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).

## References

1.
Dahi Taleghani
,
A.
, and
Olson
,
J. E.
,
2013
, “
How Natural Fractures Could Affect Hydraulic-Fracture Geometry
,”
SPE J.
,
19
(
1
), pp.
161
171
.
2.
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
.
3.
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
.
4.
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.
5.
Warren
,
J. E.
, and
Root
,
P. J.
,
1963
, “
The Behavior of Naturally Fractured Reservoirs
,”
SPE J.
,
3
(
3
), pp.
245
255
.
6.
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
.
7.
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.
8.
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
.
9.
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
.
10.
Karimi-Fard
,
M.
, and
,
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
.
11.
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
.
12.
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
.
13.
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
.
14.
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
.
15.
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
.
16.
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
.
17.
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.
18.
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
.
19.
Jia
,
P.
,
Cheng
,
L.
,
Huang
,
S.
, and
Liu
,
H.
,
2005
, “
Transient Behavior of Complex Fracture Networks
,”
J. Pet. Sci. Eng.
,
132
, pp.
1
17
.
20.
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
.
21.
Raghavan
,
R.
,
1993
,
Well Test Analysis
,
Prentice Hall
22.
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
23.
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
24.
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
.
25.
Pecher
,
R.
, and
Stanislav
,
J. F.
,
1997
, “
Boundary Element Techniques in Petroleum Reservoir Simulation
,”
J. Pet. Sci. Eng.
,
17
(
3–4
), pp.
353
366
.
26.
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
27.
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
.
28.
Pecher
,
R.
,
1999
, “Boundary Element Simulation of Petroleum Reservoirs With Hydraulically Fractured Wells,”
Ph.D. dissertation
29.
Stehfest
,
H.
,
1970
, “
Algorithm 368: Numerical Inversion of Laplace Transforms
,”
Commun. ACM
,
13
(
1
), pp.
47
49
.
30.
Ang
,
W.-T.
,
2007
,
A Beginner's Course in Boundary Element Methods
,
Universal-Publishers
, Boca Raton, FL.
31.
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
.
32.
Liggett
,
J.
, and
Liu
,
P.
,
1983
,
The Boundary Integral Equation Method for Porous Media Flow
,
George Allen and Unwin Publishers
, London.
33.
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.
34.
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
.
35.
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
.
36.
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
.