A Semianalytic Solution for Flow in Finite-Conductivity Vertical Fractures by Use of Fractal Theory
- Authors
- Manuel Cossio (Texas A&M University) | George Moridis (Lawrence Berkeley National Laboratory) | Thomas A Blasingame (Texas A&M University)
- DOI
- https://doi.org/10.2118/153715-PA
- Document ID
- SPE-153715-PA
- Publisher
- Society of Petroleum Engineers
- Source
- SPE Journal
- Volume
- 18
- Issue
- 01
- Publication Date
- January 2013
- Document Type
- Journal Paper
- Pages
- 83 - 96
- Language
- English
- ISSN
- 1086-055X
- Copyright
- 2013. Society of Petroleum Engineers
- Disciplines
- 3.2.3 Hydraulic Fracturing Design, Implementation and Optimisation, 5.8.6 Naturally Fractured Reservoir
- Downloads
- 4 in the last 30 days
- 934 since 2007
- Show more detail
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Summary
The exploitation of unconventional reservoirs complements the practice of hydraulic fracturing, and with an ever-increasing demand in energy, this practice is set to experience significant growth in the coming years. Sophisticated analytic models are needed to accurately describe fluid flow in a hydraulic fracture, and the problem has been approached from different directions in the past 3 decades--starting with the work of Gringarten et al. (1974) for an infinite-conductivity case, followed by contributions from Cinco-Ley et al. (1978), Lee and Brockenbrough (1986), Ozkan and Raghavan (1991), and Blasingame and Poe (1993) for a finite-conductivity case. This topic remains an active area of research and, for the more-complicated physical scenarios such as multiple transverse fractures in ultratight reservoirs, answers are currently being sought.
Starting with the seminal work of Chang and Yortsos (1990), fractal theory has been successfully applied to pressure-transient testing, although with an emphasis on the effects of natural fractures in pressure/rate behavior. In this paper, we begin by performing a rigorous analytical and numerical study of the fractal diffusivity equation (FDE), and we show that it is more fundamental than the classic linear and radial diffusivity equations. Thus, we combine the FDE with the trilinear flow model (Lee and Brockenbrough 1986), culminating in a new semianalytic solution for flow in a finite-conductivity vertical fracture that we name the "fractal-fracture solution (FFS)." This new solution is instantaneous and comparable in accuracy with the Blasingame and Poe solution (1993). In addition, this is the first time that fractal theory is used in fluid flow in a porous medium to address a problem not related to reservoir heterogeneity. Ultimately, this project is a demonstration of the untapped potential of fractal theory; our approach is flexible, and we believe that the same methodology could be extended to different applications.
One objective of this work is to develop a fast and accurate semianalytical solution for flow in a single vertical fracture that fully penetrates a homogeneous infinite-acting reservoir. This would be the first time that fractal theory is used to study a problem that is not related to naturally fractured reservoirs or reservoir heterogeneity. In addition, as part of the development process, we revisit the fundamentals of fractals in reservoir engineering and show that the underlying FDE possesses some interesting qualities that have not yet been comprehensively addressed in the literature.
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References
Abdassah, D. and Ershaghi, I. 1986. Triple Porosity Systems for RepresentingNaturally Fractured Reservoirs. SPE Form Eval 1 (2):113-127. http://dx.doi.org/10.2118/13409-PA.
Acuña, J.A., Ershaghi, I., and Yortsos, Y.C. 1995. Practical Application ofFractal Pressure-Transient Analysis in Naturally Fractured Reservoirs. SPEForm Eval 10 (3): 173-179. http://dx.doi.org/10.2118/24705-PA.
Beier, R.A. 1994. Pressure-Transient Model for a Vertically Fractured Wellin a Fractal Reservoir. SPE Form Eval 9 (2): 122-128. http://dx.doi.org/10.2118/20582-PA.
Blasingame, T.A. 2010a. Petroleum Engineering 620—Fluid Flow in PetroleumReservoirs- Reservoir Flow Solutions-Linear Flow Solutions: Infinite andFinite-Acting Reservoir Cases. College Station, Texas: Texas A&MUniversity.
Blasingame, T.A. 2010b. Petroleum Engineering 620—Fluid Flow in PetroleumReservoirs- Reservoir Flow Solutions-Solution of the Radial Flow DiffusivityEquation. College Station, Texas: Texas A&M University.
Blasingame, T.A. and Poe, B.D. 1993. Semi-analytic Solutions for a Well witha Single Finite-Conductivity Vertical Fracture. Paper SPE 26424 presented atthe SPE Annual Technical Conference and Exhibition, Houston, Texas, 3-6October. http://dx.doi.org/10.2118/26424-MS.
Bowman, F. 1958. Introduction to Bessel Functions. New York: DoverPublications.
Camacho-Velazquez, R., Fuentes-Cruz, G., and Vasquez-Cruz, M. 2008.Decline-Curve Analysis of Fractured Reservoirs with Fractal Geometry. SPERes Eval & Eng 11 (3): 606-619. http://dx.doi.org/10.2118/104009-PA.
Camacho-Velazquez, R., Vasquez-Cruz, M., Castrejón-Aivar, R. et al. 2005.Pressure-Transient and Decline-Curve Behaviors in Naturally Fractured VuggyCarbonate Reservoirs. SPE Res Eval & Eng 8 (2): 95-112.http://dx.doi.org/10.2118/77689-PA.
Chang, J., Yortsos, Y., and Yanis, C. 1990. Pressure Transient Analysis ofFractal Reservoirs. SPE Form Eval 5 (1): 31-38. http://dx.doi.org/10.2118/18170-PA.
Chapman, S.J. 2008. Fortran 95/2003 for Scientists and Engineers,third edition. Boston: McGraw-Hill.
Cinco-Ley, H. and Meng, H.-Z. 1988. Pressure Transient Analysis of Wellswith Finite Conductivity Vertical Fractures in Dual Porosity Reservoirs. PaperSPE 18172 presented at the SPE Annual Technical Conference and Exhibition,Houston, Texas, 2-5 October. http://dx.doi.org/10.2118/18172-MS.
Cinco-Ley, H., Samaniego, F., and Dominguez, A. 1978. Transient PressureBehavior for a Well with a Finite-Conductivity Vertical Fracture SPE J. 18 (4): 253-264. http://dx.doi.org/10.2118/6014-PA.
Clarkson, C.R., Nobakht, M., Kaviani, D. et al. 2012. Production Analysis ofTight-Gas and Shale-Gas Reservoirs Using the Dynamic-Slippage Concept. SPEJ. 17 (1): 230-242. http://dx.doi.org/10.2118/144317-PA.
Cossio, M. 2012. A Semi-Analytic Solution for Flow in Finite-ConductivityVertical Fractures Using Fractal Theory. MS Thesis, Texas A&M University,College Station, Texas.
Flamenco-Lopez, F. and Camacho-Velazquez, R. 2003. Determination of FractalParameters of Fracture Networks Using Pressure-Transient Data. SPE Res Eval& Eng 6 (1): 39-47. http://dx.doi.org/10.2118/82607-PA.
Franquet, M., Wattenbarger, R.A., Ibrahim, M. et al. 2004. Effect ofPressure-Dependent Permeability in Tight Gas Reservoirs, Transient Radial Flow.Paper PETSOC 2004-089 presented at the Canadian International PetroleumConference, Calgary, Alberta, Canada, 8-10 June. http://dx.doi.org/10.2118/2004-089.
Fuentes-Cruz, G., Camacho-Velazquez, R., and Vasquez-Cruz, M. 2010. AUnified Approach for Falloff and Buildup Tests Analysis Following a ShortInjection/Production Time. Paper SPE 133539 presented at the SPE WesternRegional Meeting, Anaheim, California, 27-29 May. http://dx.doi.org/10.2118/133539-MS.
Gringarten, A.C., Ramey, H.J., and Raghavan, R 1974. Unsteady-State PressureDistributions Created by a Well with a Single Infinite-Conductivity VerticalFracture. SPE J. 14 (4): 347-360. http://dx.doi.org/10.2118/4051-PA.
Kong, X.Y., Li, D.L., and Lu, D.T. 2009. Transient Pressure Analysis inPorous and Fractured Fractal Reservoirs. Sci China Ser E Tech Sci 52 (9): 2700-2708. http://dx.doi.org/10.1007/s11431-008-0245-z.
Lee, S.-T. and Brockenbrough, J.R. 1986. A New Approximate Analytic Solutionfor Finite-Conductivity Vertical Fractures. SPE Form Eval 1(1): 75-88. http://dx.doi.org/10.2118/12013-PA.
Lee, W.J. and Wattenbarger, R.A. 1996. Gas Reservoir Engineering,first edition. Richardson, Texas: textbook series, SPE
Mathematica, version 8.0. 2010. Champaign, Illinois: Wolfram Research,Inc.
Moré, J.J., Garbow, B.S., and Hillstrom, K.E. 1984. The MINPACK Project. InSources and Development of Mathematical Software (Prentice-Hall Series inComputational Mathematics), ed. W.J. Cowell. Englewood Cliffs, New Jersey:Prentice Hall.
Ozkan, E. and Raghavan, R. 1991. New Solutions for Well-Test-AnalysisProblems: Part 1-- Analytical Considerations. SPE Form Eval 6 (3): 359-368. http://dx.doi.org/10.2118/18615-PA.
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.
TableCurve 2D, version 5.01. 2012. San Jose, California: Systat Software,Inc.
Warren, J.E. and Root, P.J. 1963. The Behavior of Naturally FracturedReservoirs SPE J. 3 (3): 245-255. http://dx.doi.org/10.2118/426-PA.
Yun, M., Yu, B., and Cai, J. 2009. Analysis of Seepage Characters in FractalPorous Media. International Journal of Heat and Mass Transfer 52 (13-14): 3272-3278. http://dx.doi.org/10.1016/j.ijheatmasstransfer.2009.01.024.