Research Papers: Petroleum Engineering

A General Boundary Integral Solution for Fluid Flow Analysis in Reservoirs With Complex Fracture Geometries

[+] Author and Article Information
Miao Zhang, Luis F. Ayala

Department of Energy and Mineral Engineering,
The Pennsylvania State University,
University Park, PA 16802

Contributed by the Petroleum Division of ASME for publication in the JOURNAL OF ENERGY RESOURCES TECHNOLOGY. Manuscript received July 19, 2017; final manuscript received December 4, 2017; published online January 22, 2018. Assoc. Editor: Reza Sheikhi.

J. Energy Resour. Technol 140(5), 052907 (Jan 22, 2018) (15 pages) Paper No: JERT-17-1373; doi: 10.1115/1.4038845 History: Received July 19, 2017; Revised December 04, 2017

Modeling fractured reservoirs, especially those with complex, nonorthogonal fracture network, can prove to be a challenging task. This work proposes a general integral solution applicable to two-dimensional (2D) fluid flow analysis in fractured reservoirs that reduces the original 2D problem to equivalent integral equation problem written along boundary and fracture domains. The integral formulation is analytically derived from the governing partial differential equations written for the fluid flow problem in reservoirs with complex fracture geometries, and the solution is obtained via solving system of equations that combines contributions from both boundary and fracture domains. Compared to more generally used numerical simulation methods for discrete fracture modeling such as finite volume and finite element methods, this work only requires discretization along the boundary and fractures, resulting in much fewer discretized elements. The validity of proposed solution is verified using several case studies through comparison with available analytical solutions (for simplified, single-fracture cases) and finite difference/finite volume finely gridded numerical simulators (for multiple, complex, and nonorthogonal fracture network cases).

Copyright © 2018 by ASME
Your Session has timed out. Please sign back in to continue.


Dahi Taleghani, A. , and Olson, J. E. , 2013, “ How Natural Fractures Could Affect Hydraulic-Fracture Geometry,” SPE J., 19(1), pp. 161–171. [CrossRef]
Zhou, D. , Zheng, P. , Peng, J. , and He, P. , 2015, “ Induced Stress and Interaction of Fractures During Hydraulic Fracturing in Shale Formation,” ASME J. Energy Resour. Technol., 137(6), p. 062902. [CrossRef]
Rahman, M. , and Rahman, M. , 2011, “ Optimizing Hydraulic Fracture to Manage Sand Production by Predicting Critical Drawdown Pressure in Gas Well,” ASME J. Energy Resour. Technol., 134(1), p. 013101. [CrossRef]
Shakiba, M. , and Sepehrnoori, K. , 2015, “ Using Embedded Discrete Fracture Model (EDFM) and Microseismic Monitoring Data to Characterize the Complex Hydraulic Fracture Networks,” SPE Annual Technical Conference and Exhibition, Houston, TX, Sept. 28–30, SPE Paper No. SPE-175142-MS.
Warren, J. E. , and Root, P. J. , 1963, “ The Behavior of Naturally Fractured Reservoirs,” SPE J., 3(3), pp. 245–255. [CrossRef]
Kazemi, H. , Merrill, L. S. , Porterfield, K. L. , and Zeman, P. R. , 1976, “ Numerical Simulation of Water-Oil Flow in Naturally Fractured Reservoirs,” SPE J., 16(6), pp. 317–326. [CrossRef]
Hill, A. C. , and Thomas, G. W. , 1985, “ A New Approach for Simulating Complex Fractured Reservoirs,” Middle East Oil Technical Conference and Exhibition, Bahrain, Mar. 11–14, SPE Paper No. SPE-13537-MS.
Evans, R. D. , and Lekia, S. L. , 1990, “ A Reservoir Simulation Study of Naturally Fractured Lenticular Tight Gas Sand Reservoirs,” ASME J. Energy Resour. Technol., 112(4), pp. 231–238. [CrossRef]
Kim, J. ‐G. , and Deo, M. D. , 2000, “ Finite Element, Discrete‐Fracture Model for Multiphase Flow in Porous Media,” AIChE J., 46(6), pp. 1120–1130. [CrossRef]
Karimi-Fard, M. , and Firoozabadi, A. , 2003, “ Numerical Simulation of Water Injection in Fractured Media Using the Discrete-Fracture Model and the Galerkin Method,” SPE Reservoir Eval. Eng., 6(2), pp. 117–126. [CrossRef]
Karimi-Fard, M. , Durlofsky, L. J. , and Aziz, K. , 2004, “ An Efficient Discrete Fracture Model Applicable for General Purpose Reservoir Simulators,” SPE J., 9(2), pp. 227–236. [CrossRef]
Geiger, S. , Matthai, S. , Niessner, J. , and Helmig, R. , 2009, “ Black-Oil Simulations for Three-Component, Three Phase Flow in Fractured Porous Media,” SPE J., 18(4), pp. 670–684. [CrossRef]
Jiang, J. , and Younis, R. M. , 2016, “ Hybrid Coupled Discrete-Fracture/Matrix and Multi-Continuum Models for Unconventional Reservoir Simulation,” SPE J., 21(3), pp. 1009–1027. [CrossRef]
Lee, S. H. , Lough, M. F. , and Jensen, C. L. , 2001, “ Hierarchical Modeling of Flow in Naturally Fracture Formation With Multiple Length Scales,” Water Resour. Res., 37(3), pp. 443–455. [CrossRef]
Li, L. , and Lee, S. H. , 2008, “ Efficient Field-Scale Simulation of Black Oil in a Naturally Fractured Reservoir Through Discrete Fracture Networks and Homogenized Media,” SPE Reservoir Eval. Eng., 11(4), pp. 750–758. [CrossRef]
Yeh, N. S. , Davison, M. J. , and Raghavan, R. , 1986, “ Fractured Well Responses in Heterogeneous Systems—Application to Devonian Shale and Austin Chalk Reservoirs,” ASME J. Energy Resour. Technol., 108(2), pp. 120–130. [CrossRef]
Spivey, J. , and Semmelbeck, M. , 1995, “ Forecasting Long-Term Gas Production of Dewatered Coal Seams and Fractured Gas Shales,” Low Permeability Reservoirs Symposium, Denver, CO, Mar. 19–22, SPE Paper No. SPE-29580-MS.
Zhou, W. , Banerjee, R. , Poe, B. D. , Spath, J. , and Thambynayagam, M. , 2013, “ Semianalytical Production Simulation of Complex Hydraulic-Fracture Networks,” SPE J., 19(1), pp. 6–18. [CrossRef]
Jia, P. , Cheng, L. , Huang, S. , and Liu, H. , 2005, “ Transient Behavior of Complex Fracture Networks,” J. Pet. Sci. Eng., 132, pp. 1–17. [CrossRef]
Yu, W. , Wu, K. , and Sepehrnoori, K. , 2016, “ A Semianalytical Model for Production Simulation From Nonplanar Hydraulic-Fracture Geometry in Tight Oil Reservoirs,” SPE J., 21(3), pp. 1028–1040. [CrossRef]
Raghavan, R. , 1993, Well Test Analysis, Prentice Hall, Upper Saddle River, NJ.
Liggett, J. A. , 1977, “ Location of Free Surface in Porous Media,” J. Hydraul. Div., 103(4), pp. 353–365. http://cedb.asce.org/CEDBsearch/record.jsp?dockey=0007291
Kikani, K. , 1989, “Application of Boundary Element Method to Streamline Generation and Pressure Transient Testing,” Ph.D. dissertation, Stanford University, Stanford, CA. https://pangea.stanford.edu/ERE/research/geoth/publications/techreports/SGP-TR-126.pdf
Sato, K. , and Horne, R. N. , 1993, “ Perturbation Boundary Element Method for Heterogeneous Reservoirs—Part 2: Transient-Flow Problems,” SPE Form. Eval., 8(4), pp. 315–322. [CrossRef]
Pecher, R. , and Stanislav, J. F. , 1997, “ Boundary Element Techniques in Petroleum Reservoir Simulation,” J. Pet. Sci. Eng., 17(3–4), pp. 353–366. [CrossRef]
Kryuchkov, S. , and Sanger, S. , 2004, “ Asymptotic Description of Vertically Fractured Wells Within the Boundary Element Method,” J. Can. Pet. Technol., 43(3), pp. 31–36. https://www.onepetro.org/journal-paper/PETSOC-04-03-02
Fang, S. , Cheng, L. , and Ayala, L. F. , 2017, “ A Coupled Boundary Element and Finite Element Method for the Analysis of Flow Through Fractured Porous Media,” J. Pet. Sci. Eng., 152, pp. 375–390. [CrossRef]
Pecher, R. , 1999, “Boundary Element Simulation of Petroleum Reservoirs With Hydraulically Fractured Wells,” Ph.D. dissertation, University of Calgary, Calgary, AB, Canada. http://adsabs.harvard.edu/abs/1999PhDT........87P
Stehfest, H. , 1970, “ Algorithm 368: Numerical Inversion of Laplace Transforms,” Commun. ACM, 13(1), pp. 47–49. [CrossRef]
Ang, W.-T. , 2007, A Beginner's Course in Boundary Element Methods, Universal-Publishers, Boca Raton, FL.
Granet, S. , Fabrie, P. , Lemonnier, P. , and Quintard, M. , 1998, “ A Single-Phase Flow Simulation of Fractured Reservoir Using a Discrete Representation of Fractures,” European Conference on the Mathematics of Oil Recovery (ECMOR), Peebles, Scotland, Sept. 8–11, Paper No. C-12.
Liggett, J. , and Liu, P. , 1983, The Boundary Integral Equation Method for Porous Media Flow, George Allen and Unwin Publishers, London.
Blasingame, T. A. , and Poe, B. D., Jr ., 1993, “ Semianalytic Solutions for a Well With a Single Finite-Conductivity Vertical Fracture,” SPE Annual Technical Conference and Exhibition, Houston, TX, Oct. 3–6, SPE Paper No. SPE-26424-MS.
Nashawi, I. S. , and Malallah, A. H. , 2007, “ Well Test Analysis of Finite-Conductivity Fractured Wells Producing Under Constant Bottomhole Pressure,” J. Pet. Sci. Eng., 57(3–4), pp. 303–320. [CrossRef]
Gringarten, A. C. , Ramey, H. J. , and Raghavan, R. , 1974, “ Unsteady-State Pressure Distributions Created by a Well With a Single Infinite-Conductivity Vertical Fracture,” SPE J., 14(4), pp. 347–360. [CrossRef]
Xu, Y. , Cavalacante Filho, J. S. A. , Yu, W. , and Sepehrnoori, K. , 2017, “ Discrete-Fracture Modeling of Complex Hydraulic Fracture Geometries in Reservoir Simulators,” SPE Reservoir Eval. Eng., 20(2), pp. 403–422. [CrossRef]


Grahic Jump Location
Fig. 1

(a) Actual reservoir configuration and (b) BEM discretized system

Grahic Jump Location
Fig. 2

The combined BEM-FVM matrix

Grahic Jump Location
Fig. 3

Well pressure responses in infinite-large reservoirs under different fracture conductivity scenarios

Grahic Jump Location
Fig. 4

Well rate responses in various size reservoirs under finite fracture conductivity (CFD = 5)

Grahic Jump Location
Fig. 5

Well pressure responses in various size reservoirs under infinite fracture conductivity

Grahic Jump Location
Fig. 6

Reservoir and well configuration

Grahic Jump Location
Fig. 7

Production rate and cutline pressure profile comparisons

Grahic Jump Location
Fig. 8

Pressure distribution comparison for t = 1 day

Grahic Jump Location
Fig. 9

(a) Finite element discretization in comsol (2243 elements), and (b): Discretized BEM-FVM model in this work (117 elements)

Grahic Jump Location
Fig. 10

Cutline location (a) and pressure comparison (b) along the cutline

Grahic Jump Location
Fig. 11

Production rate comparison between comsol and this work

Grahic Jump Location
Fig. 12

Pressure distribution comparison under various production times

Grahic Jump Location
Fig. 13

Boundary element discretization with constant elements

Grahic Jump Location
Fig. 14

Local coordinate system for integration

Grahic Jump Location
Fig. 15

Special cells in finite volume fracture model



Some tools below are only available to our subscribers or users with an online account.

Related Content

Customize your page view by dragging and repositioning the boxes below.

Related Journal Articles
Related eBook Content
Topic Collections

Sorry! You do not have access to this content. For assistance or to subscribe, please contact us:

  • TELEPHONE: 1-800-843-2763 (Toll-free in the USA)
  • EMAIL: asmedigitalcollection@asme.org
Sign In