73758x4i(1-Siw)



’



where 4; is initial porosity. By combining this expression with the material-balance equation (Eq. 10). the recovery factor above the bubblepoint in stock-tank barrels per acre-foot may be expressed as



Np=



7375Wi(Pi-Pb)[Co



+Cf-Siw(cc~-~w)l



I (27)



Boi[lfco(Pi-Pb)l



where c,,, is the compressibility volume per volume per psi.



.. .



Example Problem 2. Zone D-7 in the Ventura Avenue field, described by E.V. Watts,” is an example of an undersaturated oil reservoir without water drive. Its reservoir characteristics are



(26)



where k = absolute permeability,



P,~ = Pa = pb =



Undersaturated Oil Reservoirs Without Water Drive Above the BubblepointVolumetric Method t7-19



o.0979



Pa



B ob =



HANDBOOK



be made for each permeability bank that is known to be continuous and the results converted into rate/time curves for each by combining Eqs. 16 and 19. The estimated ultimate recovery will then be based on a superposition of such rate/time curves for the different zones. If there is a wide divergence in permeabilities, one may find that at a time when the combined rate for all zones has reached the economic limit the more permeable banks will be depleted and have yielded their full unit recovery while the pressure depletion and the recovery from the tighter zones are still incomplete.



Boi



API Estimation of Oil and Gas Reserves



ENGINEERING



darcies, oil FVF at bubblepoint, RBLSTB, oil viscosity at bubblepoint, cp, abandonment pressure, psig, and bubblepoint pressure, psig.



The permeability distribution in most reservoirs is usually sufficiently nonuniform in vertical and horizontal directions to cause the foregoing depletion calculations on average material to be fairly representative. However, when distinct layers of high and low permeability, separated by impervious strata, are known to be present, the depletion process may advance more rapidly in high-permeability strata than in low-permeability zones. In such cases separate performance calculations should



of interstitial water in



pi = 8,300 psig at 9,200 ft,



pb = #Ii = s 1M’ = B oh = B o(1 = 70 = CO = cw = Cf = S,, = Rsb =



3,500 psig, 0.17, 0.40,



1.45, 1.15, 32 to 33”API, 13x10-6, 2.7~10-~, 1.4x10-6, 0.22, and 900 cu ft/bbl.



Solution. On the basis of these data, Watts computes the recovery by expansion above the bubblepoint at 47 bbliacre-ft and by a depletion mechanism below the bubblepoint at 110 bbl/acre-ft (see Ref. 19 for details).



ESTIMATION



40-13



OF OIL AND GAS RESERVES



Volatile Oil ReservoirsVolumetric Methods20-25 Deeper drilling, with accompanying increases in reservoir temperatures and pressures, has also revealed a class of reservoir fluids with a phase behavior between that of ordinary “black” oil and that of gas or gas condensate. These intermediate fluids are referred to as “highshrinkage” or “volatile” crude oils because of their relatively large percentage of ethane through decane components and resultant high volatility. Volatile-oil reservoirs are characterized by high formation temperatures (above 200°F) and abnormally high solution GOR and FVF (above 2). The stock-tank gravity of these volatile crudes generally exceeds 45 “API. The inherent differences in phase behavior of volatile oils are sufficiently significant to invalidate certain premises implicit in the conventional material-balance methods. In such conventional material-balance work it is assumed that all produced gas, whether solution gas or free gas, will remain in the vapor phase during the depletion process, with no liquid condensation on passage through the surface separation facilities. Furthermore, the produced oil and gas are treated as separate independent fluids, even though they are at all times in compositional equilibrium. Although these basic assumptions simplify the conventional material-balance calculations, highly inaccurate predictions of reservoir performance may result if they are applied to volatile-oil reservoirs. In highly volatile reservoirs, the stock-tank liquids recovered by condensation from the gaseous phase may actually equal or even exceed those from the associated liquid phase. This rather surprising occurrence is exemplified in a paper by Woods,24 in which the case history of an almost depleted volatile-oil reservoir is presented. Example Problem 3. Woods’ reservoir data for this volatile-oil reservoir were pi = 5,000 psig, pb = 3,940 psig, TR = 250”F, c$ = 0.198. k = 75 md, Sib,, = 0.25, R,,, = 3,200 scf/bbl, yoi = 44”API, You = 62”API, and B oh = 3.23. Solution. At 80% depletion when pR = 1,450 psig and R =23,000 scf/bbl, the percentage recovery was 2 1% of which 5% was from expansion above the bubblepoint, 9% from the depletion mechanism, and 7% from liquids condensed out of the gas phase by conventional field separation equipment (see Ref. 24 for details). In view of the increasing number and importance of volatile-oil reservoirs in recent years, appropriate techniques have been developed to provide realistic predictions of the anticipated production performance of these reservoirs. 2o-z5 The depletion processes are simulated by an incremental computation method, using multicomponent flash calculations and relative-permeability data, as indicated in the following stepwise sequence for a chosen pressure decrement:



1. The change in composition of the in-place oil and gas is determined by a flash calculation. 2. The total volume of fluids produced at bottomhole conditions is determined by a volumetric material balance. 3. The relative volumes of oil and gas produced at bottomhole conditions are determined by a trial-and-error procedure that involves simultaneously satisfying the volumetric material balance and the relative-permeability relationship. 4. This total well-stream fluid is then flashed to actual surface conditions to obtain the producing GOR and the volume of stock-tank liquid corresponding to the selected pressure decrement. When this calculation procedure is repeated for successive pressure decrements, the resultant tabulations represent the entire reservoir depletion and recovery processes. Since these stepwise calculations are rather tedious and time-consuming, the use of digital computers is recommended. This method of reservoir analysis provides compositional data on all fluid phases, including the total wellstream. This information is then readily available for separator, crude-stabilization, gasoline-plant, or related studies at any desired stage of depletion. In the case of small reservoirs with relatively limited reserves, such lengthy laboratory work and phasebehavior calculations may not be justified. An empirical correlation was developed24 for prediction of the ultimate recovery in such cases, based only on the initial producing GOR, R, the reservoir temperature, TR, and the initial stock-tank oil gravity, yO;. N,, = -0.070719+-



+O.O011807y~i,



143.50 +O.O001208OT,



R



.



. .



.



(28)



where N,, =ultimate oil production from saturation pressure ph to 500 psi, in stock-tank volume per reservoir volume of hydrocarbon pore space. It is claimed that this correlation will give values within 10% of those calculated by the more rigorous procedure previously outlined.



Oil Reservoirs With Gas-Cap DriveVolumetric Unit Recovery Computed by Frontal-Drive MethodZ628 The Buckley-Leverett frontal-drive method may be used in calculating oil recovery when the pressure is kept constant by injection of gas in a gas cap but is also applicable to a gas-cap drive mechanism without gas injection when the pressure variation is relatively small so that changes in gas density, solubility, or the reservoir volume factor may be neglected. A reservoir with a very large gas-cap volume as compared with the oil volume can sometimes be considered to meet these qualifications even though no gas is being injected. The two basic equations, Eqs. 29a and b, refer to a linear reservoir under constant pressure with a constant cross-sectional area exposed to fluid flow and with the free gas moving in at one end of the reservoir and fluids being produced at a constant rate at the other end. Interstitial water is considered as an immobile phase.



PETROLEUM



40-14



s?



I



I



I



0



I



I



I



lbfil -Al



VE A



!I



HANDBOOK



Note: Sk as used in this section is gas saturation as a fraction of the hydrocarbon-filled pore space. When N is in cubic meters, q1 is in cubic meters per day. The calculation procedure is first to calculate the fractional-flow curve (Fig. 40.7, Curve A). The average gas saturation in the swept area at breakthrough, which is equivalent to the fraction of oil in place recovered, may then be obtained from the fractional-flow curve by constructing a straight line tangent to the curve through the origin and reading Sk at fR = 1.O. The time of breakthrough at the outlet face may be computed from the slope of the curve at the point of tangency. The subsequent performance history after breakthrough may then be calculated by constructing tangents at successively higher values of Sk and obtaining Sh in a similar manner.



’



I



ENGINEERING



--i



Example Problem 4. Welge2s presents a typical calculation of gas-cap drive performance for the Mile Six Pool in Peru. Given: 0=



I



0



0.10 0.20 0.30



I



&O



I 0.50 0.60



Reservoir volume= 1,902 X lo6 cu ft, distance from original GOC to average withdrawal point = 1,540 ft,



0.70



S&GAS SATURATION, FRACTION OF HYDROCARBON FILLED PORE SPACE Fig. 40.7-Frontal-drive



method in gas-cap



1,902x IO6 average cross-sectional



drive



=1.235x106 If the capillary-pressure forces are neglected. fractional-flow equation of gas is



the



(294



E=



k sin @A@,--pR)



.



..



36%.,qr



(29b)



where fX = E = 8 = A =



fractional flow of gas, parameter, dip angle, degrees, area of cross-section normal to bedding plane, sq ft, PO = density of reservoir oil, g/cm3, ph’ = density of reservoir gas, g/cm3. and q, = total flow rate, reservoir cu ft/D.



5.615NB, q,(df,,dS;)



.



.



1,540



sq ft,



k, = 300 md, 8 = 17.50, ps = 0.0134 cp,



P”o = 1.32 cp, q, = 64,000 res cu ft!D [I8 125 res m”/d], B,, = 1.25, B, = 0.0141 N = 44~ lo6 STB [6.996x106 m”], R,, = 400 cu ft/bbl [71.245 m’/m’J, PO = 0.78 g/cm”, and Ph’ = 0.08 g/cm 3 Solution. The performance history calculations given in Table 40.5 in a slightly simplified form.



are



Oil Reservoirs Under Gravity Drainage 29-37 Occurrence of Gravity Drainage



Since the ratio of k,lk, is a function of gas saturation, and all other factors are constant, j$ can be determined by Eq. 29a as a function of gas saturation (see Fig. 40.7, Curve A). The rate-of-frontal-advance equation may be rearranged to give the time in days for a given displacing-phase saturation to reach the outlet face of the linear sand body as a function of the slope of the fractional flow vs. saturation curve (Fig. 40.7, Curve B) as follows: t=



area =



(30)



Gravity drainage is the self-propulsion of oil downward in the reservoir rock. Under favorable conditions it has been found to effect recoveries of 60% of the oil in place, which is comparable with or exceeding the recoveries normally obtained by water drive. Gravity is an ever-present force in oil fields that will drain oil from reservoir rock from higher to lower levels wherever it is not overcome by encroaching edge water or expanding gas. Gravity drainage will be most effective if a reservoir is produced under conditions that allow flow of oil only or counterflow of oil and gas. This may be attained under pressure maintenance by crestal-gas injection, which keeps the gas in solution, or it may be attained by a gradual reduction in pressure, so that the oil and gas can segregate continuously by counterflow. It also may be obtained by



ESTIMATION



OF OIL AND GAS RESERVES



40-15



first producing the reservoir under a depletion-type mechanism until the gas has been practically exhausted, then by gravity drainage. A thorough discussion of the many aspects of gravity drainage will be found in the classic paper by Lewis.32 Several investigators 33m36have attempted to formulate gravity drainage analytically, but the relationships are quite complicated and not readily adaptable to practical field problems. Most studies agree, however, that the occurrence of gravity drainage of oil will be promoted by low viscosities, p,, , high relative permeability to oil, k,, high formation dips or lack of stratification, and high density gradients (p, -p,). Thick sections of unconsolidated sand with minimal surface area, large pore sizes, low interstitial water saturation, and consequently high k, appear to be especially favorable. These factors usually are combined in a rate-of-flow equation. which states that such flow must be proportional to (k,,lp,)(p,, -p,) sin 8, in which 8 represents the angle of dip of the stratum. Smithj7 compared the values of this term for a dozen reservoirs, some of which had strong gravity-drainage characteristics and some of which lacked such characteristics. When expressing k,,, in millidarcies, p,, in centipoises, and p,, and pI: in g/cm”, it was found that for reservoirs exhibiting strong gravity-drainage characteristics the value of the term (k,,ip,)(p, -P,~) sin 0 ranged from 10 to 203 and that in reservoirs where gravity-drainage effects were not apparent, this function showed values between 0.15 and 3.4.



y(, =22.5”API, N,, for Jan. 1, 1957=44.6 million bbl of oil; estimated ultimate 47 million bbl or I, 124 bbliacreft, corresponding to 63% of the initial oil in place. During the first 20 years the oil level in the field receded almost exactly in proportion to the amount of oil produced, just as in a tank. 2. Okluhoma City Wilcox Reservoir, OK. 29~32The discovery well, Mary Sudik No. I, blew out in March 1930, and flowed wild for 11 days. The segregation of gas and development of gravity drainage began to be important in 1934, when the average pressure became less than 750 psig, and was virtually complete by 1936, when the average pressure had dropped to 50 psig. Water influx played an effective role until 1936, when it came to a halt after invading the bottom 40% of the reservoir. Gravity has been the dominant mechanism since. The Wilcox sand consists of typical round frosted sand grains, clean and poorly cemented. The average depth is 6,500 ft; the formation dip is 5 to 15”; 884 wells have been drilled on a total area of 7,080 acres. The net pay thickness is 220 ft. The 890,000 net acre-ft of Wilcox pay contained originally 1,083 million bbl of stock-tank oil, as confirmed by material balance. Reservoir data for this reservoir are pi =ph = 2,670 psi at minus 5,260 ft, TR= 132”F, $=0.22, k ranges from 200 to 3,000 md, S;,.=O.O3 (oil wet), Rt,, =735 cu ft/bbl, B,;=l.361, y,i=40”APl, yoci=38 tO 39”API. According to Katz, z9 oil saturations found in the gas zone were between 1 and 26%, while saturations between 53 and 93% were found in the oil-saturated zone below the GOC. The oil saturation below the WOC has been estimated at 43%, showing gravity to be more effective than water displacement in this reservoir. Cumulative production, N,, for Jan. 1, 1958, is estimated at 525 million bbl and the ultimate recovery at 550 million bbl. After an estimated 189 million bbl displaced by the water influx is deducted, the upper 60% of the Wilcox reservoir will yield under gravity drainage ultimately 361 million bbl or 696 bbliacre-ft, corresponding to 57% of the oil in place.



Case Histories of Gravity Drainage After Pressure Depletion The most spectacular cases of gravity drainage have been of this kind. Following are the two best known. 1. Lukeview Pool in Kern County, CA. 3’~32 The discovery well in the Lakewood gusher area blew out in March 1910, flowed wild for 544 days, and ultimately produced 8% million bbl of oil, depleting the reservoir pressure. Gravity drainage thereafter controlled this reservoir. There was no appreciable water influx. The sand is relatively clean and poorly cemented. The average depth is 2,875 ft. The formation dip is IS to 45”. There are I26 producing wells on 588 acres. The net sand thickness averages 7 1 ft, the height of the oil column is 1,285 ft. and there are 41,798 net acre-ft of pay. Reservoir data for this reservoir are pi =P/, = 1,285 psi& PR on Jan. I, l957=35 psig, r,= 115°F. 4=0.33, k ranges up to 4,800 md and averages 3.600 md (70% of samples above 100 md, 37% above 1,000 md), S,,, =0.235, R,,,=200 cu ftibbl, Boi= 1.106,



TABLE



Oil Reservoirs With Water DriveVolumetric Method9 General Discussion Natural-water influx into oil reservoirs is usually from the edge inward parallel to the bedding planes (edgewater drive) or upward from below (bottomwater drive). Bottomwater drive occurs only when the reservoir thickness exceeds the thickness of the oil column, so that the oil/water interface underlies the entire oil reservoir. It is



40.5~PERFORMANCE-HISTORY



CALCULATION



s: = S’ near Outget Face 0.30 a 35



ro k 0.197 0.140



kro’k,, 0.715 0.364



0.496 0.642



0.395 0.40 0.45 0.50



0.102 0.097 0.067 0.045



0.210 0.200 0.118 0.0715



0.739 0.752 0.829 0.885



f,



df,lds;



1 .a7 1.81 1 .25 0.94



Recover; Fraction of Oil in Place



k



Flowing GOR =



If,41 -01(&/Q x5. I?? l+R,



-



-



-



7.1 7.3 10.6 14.1



0.534 0.535 0.586 0.622



1.808 1.908 2.811 4.227



PETROLEUM



40-16



TABLE 40X-CONDITIONS FOR UNIT-RECOVERY EQUATION, WATER-DRIVE RESERVOIR



Reservoir pressure Interstitial water, bbllacre-ft Reservoir oil, bbllacre-ft Stock-tank oil, bbllacre-ft



Initial Conditions



Ultimate Conditions



Pi



Pa



7,75848,,



7,75&S,,



7.756@(1 -S,,)



7,758@,,



Recovery-Efficiency



further possible only when vertical permeabilities are high and there is little or no horizontal stratification with impervious shale laminations. In either case, water as the displacing medium moves into the oil-bearing section and replaces part of the oil originally present. The key to a volumetric estimate of recovery by water drive is in the amount of oil that is not removed by the displacing medium. This residual oil saturation (ROS) after water drive, S,,, plays a role similar to the final (residual) gas saturation, S,, , in the depletiontype reservoirs. To determine the unit-recovery factor, which is the theoretically possible ultimate recovery in stock-tank barrels from a homogeneous unit volume of 1 acre-ft of pay produced by complete waterflooding, the amount of interstitial water and oil with dissolved gas initially present will be compared with the condition at abandonment time, when the same interstitial water is still present but only the residual or nonfloodable oil is left. The remainder of the original oil has at that time been removed by water displacement. Unit-Recovery



Equation



The unit recovery for a water-drive reservoir is equal to the stock-tank oil originally in place in barrels per acrefoot minus the residual stock-tank oil at abandonment time (Table 40.6). By difference, the unit recovery by water drive, in stock-tank barrels per acre-foot, is .(31)



where N,,. is the unit recovery by water drive, in stocktank barrels, and S,, is the residual oil saturation, fraction. The ROS at abandonment time may be found by actually submitting cores in the laboratory under simulated reservoir conditions to flooding by water (flood-pot tests). Another method commonly used is to consider the oil satuTABLE



40.7-RECOVERY-EFFICIENCY



Reservoir Number 1 2 3 4 5



$I 0.179 0.170 0.153 0.192 0.196



Factor



The unit recovery should be multiplied by a permeabilitydistribution factor and a lateral-sweep factor before it may be applied to the computation of the ultimate recovery for an entire water-drive reservoir. These two factors usually are combined in a recoveryefficiency factor. Baucum and Steinle3’ have determined this recovery-efficiency factor for five water-drive reservoirs in Illinois. Table 40.7 lists the recovery efficiencies for these reservoirs, together with some other pertinent data. Average Recovery Factor From Correlation of Statistical Data In 1945, Craze and Buckley,39,40 in connection with a special API study on well spacing, collected a large amount of statistical data on the performance of 103 oil reservoirs in the U.S. Some 70 of these reservoirs produced wholly or partially under water-drive conditions. Fig, 40.8 shows the correlation between the calculated ROS under reservoir conditions and the reservoir oil viscosities for these water-drive reservoirs. The deviation of the ROS from the average trend in Fig. 40.8, vs. permeability, is given by the average trend in Fig. 40.9. The deviation of the ROS from the average trend in Fig. 40.8, vs. reservoir pressure decline, is given by the average trend in Fig. 40.10. Example Problem 5. In a case where the porosity, 4=0.20, the average permeability, k=400 md, the interstitial water content, Si,=O.25, the initial oil FVF, B,, = 1.30, the oil FVF under abandonment conditions, B, = 1.25, the initial reservoir oil viscosity, pLo= 1.O cp, and the abandonment pressure, pu =90% of the initial pressure, pi, determine the average ROS. Solution. S,, may be estimated as 0.35+0.03-0.04= 0.34 and the average water-drive recovery factor from Eq. 31 is l-O.25



N,,.=(7,758)(0.20)



0.34 >



=473 STBlacre-ft FOR WATER-DRIVE



S,,



B,



S,,’



Unit-Recovery Factor (bbl/acre-ft)



0.400 0.340 0.265 0.370 0.360



1.036 1.017 1.176 1.176 1.017



0.20 0.20 0.20 0.20 0.20



526 592 504 500 653



RESERVOIR



Actual Recovery’ (bbllacre-ft) 429 430 428 400 482



Recovery Efficiency (O/o) 82 73 85 80 74 Average = 79



‘From



flood-pot tests



HANDBOOK



ration as found by ordinary core analysis after multiplying with the oil FVF at abandonment, B,)O, as the residual oil saturation in the reservoir to be expected from flooding with water. This is based on the assumption that water from the drilling mud invades the pay section just ahead of the core bit in a manner similar to the water displacement process in the reservoir itself.



- S,,)IB,, 7,75&S~B,,



7,7584(1



ENGINEERING



ESTIMATION



OF OIL AND GAS RESERVES



40-17



lo.30 .. 5, F :: *a20 Lsk 3a LiL 1 8 lO.‘O 02 2’0 ?I+ 0 OIL h i0 g 6 -o .,o & L 4 EE -0.20 2 g 0 0.2



-0.30 0.4 06



I



2



4



6



IO



20



40



60



100



EC0



20



40



OIL VISCOSITY AT RESERVOIR CONDITIONS; CENTIPOISES



Fig. 40.8-Effect



of oil viscosity on ROS water-drive sand fields.



In another statistical study of the Craze and Buckley data and other actual water-drive recovery data on a total of 70 sand and sandstone reservoirs, the API Subcommittee on Recovery Efficiency t6 developed Eq. 32 for unit recovery for water-drive reservoirs, N,,. In stocktank barrels per acre-foot,*



-0.2159



,



.



..



(32)



where symbols and units are as previously defined except permeability, k, is in darcies, and pressure, p, is in psig. Example Problem 6. For the same water-drive reservoir used previously and assuming pwi =O.S cp, the API statistical equation yields the following unit recovery factor: (0.20)(1-0.25) N,, =4,259 1.30



1.0



x-



100



200



400



AVERAGE PERMEABILITY



lcco



EOW



4oM)



Io.ow



OF RESERVOIR; MILLIDARCIES



Fig. 40.9-Relation between deviation of ROS from average trend in Fig. 40.8 and permeability water-drive sand fields.



Water-Drive Unit Recovery Computed by Frontal-Drive Method26-28 The advance of a linear flood front can be calculated by two equations derived by Buckley and Leverettz6 and simplified by Welge** and by Pirson. ” These are known as the fractional-flow equation and the rate-of-frontaladvance equation. This method assumes that (1) a flood bank exists, (2) no water moves ahead of this front, (3) oil and water move behind the front, and (4) the relative movement of oil and water behind the front is a function of the relative permeability of the two phases. If the throughput is constant and the capillary-pressure gradient and gravity effects are neglected, the fractionalflow equation can be written as follows:



fw=



1 1 +(k,lk,,,,)(pJp,)



’



.



. (33)



1



‘.0422



-0.2159



( > 0.9



= 504 STB/acre-ft Because data were arrived at by comparing indicated recoveries from various reservoirs with the known parameters from each reservoir, the estimated residual oil and the average recovery factor based on these correlations allows for a recovery-efficiency factor (permeabilitydistribution factor times lateral-sweep factor) that is not present in the unit-recovery factor based on actual residual oil as found by flood-pot tests or in the cores. ‘because Eq 32 IS empirlcally darned, conversion to metric units jmJ/ha.m) mulbpl~cark?m of Nup by 1.2899



requires



0



20 RESERWR



40 60 SO PRESSURE DECLINE: PER CENT



100



Fig. 40.10--Relation between deviation of ROS from average trend in Fig. 40.8 and pressure-decline water-drive sand flelds.



PETROLEUM



40-18



ENGINEERING



HANDBOOK



3 1.0 5 0.9 2



k-~0.8 d 5 0.7 Iz - 0.6 ii? :



0.5



1.05 Iv..



.2



I



0.3



0.4



0.5



WATER



0.6



0.7



SATURATION,



FRACTION



0.8



0.9



S,, TIME



OF PORE SPACE



YEARS



+



Fig. 40.11-Fraction of water flowing in total stream f, and slope off, curve df,/dS,, vs. water saturation S,, (example: frontal-water-drive problem).



Fig.



wheref,, is the fraction of water flowing in the reservoir at a given point, k,. is the water relative permeability, fraction, and pn, is the reservoir water viscosity, cp. Since k,,lk,. is a function of water saturation, f,+, can be determined by Eq. 33 as a function of water saturation for a given water/oil viscosity ratio (see Fig. 40.11, Curve A). The Buckley-Leverett rate-of-frontal-advance equation may be rearranged to give the time in days for a given displacing phase saturation to reach the outlet face of the linear sand body as a function of the slope of the fractional flow vs. saturation curve (Fig. 40.1 I, Curve B) as follows:



Fig. 40.12 is a plot of the results of the performancehistory calculation from Table 40.8. If the economic limit is taken to be a WOR of 50, then it can be noted from Fig. 40.12 that the unit-recovery factor will be 575 bbllacre-ft to be recovered in 20.7 years.



5.615 NB, t= qr(df,,,dSi,*,)



(



.



..



.



(34)



where df,ldS,,. is the slope of thef, vs. Si, curve; the time, t, is in days; and the total liquid flow rate, qr, is in reservoir cubic feet per day. The average water saturation behind the flood front at breakthrough, and therefore the oil recovery, may be obtained from the fractional-flow curve by constructing a straight line tangent to the curve through S;, atf,=O, and reading S ;,, at f, = 1.O. The time of breakthrough at the producing well may be computed from the slope of the curve at the point of tangency. The subsequent performance history after breakthrough may be calculated by constructing tangents at successively higher values of S;, and obtaining Si, in a similar manner. Table 40.8 illustrates the calculation procedure for a water drive at constant pressure in a homogeneous reservoir and with a water-influx rate equal to the production rate.



40.12-Example of frontal-drive problem, unit-recovery factor, and WOR vs. time.



Effect of Permeability



Distribution ‘t41-44



In some reservoirs there may be distinct layers of higher and lower permeabilities separated by impervious strata. which appear to be more or less continuous across the reservoir. In such a case, water and oil will advance much more rapidly through the higher-permeability streaks than through the tighter zones, and therefore the recovery at the economic limit will be less than that indicated by the unit-recovery factor. Methods for computing waterflood recoveries that take into account the permeability distribution were proposed by Dykstra and Parsons,4’ Muskat. and Stiles.43 In the Dykstra-Parsons paper4’ it is assumed that individual zones of permeability are continuous from well to well, and a computation procedure as well as charts are presented for the coverage or fraction of the total volume of a linear system flooded with water for given values of (1) the mobility ratio knvpolkropw, (2) the produced WOR, and (3) the permeability variance. This permeability variance is a statistical parameter that characterizes the type of permeability distribution. It is obtained by plotting the percentage of samples “larger than” the sample being plotted vs. the logarithm of permeability for that sample on log-probability graph paper and then dividing the difference between the median or 50% permeability and the 84. I % permeability by the median permeability. Although the Dykstra-Parsons method



ESTIMATION



OF OIL AND GAS RESERVES



TABLE



40-l 9



40.8-WATER-DRIVE



PERFORMANCE-HISTORY



Time



s



1w



~ S,,



0.545 0.581 0.605 0.634 0.673 0.718



0.619 0.655 0.675 0.697 0.720 0.748



fw



df,JdS,w



(years)



0.800 0.875 0.910 0.940 0.970 0.990



2.70 1.69 1.29 0.95 0.64 0.33



3.94 6.29 8.24 11.19 16.61 32.21



Residual Oil Saturation (1 -S,,) 0.381 0.345 0.325 0.303 0.280 0.252



CALCULATION* Unit-Recovery Factor (bbl/acre-ft)



WOR = f,/l -f,



441 484 507 534 561 594



4.0 7.0 10.1 15.7 32.3 99.0



‘N = 597,000 STB, ao, = 1 30, o=o 20. S,, =0 25, and qr = 200 E/D x 5 615 cu ftlbbl = > ,222 ,esewow cu fl/D



does not allow for variations in porosity, interstitial water. and floodable oil in the different permeability groups, it has apparently been used extensively and successfully on close-spaced waterfloods. mainly in California. Johnson’4 in 1956 published a simplification of this method and presented a series of charts showing the fractional recovery of oil in place at a given produced WOR for a given permeability variance, mobility ratio, and water saturation. Reznik er al. 4s published an extension to the Dykstra-Parsons method that provides a discrete analytical solution to the permeability stratification problem on a real-time basis. In the Stiles method4” it again is assumed that individual zones of permeability are continuous from well to well and that the distance of penetration of the flood front in a linear system is proportional to the average permeability of each layer. Instead of representing the entire permeability distribution by one statistical parameter, Stiles tabulates the available samples in descending order of permeability and plots the results in terms of dimensionless permeability and cumulative capacity fraction as a function of cumulative thickness. From these data, Stiles computes the produced water cut of the entire system as the watering out progresses through the various layers, starting with those of the highest permeability. Stiles then assumes that at a given time each layer that has not had breakthrough will have been flooded out in proportion to the ratio of its average permeability to the permeability of the last zone that had just had breakthrough, and then constructs a recovery vs. thickness relationship. This then is combined with previous results to yield a recovery vs. water-cut graph. The Stiles method is used extensively and successfully, mainly in the midcontinent and Texas, for close-spaced waterfloods. It does not make allowance for the difference in mobility existing in the formation ahead of and behind the flood front. which the DykstraParsons method allows for. It also does not provide for differences in porosity, interstitial water, and floodable oil in the various permeable layers. Arps ’ introduced in 1956 a variation of the Stiles method, called the “permeability-block method.” This method handles the computations by means of a straightforward tabulation and does make allowance for the differences in porosity, interstitial water, and floodable oil existing in the various permeable layers. Since it is designed primarily for the computation of recoveries from waterdrive fields above their bubblepoint. no free-gas satura-



tion is assumed. The method further assumes that (I) no oil moves behind the front, (2) no water moves ahead of the front, (3) watering out progresses in order from zones of higher to zones of lower permeability. and (4) the advance of the flood front in a particular permeability streak is proportional to the average permeability. This method, applied to a hypothetical pay section 100 ft thick, is illustrated in Table 40.9, which is based on data from a Tensleep sand reservoir in Wyoming where good statistical averages of more than 3,000 core analyses were available. Part of these cores were taken with water-base mud that yielded the residual-oil figures on Line 6. Another portion was taken with oil-base mud and yielded the interstitial-water figures of Line 7. An oil/water viscosity ratio of 12.5 was used in calculating the WOR of Line 13. In Group I the recovery of 61.7 bbliacre-ft for WOR= 15.5 is the product of the fraction of samples in the group and the unit-recovery factor. In all other groups for WOR = 15.5 the full recovery is reduced in the proportion of its average permeability to 100 md. The total recovery at WOR= 15.5 is shown as 175.6 bbliacre-ft. The cumulative recoveries for WOR’s of 35.9, 76.5, 307.7, and infinity are calculated in a similar manner. Fig. 40.13 is a plot of WOR vs. recovery factor. From Fig. 40.13 it can be seen that, if the economic limit is taken to be a WOR of 50, the recovery factor would be 297 bbliacre-ft. It should be stressed that the permeability-block method is applicable only when the zones of different permeability are continuous across the reservoir, or between the source of the water and the producing wells. When the waterfront has to travel over large distances, nonuniformity of permeability distribution in lateral directions begins to dominate, and recoveries will approach those obtainable if the formation were entirely uniform (permeability distribution factor= 1). In such a case, an estimate based on the permeability-block method may be considered as conservative, except for the fact that one of the basic assumptions of this method is that the WOC, or front, moves in pistonlike fashion through each permeability streak, sweeping clean all recoverable oil. In reality, part of this oil will be recovered over an extended period after the initial breakthrough, which may tend to make the estimate optimistic. Those using the permeability-block method hope that these two effects are more or less compensating.



PETROLEUM



40-20



TABLE



40.9-WATER



DRIVE



PERMEABILITY-BLOCK



(15) (16) (17) (18)



Permeability range, mud Percent of samples in group Average permeability, md Capacity, darcy-ft (2) x (3) + 1,000 Average porosity fraction $ Average residual-oil fraction Sgr Average interstitial-water fractron S,, Relative water permeabrlity behind front k Relative oil permeability ahead of front k,, Unit-recovery factor (B,, = 1.07) Cumulative “wet” capacity, E(4) Cumulative “clean oil” capacity, 3.241 - (11) Water-oil ratio WOR= (~00~c)(8/9)(1 l/12) Cumulative recovery at WOR = 15.5 bbllacre-ft Min k wei =I00 md Cumulative recovery at WOR = 35.9 bbllacre-ft Min k,,, =50 md Cumulative recovery at WOR = 76.5 bbl/acre-ft Min k we, = 25 md Cumulative recovery at WOR = 307.7 bbllacre-ft Min k we, =lO md Cumulative recovery at WOR = mbbllacre-ft Min k wer=0 md



Effect of Buoyancy and Imbibition In limestone pools producing under a bottomwater drive, such as certain of the vugular D-3 reef reservoirs in Alberta, one finds an extreme range in the permeabilities, often running from microdarcies on up into the darcy range. Under those conditions the modified Stiles method heretofore described yields results that are decidedly too



400,



I



I



I



I



r



f



n /



1



200. 0 G.--



I I



~100 g



80-



5



40



I I 1



I I I



- ECONOMIC , .9 WOR=5Ojmi ’ 60kIMIT 5 50 -“T---q---



I I I



!



RECOVERY FACTOR =297 BBL/ACRE, FT@ WOR =50



20



lOI 0 RECOVERY



31 200 FACTOR,



, 400



,



I 600



BBL/ACRE-FT



Fig. 40.13-Example of modified Stiles permeability-block method WOR vs. recovery factor.



HANDBOOK



CALCULATIONS Total



2



3



4



5



>lOO 8.5 181.3 1.541 0.159 0.173 0.185 0.65 0.475 726 1.541 1.700 15.5 61.7



50 to 100 10.9 69.0 0.752 0.150 0.195 0.154 0.63 0.53 693 2.293 0.948 35.9 52.1



25 to 50 14.5 34.4 0.499 0.152 0.200 0.131 0.60 0.61 722 2.792 0.449 76.5 36.0



10 to 25 21.2 16.1 0.341 0.130 0.217 0.107 0.56 0.66 623 3.133 0.108 307.7 21.3



0 to 10 44.9 2.4 0.108 0.099 0.222 0.185 0.54 0.47 415 3.241 0 4op5



175.6



61.7



75.5



72.0



42.5



8.9



260.6



61.7



75.5



104.7



85.1



17.9



344.9



61.7



75.5



104.7



132.1



44.7



418.7



61.7



75.5



104.7



132.1



186.3



560.3



Group (1) (2) (3) (4) (5) (6) (7) (8) (9) (10) (11) (12) (13) (14)



ENGINEERING



100.0 3.241



low. The reason is that, in pools like the Redwater D-3, there is a substantial density difference between the rising salt water and the oil. While the water rises and advances through the highly permeable vugular material, it may at first bypass the low-permeability matrix material, leaving oil trapped therein. However, as soon as such bypassing occurs, a buoyancy gradient is set up across this tight material, which tends to drive the trapped oil out vertically into the vugular material and fractures. In the case of Redwater D-3, where the density difference between salt water and oil is 0.26, while the vertical permeabilities for matrix material are only a fraction of the horizontal permeabilities, a simple calculation based on Darcy’s law applied to a vertical tube shows that during the anticipated lifetime of the field very substantial additional oil recovery may be obtained because of this socalled buoyancy effect. To calculate the recovery under a buoyancy mechanism it is necessary first to determine by statistical analysis of a large number of cores the average interval between highpermeability zones or fractures. A separate computation is then made for each of the permeability ranges to determine what percentage of the matrix oil contained in a theoretical tube of such average length may be driven out during the producing life of the reservoir under the effect of the buoyancy phenomenon. Surprisingly improved recoveries are sometimes indicated by this method over what one would expect from a Stiles type of calculation, and the results from recent studies of the rise in water table of the Redwater D-3 seem to confirm the validity of this concept. In addition to this buoyancy phenomenon the effect of capillarity and preferential wetting of the reservoir rock by water also should be considered. Imbibition of water from fractures and vugular material into the lowpermeability matrix as the water advances may materially aid the buoyancy mechanism but is much more difficult to evaluate quantitatively.



ESTIMATION



OF OIL AND GAS RESERVES



TABLE



40-21



40.10-PSEUDOCRITICAL



CALCULATIONS Volume % or MO&



Component (11 Methane Ethane Propane lsobutane Normal butane



lsopentane Normal penlane Hexanes



86.02 7.70 4.26 0.57 0.87 0.11 0.14 0.33



343.5 550.1 666.2 733.2 765.6 630.0 847.0 914.6



Factor



The compressibility factor z is a dimensionless factor which, when multiplied by the reservoir volume of gas, as computed by the ideal-gas laws, yields the true reservoir volume. The reservoir volume occupied by 1 lbmmole of gas (gas weight in pounds equal to molecular weight), in cubic feet, is G=



(10.73)z(460+TR) PR



(



.



. . . (35)



where G is the total initial gas in place in reservoir, in standard cubic feet, and TR is the reservoir temperature, “F. For example, 1 lbm-mole of methane (molecular



Critical Pressure



2x3 100



(;:)



Volumetric Recovery Estimates for Nonassociated Gas Reservoirs46-53 Compressibility



Critical Temperature



(2)



100.00



FROM GAS ANALYSIS



(77 673 708 617 530 551 482 485 434



(5) 296-42.4 26.4 4.2 6.7 0.9 1.2 3.0 362.6



2x4 100



(‘3) 572 54.5 26.3 3.0 4.8 0.5 0.7 1.4 663.2



weight 16.04) under standard conditions (PR = 14.7 psia, TR=~O’F) occupies 379.4 cu ft. The compressibility factor may be determined in the following ways. 1. Experimentally by PVT analysis of a gas sample. 2. By computation from an analysis of the gas expressed in mol% or volume %. With this method a weightedaverage or pseudocritical pressure and temperature are obtained for the gas by multiplying the individual critical pressure and temperature for each component, with the corresponding mol% of such component as shown in Table 40.10. The gas whose composition is given in Table 40.10 has a pseudocritical temperature of 382.8”R and a pseudocritical pressure of 663.2 psia. The pseudoreduced temperature then is found at a temperature of 150°F as (460 + 150)/382.8 = 1.59 and its pseudoreduced pressure



PSEUOO REDUCED PRESSURE



Fig. 40.14B-Compressibility factors for natural sures of 10,000 to 20,000 psia.



PSEUDO REDUCED PRESSURE



Fig.



40.14A-Compressibility



factors for natural gases.



gases



et



pres.



40-22



PETROLEUM



RESERVOIR



PRESSURE



Fig. 40.15-Gas



FVF 8,



1 B&l



460+ T, --------Z 460+60



14.17 = ~ p,+14.7



and reciprocal



RESERVOIR



(pR] IN PSI GAUGE



Fig. 40.16-Gas



p,+14.7



460+60



1



14.7



460+T,



z



8,



40.1 I-PSEUDOCRITICAL CALCULATIONS FROM SPECIFIC GRAVITY



Specific gravity of Gas (Air=l.O)



Pseudocritical Temperature (OR) (460+ OF)



Pseudocritical Pressure (psia) (14.7+ psig)



0.55 0.60 0.70 0.80 0.90 1 .oo 1.10 1.20 1.30 1.40 1.50 1.60 1.65



348 363 392 422 451 480 510 540 570 600 629 658 673



674 672 669 665 660 654 648 641 632 623 612 600 593



460+ T, p-z 460+60



gas FVF



pR+14.7 14.7



460+60 ~460+T,



1 z



vs. pressure,



psig, and temperature, Gas gravity 0.7 (air 1 .O).



OF



at 750 psia as 7501663.2 = 1.13. These ratios are entered into the chart of Fig. 40.14A to read z=O.91. This correlation chart46 and an extended correlation chartj7 for higher-pressure gas reservoirs up to 20,000 psia, Fig. 40.14B, are designed for gaseous mixtures containing methane and other natural gases but substantially free of nitrogen. For hydrocarbon gases containing substantial amounts of hydrogen sulfide or CO1 , these correlations do not apply, and additional corrections are necessary as described in Ref. 48. (See Chap. 20 for complete coverage of gas properties and gas property correlations, some of which are specific to computer applications.)



HANDBOOK



(pR) IN PSI GAUGE



14.17 = ~ p,+14.7



FVF 8,



1 -=



vs. pressure, psig, and temperature, Gas gravity 0.6 (air 1 .O).



TABLE



PRESSURE



and reciprocal



gas FVF



ENGINEERING



OF.



3. By computation from the specific gravity of the gas. If only the specific gravity of the gas (air= 1.O) is known, another approximate correlation can be used, based on California natural gases,49 which is expressed by Table 40.11. For example, if the specific gravity of a gas is 0.66. the pseudocritical temperature can be estimated by interpolation as 381 “R and pseudocritical pressure as 670 psia. The pseudoreduced values then are found as before and the z factor read from Fig. 40.14A. Gas FVF The gas FVF, B,, is a dimensionless factor representing the volume of free gas at a reservoir temperature of T”F and a pressure of p psia per unit volume of free gas under standard conditions of 60°F and 14.7 psia. If the compressibility factor, z, is known, B,? may be computed by 14.7 460+7-, B,=pR



460+60



z=O.O2827(46O+T,)i. PR



. . . . . . . . . . . . . . . . . . . . . . . . . . . . . (36) Typical values of the gas FVF, B, , and the reciprocal gas FVF, l/B,, for different temperatures and pressures and for gases of specific gravities between 0.6 and 1.0 will be found in Figs. 40.15 through 40.19. In estimating gas reserves, the estimator should be careful to indicate clearly the pressure base at which the reserves are stated. Reserves at a base pressure of 14.4 psia will be approximately 16% greater than the same reserves stated at a base pressure of 16.7 psia.



ESTIMATION



40-23



OF OIL AND GAS RESERVES



400



0.4



as



xx)



0.2



02



01 ma



2" g



0.06



= L



a04



Y 3



a02



6 s H e



QOl o.cca 3 0.006 a004



l.ow RESERVilR



PRESSURE



Fig. 40.17-Gas



&)



14.17 = ___ p,+14.7



FVF 6,



and reciprocal 1



S,, may be estimated from a material-balance calculation on the condensate present in the reservoir gas under initial conditions, and the condensate to be recovered during the depletion of the reservoir in the surface separation equipment. Effect of Permeability



Distribution



Unless a gas reservoir is known to be permeable and homogeneous, the unit-recovery factor should be corrected for the fact that depletion may progress more rapidly in the high-permeability strata than in the low-permeability zones, particularly if these zones are separated by impervious strata. An uneconomic rate of production may be reached before the tighter zones are drained down to abandonment pressure. In many cases, nonuniformity of permeability in lateral directions provides a compensating influence. In very hard and tight formations, extensive fracturing may have the same result. A computation based on the assumption that the strata of different permeabilities are uniform and continuous across the reservoir is therefore in most cases too pessimistic. Such a computation does provide a means, however, to indicate the minimum recoverable reserves while the assumption of a completely homogeneous reservoir and the direct use of the unit-recovery factor indicate a maximum figure for the recoverable reserves. A permeability-block method to compute such minimum reserves for a nonassociated dry gas reservoir is as follows.



ESTIMATION



40-25



OF OIL AND GAS RESERVES



According to Eqs. 13 and 14 of Sec. 11.15 in Ref. 50, the boundary pressure in a closed cylindrical gas reservoir, drained by a well in the center with zero pressure against the sandface, may be approximated as



TABLE



40.13-CONDITIONS FOR UNIT RECOVERY EQUATION IN A DRY-GAS RESERVOIR



Reservoir pressure



,



.(39)



while the gas production rate is .



ql: =C*k,h,p*,



Free gas, scf/acre-ft



.



.



(41,)I Cl(k#)lP,



PI (--> PI



Ultimate Conditions



PI



P.3



43,56OQS,,



43,56O~S,,



43,5604(1 -S,,)



B,l



.(40)



in which Ct and Cz are constants and $h and h, are effective hydrocarbon porosity and effective thickness, respectively It will be assumed that a large number of core analyses are available on a gas reservoir, which are divided in permeability groups as shown on Table 40.13. The average permeability, k , for each group is then corrected to the relative gas permeability,J,, , at the given Si,V saturation. The average porosity. 4. for each group is corrected also to the effective hydrocarbon-bearing porosity, +/, =4(1 -s,,, 1. It will further be assumed that each permeability group represents a separate and distinct homogeneous layer having a relative gas permeability k, and a hydrocarbonfilled porosity 4h equal to the average for each group. Each layer is sealed off from the others and feeding into a common wellbore that is exposed to zero pressure. To keep the computations as simple as possible it will further be assumed that the ideal-gas laws are applicable. The same method may be applied by taking the deviation from the ideal-gas laws into consideration, by assuming other than zero wellbore pressure, and by taking into account liquid condensation in gas-condensate reservoirs, but such computations soon become rather unwieldy. By the time Group I, comprising the highest permeability, is bled down to a pressure p t , a time 1 has expired, which according to Eq. 39 is equal to



t=



Interstitial water, cu ftl acre-ft



Initial Conditions



43,56Oc$(i-S,) B !F



while the cumulative production from all layers, G,,, , at this time is



(44)



in which C3 and Cd are constants. The fractional production rate from all layers. fsn, with respect to the initial production rate from all layers is, therefore,



while the cumulative production from all layers as a fraction of the total gas in place in all layers is ,1



~(~~),(h,),,[l-(P,ipi)l 1



(41)



G P”



.,....... G,Z



The fractional pressure Pn/p; in any layer n at this same time t is found by substituting the t value of Eq. 41 into Eq. 39.



-I



..(42)



The combined production rate from all layers, y,, , at this time is, according to Eq. 40.



(43)



_



(46)



II



Thus a rate-cumulative relationship may be established based on Eqs. 45 and 46, whereby the rate is expressed as a fraction or percentage of the initial rate, and the cumulative as a fraction or percentage of the gas in place. By selecting an appropriate economic limit rate the recovery factor can then be found. The computation procedure is illustrated with the example in Table 40.14. Usually only three or four assumptions for the ratio p,/p t are necessary to delineate the curve, which may then be plotted on semilog paper as shown in Fig. 40.20. In this particular case, it could be estimated that the minimum recovery factor for this reservoir at a time when the production rate has declined to 1% of its initial value would be on the order of 74% of the gas in place.



PETROLEUM



40-26



TABLE



40.14-PERMEABILITY-BLOCK



METHOD



2



1



Group (n) (I) Permeability,



lO was merely contributing development. This position, while originally accepted by the Internal Revenue Service, has recently come under



PETROLEUM



ENGINEERING



HANDBOOK



c Fig. 41.2-Discounted-cash-flow method. Rate of return j’= I;P/I;C, = P/C, = constant. At abandonment time, C, = Tm, (no interest).



NET PROFIT P=,‘C 0



Fig.



41.4-Morkill method. Rate of return I’= YZPn;(C,-S) = P/C, -S=constant. At abandonment time t,,C, =S (includinginterest).



attack and is being severely restricted. The government’s current position is that in most instances, at the date of transfer, the taxpayer performing services recognizes taxable income. The issue is far from settled, and additional activity is expected to clarify the tax consequences of such transactions.



Different Concepts of Valuation



Fig. 41.3~Hoskold method. Rate of returnj’= P/C, = constant. At abandonment time t,,C, = S (includinginterest.)



The literature includes many different methods that may be used to evaluate the known or estimated future projection of net income from a given venture. ‘7-20 One of them, the discounted-cash-flow method, illustrated in Fig. 41.2. simply reduces these future income payments to present worth or present value by a chosen rate of compound interest or rate of return. It represents the banker’s approach to a stream of future income payments and is widely used in industrial work. The Hoskold method, illustratedin Fig. 41.3, was specifically designed for ventures with a limited life, such as mines or oil or gas wells, and was first used in mineevaluation work. The Morkill method, illustrated in Fig. 4 1.4. is actually a refinement of the Hoskold method and is also mainly applicable to ventures with a limited life, such as mines and oil or gas wells. The accounting method. illustrated in Fig. 41.5, represents the accounting approach to the valuation problem and takes into account the actual depletion pattern applicable to the given venture. It is particularly suited for those ventures where a specified total number of units of production is involved and where. as is the case in most extractive industries. the depletion applied to the original capital investment is on a unit-of-production basis.



VALUATION



OF OIL AND



GAS



RESERVES



41-17



Fig. 41.5-Accounting method. Rate of returnj’= Z/XC,. At abandonment time t,, C, = ED, (no interest).



Fig. 41.6-Average-annual-rate-of-return method. Rate of return j’= present worth of W/present worth of XC, = Area ABCDElArea FGHK. At abandonment time t,, C, =ZD, (no interest).



The average-annual-rate-of-return method, illustrated in Fig. 41.6, is essentially a refinement of the accounting method and, by applying the present-worth concept to both the net annual profits and the net remaining investment balances, simplifies the computations and properly weighs the time pattern of the income. A complete summary of the basic equations for these different methods and their appraisal and rate-of-return equations will be found in Table 41.10. The top part of this table shows the equations for continuous compounding and the solution for the constant-rate case. The bottom part shows the appraisal equations and the rate-of-return equations for the general case where the cash flow, I, varies from year to year.



jetted cash flow to present value by means of the desired rate of interest. The appraisal value is then



Discounted-Cash-Flow



Method



This method, also referred to as the investors method ‘* or internal-rate-of-return method, “,‘* is the one most often used in appraisal work. It is based on the principle that, in making an investment outlay, the investor is actually buying a series of future annual operating-income payments. The rate of return (with this method) is the maximum interest rate that one could pay on the capital tied up over the life of the investment and still break even. The time pattern of these future income payments is, therefore, given proper weight. No fixed amortization pattern needs to be adopted with this method because the annual amounts available for amortization are equal to the difference between the net income and the fixed profit percentage on the unreturned balance of the investment. The computations necessary for a property evaluation are, therefore, relatively simple. They usually involve only the discounting of the pro-



Cj=I,(l+i’)-“+I2(l+i’)-‘I~+.



.+Z,(l+i’)“-‘,



fl=r, C;=



C I,(l+i’)“-“, n=l



.. .



...



..



(7)



in which I,, I2 . . . I, represents the projection of the cash income in successive years and the compound-interest factor for the speculative effective interest rate i’is computed for the assumption that the entire income for each year is received at mid-year. Appropriate midyear compoundinterest factors (1 +i’)“-’ will be found in Table 4 1.11 for speculative effective interest rates from 2 to 200%. In the case of oil-producing properties, the computed earning power by this method is not necessarily the same as the average rate of return later shown on a company’s books for the net investment in the property. Most oil companies amortize their investments in producing properties in proportion to the depletion of the reserves or on a unit-of-production basis. However, no provision for such amortization pattern is made in the discounted-cashflow method. When the production rate and the income both follow constant-percentage decline and the ratio between initial and final production rates is substantial, no serious difference will result. However, when the rate of production and the income are constant for a long period of time, a substantial difference may develop and the average rate of return, as shown later on the company’s books, may be appreciably higher than the rate of return used in the evaluation by the discounted-cash-flow method.



PETROLEUM



41-18



TABLE



41.10-SUMMARY



OF EQUATIONS



APPLICABLE



TO DIFFERENT



VALUATION



Discounted Cash Flow For continuous compounding, basic equation



I df=j’C, dl-dC, where f=O C, =C, t=t, c,=o



ENGINEERING



HANDBOOK



METHODS



Hoskold (8)



(14)



I dl+jS dt=j’C, dt+dS where t=O S=O t=t, s=c,



Appraisal equation for constant annual income of I dollars per year



Rate-of-return equation for constant annual income of I dollarsper year General case: Appraisal equation



(15)



Solutionforj’which willsatisfy Eq. 9



,



“=ta



c, =



je -I’,



i’= C, l-e-“. ” = 1,



c /,(l+I’)“-” n=,



(7)



“5



/“(I+/)‘a-”



(10)



c,= 1 + r[(l +,)‘a-‘] or



FP”E/ c,=i’i’ --c J T-l



i



Rate-of-return equation



(11)



(1 +i)-‘8



Solutionfor i’that willsatisfy Eq. 7



(12)



or



i



j’=



L



FP”E’



---(l+i)y’s C, 1 -(1 +i)-‘a



The method may be illustrated with the diagram of Fig. 41.2, which shows the application of the discounted-cashflow method to a venture that is expected to yield an income of $1 OO,OOO/yrevenly over a period of 10 years and where a speculative nominal rate of return j’ of lSX/yr is desired. Time in years is plotted on the horizontal axis, while the constant income is represented by the horizontal line for $lOO,OOO/yr in the upper part of the diagram. The top portion of the diagram shows how the portion of the total income, I, allocated to amortization, mk, increases, while the net-profit portion (P) decreases with time. The bottom portion of the chart illustrates the manner in which the cumulative Cmk gradually reduces the unreturned balance of the investment, CB =C; -Cmk, from its initial value, C;, to zero at abandonment of the venture. The computation of the curves for this constant-rate case is based on the basic differential equation for discounted cash flow, Idt=j’CBdr-dCB,



(8)



1



(13)



where I = yearly net income, dollars, j’ = nominal annual speculative interest rate, fraction, and Cs = balance of unreturned portion of investment, dollars. Integration of this equation for constant-rate income between the limits r=O, CB = C; and t =t,, , Cs =0 leads to the appraisal value C, for a nominal rate of return j’=O. 15:



c;=(l-a-J”



. . . (8)



where



.fw =



where



fit= fractional flow of the displacing fluid, kc,= effective permeability to oil, md, k,,. = effective permeability to water, md A= cross-sectional area of flow, sq ft, 41 = total flow rate, (qM.+qo), BID, P,. = capillary pressure, p. -p ,,,, psi,



Ap = density difference, g/cm3, p,,, -po, a= dip angle, positive updip, !-l= phase viscosity, cp, and L= distance. ft.



l+(k,/k,,,)(p,,,/,u,)’



. . (10)



“““““‘.“”



..



.



...



(11)



where L = distance, ft, 9, = total flow rate, B/D, f$ = porosity, A = cross-sectional area, sq ft, and t = time, days.



sin 0)



~0 kw . . . . . . . . . . . . . . . I.



1



s,~)



, + CL,I’ kc,



. ..



In the case of a water drive, neglecting the effects of the capillary pressure gradient and the dip of the reservoir, the terms dP,/aL and gAp sin f3become insignificant. The fractional flow equation then reduces to



which states that the fraction of water in the flow stream is a function of the relative-permeability relationships in which p0 and CL, are constant for a given reservoir pressure. Since k,/k, is a function of saturation, Buckley and Leverett20 derived the following frontal-advance equation on the basis of relative-permeability concepts.



In practical units, the equation becomes -0.434A.p



Fig. 44.5-Permeability variation vs. mobility ratio, showing lines of constant E,(l -0.4OS,,) for a producing WOR of 100.



fw=



fit, = fraction of water in the flowing stream, k,,k,,. = effective formation permeability to the specific phase, kk, and kk,,, oil viscosity, water viscosity, fluid volumetric flow rate per unit crosssectional area, P,. = capillary pressure, p. -pn L= distance along direction of measurement, Ap = density difference between water and oil, PLI-PO> @= angle of formation dip referenced to horizontal, and acceleration caused by gravity. g= 2 1 +0.001127L!!L CJtPo



HANDBOOK



(9)



This states that the distance a plane of constant saturation (S,) advances is directly proportional to time and to the derivative (afJaS,) at that saturation. The value of the derivative may be obtained for any value of water saturation by plotting f,b, from Eq. 9 vs. S,,. and graphically taking the slopes at values of S,,. Fig. 44.6 shows a plot off,. vs. S,,, in addition to the resultant df,,,./dS, vs. S,. relationships for the S,, vs. k,/k,,, data at a viscosity ratio of water to oil of 0.50 (see Table 44.2). If the df,,ldS, values found in Fig. 44.6 are substituted into Eq. 11, the distance that a given water-saturation plane or front will advance for any time f can be calculated for the known throughput q in barrels per day, fractional porosity, and cross-sectional area (in sq ft).



WATER-INJECTION



PRESSURE MAINTENANCE



44-11



8. WATERFLOOD PROCESSES



Fig. 44.7 represents the water-saturation profile or frontal-advance curves for a bed that is 1,320 ft wide and 20 ft thick, and has a porosity of 20% and a throughput of 900 B/D for 60. 120, and 240 days with the f,,,, L3f,,,/&S,,. vs. S,,. relationship shown in Fig. 44.6 The curves shown in Fig. 44.7 are characteristically doublevalued or triple-valued. For example, the water saturation after 240 days at 400 ft is 20, 36, and 60%. The saturation can have only one value at any place and time, and the difficulty is resolved by dropping perpendiculars so that the areas to the right (A) equal the areas to the left (B). Fig. 44.8 represents the initial water and oil distributions in the example reservoir and also the distributions after 240 days. The area to the right is the flood front or “oil bank,” and the area to the left is the water-invaded zone. The area above the 240-day curve and below the 90% water-saturation curve represents oil that may be recovered by the displacement of additional volumes of water through the area. The area above the 90% water saturation curve represents unrecoverable oil because the ROS is 10%.



TABLE 44.2-S,



vs. k,/k, DATA AT A VISCOSITY OF WATER TO OIL OF 0.50 wS 0.20 0.30 0.40 0.50 0.60 0.70 0.80 0.90



RATIO



ko’kw GO 5.5 1.70



0.55 0.17 0.0055 0.0000



Welge Calculations. In 1952, We1 e*’ extended the earlier work of Buckley and Leverett 30 to derive a simplified method for calculating fractional flow and recovery performance after water breakthrough. The basic equations developed by Welge are as follows: S,*,-S,,.* =wifl, a O.O01583k,hAp , ..



I, -



. . (22)



pLw log? +0.682! -0.748 > a rH where d is the distance between rows of wells, ft and a is the distance between wells in a row, ft.



These equations allow the determination of the steadystate injectivities for the normal patterns if it is assumed the system is completely filled with liquid and has a mobility ratio of one. There are a number of papers that report the results of investigations to define the variation of injectivities for the five-spot pattern at mobility ratios that are other than one. Various techni ues were used. Deppeg4 and Aronofsky and Ramey 95 used potentiometric model techniques; Caudle and Witte8* used the X-ray shadowgraph technique and a porous model of the reservoir element. In the Caudle and Witte study, ‘* one-eighth of the fivespot pattern was modeled. Nobles and Janzen used resistance networks to simulate mobility differences, and Prats et al. *’ used an analytical solution. Qualitatively, all investigators arrived at the same conclusion-i.e., if the mobility ratio is favorable (MI l), injectivities will decline continuously during the entire operation; however, if the mobility ratio is unfavorable (M> I), injectivities will increase continuously. In their work, Caudle and Witte determined the variation in injectivity for the five-spot pattern as a function of the mobility ratio that exists before and after water breakthrough. Fig. 44.65 shows the results of their studies, in the form of the relationships between the conductance ratio, the mobility ratio, and the fractional areas of the reservoir that are contacted by the injected fluid. Craig4 points out that, subsequent to fill-up, the relationships developed by Caudle and Witte can be used along with Eq. 20 to calculate water injection rates for the five-spot pattern: ’. W’= FcXib,



Seven-spot pattern 84: O.O02051k,,hAp



. . . . . . . . . . . . (23)



I,



where . = water-injection rate, i: = Caudle and Witte conductance ratio, and ib = the injection rate of fluid that has the same mobility as the reservoir oil in a liquidfilled (base) pattern, as calculated from Eq. 20.



Inverted nine-spot patterns4 :



.....



..



........



. . . . (25)



As the intake rate declines in the early stages of injection, it is important to be able to tell whether the decline results from the plugging of the sand (a situation that requires remedial work), from natural reservoir fill-up, or from mobility ratio effects. Consequently, a method is required to determine the intake capacity of the well itself without regard to the conductivity of the well system surrounding it. Such a method would be achieved by conducting periodic tests on certain selected wells scattered across the flood area. A close check of the efflclency of the input of the wells could then be maintained. One practical method of determining the efficiency of the input wells is to use the calculated injectivity index



WATER-INJECTION



Fig. 44.66-Composite



PRESSURE MAINTENANCE



& WATERFLOOD PROCESSES



of the well. The injectivity index is defined as the number of barrels per day of gross liquid that is pumped into an injection well per pound per square inch of pressure differential between the mean injection pressure and the mean formation pressure associated with a specific subsurface datum, usually the mean formation depth. To be most valuable in a study of the behavior of the individual input wells, use of the injectivity-index concept should be restricted to defining the conductivity of an individual well; it should not be used to determine the general conductivity of the well system. This restricted injectivity index, which may be called a “localized injectivity index,” is best used in measuring the conductivity of the cylinder of sand surrounding the well-most of the pressure drop takes place in this cylinder, whose inside wall is the sandface. The localized injectivity index can be calculated from a modified Eq. 16:



I &,=



0.@-)7@%h(Pi,~-Pbp) cLw ln(ri,r,)



3 . .



‘. . . .



(26)



where p@ is the transient backpressure and ri is the distance from the well to the point of pressure equalization at Pbp.



Differentiating, the localized injectivity index, with pbP being constant, is expressed as I-di,-



Flg. 44.67-Hewitt



type log, Hewitt field, Carter County, OK.



O.O0708k,h y, ln(rj,r,).



..



..



. . (27)



dp,w Experimentally, it has been found that, for small volumes of injected water, di ,,A$ iw is constant. If small volumes of water are injected during the course of a test, r; changes only slightly; r; is considerably greater than rw and the logarithm of ri/rw is practically constant. If, however, larger volumes are injected during the course of the test, the In r,/r, will no longer be constant and the localized injectivity index, di,ldpi,, will not be constant. If large enough volumes are injected so that equi-



unit, Chubbee structure map.



librium conditions are obtained, the corresponding pattern formula is applicable. In the case of a five-spot pattern, the change in intake rate for each change in pressure can be approximated by I-



diw dpiw



O.O03541k,,,h pW [ln(d/r,)-0.61901



(28) ’ ’



..



where d is the distance between unlike wells. The transient backpressure, pbP, is a pressure phenomenon that occurs when the intake rate of an injection well is changed. Theoretically, the flow of water from a well into the surrounding formation will continue until the intensity of the sandface pressure is reduced to that of the reservoir pressure. In practice, if the pressure on an input well suddenly is reduced to the atmospheric pressure at the surface, the well backflows for a period of time that varies from a few minutes to several hours. The pressure that caused the backflow of water from the well is defined as the transient backpressure. This pressure, which occurs near the wellbore, is greater than the average reservoir pressure and has been attributed qualitatively to the compressibility of water and gas near the wellbore. When the injection is terminated, the backflow is caused by the expansion of the water and gas that results from the decrease in pressure. Quantitative treatment of this phenomenon has been given by Nowak and Lester ’’’ and Hazebroek et aZ. ’I2 The transient backpressure gradually dissipates and approaches the reservoir pressure. The localized injectivity index should be determined after the transient pressure has started falling very slowly or is in equilibrium with the reservoir pressure. A comparison of the injectivity indices for injection wells in the waterflood will give an indication as to the wells that are not performing satisfactorily, and investigations should be made to determine whether the remedial measures are necessary to improve the injectivity rate. The intake rate of a normal well declines during its life, at least until a constant steady-state pressure distribution is established in the part of the reservoir affected by the well. In addition to the normal well decline, the sandface



44-36



PETROLEUM ENGINEERING



HANDBOOK



TABLE 44.16-SUMMARY OF ROCK AND FLUID PROPERTIES, RESERVOIR PROPERTIES, AND PRODUCTION-INJECTION DATA, JAY/LITTLE ESCAMBIA CREEK (LEC) WATERFLOOD TABLE 44.15-HEWITT



UNIT RESERVOIR DATA



Rock and Fluid Properties



General Unit area, acre Floodable net sand volume, acre-ft Average composite thickness, ft Original oil in place, MMbbl Rock Properties Permeability, md Porosity, % Interstitial water, % Lorenz coefficient Permeability variation Fluid Properties Mobility ratio Original reservoir pressure, psig Reservoir temperature, “F Original FVF, RBlSTB Flood start FVF, RBlSTB Oil stock-tank gravity, “API Oil viscosity, cp Original dissolved GOR, cu ft/STB Primary recovery mechanism



2,610 284,700 109 350.8 184 21 .o 23.0 0.49 0.726 4.0 905 96 1.13 1.02 35 8.7 253 solution gas drive gravity drainage



gradually becomes plugged by suspended solids in the injected water. These suspended solids include materials like clay, silt, iron oxide, and hydroxides. In addition to suspended solids, dissolved and organic growths may contribute to the plugging of the formation sandface. Plugging of the sandface by these materials may be minimized with the proper treatment of the injection water. This treatment is covered in this chapter under the heading Water Treating. By means of rate/pressure curves established at intervals of a few months, it is possible to distinguish between the decrease in intake rate caused by plugging and that caused by fill-up of the reservoir as mobility ratio effects. Rate/pressure curves are helpful also in indicating the value of the critical breakthrough pressure at which rupture of the formation occurs. If plugging is occurring and the injection rate declines, backflow of the well may be induced to remove the material from the sandface. Or if the plugging material on the sandface cannot be removed by backflowing, then perhaps it can be dissolved through the use of various types of acids. If necessary, fracturing may be used to increase the injectivity rate in the well.



Water-Injection Case Histories Many examples of field case histories of water-injection projects can be found in the literature. Seven case histories of waterfloods in both sandstone and limestone reservoirs, using pattern as well as peripheral injection, are detailed in SPE Reprint Series No. Za, Waterflooding (1973). SPEReprintSeriesNo~.4(1962)and4a(1975), Field Case Histories and Oil and Gas Reservoirs, also describe the history of several typical waterflood and pressure-maintenance projects. For this chapter, three recently reported water-injectionproject case histories were selected from the literature as a means of illustrating the use of contemporary technolo-



Porosity, O/O Permeability, md Water saturation, O/O Oil FVF, RBlSTB Oil viscosity, cp Oil gravity, OAPI Sol&on &OR, scf/STB Hydrogen sulfide content, mol% Mobility ratio (water/oil) Reservoir Properties



14.0 35.4 12.7 1.76 0.18 51 1,806 8.8 0.3



Datum, ft subsea Original pressure, psia Current pressure, psia Saturation pressure, psia Temperature, OF Production area, acres Net thickness, ft OOIP, MMSTB Production/Injection (Jan. 1, 1981)



15,400 7,850 5,750 2,830 285 14,415 95 728



Oil production rate, MSTBlD Cumulative oil production, MMSTB Water injection rate, Mbbl Cumulative water injection, MMbbl



90 296 250 524



gy and reservoir engineering methods to solve some of the more complex problems encountered in many oil fields today. Summarized in the following discussion are results of projects involving (1) an older field with multiple sands, (2) a deep carbonate reservoir, and (3) an offshore field. The effects of extensive waterflooding operations in the Hewitt field unit, Carter County, OK, were reported in 1982 by Ruble. ‘I3 The project described in that paper is a pattern waterflood in multiple sands that had been essentially depleted through 50 years of primary operations. The project is a good example of a simultaneous waterflooding of numerous sands containing relatively high-viscosity oil at shallow depths, as shown in Fig. 44.66. A structure map of the Hewitt unit is shown in Fig. 44.67. A summary of the reservoir performance data is given in Table 44.15. The additional oil recovery by waterflooding has been estimated to be 34.9~ lo6 STB (123 bbllacre-ft) as compared to a primary recovery of 109.6~ lo6 STB (385 bbl/acre-ft). These numbers represent approximately 10 and 31% of the OOIP, respectively. Among the outstanding features of this project are (1) the use of triple completion injection wells with tubing and packer installations for control of the water that is injected into as many as 22 individual sands, (2) the plugging of 680 old wells and drilling of 149 new wells, and (3) the use of surveillance and selective injection programs to optimize oil recovery. Langston et al. ’I4 have reported on a large-scale waterinjection project in the Jay/Little Escambia Creek field in Florida and Alabama. The project is a good example of a pressure- and rate-maintained project in a deep, undersaturated, carbonate reservoir. A summary of the production performance data for the field is presented in Table 44.16. The injection pattern is a 3 : 1 staggered line drive, as shown in Fig. 44.68. Reservoir pressure and



WATER-INJECTION



PRESSURE MAINTENANCE



l? WATERFLOOD PROCESSES



oil production rates, shown in Fig. 44.69, were maintained at constant levels for 6 years before they began to decline. Ultimate oil recovery is expected to be 346 x lo6 STB, or 47.5% of the OOIP. This represents 222~ lo6 STB more recovery than from primary operations-i.e., water-injection procedures will account for 64% of the total anticipated recovery. A great number of rock and fluid property data were acquired during the early development phase of the field. Use of these data provided the basis for decisions concerning unitization and the subsequent injection program. Although water injection programs are being carried out in many offshore fields, primarily in the Persian Gulf area, in the North Sea, on the Louisiana-Texas gulf coast, and on the California coast, case histories have been reported on only a few. Jordan et al. ‘I5 reported on injection operations in the Bay Marchand field, offshore Louisiana, in April 1969. Initial reservoir pressures in individual sands of the Bay Marchand field ranged from 4,600 to 5,29 1 psig. Reservoir temperatures varied from 182 to 197°F. Initial GOR’s averaged 450 scf/STB and oil gravities were between 21 and 30”API. PVT properties varied with depth and the oil columns were undersaturated at their volumetric midpoints. Oil viscosities ranged from 1.1 to 1.9 cp, indicating favorable mobility ratios. Porosities were rather uniform and averaged 29%. However, permeabilities exhibited wide variations; three reservoirs had geometric-mean air permeabilities of less than 100 md, while the remaining sands had values up to 2,000 md. Initial water saturations exhibited a corresponding variation, from 40 to 15 % Pressure maintenance using seawater for injection began in 1963. According to McCune, ‘I6 who reported on operations in the Bay Marchand field in Oct. 1982, successful injection operations have been carried out over a 20-year period in six major sand reservoirs. A typical sand unit structure map and pressure-production history are illustrated in Figs. 44.70a and 44.70b, respectively. The techniques used to test, treat, filter, and pump seawater are discussed in detail in the papers by Jordan et al. ‘I5 and McCune. ‘I6 The basic methods used in the Bay Marchand field, which include both coarse and fine filtration of solids, oxygen removal, and chemical treatment for control of corrosion and bacteria, have since been adopted in many other seawater injection projects.



Pilot Floods A pilot waterflood is conducted to provide a means of evaluating the feasibility of a full-field implementation of the waterflood process. Both reservoir performance and’ operational procedures can be evaluated during the pilot flood. This experience is helpful in performing the engineering and economic studies that are necessary in deciding whether expanded waterflood operations should be carried out. It is important to understand that a pilot flood should be designed to assure engineering success rather than economic success. Any small economic loss sustained by the pilot flood can be weighed directly against the much greater economic loss that would result from expanded waterflood operations that are undertaken without accurate pilot performance data. Such economic losses can result



44-37



o PRODUCING WELL WELL



Fig. 44.68-Jay/Little map.



Fig. 44.68-Jay/Little



Escambia Creek waterflood well location



Escambia Creek unit performance.



from the project capital investments or from a reduction in the ultimately recoverable oil reserves. Caudle and Loncaric 5o has suggested several aspects of field pilot operations that need to be considered to achieve the greatest amount of useful data from the project. Fluid movement is most critical; one cannot isolate a segment (pilot area) of a reservoir and confine assessments of fluid movement to that segment. A commonly used pilot flood pattern is the inverted fivespot, in which there is one injection well and four producing wells; all other nearby wells are shut in. The popularity of this pattern is mostly because only one injection well is required. The inherent problem with this pattern is that



PETROLEUM ENGINEERING



44-38



WEIORLEMS



HANDBOOK



‘\r, ‘:I :r II I: i



80



0



structure map, Bay Marchand field.



Fig. 44.70B-Pressure production history vs. time. Typical unit reservoir, Bay Marchand field.



three-fourths of the produced fluid comes from outside the “pilot area” while, at the same time, fluid leaves the pilot area from the regions between the producing wells. The re!ative volumes are affected by the ratio of production rates to injection rates. A “volumetric balance” can be maintained in the pilot area by allocating only one-fourth the rate of the injection well to each production well. Although the volumes are balanced, the production history will still reflect the fact that only one-fourth of the oil that is produced actually comes from inside the pilot area. Therefore, no reliable estimate of the amount of recoverable oil in the pilot area can be made. Computer model studies show that the production history for this pilot pattern is so greatly affected by conditions outside the pilot area that correction factors are probably inadequate to compensate for the errors. This is especially true if there is a gas saturation in the reservoir at the start of injection. The considerations noted previously suggest that a reversal of that pattern, in which one producing well is surrounded by four injection wells, could be a more accurate mechanism for evaluating the performance of a pilot flood. This pattern would minimize the escape of the oil originally contained in the pilot area as well as the entry of outside oil into the pilot area. The conventional tivespot pattern, as it is known, is probably the most simple and useful pilot pattern. While it is true that three-fourths of the injected fluid will not enter the pilot area, the production from the center producer will be much more useful for predicting total fluid recoveries.



The purpose of the pilot flood is to facilitate an evaluation of the performance of a small section of the reservoir so that the resulting information can be used to estimate the behavior of a much more extensive operation. If the production history of the individual pilot well does not generate data that are representative of the entire area to be flooded, a correction factor can be used to adjust the actual production history in order that the potential production of a fully developed or “confined” pattern flood element can be estimated. Such a pilot (or pilot production well) must operate as if it were in a confined area (i.e., in one that is surrounded by many similar areas). In reality, such a situation could occur only if the pilot area composed the entire proposed flood project. However, if a sufficient number of similar elements are operated around the pilot, results that would closely approximate those of the confined case could be achieved. The number of similar elements around the pilot area that are necessary to generate results that are usable without correction depends on the mobility ratio and initial gas saturation. Model studies”‘,“* have shown that, in general, the single conventional five-spot pilot is adequate for mobility ratios below one. More complex pilot patterns are necessary at higher mobility ratios. Certain considerations should be weighed in deciding the location of the pilot area. Knowledge of the reservoir’s geometrical configuration, its structural data, and its stratigraphic data are necessary to make the selection. A partially confined or bounded area will increase the value



Fig. 44.70A-Typical



unit



WATER-INJECTION



PRESSURE MAINTENANCE



8 WATERFLOOD PROCESSES



of the pilot in predicting the behavior of an expanded flood. The boundaries to be sought are as follows: (1) oil/water contacts with respect to monoclinal or anticlinal structures, (2) fault planes, (3) small fault blocks, (4) structural or permeability pinchouts, and (5) shale-outs to the side. Reservoir and well conditions must be evaluated before initiation of the pilot flood. In selecting the portion of the reservoir in which the pilot flood is to begin, it is important to be informed concerning these elements: (1) the pattern and spacing of injection and producing wells with respect to the formation structure and the distribution of formation properties, (2) the type of well completions, completion intervals, and the repair and workovers that have occurred in the past, and (3) the productivity factors that have been measured for producing oil wells. Reservoir conditions and other related data provide information that is necessary before, and at the initiation of, the pilot flood. Some characteristics and categories of data that are valuable in determining the magnitude and distribution of oil, water, and gas saturations before the start of the pilot flood are (1) the development and production history, (2) total oil recoveries during primary operations, (3) encroachment of water or gas, (4) reservoir pressures within and surrounding the selected pilot flood area, and (5) distribution of fluids through gravity drainage. The behavior of the reservoir and the wells should be evaluated continually throughout the life of the pilot flood. The records of this monitpring should include information about the following matters: (1) water-injection history on each well, including the time the injection began; (2) cumulative volumes of water and the rate of injection, by well, for the flood; (3) injection pressures and the identities of the sections taking water; (4) fluid production history, by well, for the total area within the flood region and for wells in the surrounding area; included should be the rate of production and the cumulative volumes of oil, water, and gas; (5) WOR and GOR trends; (6) reservoir pressure distribution inside, and surrounding, the flood area; (7) the frontal advance and associated displacement efficiency of water, as evidenced by the time and location at which water appears in individual wells; (8) workover history of both injection and producing wells; and (9) any pertinent changes in the pilot flood program. There are two efficiency factors that may be calculated and used in evaluating the effectiveness of the pilot flood. One involves a displacement efficiency, determined on the basis of the ratio of the volume of total fluids produced to the volume of water injected. This ratio will indicate whether the injected water is effectively moving fluids from the injection well to the producing well (or wells) within the pilot area. The second factor involves the sweep efficiency within the flood pattern and the fractional depletion of the oil zone, which determine the economic life of the reservoir as well as the ultimate oil recovery. Production data in the form of production-decline curves may be used to evaluate the pilot flood performance. The usual procedure in presenting the history of oil production in pilot flood operations has been to plot the logarithm of oil production vs. time or the logarithm of time. The advantages of using production-decline curves are that



44-39



they indicate the time of fill-up and the current oilproduction response with respect to the injection program. However, there are limitations in using production-decline curves to evaluate injection efficiencies and the future behavior of the pilot. Among these limitations is the fact that true decline conditions seldom exist because fluid production is controlled by water-injection rates. There is no basis for assuming any particular shape with regard to a production-decline curve because the oil rate does not necessarily vary with time; the oil production rate is directly dependent on the rate at which water is injected and on the physical characteristics of the reservoir rock and the fluids it contains. During the development and operation of the pilot test, certain conclusions regarding the performance of an expanded waterflood may be drawn. For example, if the reservoir has a high water saturation, the water may be more mobile than the oil, which would soon result in a high WOR in the pilot area. Because of the permeability reductions around the wellbores of the input wells, the formation itself might not take a satisfactory injection rate without exceeding the maximum pressure. Again, excessive pressure would produce adverse conditions. Watercut data, used in conjunction with the Stiles calculation I9 or other similar conformance calculations, will indicate whether the pilot is performing as expected.



Surface-Active Agents in Waterflooding Surface-active agents in waterflooding are used to improve oil recovery by (1) improving mobility, (2) reducing interfacial tension, and (3) altering the rock wettability. Laboratory investigations and field tests in which various surface-active agents and other chemicals are used will be discussed in Chap. 45, “Miscible Displacement,” and Chap. 47, “Chemical Flooding.” The large number of technological advances that have taken place during the past decade and the voluminous publications on the use of surface-active agents allow only a brief reference to the subject in this chapter. Mobility Improvement Control of the mobility of the injected water, along with the use of surface-active agents and chemicals to alter the wettability characteristics of the reservoir rock, are among the techniques now being used in certain waterflood projects to improve oil displacement efficiencies. The addition of an acrylamide polymer or some similar chemical to increase the viscosity of water causes area1 and vertical coverage in the reservoir to be increased as a result of a reduction in the mobility ratio between the displaced and displacing fluids. This addition of a polymer also reduces the volume of injected fluids required in the oil displacement process that lowers the saturation in the swept portion of the reservoir to its residual value. The first field studies involving the use of polymers for mobility control were reported by Sandiford in 1964. ‘I9 The injection of a high-molecular-weight polyacrylamide polymer to increase waterflood sweep efficiencies through improved mobility ratios was considered to be unprofitable in two reported case histories ‘20.‘2’that are summarized below. In the Wilmington field, CA, ‘*O a large-scale injection program was initiated during 1969 in relatively unconsolidated sands that contained an



44-40



18”API gravity crude oil with a reservoir oil viscosity of 30.8 cp, The mobility ratio of brine/oil was 14.2, compared to a mobility ratio of 1.33 for a 250-ppm polymer/ oil. After injection of 1,300,OOO Ibm of polymer over a period of 2.5 years at an average concentration of 213 ppm, the injection of polymer was discontinued because no increase in oil recovery could be attributed to the polymer injection. The poor response was believed to be caused by (1) a polymer concentration that was too low; (2) injection rates that decreased by an average of 25% (as a result of scale formation), accumulation of undissolved polymer on the face of the formation, and possible reduction in the reservoir permeability from adsorption of the polymer (85 lbm/acre-ft); and (3) a premature breakthrough of the polymer solution through highly permeable intervals. A pilot project 12’ in the Pembina field of Alberta, Canada, was started in Nov. 1971 with two la-acre, fivespot patterns composed of six injection wells and two producing wells. The producing interval consisted of a conglomerate zone underlain by a sandstone, and these zones had average permeabilities of 63.6 and 25.3 md, respectively. The viscosity of the 37”API crude oil, at reservoir conditions, was 1.05 cp. A total of 217,400 lbm of polymer was injected, with the first 124,750 lbm being injected at a concentration of 1,000 ppm and the remaining 92,650 Ibm being injected at decreasing concentrations from 1,000 to 100 ppm. The conclusions reached from the Pembina pilot project were as follows. 1. The overall performance of the producing wells in the pilot area showed no permanent improvement. 2. Early breakthrough of polymer through the conglomerate zone indicated that the polymer did not significantly reduce the effects of the highly permeable interval. 3. Water/rock interaction and formation water commingling reduced the effective viscosity level of the polymer solution to approximately 25 % of the designed value. 4. There was a significant reduction in the injection rates of two injection wells during polymer injection. 5. Adsorption of the polyacrylamide polymer was about 2 mg/m’ of surface area. The injection of polymer solutions to improve oil recovery through mobility control has not yet been well established for general application. Laboratory displacement tests should be performed on reservoir rock samples, and the reservoir crude oil and formation water should be used as a guide in selecting the type of polymer and the concentrations necessary for scaling the formulation to field conditions. Of particular significance is the effect of the formation water’s salinity on reducing both the viscosity of the polymer solution and its adsorption by the reservoir rock. Published reports “‘-‘24 about various field applications of polymer solutions have indicated improvements in oil recovery efficiencies of 5 to 15% above recoveries from conventional waterfloods. Reduction in Interfacial Tension Early laboratory tests ‘25m’27indicated that dilute solutions of surfactants would remove more oil from sandstone cores than would untreated water. The economic feasibility of using this process in a waterflood has been ques-



PETROLEUM ENGINEERING



HANDBOOK



tioned because of the loss of the surfactant by adsorption at the rock/liquid interfaces. The adsorption is especially problematical with both anionic and cationic surfactants, and it occurs to a lesser degree with nonionic surfactants. In one field project, the results of which were published’28 in 1968, a nonionic surfactant was injected at concentrations of 25 to 250 ppm into a sandstone reservoir at an advanced stage of waterflooding; an additional oil recovery of approximately 9% was attributed to the use of the surfactant. Alteration of Rock Wettability Recognition of the use of alkaline salts to improve oil recovery was first disclosed by Squires ‘29 and patented by Atkinson I30 in 1927. Wagner and Leach, 13’in 1959, presented laboratory results that showed improved oil recovery through the injection of water containing chemicals that altered the pH of the injected water. Acidic injection water resulted in an improvement in WOR and a corresponding increase in recovery; however, its use as an injection medium has not proved practical because of chemical reactions with most reservoir rocks. Subsequent laboratory tests 13* established similarly improved oil recovery results with sodium hydroxide. Laboratory tests have indicated that the injection of caustic solutions can result in improved oil recovery, primarily as a result of lowering the water relative permeability, ‘33 pH control ‘34 and the oil/water interfacial tension. ‘35 These effect;, though, are dependent on the water salinity, ‘34 the temperature, ‘36 and the type of crude oil. In 1974, there was a report ‘34 of a field trial in which a solution containing 3.2 wt% sodium carbonate was injected into a previously waterflooded Miocene sand in southeast Texas. The test involved two wells located 36 ft apart. Some improvement in oil cuts was noted at the producing well before alkaline water breakthrough, suggesting the formation of a low-mobility water-in-oil emulsion bank. No economic evaluation of the test was reported. The first field test of the caustic flooding recess was mentioned by Nutting 13’in 1925. A report’& published in 1962 of a field trial in which sodium hydroxide was used in the Muddy “J” sand, Harrisburg field. West Harrisburg Unit, Banner County, NE. The injection of a 40,000-bbl slug of 2.0 wt% sodium hydroxide resulted in a recovery of approximately 8,700 bbl of oil from an area that previously had been flooded out by normal water injection operations. In another case, an 8% PV slug of 2.0 wt% sodium hydroxide was injected into a portion of the Singleton field, Banner County, NE. The test was in an area under waterflood that had not been completely watered out. Increased oil recovery, reported ‘38 in 1970, amounted to 17,600 bbl, or 2.34% PV. The only description of a large-scale field trial of caustic flooding that has been published ‘39 involved a 63-acre area in the Whittier field, CA. The area had been under waterflood for 2.5 years before caustic was injected. A 0.2 wt% sodium hydroxide slug, equal in volume to 23% PV, was injected. The slug was followed by plain water. The increase in oil recovery beyond that by waterflooding was estimated to be from 350,ooO to 470,000 bbl, or 5.03 to 6.75% PV.



WATER-INJECTION



PRESSURE MAINTENANCE



& WATERFLOOD PROCESSES



Water Source and Requirements During the planning stages of a waterflood program, these basic steps must be taken: (1) the water requirements should be determined as accurately as the data will permit; (2) all possible water sources should be surveyed with special attention given to satisfying the quantitative requirements: and (3) the selected source should be developed in the most economical manner permitted by good engineering practice. Waterflood Requirements Daily Water-Injection Rates. The largest daily demand for water from the water source occurs during the fill-up period when there is no return water available. During the early life of the reservoir’s injection program, or during the fill-up period, it is usually advantageous to maintain a high rate of injection so as to accomplish an early fill-up (a rate between 1 and 2 B/D/acre-f1 is desirable). One author I40 states that after fill-up has been achieved, the injection rate should be maintained at about 1 B/D and not less than % B/D/acre-ft. Flood pattern, well spacing, and injection pressures should be designed to meet these requirements. Ultimate Water Requirements. The PV method has been found to give a good approximation of the ultimate water requirements for a waterflood. The volume of water required should range from 150 to 170% of the total pore space, and the measurement of such space should include the PV of any adjacent overlying gas sand or basal water sand. The ultimate water requirements, together with the average water-injection rate, will serve as a basis for estimating the total life of the waterflood. Makeup Water. The volume of return water becomes an increasingly significant percentage of the required injection rate as a flood progresses; therefore, it is an economic necessity that produced water be injected unless the treating cost of the produced water is higher than that of the makeup water. If gas or water sands are not present, the produced water will compose 40 to 50% of the ultimate water requirements. If gas or water sands are present, less return water will be available-thus, the ultimate makeup water requirement will increase to as much as 60 to 70% of the total quantity of water that is injected. In recent years, federal and state agencies have enacted regulations that limit or prohibit disposal of oilfield waters in surface systems. Environmental regulations should be reviewed carefully when studies of the treatment and disposal of produced water are being made. Water Sources There are three principal freshwater sources and two sources of salt water that can be used for waterflooding purposes. Freshwater supplies include surface waters, municipal water. waters from alluvium beds, and some subsurface waters. Saltwater sources include some subsurface waters and the oceans. Where economically permitted, salt water usually is preferable to fresh water. Fresh Water-Surface Sources. Surface waters, including ponds, lakes, streams, and rivers, have been used throughout the history of oilfield waterflooding projects,



44-41



and these are the sources for which competition from other industries and from municipalities is highest. There are a number of other factors that limit the availability of this resource. For example, there is a continuing growth in the demand for fresh water, and droughts have resulted in water shortages in some areas during recent years. In addition, some states have taken legislative action to control freshwater supplies. Therefore, when fresh water is to be used in a waterflood project, it may be necessary to obtain approval from the appropriate state agency before proceeding with development of such a source. If salt water is chosen as the injection medium, legal approval for the withdrawal of the water may not be necessary Small ponds and streams are very unreliable as a constant source of supply for all seasons of the year. Large lakes and rivers are preferable; however, these also may have limited capacity during drought periods. The principal disadvantages of surface sources are the unreliability of their quality and quantity, the high cost of treating equipment, and the cost of the chemicals that are necessary to obtain a satisfactory water. Fresh Water-Alluvium Beds. A more favored method of using river or stream waters calls for the alluvium beds near the river to be tested with shallow wells. Use of this source in some of the world’s largest waterfloods-the Salem unit in Illinois, 14’ rhe Burbank unit, I42 and the Olympic pool in Oklahoma-indicates the high productivity that can be achieved from alluvium beds. If closed injection systems are used, chemical treatment (with the possible exception of a bactericide) normally is not required. Filtration usually is unnecessary because of the natural filtration of the alluvium beds. Sulfate-reducing bacteria are anaerobic and thrive within a few feet of the surface, so waters from alluvium beds frequently can be highly contaminated with these bacteria. However, low-cost chemical treatment can control these organisms. Having noted this minor problem, it is safe to say that the quality of water from alluvium wells is more dependable than that from direct surface sources. Wells are not subject to extreme turbidity changes during rainy seasons or to the variable organic content of the surface waters. The reliability of alluvium beds as a continuing source of water is slightly better than the reliability of an adjacent river or stream. The water table will drop steadily when a river dries up, but wells should go on supplying water for some time after the surface waters are depleted. The principal advantages of alluvium-bed sources are their low development cost, low pumping cost, and the possibility that they will not need filtration. If bacteria are not a problem, corrosion rates should be low and chemical treatment unnecessary. Fresh Water-Subsurface Formations. In certain areas, subsurface sand or carbonate formations may be tested for water production with good results. Good-quality water often is produced from certain formations whose depths range from close to the surface to 1,000 ft or more. As in the case of the alluvium wells, closed systems usually are used, thus eliminating chemical treatment and filtration requirements. When a well is completed in a freshwater subsurface formation, drawdown tests should



PETROLEUM ENGINEERING



44-42



TABLE 44.17--RESERVOIR



HANDBOOK



ENGINEERING



Effects



Remedial Treatment



pH Control



Tolerance Suggested



Hydrogen sullide. H,S



Odor or taste. If lab analysis desired, sample is preserved by addihon of zinc acetate and sodium hydroxide



Very corrosive in the presence of motslure. parttcularly if oxygen is present.



1 Open aeration (poor) 2 Synthebc or natural combustton exhaust gases flowtng countercurrent to water WIpacked towers. 3 Forced-draft aerators.



A decrease in DH will increase rate df corrosion, but the corrosion rate also depends on the composition of the contacted metal and the alkalinity of the solution.



50 ppm. “’ Corrosron rate is rapid up to 15 ppm. Hugh H,S concentrahons,may act to tnhrbil corroston.



Carbon dioxide, CO,



Determine the slabillty of the carbonatebicarbonate balance, titrate for free CO, at source point.



1 Aeratton by the three methods mentioned above. 2 Increase the alkalinity. 3. Chemtcal inhibitors.



An increase in pH also will decrease the free CO, that IS present. Free CO, may not exisl in water wtth pH values which are greater than 8 3.



Oxygen



Determine if the Fe + + ion is being oxidized. Dissolved 0, meter and membrane probe is used when H,S is absent.



1. Corroston Increases wtth Increasing percentages of co,. 2 Removal of CO, may cause preclpltatton of metalltc carbonates or bicarbonates. 1. Ii is largely responsible for corrosion of equipment. 2 Its reaction with metallic tons (Fe + + mostly) wtll cause plugging in the reservotr



1. Use of closed systems will minimize oxygen use. 2. Open systemsvacuum aeration has been used. 3. Counterflow (in bubble lower) of natural gas with low oxygen content.



No effect is to be found tn either acidtc or alkaline water.



Funclton of the carbonate and bicarbonate stabtlity vs. corrosive activtty IS caused directly by the CO,. Not as corrosive as equal porttons of O2 or H,S. Limtts of detectionI.e., 10 ppb (Note: iron bacteria can grow in waters contatntng 0.3 pm. ‘53 SRB can also live in aerobic conditions.) Soluble 0, IS approxtmately four times as corrosive as equal mole volumes of COP.



Dtssolved Gas



Test



be made to determine the initial productivity. The test should be conducted for a sufficient length of time to determine the static working fluid level, which will indicate the rate at which the well can be produced. Optimal spacing in the water-supply wells may vary from 25 ft for sand points to as much as 1,320 ft for deep wells. The productivity will indicate how many wells are necessary to meet the daily water requirements. Where a number of deep wells are required to develop the freshwater source, the economic viability of drilling the additional wells should be carefully considered. Pumping equipment for water wells may include surface-driven or submersible, centrihtgal (or rod) pumps. If a high-pressure gas source is available, gas-lifting methods should be considered also. Selection of the pumps should be governed by economic considerations, and these are influenced by the static fluid level, the drawdown, and the desired productivity. The advantages of freshwater wells in subsurface formations include low corrosion rates and the possible elimination of the need for chemical treating and filtration. Salt Water-Subsurface. In most oil fields, either above or below the oil zones, there are saltwater formations that are potential sources of water supply. 143 The relatively shallow saltwater wells are similar in most respects to the shallow freshwater wells. 144.‘45The saltwater wells are completed in the same manner and have the same advantages of being adaptable to closed injection systems. Many producing areas have deep saltwater formations that have extensive area1 coverage and a thickness of up to several hundred feet. These prolific saltwater-producing formations frequently have high working fluid levels. Such formations may contain waters with high mineral content, and have wellhead temperatures in the range of 100 to



173°F. Hydrogen sulfide may or may not be present. If the water contains significant amounts of hydrogen sulfide, open systems that incorporate aeration, sedimentation, and filtration capabilities should be used. Examples of prolific formations are the Arbuckle 146and Mississippi limestones in Kansas and Oklahoma, the Ellenburger lime in Texas, the Tar Springs in Illinois, and the Madison lime in Wyoming. The drilling and completion costs of deep supply wells may range up to, and exceed, $500,000; however, they frequently are the most economical source of large volumes of water because of small fluid-level drawdowns. The advantages of the deep saltwater wells include their adaptability to closed systems, their high and reliable productivity, the compatibility of salt water with the oil sand, and, where high hydrostatic fluid levels are found, the relatively low lifting costs. Salt Water-Ocean. Use of ocean water for injection purposes is confined to coastal regions and offshore fields. “6*‘47-149Closed systems in which shallow wells on the shore are used as the source of supply are preferred. A moderately high corrosion rate should be expected, and ocean water probably will require a bactericide. The advantages of oceanwater supply include an inexhaustible source and low development and pumping costs. Salt Water-Return Water. During the life of a flood, the return water may represent a total volume of from 30 to 60% of the injection requirements. The use of the return water for injection may improve the economic condition of the overall project. In open systems, return water generally is added to the makeup water and injected. The mixing of the waters in a pond or settling tank permits precipitation and sedimentation of the incompatible constituents. In recent years, however, it has been determined



WATER-INJECTION



PRESSURE MAINTENANCE



8 WATERFLOOD PROCESSES



TABLE 44.18-WATER-INJECTION Organisms Sulfate reducers



Organisms



Iron bacteria



GWlUS Desulfouibrio



-



Pseudomonas



-



Leptothrix Crenothrix Gallienella



-



Thallophyta



Algae



Thallophyia



Fungi



‘Llmlled “Mercuric



Phylum



to Iron-free waters and phenolic compounds:



Enwonment



Anaerobic (though they cannot grow in the presence of free oxygen, they can live; will not grow In highly saline waters.) Low-pH waters also stifle growth. Aerobic or facullallve (usually require free oxygen for growth) Bactena withdraw ferrous Iron (Fe + + ) that is present in their aqueous habitat and deposit It in the form of Fe(OH). Chlorophyll-contalnmg plants (require presence of sunlight and moislure for growth). Oxvaen (reauire presence of’free oxygen).



44-43



PRESSURE MAINTENANCE Agents Used for Treating



Effect of Agents m Reducing Growth



Purpose in Treating



Chlortne* Quaternary ammonium compounds Other bactericides”



Partiallv effective Effective



Chlorine’ Quaternary ammonium compounds Bactencides Chlorine’



Effective Effective. (Note, change bactericide 11 immunization occurs.) Effective Effective. (Note: slug injection is usually sufflcient.1



Copper sulfate



Effective, depending 1. To prevent pluggmg on water alkalinity. of equipmenl. Effective, depending on 2. To prevent plugging of sandface. water alkalmlty. Effective Effective 1. To prevent plugging Effective of equipment. 2. To prevent plugging of sandface.



Sodium pendachlorphenate Closed system Chlorine’ Closed system



Effective



1. To orevent coriosive activity as a result of H 2S formation. 2. To prevent pluggmg of sandface.



1. To prevent plugging of equipment 2. To prevent plugging of sandface. 1. To prevent plugging of equipment. 2. To prevent pluggmg of sandface



fatly and resin amines: formaldehyde



that the mixing of the produced water and makeup water results in increased scale deposition and corrosion in the surface system and injection wells. Also, scale deposition in the perforations, and the transport of suspended solids (a product of corrosion) into the formation, reduce the well injectivity and necessitate frequent backwashing and acid treatments. Therefore, in many of the major waterfloods, the waters are isolated in the surface system and are injected separately into the reservoir. In closed systems, the compatibility of the return and makeup waters is more critical than it is in an open system, but the two waters can be mixed satisfactorily in most cases. Complete analysis of the water should be made, with special attention being given to the detection of any combinations of ions that may precipitate on being mixed. The effect of the more common precipitates and the treatment of them is covered in this chapter under Water Treating.



Water Treating During the early days of waterflooding, only the quantity, not the quality, of the water was given consideration. How-ever, it was soon noted that when the quality was poor, higher injection pressures were required to maintain suitable injection rates and corrosion problems mounted. As a result, the operators of the early waterfloods began to realize that the quality of the water was equally as important as the quantity, and that poor water treating was proving disastrous to waterfloods that otherwise might have been successful. Water-treating practices have improved greatly as the waterflood industry has matured, a point that is substantiated in the literature by the many contributions on this subject. ‘45,‘50-‘64API has published recommendations for analysis of oilfield waters I50 and biological analysis of injection waters. 15’Successful results normally can be achieved when these recommended



procedures are followed. Standardized procedures for membrane-filterability tests, ‘52 a useful tool in water testing, also have been adopted by the industry. After the water source is known, a water analysis is required to determine these matters: (1) compatibility of the injection water with the reservoir water (the test should include actual blends as well as theoretical combinations); (2) whether an open or closed injection facility would be the most suitable; and (3) what treatment is necessary to have an acceptable water for the reservoir and to minimize corrosion of the equipment. Prudent operation of the waterflood requires that water analyses be conducted periodically to determine the presence of dissolved gases, certain minerals (discussed later), and microbiological growth-undesirable constituents of water. Samples of the injected water should be collected at several points in the system-for example, at any point in the system where a change in water quality could or should occur, and at the injection wells. Sampling The importance of good sampling practices cannot be overemphasized. An extremely acccurate chemical analysis of a water sample followed by a brilliant assessment of the problems indicated by the analysis is worthless if the sample does not represent the water in the system. Dissolved Gases To eliminate the loss of dissolved gases through changes in tempeature and pressure, testing of such gases should be carried out in the field soon after a water sample is taken. The three dissolved gases to be considered are hydrogen sulfide, CO2, and oxygen. Table 44.17 lists the test, the effects of the gas when present, remedial treatment, pH control, and tolerance permitted in ppm.



44-44



Microbiological



PETROLEUM ENGINEERING



Growth



Static control of colonies of one-celled animals and plants is of much concern to operators attempting to maintain a suitable water for injection. Aerobic, anaerobic. fungal, and algal growths will cause reservoir and equipment plugging and corrosion. Table 44.18 lists the various organisms, their environment, the various treating agents that have been used, the results that may be expected, and the purpose of the treatment. Special attention and control are required for sulfatereducing bacteria (SRB). The presence of the sulfate ion is essential to the growth and reproduction of these particular bacteria. Sulfate, in turn, causes plugging. The reaction of the sulfate ion with the SRB forms the sulfide ion. which then reacts with iron. Iron sulfide is serious plugging agent and H 2 S is an extremely corrosive agent. Early studies of SRB involved the the plate-count method, 153~1s4a clinical practice derived for the purpose of isolating and identifying bacteria. But this technique is of little value in assessing sulfate-reducing bacteria activity, which is what really counts. The objective of studies of SRB in a water system is to determine whether practical problems exist, and to be able to execute effective countermeasures if such problems are found. The concept of bacterial activity was developed to meet this objective. The procedures for conducting these studies are presented in the API RP 38 publication. Is’ Many organic and inorganic bactericides are now available to control this problem. Minerals Appearance. A notation concerning the appearance of the water at the time it is sampled is important for future reference. Frequently, organic growths and precipitated material can be detected visually. Temperature. The temperature of the water sample is important in estimating the solubilities of various materials. For example, calcium carbonate solubility decreases with increasing temperature, as does calcium sulfate and all sulfates. Significance of pH. Simply put, pH is a measure of the acidity or caustic intensity of water. Two important points to remember are that calcium carbonate and iron solubilities both decrease with increasing pH value; therefore, the higher the pH the more difficult it is to hold iron in solution and to keep calcium scale from forming. However, if iron is being removed in the water-treating program, then a high pH may be beneficial. The pH value is very important when corrosion control is considered. Turbidity. A turbidity test measures the suspended material in a water and it is based on the intensity of light scattered by the sample in comparison with light scattered by a known concentration of a standard solution. The higher the scattered light, the higher the turbidity. Standards are compared to Formazin polymer, which has gained acceptance as the turbidity reference standard suspension for water. The generally accepted method of measurement is conducted with a nephelometer. Results are reported in nephelometric turbidity units (NTU), which correspond



HANDBOOK



with Formazin turbidity units (FTU) and Jackson candle units (JCU). Normal turbidity measurements are within the 0- to 50-NTU range. Iron. Some form of iron is probably the most common plugging agent encountered in injection wells. Ferrous ;‘Foent(tF;+’ 1 .IS soluble to 100-t ppm, while ferric iron ) is insoluble except at low pH levels (3 ppm or less). Low iron contents are desirable in any water. The retention of soluble iron in solution is the prime objective in closed systems. In properly operated iron-removal plants, the iron content in the finished water should be less than 0.2 ppm. In many cases, it is possible to reduce the iron so that it is consistently less than 0.1 ppm. There should be no significant increase in iron content as the water travels from the pressure source to the injection wells. Manganese. Soluble manganese in water reacts somewhat as iron does, except that it is more difficult to remove. In most waters, good manganese removal requires a pH level of 9.5 to 10 ppm. Manganese problems in the Appalachian oil fields have been very severe. Only in a few isolated cases has it been troublesome in the Illinois basin; it has been of little concern in most floods in that area, or farther west. Low to moderate manganese contents are found in many waters and can be tolerated as long as the pH values remain low enough to keep it in solution. Alkalinity. The alkalinity of water is defined by the measure of its acid-neutralizing capacity. Since the occurrence of hydroxide is quite unusual in flood waters, alkalinity generally can be taken as a measure of carbonates and bicarbonates. Calcium carbonate solubility depends on alkalinity; however, other factors, such as pH, calcium content, temperature, and total dissolved solids, influence the reaction. Sulfates. Sulfates are of most interest from a deposition standpoint. Three generalizations may be made with regard to this class of substances. 1. An abnormally low or zero sulfate value in a brine suggests the possibility of the presence of barium and strontium. It requires practice and experience to evaluate a low-sulfate-content water. 2. In general, high-sulfate water should not be mixed with water containing appreciable amounts of barium or strontium. 3. A high-sulfate brine indicates there is a possibility of exceeding the calcium sulfate solubility. The solubility of SrS04 or CaS04 is governed by the limiting factor of either SO4 or Ca or Sr and the ionic strength or foreign salt concentration of the brine. Chlorides. Chlorides are the primary indication of the salinity of a water, or the ionic strength of a brine, or the presence of a fresh water. Chloride tests can be useful in tracing the progress of a waterflood. Hardness. The term hardness refers to a measure of the amounts of calcium and magnesium that are present in the water and is expressed in ppm of calcium carbonate. Since calcium is involved, the hardness of the water is of importance in relation to calcium carbonate stability.



WATER-INJECTION



PRESSURE MAINTENANCE



& WATERFLOOD PROCESSES



Calcium and Magnesium. These two minerals are grouped together because they are the principal contributors to a water’s hardness. The calcium salts are less soluble than magnesium under most practical conditions. Also. the presence of an appreciable quantity of calcium is necessary for calcium sulfate and calcium carbonate scale to form. It is important to note that other factors, beyond the calcium value, must be considered in assessing calciurn carbonate formation. Suspended Solids. Suspended solids are a mixture of line, nonsettling particles, or precipitated material in the water. Unless suspended solids are removed, difficulties involving plugging of the injection or disposal wells can be expected. Dissolved Solids. It is necessary to prevent precipitation of those soluble salts that are dissolved in the water, so that there will be no plugging of the sandface. Total Solids. Technically, the term “total solids” means the combination of dissolved and suspended solids. Long experience in operating water injection systems has established that good water-quality control requires knowledge of not only the general content of the water but the constituents of the undissolved (suspended) material that exist under in-line conditions. It is this suspended material that may cause well and reservoir plugging. The suspended solids often are the result of the precipitation of constituents of the water, but the quantity and type of solids that actually are precipitating cannot be ascertained from the water analysis alone. The MilliporeTM filter test has been developed to provide a means of measuring suspended material under injection system conditions. This test is conducted with the MF-Millipore filter of mixed esters of cellulose and a uniform pore size of, generally, 0.45pm opening. The filter diameter may be of several sizes; however, 90-mmdiameter filters are recommended because a greater volume of throughput water can be handled, thus giving a more representative test for the system being examined. A small stream of water is taken, through suitable connections and the test apparatus, from the selected point in a system. The test apparatus that holds the filter will trap all the suspended material flowing through the sample line. The water effluent that passes through the filter is measured and recorded, for use in the later analysis, as volume throughput in milliliters of water. After sufficient water has passed through it and/or the initial pressure of about 10 psig has increased sufficiently to indicate plugging, the filter is removed and placed in a protective screwcap tube (preferably containing distilled water to prevent the drying out of the filter) and submitted to the laboratory for either comprehensive or selective analysis. As a safety precaution, it is highly recommended that duplicate samples be obtained through the use of a parallelapparatus hookup. Identification of the solids and particle size distribution (with Coulter counter) is useful for designing facilities to treat and to remove solids from the water. Barium. Barium ions have been quite troublesome in many cases because of the extremely low solubility of the most common form of their deposition, barium sulfate.



44-45



It is generally undesirable to mix a water with appreciable amounts of barium with a water containing high sulfates or strontium. Strontium. This is another alkaline earth metal that occurs in small quantities and is associated with calcium and barium minerals. It is found principally in the form of celestite (SrS04) and strontianite (SrC03) ores; its solubility in both forms is considerably greater than its barium counterpart but much less than CaS04. Sequestering and Chelating Agents. The use of sequestering and chelating agents in injection waters plays an important role in preventing the precipitation of salts of calcium, barium, strontium. iron, copper, nickel, manganese, etc. ‘55 The definition of each term is given as: (1) sequester: to set apart, to put aside, or to separate, and (2) chelute: pertaining to or designating a group or compound which, by means of two valences (principal or residual, or both), attaches itself to a central metallic atom so as to form a heterocyclic ring. The sequestering agent will separate the metallic cation from the anion by chelation. This will prevent the metallic ion from reacting with the anions to form precipitates that will cause plugging of the reservoir. If precipitation of the metallic salt ions does occur, reverse flow of the injection well and acid treatments usually will correct the situation so that normal injection rates can be continued and maintained. The requirements for desirable sequestering agents are that they Is5 (1) form chelates in the presence of other ions such as calcium, magnesium, strontium, barium, and others that are common to waters used for secondary recovery, (2) form stable watersoluble chelates or complexes with iron, (3) be compatible with other chemical compounds used for water treatment, (4) be economically feasible, and (5) be easy and safe to handle. The most widely used sequestering agents are “Versentates” (trademark for certain salts of ethylenediaminetetraacetic acid and related compounds), citric acid, gluconic acid, organic phosphonates, and the polyphosphates. Of these, the citric acid sequestrants have been most successful. Corrosion Inhibitors. Corrosion inhibitors are chemicals that are used to control the corrosive activity between the metallic alloys and water. “The current interest in chemical inhibition is largely a result of the availability of organic treating compounds that possess both corrosioninhibiting and biocidal properties. Field and laboratory tests made with organic inhibitors such as quaternary, rosin, and fatty amine compounds have indicated favorable results in minimizing corrosion caused by dissolved acidic gases.” ‘56



Selection and Sizing of Waterflood Plants The selection and the sizing of waterflood plant facilities normally are unique to each waterflood because of the many variable parameters. The primary parameters might be the volume and pressure, while secondary parameters might include the treating requirements and the economic position of the investor. A variation in any single one of these parameters might drastically modify or completely change the selection and sizing of a waterflood plant.



44-46



The volumes of injection water to be handled will, of course, be the most important basic item of information to learn for determining the size of the plant. Here, too, there are several parameters on which the calculation is based. Essentially, the water volume is a function of the gross size of the reservoir to be flooded, the porosity of the reservoir rock, the anticipated conformity or efficiency of the flood, and the ROS at both the initiation and completion of the flood. These data will be applied to the actual reservoir calculations, and only the final gross volume and the required daily injection rate must be known by the plant designer. As a general rule of thumb, 8 to 15 bbl of injected water per barrel of secondary oil, or I % to 2 PV of injected water, will provide a reasonable estimate of the ultimate water-handling requirements. Daily injection rates may vary from 5 to 25 bblift of pay. The producing-equipment capacity may be a limiting factor in determining the maximum injection rates. A relatively high ratio between the amount of fluid that is injected and the amount of fluid that is produced can be anticipated before fill-up. There are certain other factors that should be considered in designing the proper capacity of the plant facilities. If the available quantity of supply water is relatively small, it is usually necessary to consider produced brine along with other supply waters so that an adequate injection volume is provided. Where the original source water is not compatible with the produced water, or where the produced water is best handled in a closed system and original source water is best handled in an open or semiopen system, flexibility in capacity design will be required. This flexibility is necessary to adjust or to balance capacities between two separate injection systems (one with a constantly increasing load, the other with a constantly diminishing load). The pressure required to inject water into a formation is a function of formation depth, rock permeability, water quality, and the injection rate that is required. The basic reservoir data and secondary-recovery study will have defined the rock properties so that the anticipated surface pressures can be defined closely, if no adverse effects are anticipated as a result of poor-quality or incompatible water. Poor quality might be because the water contains a large quantity of solids as a result of poor filtration, inadequate settling, precipitation in an unstable water, or the growth of bacteria. Incompatibility might result from mixing injection water with formation water, from the swelling of clay particles, or from chemical reactions between the rock minerals and the injected water. In general. it has been found that the pressures than initially are encountered are less than might be anticipated when the only governing factors are depth and permeability; however, increasing pressures should be expected if there is no plan to reduce the injection rate as fill-up is approached. A final factor in predicting injection rates is the method of production. If the reservoir is to be produced by natural flow, the injection pressure must be sufficient to overcome dynamic hydraulic forces and to support a flowing rate of production. If, on the other hand, production is to be by mechanical means, with producing fluid levels at or near reservoir depth, a considerable reduction in injection-pressure requirements is possible. Consideration should be given to what the maximum allowable injection pressures should be. As a rule of thumb, pres-



PETROLEUM ENGINEERING



HANDBOOK



sure at the surface should not exceed 0.5 psi for every foot of reservoir depth. The maximum wellhead injection pressure will limit the resulting pressure at the perforations, which is less than the parting or fracture pressure. This pressure can be determined by an injectivity test conducted before or during pilot flood operations. Breakdown pressures are often encountered below the 0.5-psi value, and in such circumstances the maximum pressure will be defined by the breakdown pressure. In older fields, or in reservoirs located at considerable depth, the mechanical strength of the injection-well casing may be the deciding factor concerning the pressure limit. This limitation can be overcome by installation of competent tubing set on a packer. The source and the condition of the supply water will be the most important factors in determining a treating method. It is generally good practice to plan originally on using a closed system that requires little or no treating. Subsequently, the closed system may evolve into one in which the mixing of produced water will require custom-tailoring for conditions that are unique to the particular flood being considered. By starting with a basic treating system, the unit may be expanded into a complete version that may include aeration, chemical treating, flocculation, settling, corrosion inhibition, and bacteria control. In developing the proper treating system for a particular plant, the economic factors that are unique to the situation should be given close attention. If the flood is to be of relatively short duration, it may be profitable to use a system that is less than adequate and to anticipate more than normal maintenance demands. In other circumstances, it might prove most profitable to install corrosionresistant equipment and to reduce the use of corrosioninhibiting treatment. Consideration should be given to installing fiberglass tubing or internally plastic-lined tubing in injection wells. Also, if new injection wells are to be drilled, a full or partial string of fiberglass casing should be considered to minimize corrosion and scale buildup, especially in the area across the producing formation. A paper published in 1980 discusses the use of fiberglass liners and injection tubing in a west Texas waterflood. I65 Possibly the last item to be considered by many design engineers, and yet the most important item in many companies, is the financial position of the investor. It is quite possible that a particular operator may have limited investment capital and would find it desirable to keep this sum to a minimum, at the expense of higher future operation costs or additional future investment. The capital investment situation might also affect the choice of injection rate. The operator might be in a financial position in which a low, long-term, constant income would be most advantageous; in other circumstances, a short-term, highincome situation might be most desirable. Under either of these conditions, the normal approach to determining injection rates and plant design would be modified to produce the most desirable income vis-8-vis investment conditions. When the most desirable injection rate as well as the pressure and treating technique have been determined, the plant must be designed to f’it the prescribed conditions. F0r.a closed system, the plant design may be extremely simple and yet completely automatic. With in-line, high-



WATER-INJECTION



PRESSURE MAINTENANCE



& WATERFLOOD PROCESSES



pressure filtration equipment and a relatively highdischarge head source well pump, it is possible to use the supply pump as the injection pump and to inject directly from the supply well to the injection well. In this plan, individual cartridge-type well filters may be used if the supply water is relatively free of solids. The next stage in increasing the capacity of the injection plant would be to install a booster pump downstream from the filters, so that the supply pump and filters would not have to operate at injection pressures. The step after that would be to place a gas- or oil-blanketed water surge tank between the supply and filter system and the injection pumps. With this arrangement, low-pressure equipment can be used for supply and filtration; if the supply water and produced water are found compatible, produced water can be commingled in the surge tank. Where the systems are separated, it is also possible to use injection pumps with maximum pressure capacities. Further flexibility is also possible in that both source and injection rates can be varied independently, as long as the supply rate is at least as great as the injection rate. Corrosion frequently is minimized in the low-pressure side of this type of system by use of plastics, which also results in reduced fabrication costs. If a supply water is naturally aerated, the operation of a closed system becomes pointless. Also, because of excessive amounts of dissolved acid gases and/or a high content of dissolved iron, it may be desirable to aerate the water as a treating technique. When an open treating system is being designed, consideration should be given to using natural elevation or substructures to obtain gravity flow through the system. Under these circumstances, open gravity filters are often the most economical and practical. When a complete chemical-treating program is planned, the most common approach is to have the prefabricated mixing and sludge tank placed immediately ahead of the filters. In certain circumstances, it has been found desirble to deaerate the treated water before using it for injection. Chemical treatments can be used; however, chemicals are too costly except for the removal of very small quantities of oxygen. Counterflow, bubble-tray towers that use natural gas or a vacuum are sometimes used for oxygen removal. However, oxygen is not removed if it can be avoided, because of the relatively high cost of the process; the price must be weighed against the deleterious effects of the entrained oxygen. Centrifugal pumps have proved most satisfactory for low-pressure supply water and for injection at low pressures. Among the advantages of this type of pump are the small number of its moving parts and its excellent adaptability to volume control; however, in cases in which an appreciable amount of power is to be used, the relatively low efficiencies of centrifugal pumps (particularly when they are operated at other than design conditions) may preclude their use. In selecting centrifugal pumps, the proper metals should be chosen carefully for both the case and the trim to ensure the best performance. The greatest economy may be achieved with a cheaper pump that is subject to some corrosion rather than with a much more expensive pump, even though it might not be susceptible to corrosion. The positive-displacement type of injection pump is the most common one in use. Some use has been made of multistage centrifugal pumps; however,



44-47



they have not yet been widely accepted because of some limitations in flexibility and efficiency. The most generally accepted type of pump for mediumto high-pressure water injection is of either vertical or horizontal multicylinder design. These pumps are relatively simple to operate and to maintain, and they can be purchased with a variety of corrosion-resistant parts and accessories. The selection of the proper number of pumps and their capacity is contingent on the present and future requirements for the project. It is, of course, a good practice to provide a standby capacity that is sufficient to maintain continuous injection in case one pump has a mechanical failure. This can be accomplished by distributing the maximum design load over two or more units so that at least half the injection capacity can be maintained. A considerable number of filtering techniques are now used in the oil field. These involve ceramic-, metallic-, paper-, and cloth-element pressure filters with sand, gravel, or coal media; and rapid sand pressure filters with sand, coal, or graphite media. The choice of filters is a function of the raw water quality and volume of water required for injection. If solids in the water must be reduced to submicrometer size, one of the element-type or diatomaceous-earth filters, or a combination of the two, is recommended. For less rigorous filtration, the gravity or rapid sand pressure filters are most widely used. In general, filtration rates are considered normal at about 2 gal/min-sq ft of filter area; however, this figure will vary considerably depending on the quality of the influent and the desired quality of the effluent. Decreased rates also may be desirable if very frequent backwashing is netessary. The rates and techniques for backwashing are prescribed by the manufacturers of the various types of filters; this function should be considered in plant design to ensure adequate clear-water storage for both backwashing and continuous injection. It may be desirable to install additional filter capacity so that filtration will not stop during backwashing. The addition of standby filtration facilities also offers a guarantee against a total shutdown in which a filter requires a complete change of the filter medium. Refs. 116, 144, 145, 147, 148, and 149 discuss waterflood plant facilities. Also, Ref. 163 discusses waterflood plant facilities for a North Sea waterflood project. For a more derailed discussion on plant design criteria, design calculations, etc., the reader is directed to Chap. 15, Surface Facilities for Waterflooding and Saltwater Disposal.



Nomenclature a A B 3, B,,, B (,R



= = = = = =



distance between wells in a row, ft cross-sectional area, sq ft FVF, RBISTB 011 FVF, RBISTB initial oil FVF, RBlSTB oil FVF at current reservoir conditions, RBiSTB C,, = correction for gas expansion d = distance between rows of wells, ft EC = fractional coverage or conformance efficiency



44-48



E;y = efficiency of permeability variation, fraction ER = oil recovery efficiency, fraction f,,.,, = corner well producing water cut, fraction f,,,,. = side well producing water cut, fraction f(,z = fraction of oil flowing at the producing end of the system f, = fraction of total flow coming from the swept portion of the pattern f,,. = fractional flow of water F, = Caudle and Witte conductance ratio FF = ratio of viscous to gravity forces F C’S= oil/gas saturation ratio Fp = cornerto~side~well producing-rate ratio F,,.,, = WOR g = acceleration caused by gravity, ft/sec2 h = formation thickness, ft it, = injection rate of fluid that has same mobility as the reservoir oil in a liquidfilled (base) pattern, as calculated from Eq. 20, RB/D i,,.= water-injection rate, RBiD k,, = effective permeability to oil, md k,,. = effective permeability to water, md k.,= permeability of x layer, or the layer that has just been flooded, md k = mean permeability, md k, = permeability value at 84. I % of cumulative sample, md L = distance, ft M = mobility ratio M,,,, = water/oil mobility ratio multiplied by the FVF of the reservoir oil at the time of flooding n = number of layers IIBT = number of layers in which water has broken through (varies from 1 to n) N = initial oil in place, STB, or ratio of square root of production rates N,, = oil produced, STB N ,“I = recovery to depletion (abandonment), fraction = pressure at depletion (abandonment), psi PO = transient backpressure. psi P II,, = effective reservoir pressure (external PC boundary pressure), psi p, = initial pressure, psi Api