This paper addresses the model, solution, and analysis of fluid flow behavior in fractal reservoirs considering wellbore storage and skin effects (WS–SE). In the light of the fractional calculus (FC), the general form of fluid flow model considering the history of flow in all stages of production is presented. On the basis of Bessel functions theory, analytical solutions in the Laplace transform domain under three outer-boundary conditions, assuming the well is producing at a constant rate, are obtained. Based on the analytical solutions, various examples, discussing the pressure-transient behavior of a well in a fractal reservoir, are presented.
Issue Section:
Research Papers
References
1.
Beier
, R. A.
, 1994
, “Pressure-Transient Model for a Vertically Fractured Well in a Fractal Reservoir
,” SPE Form. Eval.
, 9
(2
), pp. 122
–128
.10.2118/20582-PA2.
Camacho-Velázquez
, R.
, Fuentes-Cruz
, G.
, and Vasquez-Cruz
, M.
, 2008
, “Decline-Curve Analysis of Fractured Reservoirs With Fractal Geometry
,” SPE Reservoir Eval. Eng.
, 11
(3
), pp. 606
–619
.10.2118/104009-PA3.
Park
, H. W.
, Choe
, J.
, and Kang
, J. M.
, 2000
, “Pressure Behavior of Transport in Fractal Porous Media Using a Fractional Calculus Approach
,” Energy Sources
, 22
(10
), pp. 881
–890
.4.
Flamenco-Lopez
, F.
, and Camacho-Velazquez
, R.
, 2003
, “Determination of Fractal Parameters of Fracture Networks Using Pressure-Transient Data
,” SPE Reservoir Eval. Eng.
, 6
(1
), pp. 39
–47
.10.2118/82607-PA5.
Cossio
, M.
, Moridis
, G. J.
, and Blasingame
, T. A.
, 2013
, “A Semianalytic Solution for Flow in Finite-Conductivity Vertical Fractures by Use of Fractal Theory
,” SPE J.
, 18
(1
), pp. 83
–96
.10.2118/153715-PA6.
Mishra
, S.
, Brigham
, W. E.
, and Orr
, F. M.
, Jr., 1991
, “Tracer and Pressure-Test Analysis for Characterization of Areally Heterogeneous Reservoirs
,” SPE Form. Eval.
, 6
(1
), pp. 45
–54
.10.2118/17365-PA7.
Hewett
, T. A.
, 1986
, “Fractal Distributions of Reservoir Heterogeneity and Their Influence on Fluid Transport
,” SPE
Annual Technical Conference and Exhibition
, New Orlean, LA
, Oct. 5–8, Paper No. SPE-15386-MS.10.2118/15386-MS8.
Barker
, J. A.
, 1988
, “A Generalized Radial Flow Model for Hydraulic Tests in Fractured Rock
,” Water Resour. Res.
, 24
(10
), pp. 1796
–1804
.10.1029/WR024i010p017969.
Kim
, T. H.
, and Schechter
, D. S.
, 2009
, “Estimation of Fracture Porosity of Naturally Fractured Reservoirs With No Matrix Porosity Using Fractal Discrete Fracture Networks
,” SPE Reservoir Eval. Eng.
, 12
(2
), pp. 232
–242
.10.2118/110720-PA10.
Raghavan
, R.
, and Chen
, C.
, 2013
, “Fractured-Well Performance Under Anomalous Diffusion
,” SPE Reservoir Eval. Eng.
, 16
(3
), pp. 237
–245
.10.2118/165584-PA11.
Chang
, J.
, and Yortsos
, Y. C.
, 1990
, “Pressure Transient Analysis of Fractal Reservoirs
,” SPE Form. Eval.
, 5
(1
), pp. 31
–38
.10.2118/18170-PA12.
Park
, H. W.
, Jang
, I. S.
, and Kang
, J. M.
, 1998
, “An Analytic Approach for Pressure Transients of Fractally Fractured Reservoirs With Variable Apertures
,” In Situ
, 22
(3
), pp. 321
–337
.13.
Metzler
, R.
, Glöckle
, W. G.
, and Nonnenmacher
, T. F.
, 1994
, “Fractional Model Equation for Anomalous Diffusion
,” Phys. A
, 211
(1
), pp. 13
–24
.10.1016/0378-4371(94)90064-714.
Mainardi
, F.
, and Pagnini
, G.
, 2003
, “The Wright Functions as Solutions of the Time-Fractional Diffusion Equation
,” Appl. Math. Comput.
, 141
(1
), pp. 51
–62
.10.1016/S0096-3003(02)00320-X15.
Bhalekar
, S.
, and Daftardar-Gejji
, V.
, 2013
, “Corrigendum to ‘Solving Multi-Term Linear and Non-Linear Diffusion-Wave Equations of Fractional Order by Adomian Decomposition Method’ [Applied Mathematics and Computation 202 (2008) 113–120]
,” Appl. Math. Comput.
, 219
(16
), pp. 8413
–8415
.10.1016/j.amc.2013.02.07216.
Kumar
, S.
, Kumar
, D.
, Abbasbandy
, S.
, and Rashidi
, M. M.
, 2014
, “Analytical Solution of Fractional Navier–Stokes Equation by Using Modified Laplace Decomposition Method
,” Ain Shams Eng. J.
, 5
(2
), pp. 569
–574
.10.1016/j.asej.2013.11.00417.
Ye
, H.
, Liu
, F.
, Anh
, V.
, and Turner
, I.
, 2014
, “Maximum Principle and Numerical Method for the Multi-Term Time–Space Riesz–Caputo Fractional Differential Equations
,” Appl. Math. Comput.
, 227
, pp. 531
–540
.10.1016/j.amc.2013.11.01518.
Razminia
, K.
, Razminia
, A.
, and Machado
, J. A. T.
, 2014
, “Analysis of Diffusion Process in Fractured Reservoirs Using Fractional Derivative Approach
,” Commun. Nonlinear Sci. Numer. Simul.
, 19
(9
), pp. 3161
–3170
.10.1016/j.cnsns.2014.01.02519.
Razminia
, K.
, Razminia
, A.
, and Trujillo
, J. J.
, 2015
, “Analysis of Radial Composite Systems Based on Fractal Theory and Fractional Calculus
,” Signal Process.
, 107
, pp. 378
–388
.10.1016/j.sigpro.2014.05.00820.
Razminia
, K.
, Razminia
, A.
, and Torres
, D. F. M.
, 2015
, “Pressure Responses of a Vertically Hydraulic Fractured Well in a Reservoir With Fractal Structure
,” Appl. Math. Comput.
, 257
, pp. 374
–380
.10.1016/j.amc.2014.12.12421.
Cafagna
, D.
, 2007
, “Fractional Calculus: A Mathematical Tool From the Past for Present Engineers [Past and Present]
,” IEEE Ind. Electron. Mag.
, 1
(2
), pp. 35
–40
.10.1109/MIE.2007.90147922.
Zhao
, Y.
, and Zhang
, L.
, 2011
, “Solution and Type Curve Analysis of Fluid Flow Model for Fractal Reservoir
,” World J. Mech.
, 1
(5
), pp. 209
–216
.10.4236/wjm.2011.1502723.
Kong
, X. Y.
, 1999
, Advanced Mechanics of Fluids in Porous Media
, University of Science and Technology of China Press
, Hefei, China
(in Chinese).24.
Li
, S. C.
, 2002
, “A Solution of Fractal Dual Porosity Reservoir Model in Well Testing Analysis
,” Prog. Explor. Geophys.
, 25
(5
), pp. 60
–62
(in Chinese).25.
Razminia
, K.
, Razminia
, A.
, and Baleanu
, D.
, 2015
, “Investigation of Fractional Diffusion Equation Based on Generalized Integral Quadrature Technique
,” Appl. Math. Modell.
, 39
(1
), pp. 86
–98
.10.1016/j.apm.2014.04.05626.
O'Shaughnessy
, B.
, and Procaccia
, I.
, 1985
, “Diffusion on Fractals
,” Phys. Rev. A.
, 32
(5
), pp. 3073
–3083
.10.1103/PhysRevA.32.307327.
Raghavan
, R.
, 2011
, “Fractional Derivatives: Application to Transient Flow
,” J. Pet. Sci. Eng.
, 80
(1
), pp. 7
–13
.10.1016/j.petrol.2011.10.00328.
van Everdingen
, A. F.
, and Hurst
, W.
, 1949
, “The Application of the Laplace Transformation to Flow Problems in Reservoirs
,” J. Pet. Technol.
, 1
(12
), pp. 305
–324
.10.2118/949305-G29.
Agarwal
, R. G.
, Al-Hussainy
, R.
, and Ramey
, H. J.
, Jr., 1970
, “An Investigation of Wellbore Storage and Skin Effect in Unsteady Liquid Flow: I. Analytical Treatment
,” SPE J.
, 10
(3
), pp. 279
–290
.10.2118/2466-PA30.
Stehfest
, H.
, 1970
, “Algorithm 368: Numerical Inversion of Laplace Transforms [D5]
,” Commun. ACM
, 13
(1
), pp. 47
–49
.10.1145/361953.36196931.
Bourdet
, D. P.
, Whittle
, T. M.
, Douglas
, A. A.
, and Pirard
, Y. M.
, 1983
, A New Set of Type Curves Simplifies Well Test Analysis
, World Oil, Gulf Publishing Company
, Houston, TX
, pp. 95
–106
.Copyright © 2016 by ASME
You do not currently have access to this content.
