Fluid flow inside heterogeneous structure of dual porosity reservoirs is presented by two coupled partial differential equations (PDE). Finding an analytical solution for the diffusivity equations is tedious or even impossible in some circumstances due to the heterogeneity of dual porosity reservoirs. Therefore, in this study, orthogonal collocation method (OCM) is proposed for solving the governing equations in dual porosity reservoirs with constant pressure outer boundary. Since no analytical solution has been proposed for this system, validation is carried out by comparing the OCM-obtained results for “dual porosity reservoirs with circular no-flow outer boundary” with both exact analytical solution and real field data. Sensitivity analyses reveal that the OCM with 13 collocation points is a good candidate for prediction of pressure transient response (PTR) in dual porosity reservoirs. OCM predicts the PTR of a real field draw-down test with an absolute average relative deviation (AARD) of 0.9%. Moreover, OCM shows a good agreement with the analytical solution obtained by Laplace transform (AARD = 0.16%). It is worth noting that OCM requires a smaller computational effort. Thereafter, PTR of dual porosity reservoirs with a constant production rate in the wellbore and constant pressure outer boundary is simulated by OCM for wide ranges of operating conditions. Accuracy of OCM and its low required computational time justifies that this approximate method can be considered as a practical candidate for pressure transient analysis in dual porosity reservoirs.

References

References
1.
Vaferi
,
B.
,
Salimi
,
V.
,
Dehghan Baniani
,
D.
,
Jahanmiri
,
A.
, and
Khedri
,
S.
,
2012
, “
Prediction of Transient Pressure Response in the Petroleum Reservoirs Using Orthogonal Collocation
,”
J. Pet. Sci. Eng.
,
98–99
, pp.
156
163
.
2.
Horne
,
R. N.
,
1990
,
Modern Well Test Analysis
, Petroway, Inc., Palo Alto, CA, p.
183
.
3.
Bourdet
,
D.
, and
Gringarten
,
A. C.
,
1980
, “
Determination of Fissure Volume and Block Size in Fractured Reservoirs by Type-Curve Analysis
,”
SPE Annual Technical Conference and Exhibition
, Dallas, TX, Sept. 21–24,
SPE
Paper No. SPE-9293-MS.
4.
Vaferi
,
B.
, and
Eslamloueyan
,
R.
,
2015
, “
Hydrocarbon Reservoirs Characterization by co-Interpretation of Pressure and Flow Rate Data of the Multi-Rate Well Testing
,”
J. Pet. Sci. Eng.
,
135
, pp.
59
72
.
5.
Qin
,
J.
,
Cheng
,
S.
,
He
,
Y.
,
Wang
,
Y.
,
Feng
,
D.
,
Yang
,
Z.
,
Li
,
D.
, and
Yu
,
H.
,
2018
, “
Decline Curve Analysis of Fractured Horizontal Wells Through a Segmented Fracture Model
,”
ASME J. Energy Resour. Technol.
,
141
(1), p. 012903.
6.
Gringarten
,
A. C.
, and
Ramey
,
H. J.
, Jr.
,
1973
, “
The Use of Source and Green's Functions in Solving Unsteady-Flow Problems in Reservoirs
,”
SPE J.
,
13
(
5
), pp.
285
296
.
7.
Mohamed
,
E.-M. M. H.
,
Ahmad
,
M. A. H. M.
,
Asia Osman Ali
,
T.
, and
Al-Hassan
,
M. M. A.-A.
,
2017
,
Numerical Solution by Finite Difference Approach for Homogenous Finite and Infinite Redial Reservoir by Computer Programing
,
Sudan University of Science and Technology
, Khartoum, Sudan.
8.
Khoei
,
A. R.
,
Hosseini
,
N.
, and
Mohammadnejad
,
T.
,
2016
, “
Numerical Modeling of Two-Phase Fluid Flow in Deformable Fractured Porous Media Using the Extended Finite Element Method and an Equivalent Continuum Model
,”
Adv. Water Resour.
,
94
, pp.
510
528
.
9.
Amaziane
,
B.
,
Bourgeois
,
M.
, and
El Fatini
,
M.
,
2014
, “
Adaptive Mesh Refinement for a Finite Volume Method for Flow and Transport of Radionuclides in Heterogeneous Porous Media
,”
Oil Gas Sci. Technol.-Revue d'IFP Energies Nouvelles
,
69
(
4
), pp.
687
699
.
10.
Siavashi
,
M.
,
Blunt
,
M. J.
,
Raisee
,
M.
, and
Pourafshary
,
P.
,
2014
, “
Three-Dimensional Streamline-Based Simulation of Non-Isothermal Two-Phase Flow in Heterogeneous Porous Media
,”
Comput. Fluids
,
103
, pp.
116
131
.
11.
Ahmadpour
,
M.
,
Siavashi
,
M.
, and
Moghimi
,
M.
,
2018
, “
Numerical Simulation of Two-Phase Mass Transport in Three-Dimensional Naturally Fractured Reservoirs Using Discrete Streamlines
,”
Numer. Heat Transfer, Part A: Appl.
,
73
(
7
), pp.
482
500
.
12.
Ahmadpour
,
M.
,
Siavashi
,
M.
, and
Doranehgard
,
M. H.
,
2016
, “
Numerical Simulation of Two-Phase Flow in Fractured Porous Media Using Streamline Simulation and IMPES Methods and Comparing Results With a Commercial Software
,”
J. Central South Univ.
,
23
(
10
), pp.
2630
2637
.
13.
Doranehgard
,
M. H.
, and
Siavashi
,
M.
,
2018
, “
The Effect of Temperature Dependent Relative Permeability on Heavy Oil Recovery During Hot Water Injection Process Using Streamline-Based Simulation
,”
Appl. Therm. Eng.
,
129
, pp.
106
116
.
14.
Jinasena
,
A.
,
Kaasa
,
G. O.
, and
Sharma
,
R.
,
2017
, “
Use of Orthogonal Collocation Method for a Dynamic Model of the Flow in a Prismatic Open Channel: For Estimation Purposes
,”
58th Conference on Simulation and Modelling
(
SIMS 58
), Reykjavik, Iceland, Sept. 25–27.
15.
Shelly Arora
,
S.
, and
Kaur
,
I.
,
2015
, “
Numerical Solution of Heat Conduction Problems Using Orthogonal Collocation on Finite Elements
,”
J. Nigerian Math. Soc.
,
34
(
3
), pp.
286
302
.
16.
Finlayson
,
B. A.
,
1974
, “
Orthogonal Collocation in Chemical Reaction Engineering
,”
Catal. Rev.: Sci. Eng.
,
10
(
1
), pp.
69
138
.
17.
Villadsen
,
J. V.
, and
Stewart
,
W. E.
,
1995
, “
Solution of Boundary-Value Problems by Orthogonal Collocation
,”
Chem. Eng. Sci.
,
50
(
24
), p.
3979
.
18.
Tien
,
C.
,
2013
, “
Adsorption Calculations and Modelling
,” Butterworth-Heinemann, Oxford, UK.
19.
Vaferi
,
B.
, and
Eslamloueyan
,
R.
,
2015
, “
Simulation of Dynamic Pressure Response of Finite Gas Reservoirs Experiencing Time Varying Flux in the External Boundary
,”
J. Natural Gas Sci. Eng.
,
26
, pp.
240
252
.
20.
Zhang
,
M.
, and
Ayala
,
L. F.
,
2018
, “
A General Boundary Integral Solution for Fluid Flow Analysis in Reservoirs With Complex Fracture Geometries
,”
ASME J. Energy Resour. Technol.
,
140
(
5
), p.
052907
.
21.
Da Prat
,
G.
,
1990
, “
Well Test Analysis for Naturally Fractured Reservoirs
,” Elsevier, Amsterdam, The Netherlands.
22.
Hassanzadeh
,
H.
, and
Pooladi-Darvish
,
M.
,
2006
, “
Effects of Fracture Boundary Conditions on Matrix-Fracture Transfer Shape Factor
,”
Transp. Porous Media
,
64
(
1
), pp.
51
71
.
23.
Ordonez
,
A.
,
Penuela
,
G.
,
Idrobo
,
E. A.
, and
Medina
,
C. E.
,
2001
, “
Recent Advances in Naturally Fractured Reservoir Modeling
,” CT&F—Ciencia, Tecnología y Futuro, Bucaramanga, CO.
24.
Sarma
,
P.
, and
Aziz
,
K.
,
2003
, “
New Transfer Functions for Simulation of Naturally Fractured Reservoirs With Dual Porosity Models
,”
SPE J.
,
11
(
3
), p.
13
.http://lya.fciencias.unam.mx/pablo/misdocs/reservoir/transfer.pdf
25.
Stewart
,
G.
,
2011
,
Well Test, Design and Analysis
,
PennWell Corporation
,
Tulsa, OK
.
26.
Zimmerman
,
R. W.
,
Chen
,
G. S.
, and
Bodvarsson
,
G.
,
1992
, “
A Dual-Porosity Reservoir Model With an Improved Coupling Term
,” 17th Workshop on Geothermal Reservoir Engineering, Stanford Geothermal Workshop, Stanford, CA, Jan. 29–31.
27.
Ahn
,
C.
,
Dilmore
,
R.
, and
Wang
,
J.
,
2016
, “
Modeling of Hydraulic Fracture Propagation in Shale Gas Reservoirs: A Three-Dimensional, Two-Phase Model
,”
ASME J. Energy Resour. Technol.
,
139
(
1
), p.
012903
.
28.
Bai
,
M.
,
Ma
,
Q.
, and
Roegiers
,
J.-C.
,
1994
, “
Dual-Porosity Behaviour of Naturally Fractured Reservoirs
,”
Int. J. Numer. Anal. Methods Geomech.
,
18
(
6
), pp.
359
376
.
29.
Barenblatt
,
G. I.
,
Zheltov
,
I.
, and
Kochina
,
I. N.
,
1960
, “
Basic Concepts in the Theory of Seepage of Homogeneous Liquids in Fissured Rocks [Strata]
,”
J. Appl. Math. Mech.
,
24
(
5
), pp.
1286
1303
.
30.
Kazemi
,
H.
,
Seth
,
M. S.
, and
Thomas
,
G. W.
,
1969
, “
The Interpretation of Interference Tests in Naturally Fractured Reservoirs With Uniform Fracture Distribution
,”
SPE J.
,
9
(
4
), p.
10
.https://www.onepetro.org/journal-paper/SPE-2156-B
31.
Mavor
,
M. J.
, and
Cinco-Ley
,
H.
,
1979
, “
Transient Pressure Behavior of Naturally Fractured Reservoirs
,” SPE California Regional Meeting, Ventura, CA, Apr. 18–20,
SPE
Paper No. SPE-7977-MS.
32.
Warren
,
J. E.
, and
Root
,
P. J.
,
1963
, “
The Behavior of Naturally Fractured Reservoirs
,”
SPE J.
,
3
(
3
), pp.
245
255
.
33.
Obinna
,
E. D.
, and
Dehghanpour
,
H.
,
2016
, “
Characterizing Tight Oil Reservoirs With Dual- and Triple-Porosity Models
,”
ASME J. Energy Resour. Technol.
,
138
(
3
), p.
032801
.
34.
Chen
,
C.-C.
,
Serra
,
K.
,
Reynolds
,
A. C.
, and
Raghavan
,
R.
,
1985
, “
Pressure Transient Analysis Methods for Bounded Naturally Fractured Reservoirs
,”
SPE J.
,
25
(
3
), pp.
451
464
.
35.
de Swaan O
,
A.
,
1976
, “
Analytic Solutions for Determining Naturally Fractured Reservoir Properties by Well Testing
,”
SPE J.
,
16
(
3
), pp.
117
122
.
36.
Najurieta
,
H. L.
,
1980
, “
A Theory for Pressure Transient Analysis in Naturally Fractured Reservoirs
,”
J. Pet. Technol.
,
32
(
7
), pp.
1241
1250
.
37.
Serra
,
K.
,
Reynolds
,
A. C.
, and
Raghavan
,
R.
,
1983
, “
New Pressure Transient Analysis Methods for Naturally Fractured Reservoirs
,”
J. Pet. Technol.
,
35
(
12
), pp.
2271
2288
.
38.
Streltsova
,
T. D.
,
1983
, “
Well Pressure Behavior of a Naturally Fractured Reservoir
,”
SPE J.
,
23
(
5
), pp.
769
780
.
39.
Bai
,
M.
,
Ma
,
Q.
, and
Roegiers
,
J.-C.
,
1994
, “
A Nonlinear Dual-Porosity Model
,”
Appl. Math. Modell.
,
18
(
11
), pp.
602
610
.
40.
Bai
,
M.
,
Roegiers
,
J.-C.
, and
Elsworth
,
D.
,
1995
, “
Poromechanical Response of Fractured-Porous Rock Masses
,”
J. Pet. Sci. Eng.
,
13
(
3–4
), pp.
155
168
.
41.
Chen
,
Z.-X.
, and
You
,
J.
,
1987
, “
The Behavior of Naturally Fractured Reservoirs Including Fluid Flow in Matrix Blocks
,”
Transp. Porous Media
,
2
(
2
), pp.
145
163
.https://link.springer.com/article/10.1007/BF00142656
42.
Ge
,
J.-L.
, and
Wu
,
Y.-S.
,
1982
, “
The Behavior of Naturally Fractured Reservoirs and the Technique for Well Test Analysis at Constant Pressure Condition
,”
Pet. Explor. Develop.
,
9
, pp.
53
65
. (in Chinese)
43.
Chen
,
Z.-X.
,
1990
, “
Analytical Solutions for Double-Porosity, Double-Permeability and Layered Systems
,”
J. Pet. Sci. Eng.
,
5
(
1
), pp.
1
24
.
44.
Mesbah
,
M.
,
Vatani
,
A.
, and
Siavashi
,
M.
,
2018
, “
Streamline Simulation of Water-Oil Displacement in a Heterogeneous Fractured Reservoir Using Different Transfer Functions
,”
Oil Gas Sci. Technol.-Rev. IFP Energies Nouvelles
,
73
(2018), p. 14.
45.
Herrera-Hernández
,
E. C.
,
Aguilar-Madera
,
C. G.
,
Hernández
,
D.
,
Luisa, D. P.
,
Camacho-Velázquezd, R. G.
,
2018
,
Comput. Appl. Math.
,
37
(
4
), pp.
4342
4356
.
46.
Rice
,
R. G.
, and
Do
,
D. D.
,
2012
,
Applied Mathematics and Modeling for Chemical Engineers
,
2nd ed.
,
Wiley
,
Hoboken, NJ
.
47.
Da Prat
,
G.
,
1990
,
Well Test Analysis for Fractured Reservoir Evaluation
,
1st ed.
,
Elsevier Science
, Amsterdam, The Netherlands.
48.
Ahmed
,
T.
,
2010
,
Reservoir Engineering Handbook
,
4th ed.
,
Gulf Professional Publishing
,
Burlington, NJ
.
49.
Yarveicy
,
H.
, and
Ghiasi
,
M. M.
,
2017
, “
Modeling of Gas Hydrate Phase Equilibria: Extremely Randomized Trees and LSSVM Approaches
,”
J. Mol. Liq.
,
243
, pp.
533
541
.
50.
Yarveicy
,
H.
,
Ghiasi
,
M. M.
, and
Mohammadi
,
A. H.
, 2018, “
Determination of the Gas Hydrate Limits to Isenthalpic Joule–Thomson Expansions
,”
Chem. Eng. Res. Des., Artic. Press
,
132
, pp.
208
214
.
51.
Eslamloueyan
,
R.
,
Vaferi
,
B.
, and
Ayatollahi
,
S.
,
2010
, “
Fracture Characterizations From Well Testing Data Using Artificial Neural Networks
,”
72nd EAGE Conference and Exhibition Incorporating SPE EUROPEC
, Barcelona, Spain, June 14–17.
52.
Yarveicy
,
H.
,
Moghaddam
,
A. K.
, and
Ghiasi
,
M. M.
,
2014
, “
Practical Use of Statistical Learning Theory for Modeling Freezing Point Depression of Electrolyte Solutions: LSSVM Model
,”
J. Natural Gas Sci. Eng.
,
20
, pp.
414
421
.
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