Carter-Tracy Water Influx Model [PDF]

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[CARTER TRACY METHOD]



Petroleum Seminar



CONTENTS INTRODUCTION.................................................................................................................................................. 2 CLASSIFICATION OF AQUIFERS............................................................................................................................ 3 RECOGNITION OF NATURAL WATER INFLUX........................................................................................................ 4 WATER INFLUX MODELS..................................................................................................................................... 6 THE CARTER-TRACY WATER INFLUX MODEL........................................................................................................ 8 CASE STUDY...................................................................................................................................................... 12 IDEAL CASE....................................................................................................................................................... 12 REAL CASE......................................................................................................................................................... 15 BELHEDAN OIL FIELD......................................................................................................................................... 15 APPENDIX......................................................................................................................................................... 21 REFERENCES...................................................................................................................................................... 27



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[CARTER TRACY METHOD]



Petroleum Seminar



INTRODUCTION The water influx is the replacement of produced fluids by formation water. Most petroleum reservoirs are underlain by water, and water influx into a reservoir almost always takes place at some rate when gas or oil is produced. Whether appreciable water is produced along with gas or oil depends on the proximity of the productive interval to the oil-water contact or gas-water contact and whether the well is coning (vertical well) or cresting (horizontal well). Nearly all hydrocarbon reservoirs are surrounded by water-bearing rocks called aquifers. These aquifers may be substantially larger than the oil or gas reservoirs they adjoin as to appear infinite in size, or they may be as small in size as to be negligible in their effect on reservoir performance. As reservoir fluids are produced and reservoir pressure declines, a pressure differential develops from the surrounding aquifer into the reservoir. Following the basic law of fluid flow in porous media, the aquifer reacts by encroaching across the original hydrocarbon-water contact. In some cases, water encroachment occurs due to hydrodynamic conditions and recharge of the formation by surface waters at an outcrop. In many cases, the pore volume of the aquifer is not significantly larger than the pore volume of the reservoir itself. Thus, the expansion of the water in the aquifer is negligible relative to the overall energy system, and the reservoir behaves



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volumetrically.



[CARTER TRACY METHOD]



Petroleum Seminar



In this case, the effects of water influx can be ignored. In other cases, the aquifer permeability may be sufficiently low such that a very large pressure differential is required before an appreciable amount of water can encroach into the reservoir. In this instance, the effects of water influx can be ignored as well. This seminar paper provides water influx calculation models specially the CarterTracy method and a detailed description of the computational steps involved in applying these models.



CLASSIFICATION OF AQUIFERS Many gas and oil reservoirs produced by a mechanism termed water drive. Often this is called natural water drive to distinguish it from artificial water drive that involves the injection of water into the formation. Hydrocarbon production from the reservoir and the subsequent pressure drop prompt a response from the aquifer to offset the pressure decline. This response comes in a form of water influx, commonly called water encroachment, which is attributed to:



Expansion of the water in the aquifer



Compressibility of the aquifer rock Artesian flow where the water-bearing formation outcrop is located structurally higher than the pay zone



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Reservoir-aquifer systems are commonly classified on the basis of:



Degree of pressure maintenance



Flow regimes



Outer boundary conditions



Flow geometries



RECOGNITION OF NATURAL WATER INFLUX Normally very little information is obtained during the exploration-development period of a reservoir concerning the presence or characteristics of an aquifer that could provide a source of water influx during the depletion period. Natural water drive may be assumed by analogy with nearby producing reservoirs, but early reservoir performance trends can provide clues. A comparatively low, and decreasing, rate of reservoir pressure decline with increasing cumulative withdrawals is indicative of fluid influx. Successive calculations of barrels withdrawn per psi change in reservoir pressure can supplement performance graphs. If the reservoir limits have not been delineated by developed dry holes, however, the influx could be from an undeveloped area of the reservoir not accounted for in averaging reservoir



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[CARTER TRACY METHOD]



Petroleum Seminar



pressure. If the reservoir pressure is below the oil saturation pressure, a low rate of increase in produced gas-oil ratio is also indicative of fluid influx.



Figure (1) The flow geometries.



Early water production from edge wells is indicative of water encroachment. Such observations must be tempered by the possibility that the early water production is due to formation fractures; thin, high permeability streaks; or to coning in connection with a limited aquifer. The water production may be due to casing leaks.



Calculation of increasing original oil-in-place from successive



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[CARTER TRACY METHOD]



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reservoir pressure surveys by using the material balance assuming no water influx is also indicative of fluid influx.



WATER INFLUX MODELS It should be appreciated that in reservoir engineering there are more uncertainties attached to this subject than to any other. This is simply because one seldom drills wells into an aquifer to gain the necessary information about the porosity, permeability, thickness and fluid properties. Instead, these properties frequently have to be inferred from what has been observed in the reservoir. Even more uncertain, however, is the geometry and areal continuity of the aquifer itself. Several models have been developed for estimating water influx that are based on assumptions that describe the characteristics of the aquifer. Due to the inherent uncertainties in the aquifer characteristics, all of the proposed models require historical reservoir performance data to evaluate constants representing aquifer property parameters since these are rarely known from exploration-development drilling with sufficient accuracy for direct application. The material balance equation can be used to determine historical water influx provided original oil-inplace is known from pore volume estimates. This permits evaluation of the constants in the influx equations so that future water influx rate can be forecasted. The mathematical water influx models that are commonly used in the petroleum industry include:



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Water Influx Models



[CARTER TRACY METHOD]



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Pot aquifer Schilthuis’ steady-state Hurst’s modified steady-state The Van Everdingen-Hurst unsteadystate The Carter-Tracy unsteady-state



Edge-water drive Bottom-water drive Radial aquifer



Fetkovich’s method Linear aquifer Figure (2) The water influx models.



THE CARTER-TRACY WATER INFLUX MODEL Van Everdingen-Hurst methodology provides the exact solution to the radial diffusivity equation and therefore is considered the correct technique for calculating water influx. However, because superposition of solutions is required, their method involves tedious calculations. To reduce the complexity of water influx calculations, Carter and Tracy (1960) proposed a calculation technique that does not require superposition and allows direct calculation of water influx. The primary difference between the Carter-Tracy technique and the van Everdingen-Hurst technique is that the Carter-Tracy technique assumes constant water influx rates over each finite time interval. The two dimensionless parameters t D and r D are given by:



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[CARTER TRACY METHOD]



−3



t D =6.328 ×10



kt ∅ μ w c t r 2e



Petroleum Seminar



Equation (1)



Where K



= average permeability, md



t



= Time, days



∅



= average porosity, dimensionless



μw = viscosity, cp



re



= radius of the reservoir, ft 2



B=1.119 ∅ c t r e h f



Equation (2)



θ is the angle subtended by the reservoir circumference, i.e., for a full circle θ = 360° and for semicircle reservoir against a fault θ =180° and f = ∅ /360. Using the Carter-Tracy technique, the cumulative water influx at any time t n, can be calculated directly from the previous value obtained at tn − 1, or:



( W e )n =( W e )n−1 + [ ( t D )n−( t D )n−1 ]



[



B ∆ pn −( W e )n−1 ( p'D )n ( p D )n−( t D )n−1 ( p 'D )n



]



Equation (3)



Where B



= the water influx constant as defined by Equation (2).



tD



= the dimensionless time as defined by Equation (1).



n



= refers to the current time step.



n-1 = refers to the previous time step. = dimensionless pressure.



PD ’



= dimensionless pressure derivative.



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PD



[CARTER TRACY METHOD]



Petroleum Seminar



Values of the dimensionless pressure p D as a function of t D and r D are in Table (1). Edwardson and coauthors (1962) developed the following approximation of p D for an infinite- acting aquifer.



p D=



370.529 √ t D +137.582 t D +5.69549 ( t D )1.5



328.834+265.488 √ t D + 45.2157 t D + ( t D )



1.5



Equation (4)



The dimensionless pressure derivative can then be approximated by '



p D=



E F



Equation (5)



Where E=716.441+ 46.7984 ( t D )0.5 +270.038 t D +71.0098 ( t D )1.5



F=1296.86 ( t D ) 0.5 +1204.73 t D +618.618 ( t D ) 1.5+ 538.072 ( t D )2 +142.41 ( t D )2.5



The following approximation could also be used betweent D > 100: p D=0.5 [ ln ( t D ) +0.80907 ] Equation (6)



With the derivative as given by: p D=1/(2t D) Equation (7) '



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It should be noted that the Carter-Tracy method is not an exact solution to the diffusivity equation and should be considered an approximation. Table (1) PD vs. tD (Infinite-Radial System, Constant-Rate at the Inner Boundary). PD



tD



PD



tD



PD



0 0.0005 0.001 0.002 0.003 0.004 0.005 0.006 0.007 0.008 0.009 0.01 0.015 0.02 0.025 0.03 0.04 0.05 0.06 0.07 0.08 0.09



0 0.025 0.0352 0.0495 0.0603 0.0694 0.0774 0.0845 0.0911 0.0971 0.1028 0.1081 0.1312 0.1503 0.1669 0.1818 0.2077 0.2301 0.25 0.268 0.2845 0.2999



0.15 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 1.2 1.4 2 3 4 5 6 7 8 9 10 15



0.375 0.4241 0.5024 0.5645 0.6167 0.6622 0.7024 0.7387 0.7716 0.8019 0.8672 0.916 1.0195 1.1665 1.275 1.3625 1.4362 1.4997 1.5557 1.6057 1.6509 1.8294



2.4758 2.5501 2.6147 2.6718 2.7233 2.9212 3.0636 3.1726 3.263 3.3394 3.4057 3.4641 3.5164 3.5643 3.6076 3.6476 3.6842 3.7184 3.7505 3.7805 3.8088 3.8355



0.1



0.3144



20



1.9601



60 70 80 90 100 150 200 250 300 350 400 450 500 550 600 650 700 750 800 850 900 950 1,000.0 0



30 40 50



2.147 2.2824 2.3884



3.8584



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CASE STUDY



tD



[CARTER TRACY METHOD]



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In this seminar, two case studies was obtained the first one is for an ideal case and the second one for real data from oil field in Libya (Belhedan field -NC59).



Case Study



Ideal Case



Real Case



IDEAL CASE The initial and current reservoir pressures are 2500 and 2490 psi, respectively. The reservoir-aquifer system has the following properties. Data



Reservoir



Aquifer



Radius, ft



2000



∞



h, ft



20



25



K, md



50



100



∅%



15



20



μw , cp



0.5



0.8



Cw,Psi-1



1X10-6



0.7X10-6



Cf,Psi-1



2X10-6



0.3X10-6



Calculate the cumulative water influx at the end of 6, 12, 18, and 24 months. The predicted boundary pressure at the end of each specified time period is given below: Time, months



Boundary pressure, psi



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[CARTER TRACY METHOD]



0



2500



6



2490



12



2472



18



2444



24



2408



Petroleum Seminar



Solution Step (1): Determine water influx constant B B = 20.4 bbl/psi Step (2): Calculate the dimensionless time tD tD = 0.9888t Step (3): For each time step (n), calculate the total pressure drop ∆ Pn=Pi-Pn and the corresponding tD N 0 1 2 3 4



t,days 0 182.5 365 547.5 730



Pn 2500 2490 2472 2444 2408



ΔPn 0 10 28 56 92



tD 0 180.5 361 541.5 722



Step (4): Since values of tD are greater than 100 by using Equation (6) and (7) to calculate PD and its derivative P’D.



t 0 182.5



tD 0 180.5



PD — 3.002



PD′ × 10−3 — 2.77



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N 0 1



[CARTER TRACY METHOD]



2 3 4



365 547.5 730



361 541.5 722



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3.349 3.552 3.696



1.385 0.923 0.693



Step (5): Calculate cumulative water influx by applying Equation (3). We after 182.5 days:



[



(20.4)(10)−(0)(2.77 X 10−3 ) W e =0+ [ 180.5−0 ] −3 3.002−(0)(2.77 X 10 )



]



W e =12,266 bbl



We after 365 days:



[



−3



(20.4)(28)−(12,266)(1,385 X 10 ) W e =12,266+ [ 361−180.5 ] 3.349−(180.5)(1,385 X 10−3 )



]



W e =42,546 bbl



We after 547.5 days: W e =42,546+ [ 541.5−361 ]



[



−3



(20.4)(56)−(42,546)( 0.923 X 10 ) 3.552−(361)(0.923 X 10−3 )



]



W e =104,406 bbl



We after 720 days: W e =104,406+ [ 722−541.5 ]



]



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[



−3



(20.4)(92)−(104,406)(0.693 X 10 ) −3 3.696−(541.5)(0.693 X 10 )



[CARTER TRACY METHOD]



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W e =202,477 bbl



The following table (2) shown results of the Carter-Tracy water influx calculations method at the end of 6, 12, 18, and 24 months. Table (2) the result of the Carter-Tracy water influx method.



Time,month 0 6 12 18 24



We,bbl 0 12,266 42,546 104,400 202,477



REAL CASE In this seminar, the real case was obtained from Waha Company for Belhedan oil field in NC59 located in the Sirte Basin.



BELHEDAN OIL FIELD The Belhedan oil Field is located in the Sirte Basin .The Belhedan structure, located on the south-eastern edge of the Beda Platform, is a small horst block which trends north-northeast, the primary zone of interest is the Belhedan Gargaf reservoir which was discovered in 1962 by Waha Oil Company when the first exploratory Well V1 was drilled in this field, development drilling which was started immediately after the first discovery. The reservoir development started early after first discovery in 1962 when the well V-1 tested commercial oil production, Belhedan field started commercial



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production in October 1963. The reservoir was developed to produce under the mechanism of natural bottom water drive.



Belhedan Oil Field



Figure (3) The location of Belhedan oil field.



Field development has continued since its discovery, as of December 31, 2009, a total of 40 wells have been drilled in Belhedan field and the field had produced 179.840 million barrels, with substantial reserves are still remaining in the main development area of the field. Presently, all wells are being produced by ESP. As of December 31st 2008, The field average oil production rate was 24691 BOPD, with 27.4% WC. Table (3) the average fluid properties of Belhedan oil field.



Symbol



Value



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Fluid Properties



[CARTER TRACY METHOD]



Saturation Pressure Gas Oil Ratio Oil Formation Factor Oil Viscosity Oil Gravity @ 60ºF



Pb GOR Boi µo ºAPI



Petroleum Seminar



Psig 536 scf/stb 110 bbl/stb 1.135 cp 1.575 36.0



Table (4) the average rock properties of Belhedan oil field.



Rock properties Porosity Horizontal Permeability Water Saturation



Symbol φ K Swi



Value 8.0% md 10-100 36%



Figure (4) The Production History of Belhedan oil field.



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Petroleum Seminar



Figure (5) The Well Location Map of Belhedan oil field.



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Since it was proved, that the reservoir is produced under water drive mechanism as shown in previous section, the next stage is to evaluate the carter tracy method water influx model for Belhedan field. Where the production rate of the first period of the reservoir life was 12000 BOPD up to end of 1982, after that the second period up to end of 1992 the production decreased due to the production cut and the development of the reservoir facilities, these two periods of production cannot considered in this study because of the low production rate and cumulative production, so the last 15 years represent the high production rate and cumulative production after the development, and for the accuracy of MBE it was considered in this study. The calculation of water influx using this model directly depends on the properties of aquifer such aquifer radius, aquifer thickness and the ratio of aquifer vertical to horizontal permeability. Unfortunately, all of these are unknown therefore, the following tests were made:  Dimensionless radius rD: from rD= 2.0 to rD= 8.0  Ratio of Kv/Kh : from 0.05 to 1.0  Assumed four aquifer thickness (h= 203 ft, 290 ft, 347 ft and 500 ft) By applying previous equations, the water influx values were calculated using different tests, the closest optimum value of OIIP is (1,107 MMSTB) was obtained with rD=8.0 and h=347ft, as illustrated in Figure (6). The plots of all tests are shown in Appendix. Water influx constant (B) = 53,207 bbl/d/psi.



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Water influx (We) = 2308.73MMbbl



[CARTER TRACY METHOD]



Petroleum Seminar



B=53206.9 25000



F/(Eo + Efw), MMSTB



20000



15000



10000



5000



0



0



5000



10000



15000



20000



25000



We/(Eo + Efw), MMSTB



Figure (6) OIIP Calculation Using Carter Tracy method.



Figure (7) Using Excel to calculate Water influx.



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Petroleum Seminar



APPENDIX Part of the graphics for some of the relationships to find the closest initial oil in place that calculated by using volumetric. 25000



F/(Eo + Efw), MMSTB



20000



15000



10000



5000



0



0



5000



10000



h = 500 ft. rD = 3.0 B = 76667 B/D/PSI 20000 25000 OIIP = 2,975 MMSTB



15000



We/(Eo + Efw), MMSTB



25000



F/(Eo + Efw), MMSTB



20000



15000



10000 h = 347 ft. rD = 3.5 B = 53206.9 B/D/PSI OIIP = 5,266 MMSTB



5000



0



0



5000



10000



15000



20000



25000



We/(Eo + Efw), MMSTB



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[CARTER TRACY METHOD] 25000000000



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f(x) = 563391.806530817 x − 127521798998.867



20000000000



F/Et



15000000000 10000000000 5000000000 0



rD = 3.5 B = 56339 B/D/PSI



0



50000



100000



150000



200000



250000



300000



∑Wed*∆P/Et



25000



F/(Eo + Efw), MMSTB



20000



15000



10000



5000



0



rD = 3.5 B = 56339 B/D/PSI (Graphically) OIIP = 4,451 MMSTB



0



5000



10000



15000



20000



25000



We/(Eo + Efw), MMSTB



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25000



F/(Eo + Efw), MMSTB



20000



15000



10000 h = 290 ft. rD = 4.0 B = 44432 B/D/PSI OIIP = 5,837 MMSTB



5000



0



0



5000



10000



15000



20000



25000



We/(Eo + Efw), MMSTB



25000



F/(Eo + Efw), MMSTB



20000



15000



10000 h = 347 ft. rD = 4.0 B = 53206.9 B/D/PSI OIIP = 3,215 MMSTB



5000



0



0



5000



10000



15000



20000



25000



We/(Eo + Efw), MMSTB



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25000



F/(Eo + Efw), MMSTB



20000



15000



10000 h = 290 ft. rD = 4.5 B = 44432 B/D/PSI OIIP = 4,838 MMSTB



5000



0



0



5000



10000



15000



20000



25000



We/(Eo + Efw), MMSTB



25000



F/(Eo + Efw), MMSTB



20000



15000



10000 h = 347 ft. rD = 4.5 B = 53206.9 B/D/PSI OIIP = 2,019 MMSTB



5000



0



0



5000



10000



15000



20000



25000



We/(Eo + Efw), MMSTB



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25000



F/(Eo + Efw), MMSTB



20000 15000 10000 h = 290 ft. rD = 5.0 B = 44432 B/D/PSI OIIP = 4,195 MMSTB



5000 0



0



5000



10000



15000



20000



25000



We/(Eo + Efw), MMSTB



25000



F/(Eo + Efw), MMSTB



20000 15000



10000 h = 290 ft. rD = 6.0 B = 44432 B/D/PSI OIIP = 1,560 MMSTB



5000



0



0



5000



10000



15000



20000



25000



We/(Eo + Efw), MMSTB



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25000



F/(Eo + Efw), MMSTB



20000



15000 10000 h = 203 ft. rD = 7.0 B = 31102.4 B/D/PSI OIIP = 4,487 MMSTB



5000



0



0



5000



10000



15000



20000



25000



We/(Eo + Efw), MMSTB



25000



F/(Eo + Efw), MMSTB



20000



15000



10000 h = 203 ft. rD = 8.0 B = 31102.4 B/D/PSI OIIP = 4,478 MMSTB



5000



0



0



5000



10000



15000



20000



25000



We/(Eo + Efw), MMSTB



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REFERENCES 1) Bill Bailey Mike Crabtree & others, Water Control, Oilfield Review, (2000). 2) T.Ahmed & P.D.Mckinney, Advanced Reservoir Engineering, Elsevier inc, Oxford, (2005). 3) Waha Oil Company, (2010), Intra-Company Unpublished Geological Report. 4)



Waha Oil Company, (2009), Unpublished intra-Company Production Report.



5)



Hamza.M.Shengaru, Saad .M. Sasi, Investigation Of Water Influx Models For The Application Of Material Balance Equation (Belhedan Oil Field), Supervised By Dr.Shaban.A.El Usta ,Tripoli university, petroleum engineering department.



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