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Rate Transient Analysis 1-4: TRADITIONAL DECLINE ANALYSIS



1. Traditional (Arps) Decline Curves



23-32: RADIAL TYPE CURVES



23. Calculations for Oil (Agarwal-Gardner Type Curves)



3. Exponential Decline



EXPONENTIAL DECLINE:



• • • •



23-24: RADIAL FLOW MODEL: TYPE CURVE ANALYSIS



Decline rate is constant.



All radial flow type curves are based on the same reservoir model:



Log flow rate vs. time is a straight line. Flow rate vs. cumulative production is a straight line.



Well in centre of cylindrical homogeneous reservoir.



Decline rate is not constant (D=Kqb).



• • • •



Straight line plots are NOT practical and b is determined by nonlinear curve fit.



•



The shapes are different because of different plotting formats.



•



Each format represents a different “look” at the data and emphasizes different aspects.



Provides minimum EUR (Expected Ultimate Recovery).



HYPERBOLIC DECLINE:



• •



b value 0 0.1-0.4 0.4-0.5 0.5 0.5-1.0



2. Decline Rate Definitions



Reservoir Drive Mechanism Single phase liquid (oil above bubble point) Single phase gas at high pressure Solution gas drive Single phase gas Effective edge water drive Commingled layered reservoirs



Information content of all type curves (Figures 25-32) is the same.



rwa



26. Blasingame: Integral-Derivative



1. qDd vs. tDd (Figure 25). 2. Rate integral (qDdi) vs. tDd (has the same shape as qDd). 3. Rate integral-derivative (qDdid) vs. tDd (Figure 26).



Decline rate is directly proportional to flow rate (b=1). Log flow rate vs. cumulative production is a straight line.



•



Boundary-dominated flow only. Constant operating conditions.



In general: qDd



qD bDpss , tDd



2 t b Dpss DA



• bDpss is a constant for a particular well / reservoir



Developed using empirical relationships.



configuration.



Quick and simple to determine EUR. EUR depends on operating conditions. Does NOT use pressure data. b depends on drive mechanism.



27. Agarwal-Gardner: Rate (Normalized)



5-10: FETKOVICH ANALYSIS



5. Analytical: Constant Flowing Pressure



6. Analytical: Constant Flowing Pressure



27-28: AGARWAL-GARDNER



•



Notes: 1. Pressure derivative is defined as pDd



d ( pD ) d (ln t DA )



2. Inverse of pressure derivative is usually too noisy and inverse of pressure integral-derivative is used instead.



boundary-dominated stems.



• qDd and tDd definitions are convenient for production data analysis.



• •



28. Agarwal-Gardner: Integral-Derivative



• qD and tDA definitions are similar to well testing. • Normalized rate (q/ p or q/ pp) is plotted. • Three sets of type curves: 1. qD vs. tDA (Figure 27). 2. Inverse of pressure derivative (1 / pDd) vs. tDA (not shown). 3. Inverse of pressure integral-derivative (1 / pDid) vs. tDA (Figure 28).



• qD and tD definitions are similar to well test. • Convenient for transient flow. • Results in single transient stem but multiple



7. Empirical: Arps Depletion Stems



re



Skin factor represented by rwa.



• qDd and tDd definitions are similar to Fetkovich. • Normalized rate (q/ p or q/ pp) is plotted. • Three sets of type curves:



SUMMARY:



• • • • • • •



No flow outer boundary.



25-26: BLASINGAME



25. Blasingame: Rate (Normalized)



4. Harmonic Decline



HARMONIC DECLINE:



• •



24. Calculations for Gas (Agarwal-Gardner Type Curves)



29-30: NORMALIZED PRESSURE INTEGRAL (NPI)



29. NPI: Pressure (Normalized)



Convenient for boundary-dominated flow. Results in single boundary-dominated stem but multiple transient stems.



30. NPI: Integral-Derivative



• pD and tDA definitions are similar to well testing. • Normalized Pressure ( p/q or pp /q) is plotted



8. Empirical: Arps-Fetkovich Depletion Stems



rather than normalized rate (q/ p or q/ pp).



•



Three sets of type curves: 1. pD vs. tDA (Figure 29). 2. Pressure integral (pDi) vs. tDA (has the same shape as pD). 3. Pressure integral-derivative (pDid) vs. tDA (Figure 30).



Replot on Log-Log Scale



31-32: TRANSIENT-DOMINATED DATA



31. Rate (Normalized) 10. Fetkovich/Cumulative Type Curves



•



9. Fetkovich Type Curves



• qD and tD definitions are similar to well testing. • Normalized rate (q/ p or q/ pp) is plotted. • Three sets of type curves:



SUMMARY:



• • •



Combines transient with boundary-dominated flow.



• • • • • •



Constant operating conditions.



Transient: Analytical, constant pressure solution.



1. qD vs. tD (Figure 31). 2. Inverse of pressure integral (1 / pDi) vs. tD (not shown). 3. Inverse of pressure integral-derivative (1 / pDid) vs. tD (Figure 32).



Boundary-Dominated: Empirical, identical to traditional (Arps). Used to estimate EUR, skin and permeability. EUR depends on operating conditions. Does NOT use pressure data. Cumulative curves are smoother than rate curves. Combined cumulative and rate type curves give more unique match (Figure 10).



33-40: FRACTURE TYPE CURVES



33. Rate



11. Comparison of



qD and 1/pD



32. Integral-Derivative



Similar to Figures 27 & 28 but uses tD instead of tDA. This format is useful when most of the data are in TRANSIENT flow.



11-14: MODERN DECLINE ANALYSIS: BASIC CONCEPTS



34. Integral-Derivative



33-37: FINITE CONDUCTIVITY FRACTURE



•



Fracture with finite conductivity results in bilinear flow (quarter slope).



Material Balance Time (tc) effectively converts constant pressure solution to the corresponding constant rate solution.



•



Dimensionless Fracture Conductivity is defined as:



•



Exponential curve plotted using Material Balance Time becomes harmonic.



•



•



Fracture with infinite conductivity results in linear flow (half slope).



Material Balance Time is rigorous during boundary-dominated flow.



•



For FCD>50, the fracture is assumed to have infinite conductivity.



12. Equivalence of



qD and 1/pD



11-12: MATERIAL BALANCE TIME



•



Actual Rate Decline



FCD



Constant Rate Q q



tc



Q



1 q



t 0



qdt



35. Elliptical Flow: Integral-Derivative



Q



Actual Time (t)



kf w kxf



36. Elliptical Flow: Integral-Derivative



37. Elliptical Flow: Integral-Derivative



Material Balance Time



13. Concept of Rate Integral



(t c) = Q /q



14. Derivative and Integral-Derivative



13-14: TYPE CURVE INTERPRETATION AIDS Rate (Normalized)



•



Combines rate with flowing pressure.



Integral (Normalized Rate)



•



Smoothes noisy data but attenuates the reservoir signal.



Derivative (Normalized Rate)



•



Amplifies reservoir signal but amplifies noise as well.



Integral-Derivative (Normalized Rate)



•



15-18: GAS FLOW CONSIDERATIONS



15. Darcy’s Law



38-40: INFINITE CONDUCTIVITY FRACTURE



Smoothes the scatter of the derivative.



38. Blasingame: Rate and Integral-Derivative



39. NPI: Pressure and Integral-Derivative



40. Wattenbarger: Rate



16. Pseudo-Pressure (pp)



15-16: PSEUDO-PRESSURE Gas properties vary with pressure:



• •



Z-factor (Pseudo-Pressure, Figures 15 & 16)



•



Compressibility (Pseudo-Time, Figures 17 & 18)



•



Pseudo-pressure corrects for changing viscosity and Z-factor with pressure.



•



In all equations for liquid, replace pressure (p) with pseudo-pressure (pp).



Viscosity (Pseudo-Pressure & Pseudo-Time, Figures 15, 16 & 18)



Note: For gas,



41-43: HORIZONTAL WELL TYPE CURVES 17-18: PSEUDO-TIME



17. Gas Compressibility Variation



• •



Compressibility represents energy in reservoir.



•



Ignoring compressibility variation can result in significant error in original gas-in-place (G) calculation.



•



Pseudo-time(ta) corrects for changing viscosity and compressibility with pressure.



•



Pseudo-time calculation is ITERATIVE because it depends on μg and ct at average reservoir pressure, and average reservoir pressure depends on G (usually known).



18. Pseudo-Time (ta)



41. Blasingame: Integral-Derivative



42. Blasingame: Integral-Derivative



43. Blasingame: Integral-Derivative



Gas compressibility is strong function of pressure (especially at LOW PRESSURES).



19-22: FLOWING MATERIAL BALANCE



19. Oil: Flowing Material Balance



20. Gas: Determination of



bpss



44-45: WATER-DRIVE TYPE CURVES



44. Blasingame: Rate



45. Agarwal-Gardner: Rate



Infinite Aquifer



Oil



Reservoir



Mobility ratio (M) represents the strength of the aquifer.



M



Gas



k aq μ res kres μ aq



Copyright



• Note: bpss is the inverse of productivity index and is constant during boundary-dominated flow.



• M = 0 is equivalent to Radial Type Curves (Figures 25-32).



21. Gas: Flowing Material Balance



22. Procedure to Calculate Gas-In-Place



SUMMARY:



• • • • • • • • •



Uses flowing data. No shut-in required. Applicable to oil and gas. Determines hydrocarbon-in-place, N or G. Oil (N): Direct calculation. Gas (G): Iterative calculation because of pseudo-time. Simple yet powerful. Data readily available (wellhead pressure can be converted to bottomhole pressure). Supplements static material balance. Ideal for low permeability reservoirs.



a A b b b B B B B c c c D D D F G G G h k k k



Dpss



pss



gi o oi



g t t



e i



CD



p pa



aq f



semi-major axis of ellipse area hyperbolic decline exponent or semi-minor axis of ellipse dimensionless parameter inverse of productivity index formation volume factor initial gas formation volume factor oil formation volume factor initial oil formation volume factor gas compressibility total compressibility total compressibility at average reservoir pressure nominal decline rate effective decline rate initial nominal decline rate dimensionless fracture conductivity original gas-in-place gas cumulative production pseudo-cumulative production net pay permeability aquifer permeability fracture permeability



k k k K L M N N p p p p p p p p p p p p p q q q h



res v



p



O D



Dd Di Did i



p p



pi



pwf wf



D



Dd



horizontal permeability reservoir permeability vertical permeability constant horizontal well length mobility ratio original oil-in-place oil cumulative production pressure average reservoir pressure reference pressure dimensionless pressure dimensionless pressure derivative dimensionless pressure integral dimensionless pressure integral-derivative initial reservoir pressure pseudo-pressure pseudo-pressure at average reservoir pressure initial pseudo-pressure pseudo-pressure at well flowing pressure well flowing pressure flow rate dimensionless rate dimensionless rate



q q q Q Q r r r r s S S t t t t t t t t t T w x



Ddi Ddid i



Dd



e



eD w



wa



gi oi



a c



ca D



DA



Dd



Dxf Dye



e



dimensionless rate integral xf dimensionless rate integral-derivative ye initial flow rate yw cumulative production Z Z dimensionless cumulative production exterior radius of reservoir Zi dimensionless exterior radius of reservoir α wellbore radius φ apparent wellbore radius μ skin μaq initial gas saturation μg initial oil saturation μg flow time pseudo-time μo material balance time μres material balance pseudo-time dimensionless time Oil field units; dimensionless time dimensionless time dimensionless time dimensionless time reservoir temperature fracture width reservoir length



2008 Fekete Associates Inc. Printed in Canada



Note: Pseudo-time in build-up testing is evaluated at well flowing pressure NOT at average reservoir pressure.



fracture half length reservoir width well location in y-direction gas deviation factor gas deviation factor at average reservoir pressure initial gas deviation factor constant porosity viscosity aquifer fluid viscosity gas viscosity gas viscosity at average reservoir pressure oil viscosity reservoir fluid viscosity



q (MMSCFD); t (days) g



All analyses described can be performed using Fekete’s Rate Transient Analysis software