Abstract

Layered zones with vertical fracture are often found in analyzing well-test data of deep/ultra-deep reservoirs and tight reservoirs. Analyzing and modeling the well-test data in a computation-accurate and easy-program manner have been a challenge for these problems due to the lack of suitable solutions. This work thus presents the generalized analytical well-test solutions for vertically fractured wells in infinite and bounded commingled reservoirs with computation accuracy and functional simplicity. These solutions are derived based on the early-time approximate solution of the infinite/finite-conductivity fracture model, Laplace and Fourier cosine transformation, pressure superposition principle, and Duhamel principle. Subsequently, model validation is carried out by comparing the pressure and derivative results with those of commercial saphir software. The results show that the average absolute percent deviation between the presented analytical solutions and saphir for three kinds of outer boundaries is ∼2% for pressure results and ∼4% for pressure derivative results. Finally, a field case in Xinjiang oilfield is interpreted, indicating that the proposed analytical well-test solutions are feasible to interpret the parameters of commingled reservoirs.

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References

1.
Jackson
,
R. R.
, and
Banerjee
,
R.
,
2000
, “
Advances in Multilayer Reservoir Testing and Analysis Using Numerical Well Testing and Reservoir Simulation
,”
SPE Annual Technical Conference and Exhibition, Dallas
,
TX
,
Oct. 1–4
, SPE-62917-MS.
2.
Eissa
,
M.
,
Joshi
,
S.
,
Singh
,
K.
,
Bahuguna
,
A.
, and
Elbadri
,
M.
,
2008
, “
Identifying Layer Permeabilities and Skin Using a Multi-layer Transient Testing Approach in a Complex Reservoir Environment
,”
All Days, SPE
,
Abu Dhabi, UAE
, p.
SPE-116969-MS
.
3.
Shi
,
W.
,
Liu
,
X.
,
Gao
,
M.
,
Tao
,
L.
,
Bai
,
J.
, and
Zhu
,
Q.
,
2023
, “
Pressure Drop Response Characteristics for Multi-injection Well Interfered Vertical Well in Heterogeneous Fractured Anticline Reservoirs
,”
ASME J. Energy Resour. Technol.
,
145
(
9
), p.
092902
.
4.
Qin
,
J.
,
Xu
,
Y.
,
Tang
,
Y.
,
Liang
,
R.
,
Zhong
,
Q.
,
Yu
,
W.
, and
Sepehrnoori
,
K.
,
2022
, “
Impact of Complex Fracture Networks on Rate Transient Behavior of Wells in Unconventional Reservoirs Based on Embedded Discrete Fracture Model
,”
ASME J. Energy Resour. Technol.
,
144
(
8
), p.
083007
.
5.
Jiang
,
R.
,
Liu
,
X.
,
Cui
,
Y.
,
Wang
,
X.
,
Gao
,
Y.
,
Mao
,
N.
, and
Yan
,
X.
,
2020
, “
Production Performance Analysis for Multi-branched Horizontal Wells in Composite Coal Bed Methane Reservoir Considering Stress Sensitivity
,”
ASME J. Energy Resour. Technol.
,
142
(
7
), p.
073001
.
6.
He
,
Y.
,
Cheng
,
S.
,
Sun
,
Z.
,
Chai
,
Z.
, and
Rui
,
Z.
,
2020
, “
Improving Oil Recovery Through Fracture Injection and Production of Multiple Fractured Horizontal Wells
,”
ASME J. Energy Resour. Technol.
,
142
(
5
), p.
053002
.
7.
He
,
Y.
,
Xu
,
Y.
,
Tang
,
Y.
,
Qiao
,
Y.
,
Yu
,
W.
, and
Sepehrnoori
,
K.
,
2022
, “
Multi-phase Rate Transient Behaviors of the Multi-fractured Horizontal Well With Complex Fracture Networks
,”
ASME J. Energy Resour. Technol.
,
144
(
4
), p.
043006
.
8.
Chu
,
H.
,
Ma
,
T.
,
Zhu
,
W.
,
Gao
,
Y.
,
Zhang
,
J.
, and
John Lee
,
W.
,
2023
, “
A Novel Semi-Analytical Monitoring Model for Multi-horizontal Well System in Large-Scale Underground Natural Gas Storage: Methodology and Case Study
,”
Fuel
,
334
(
2
), p.
126807
.
9.
Lefkovits
,
H. C.
,
Hazebroek
,
P.
,
Allen
,
E. E.
, and
Matthews
,
C. S.
,
1961
, “
A Study of the Behavior of Bounded Reservoirs Composed of Stratified Layers
,”
Soc. Pet. Eng. J.
,
1
(
1
), pp.
43
58
.
10.
Ehlig-Economides
,
C. A.
, and
Joseph
,
J.
,
1987
, “
A New Test for Determination of Individual Layer Properties in a Multilayered Reservoir
,”
SPE Form. Eval.
,
2
(
3
), pp.
261
283
.
11.
Gao
,
C.-T.
, and
Deans
,
H. A.
,
1988
, “
Pressure Transients and Crossflow Caused by Diffusivities in Multilayer Reservoirs
,”
SPE Form. Eval.
,
3
(
2
), pp.
438
448
.
12.
Kuchuk
,
F. J.
, and
Wilkinson
,
D. J.
,
1991
, “
Transient Pressure Behavior of Commingled Reservoirs
,”
SPE Form. Eval.
,
6
(
1
), pp.
111
120
.
13.
Spath
,
J. B.
,
Ozkan
,
E.
, and
Raghavan
,
R.
,
1994
, “
An Efficient Algorithm for Computation of Well Responses in Commingled Reservoirs
,”
SPE Form. Eval.
,
9
(
2
), pp.
115
121
.
14.
Anisur Rahman
,
N. M.
, and
Mattar
,
L.
,
2007
, “
New Analytical Solution to Pressure Transient Problems in Commingled, Layered Zones With Unequal Initial Pressures Subject to Step Changes in Production Rates
,”
J. Pet. Sci. Eng.
,
56
(
4
), pp.
283
295
.
15.
Sui
,
W.
, and
Zhu
,
D.
,
2012
, “
Determining Multilayer Formation Properties From Transient Temperature and Pressure Measurements in Gas Wells With Commingled Zones
,”
J. Nat. Gas Sci. Eng.
,
9
, pp.
60
72
.
16.
Shi
,
W.
,
Cheng
,
S.
,
Meng
,
L.
,
Gao
,
M.
,
Zhang
,
J.
,
Shi
,
Z.
,
Wang
,
F.
, and
Duan
,
L.
,
2020
, “
Pressure Transient Behavior of Layered Commingled Reservoir With Vertical Inhomogeneous Closed Boundary
,”
J. Pet. Sci. Eng.
,
189
, p.
106995
.
17.
Bennett
,
C. O.
,
Raghavan
,
R.
, and
Reynolds
,
A. C.
,
1986
, “
Analysis of Finite-Conductivity Fractures Intercepting Multilayer Commingled Reservoirs
,”
SPE Form. Eval.
,
1
(
3
), pp.
259
274
.
18.
Bennett
,
C. O.
,
Camacho-V.
R. G.
,
Reynolds
,
A. C.
, and
Raghavan
,
R.
,
1985
, “
Approximate Solutions for Fractured Wells Producing Layered Reservoirs
,”
Soc. Pet. Eng. J.
,
25
(
5
), pp.
729
742
.
19.
Camacho-V.
R. G.
,
Raghavan
,
R.
, and
Reynolds
,
A. C.
,
1987
, “
Response of Wells Producing Layered Reservoirs: Unequal Fracture Length
,”
SPE Form. Eval.
,
2
(
1
), pp.
9
28
.
20.
Osman
,
M. E.
,
1993
, “
Pressure Analysis of a Fractured Well in Multilayered Reservoirs
,”
J. Pet. Sci. Eng.
,
9
(
1
), pp.
49
66
.
21.
Osman
,
M. E.
, and
Abou-Kassem
,
J. H.
,
1997
, “
A New Method for Pressure Test Analysis of a Vertically Fractured Well Producing Commingled Zones in Bounded Square Reservoirs
,”
J. Pet. Sci. Eng.
,
18
(
1–2
), pp.
131
145
.
22.
Osman
,
M. E.
, and
Abou-Kassem
,
J. H.
,
1996
, “
Effect of Boundary Conditions on Pressure Behavior of Finite-Conductivity Fractures in Bounded Stratified Reservoirs
,”
J. Pet. Sci. Eng.
,
15
(
2–4
), pp.
291
307
.
23.
Li
,
J.
,
Zhang
,
C.
,
Xia
,
Y.
,
Wang
,
F.
,
Shi
,
D.
, and
Cheng
,
S.
,
2022
, “
Pressure Transient Analysis for Hydraulically Fractured Wells With Changing Conductivity in Stratified Reservoirs: Case Study in Xinjiang Oilfield
,”
ACS Omega
,
7
(
34
), pp.
30313
30320
.
24.
Wei
,
C.
,
Li
,
H.
,
Luo
,
H.
,
Li
,
Y.
, and
Cheng
,
S.
,
2023
, “
Generalized Analytical Solutions of Vertically Fractured Wells in Commingled Reservoirs: Field Case Study
,”
SPE J.
, pp.
1
15
.
25.
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
.
26.
Van Everdingen
,
A. F.
,
1953
, “
The Skin Effect and Its Influence on the Productive Capacity of a Well
,”
J. Pet. Technol.
,
5
(
6
), pp.
171
176
.
27.
Gringarten
,
A. C.
, and
Ramey
,
H. J.
,
1973
, “
The Use of Source and Green's Functions in Solving Unsteady-Flow Problems in Reservoirs
,”
Soc. Pet. Eng. J.
,
13
(
5
), pp.
285
296
.
28.
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
,”
Soc. Pet. Eng. J.
,
14
(
4
), pp.
347
360
.
29.
Luo
,
W.
,
Wang
,
X.
,
Liu
,
P.
, and
Tian
,
Q.
,
2018
, “
A Simple and Accurate Calculation Method for Finite-Conductivity Fracture
,”
J. Pet. Sci. Eng.
,
161
, pp.
590
598
.
30.
Wilkinson
,
D. J.
,
1989
, “
New Results for Pressure Transient Behavior of Hydraulically Fractured Wells
,”
The Low Permeability Reservoirs Symposium
,
Denver, CO
,
Mar. 6–8
, SPE-18950-MS.
31.
Stehfest
,
H.
,
1970
, “
Algorithm 368: Numerical Inversion of Laplace Transforms [D5]
,”
Commun. ACM
,
13
(
1
), pp.
47
49
.
32.
Xu
,
W. L.
,
2018
, “
The Study on Transient Flow Problems of Multifractures in Anisotropic Reservoirs
,”
Doctoral Dissertation
,
China University of Geosciences
,
Beijing, China
.
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