A Variable-Rate Solution to the Nonlinear Diffusivity Gas Equation by Use of Green's-Function Method
- Abelardo B. Barreto Jr. (Petrobras) | Alvaro Peres (Petrobras) | Adolfo P. Pires (Universidade Estadual do Norte Fluminense)
- Document ID
- Society of Petroleum Engineers
- SPE Journal
- Publication Date
- December 2012
- Document Type
- Journal Paper
- 57 - 68
- 2012. Society of Petroleum Engineers
- 5.6.4 Drillstem/Well Testing, 5.3.1 Flow in porous media
- 2 in the last 30 days
- 507 since 2007
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The hydraulic diffusivity equation that governs the flow of compressible fluids in porous media is nonlinear. Although the gas-well test analysis by means of the pseudopressure function has become a standard field practice, the effect of viscosity and gas-compressibility variation with pressure is often neglected. Moreover, in field operations, the gas well is submitted to a variable rate production to determine well/reservoir properties and an estimation of the absolute open flow (AOF). For slightly compressible fluids, variable rate can be properly handled by superposition in time. Unfortunately, superposition cannot be casually justified for gas reservoirs because of its nonlinear behavior. In this paper, a general solution that properly accounts for both fluid property behavior and variable rate is presented. The proposed solution, which is derived from the Green's-function method by recasting the effect of the viscosity-compressibility product variation as a nonlinear source term, can handle variable gas rate for several well/reservoir geometries of practical interest. From the general solution, an analytical expression for variable-rate tests of a fully penetrating vertical well in an infinite gas reservoir is derived. This expression is applied to a synthetic data set to calculate the pressure response for a buildup test in an infinite homogeneous reservoir. The results compared with a commercial finite-difference numerical simulator show close agreement for both drawdown and buildup periods. It is also shown that the dimensionless pseudopressure converges to the slightly compressible fluid solution for long shut-in times. Thus, during those long times, Horner analysis and log-log derivative plot can be applied to obtain good estimation of reservoir parameters, as discussed previously in literature.
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Abramowitz, M. and Stegun, I.A. 1972. Handbook of MathematicalFunctions. New York: Dover Publications.
Agarwal, R.G. 1980. A New Method To Account for Producing Time Effects WhenDrawdown Type Curves Are Used To Analyze Pressure Buildup and Other Test Data.Paper SPE 9289 presented at the SPE Annual Technical Conference and Exhibition,Dallas, Texas, 21?24 September 1980. http://dx.doi.org/10.2118/9289-MS.
Al-Hussainy, R., Ramey Jr., H.J. and Crawford, P.B. 1966. The Flow of RealGases Through Porous Media. J. Pet. Tech. 18 (5): 624?636.http://dx.doi.org/10.2118/1243-A-PA.
Barreto, Jr., A.B. 2011a. Non-Linear Gas Diffusivity Equation Solution byGreen's Functions. PhD Thesis (in Portuguese). Universidade Estadual NorteFluminense, Macaé, Brazil (2011).
Barreto, Jr., A. B. 2011b. Nonlinear Gas Well Test Problems: A GeneralizedPerturbative Solution Applied to a Vertical Well near a Sealing Fault. PaperSPE 152358 presented at the SPE International Student Paper Contest at the SPEAnnual Technical Conference and Exhibition, Denver, Colorado, 30 October?2November. http://dx.doi.org/10.2118/152358-STU.
Barreto Jr., A.B., Pires, A.P., and Peres, A.M.M. 2012. A New RigorousAnalytical Solution for a Vertical Fractured Well in Gas Reservoirs. Paper SPE150663 presented at the SPE Latin America and Caribbean Petroleum EngineeringConference, Mexico City, Mexico, 16?18 April. http://dx.doi.org/10.2118/150663-MS.
Beck, J.V., Cole, K.D., Haji-Sheikh, A. et al. 1992. Heat ConductionUsing Green's Functions. Philadelphia, Pennsylvania: HemispherePublishing Corporation.
Bourdet, D. 2002. Well Test Analysis: The Use of Advanced InterpretationModels. New York: Elsevier Sciences.
Carslaw, H.S. and Jaeger, J.C. 1959. Conduction of Heat in Solids.New York: Oxford Science Publications.
Duffy, D.G. 2001. Green's Functions with Applications. London:Chapman & Hall/CRC.
Gupta, K.C. and Andsager, R.L. 1967. Application of Variable Rate AnalysisTechnique to Gas Wells. Paper SPE 1836 presented at the Fall Meeting of theSociety of Petroleum Engineers of AIME, New Orleans, Louisiana, 1?4 October. http://dx.doi.org/10.2118/1836-MS.
Hahn, T. 2005a. Cuba--A Library for Multidimensional Numerical Integration.Computer Physics Communications. 168 (2): 78?95. http://dx.doi.org/10.1016/j.cpc.2005.01-010.
Hahn, T. 2005b. Cuba--A Library for Multidimensional Numerical Integration,report, Max-Planck-Institut für Physik. Available for download at http://cdsweb.cern.ch/record/728672.
Kale, D. and Mattar, L. 1980. Solution of a Non-Linear Gas Flow Equation bythe Perturbation Technique. J. Cdn. Pet. Tech. 19 (4):63?67. http://dx.doi.org/10.2118/80-04-06.
Koval'chuk, B.V. and Lopushanskaya, G.P. 1993. The Green's Function Methodin Stationary and Nonstationary Nonlinear Heat-Conduction Problems. J. Math.Sci. 66 (6): 2587?2591. http://dx.doi.org/10.1007/BF01097863.
Lee, W.J., Rollins, J.B., and Spivey, J.P. 2003. Pressure TransientTesting. Richardson, Texas: Society of Petroleum Engineers, Inc.
Levitan, M.M. and Wilson, M. R. 2010. Deconvolution of Pressure and RateData from Gas Reservoirs with Significant Pressure Depletion. Paper SPE 134261presented at the SPE Annual Technical Conference and Exhibition, Florence,Italy, 19?22 September. http://dx.doi.org/10.2118/134261-MS.
Peres, A.M.M., Serra, K.V., and Reynolds, A.C. 1989. Supplement to SPE18113, Toward a Unified Theory of Well Testing for Nonlinear-Radial-FlowProblems with Application to Interference Tests. SPE Form Eval.
Peres, A.M.M., Serra, K.V., and Reynolds, A.C. 1990. Toward a Unified Theoryof Well Testing for Nonlinear-Radial-Flow Problems with Application toInterference Tests. SPE Form Eval 5 (2): 151?160. http://dx.doi.org/10.2118/18113-PA.
Samaniego, F. and Cinco-Ley, H. 1991. Transient Pressure Analysis forVariable Rate Testing of Gas Wells. Paper SPE 21831 presented at the LowPermeability Reservoirs Symposium, Denver, Colorado, 15-17 April. http://dx.doi.org/10.2118/21831-MS.
Stehfest, H. 1970. Algorithm 368, Numerical Inversion of Laplace Transforms[D5]. Communications of the ACM 13 (1): 47?49. http://dx.doi.org/10.1145/361953.361969.
Thompson, L. and Reynolds, A. 1986. Analysis of Variable-Rate Well-TestPressure Data Using Duhamel's Principle. SPE Form Eval 1(5): 453?469. http://dx.doi.org/10.2118/13080-PA.
von Schroeter, T. and Gringarten, A. C. 2009. Superposition Principle andReciprocity for Pressure Transient Analysis of Data from Interfering Wells.SPE J. 14 (3): 488?495. http://dx.doi.org/10.2118/110465-PA.