Abstract

Modeling the propagation of hydraulic fractures is a complex problem that involves solving the equations that describe the intimate coupling between the cracked and porous solid phase and the fracturing fluid that drives the process according to established crack extension criteria. Most of the existing computational techniques become prohibitively expensive if applied to three-dimensional configurations, especially those involving multiple fractures because their accuracy relies on extremely fine discretizations of the geometry. These include the standard finite element method and standard displacement discontinuity method, which require substantial mesh densities in the vicinity of the crack front to capture the complex nature of the singular fields associated with fluid-driven cracks. Models have recently been proposed that offer sufficient accuracy and a significant increase in computational efficiency. These include the enhanced pseudo-3D model that explicitly incorporates a non-local elasticity relation and known crack-front asymptotic solutions that capture the multiscale near-front behavior. Here, we present an enhanced pseudo-3D model for simulating the propagation of multiple planar and nonplanar hydraulic fractures in a linear elastic medium. The algorithm is built on a fixed mesh with an adaptive tip element, and it incorporates leak-off, as well as stress interaction between the fractures. The model is validated against reference solutions of plane strain, radial, and multi-planar fracture geometries, for various sets of parameters that cover the majority of the problem parametric space.

References

References
1.
Montgomery
,
C. T.
, and
Smith
,
M. B.
,
2010
, “
Hydraulic Fracturing: History of An Enduring Technology
,”
J. Petroleum Technol.
,
62
(
12
), pp.
26
40
. 10.2118/1210-0026-JPT
2.
Economides
,
M. J.
, and
Nolte
,
K. G.
, eds.,
2000
,
Reservoir Stimulation
, 3rd ed.,
John Wiley & Sons
,
Chichester, UK
.
3.
Khristianovic
,
S. A.
, and
Zheltov
,
Y. P.
,
1955
, “
Formation of Vertical Fractures by Means of Highly Viscous Fluids
,”
4th World Petroleum Congress
,
Rome, Italy
, Vol.
2
, pp.
579
586
.
4.
Geertsma
,
J.
, and
de Klerk
,
F.
,
1969
, “
A Rapid Method of Predicting Width and Extent of Hydraulically Induced Fractures
,”
J. Petroleum Technol.
,
21
(
12
), pp.
1
571
. 10.2118/2458-PA
5.
Perkins
,
T. K.
, and
Kern
,
L. R.
,
1961
, “
Widths of Hydraulic Fractures
,”
J. Petroleum Technol.
,
13
(
9
), pp.
937
949
. 10.2118/89-PA
6.
Nordgren
,
R.
, et al
,
1972
, “
Propagation of a Vertical Hydraulic Fracture
,”
Soc. Petroleum Eng. J.
,
12
(
4
), pp.
306
314
. 10.2118/3009-PA
7.
Abe
,
H.
,
Keer
,
L. M.
, and
Mura
,
T.
,
1976
, “
Growth Rate of a Penny-Shaped Crack in Hydraulic Fracturing of Rocks, 2
,”
J. Geophys. Res.
,
81
(
35
), pp.
6292
6298
. 10.1029/JB081i035p06292
8.
Settari
,
A.
, and
Cleary
,
M. P.
,
1986
, “
Development and Testing of a Pseudo-Three-Dimensional Model of Hydraulic Fracture Geometry
,”
SPE Production Eng.
,
1
(
6
), pp.
449
466
. 10.2118/10505-PA
9.
McLennan
,
J. D.
, and
Picardy
,
J. C.
,
1985
, “
Pseudo-Three-Dimensional Fracture Growth Modeling
,”
26th US Symposium on Rock Mechanics
,
Rapid City, SD
, American Rock Mechanics Association.
10.
Warpinski
,
N. R.
,
Abou-Sayed
,
I. S.
,
Moschovidis
,
Z.
, and
Parker
,
C.
,
1993
.
Hydraulic Fracture Model Comparison Study, Complete Results
.
Technical Report
,
Sandia National Laboratories
.
11.
Vandamme
,
L.
, and
Curran
,
J. H.
,
1989
, “
A Three-Dimensional Hydraulic Fracturing Simulator
,”
Int. J. Numer. Methods Eng.
,
28
(
4
), pp.
909
927
. 10.1002/nme.1620280413
12.
Sherman
,
C. S.
,
Aarons
,
L. R.
,
Morris
,
J. P.
,
Johnson
,
S.
,
Savitski
,
A. A.
, and
Geilikman
,
M. B.
,
2015
, “
Finite Element Modeling of Curving Hydraulic Fractures and Near-Wellbore Hydraulic Fracture Complexity
,”
49th US Rock Mechanics/Geomechanics Symposium
,
San Francisco, CA
, American Rock Mechanics Association.
13.
Kumar
,
D.
, and
Ghassemi
,
A.
,
2015
, “
3D Simulation of Multiple Fracture Propagation From Horizontal Wells
,”
49th US Rock Mechanics/Geomechanics Symposium
,
San Francisco, CA
, American Rock Mechanics Association.
14.
Dontsov
,
E. V.
, and
Peirce
,
A. P.
,
2017
, “
A Multiscale Implicit Level Set Algorithm (ILSA) to Model Hydraulic Fracture Propagation Incorporating Combined Viscous, Toughness, and Leak-off Asymptotics
,”
Comput. Methods Appl. Mech. Eng.
,
313
, pp.
53
84
. 10.1016/j.cma.2016.09.017
15.
Kresse
,
O.
,
Weng
,
X.
,
Gu
,
H.
, and
Wu
,
R.
,
2013
, “
Numerical Modeling of Hydraulic Fracture Interaction in Complex Naturally Fractured Formations
,”
Rock Mech. Rock Eng.
,
46
, pp.
555
558
. 10.1007/s00603-012-0359-2
16.
Damjanac
,
B.
,
Detournay
,
C.
,
Cundall
,
P. A.
, and
Varun
,
2013
, “Three-Dimensional Numerical Model of Hydraulic Fracturing in Fractured Rock Masses,”
Effective and Sustainable Hydraulic Fracturing
,
Bunger
,
A. P.
,
McLennan
,
J.
, and
Jeffrey
,
R.
, eds.,
Intech
, pp.
819
830
.
17.
Wu
,
K.
,
Olson
,
J.
,
Balhoff
,
M. T.
, and
Yu
,
W.
,
2017
, “
Numerical Analysis for Promoting Uniform Development of Simultaneous Multiple-Fracture Propagation in Horizontal Wells
,”
SPE. Prod. Oper.
,
32
(
1
), pp.
41
50
.
18.
Dontsov
,
E. V.
, and
Peirce
,
A. P.
,
2016
, “
Implementing a Universal Tip Asymptotic Solution Into An Implicit Level Set Algorithm (ILSA) for Multiple Parallel Hydraulic Fractures
,”
50th US Rock Mechanics/Geomechanics Symposium
,
Houston, TX
, American Rock Mechanics Association.
19.
Adachi
,
J.
,
Siebrits
,
E.
,
Peirce
,
A.
, and
Desroches
,
J.
,
2007
, “
Computer Simulation of Hydraulic Fractures
,”
Int. J. Rock Mech. Mining Sci.
,
44
(
5
), pp.
739
757
. 10.1016/j.ijrmms.2006.11.006
20.
Lecampion
,
B.
,
Bunger
,
A.
, and
Zhang
,
X.
,
2018
, “
Numerical Methods for Hydraulic Fracture Propagation: A Review of Recent Trends
,”
J. Nat. Gas Sci. Eng.
,
49
, pp.
66
83
. 10.1016/j.jngse.2017.10.012
21.
Detournay
,
E.
,
2004
, “
Propagation Regimes of Fluid-Driven Fractures in Impermeable Rocks
,”
Int. J. Geomech.
,
4
(
1
), pp.
35
45
. 10.1061/(ASCE)1532-3641(2004)4:1(35)
22.
Detournay
,
E.
,
2016
, “
Mechanics of Hydraulic Fractures
,”
Ann. Rev. Fluid Mech.
,
48
, pp.
311
339
. 10.1146/annurev-fluid-010814-014736
23.
Dontsov
,
E. V.
, and
Peirce
,
A. P.
,
2015
, “
An Enhanced Pseudo-3D Model for Hydraulic Fracturing Accounting for Viscous Height Growth, Non-local Elasticity, and Lateral Toughness
,”
Eng. Fract. Mech.
,
142
, pp.
116
139
. 10.1016/j.engfracmech.2015.05.043
24.
Dontsov
,
E. V.
, and
Peirce
,
A. P.
,
2016
, “
Comparison of Toughness Propagation Criteria for Blade-Like and Pseudo-3D Hydraulic Fractures
,”
Eng. Fract. Mech.
,
160
, pp.
238
247
. 10.1016/j.engfracmech.2016.04.023
25.
Linkov
,
A. M.
, and
Markov
,
N. S.
,
2020
, “
Improved Pseudo Three-Dimensional Model for Hydraulic Fractures Under Stress Contrast
,”
Int. J. Rock Mech. Mining Sci.
,
130
, p.
104316
. 10.1016/j.ijrmms.2020.104316
26.
Skopintsev
,
A. M.
,
Dontsov
,
E. V.
,
Kovtunenko
,
P. V.
,
Baykin
,
A. N.
, and
Golovin
,
S. V.
,
2020
, “
The Coupling of an Enhanced Pseudo-3D Model for Hydraulic Fracturing With a Proppant Transport Model
,”
Eng. Fract. Mech.
,
236
.
27.
Peirce
,
A.
, and
Detournay
,
E.
,
2008
, “
An Implicit Level Set Method for Modeling Hydraulically Driven Fractures
,”
Comput. Methods Appl. Mech. Eng.
,
197
(
33–40
), pp.
2858
2885
. 10.1016/j.cma.2008.01.013
28.
Gordeliy
,
E.
, and
Peirce
,
A.
,
2013
, “
Implicit Level Set Schemes for Modeling Hydraulic Fractures Using the XFEM
,”
Comput. Methods Appl. Mech. Eng.
,
266
, pp.
125
143
. 10.1016/j.cma.2013.07.016
29.
Zia
,
H.
,
Lecampion
,
B.
,
Zia
,
H.
, and
Lecampion
,
B.
,
2020
, “
PyFrac: A Planar 3D Hydraulic Fracture Simulator
,”
Comput. Phys. Commun.
,
255
, p.
107368
. 10.1016/j.cpc.2020.107368
30.
Garagash
,
D. I.
,
Detournay
,
E.
, and
Adachi
,
J. I.
,
2011
, “
Multiscale Tip Asymptotics in Hydraulic Fracture With Leak-Off
,”
J. Fluid. Mech.
,
669
, pp.
260
297
. 10.1017/S002211201000501X
31.
Dontsov
,
E. V.
, and
Peirce
,
A. P.
,
2015
, “
A Non-Singular Integral Equation Formulation to Analyze Multiscale Behaviour in Semi-Infinite Hydraulic Fractures
,”
J. Fluid. Mech.
,
781
, p.
R1
. 10.1017/jfm.2015.451
32.
Dontsov
,
E. V.
, and
Kresse
,
O.
,
2018
, “
A Semi-Infinite Hydraulic Fracture With Leak-Off Driven by a Power-Law Fluid
,”
J. Fluid. Mech.
,
837
, pp.
210
229
. 10.1017/jfm.2017.856
33.
Moukhtari
,
F.-E.
, and
Lecampion
,
B.
,
2018
, “
A Semi-Infinite Hydraulic Fracture Driven by a Shear-Thinning Fluid
,”
J. Fluid. Mech.
,
838
, pp.
573
605
. 10.1017/jfm.2017.900
34.
Bessmertnykh
,
A. O.
, and
Dontsov
,
E. V.
,
2019
, “
A Semi-Infinite Hydraulic Fracture Driven by a Herschel-Bulkley Fluid
,”
ASME J. Appl. Mech.
,
86
(
12
), p.
121008
. 10.1115/1.4044815
35.
Palmer
,
I. D.
, and
Carroll Jr
,
H. B.
,
1983
, “
Three-Dimensional Hydraulic Fracture Propagation in the Presence of Stress Variations
,”
Soc. Petroleum Eng. J.
,
23
(
6
), pp.
870
878
. 10.2118/10849-PA
36.
Palmer
,
I. D.
, and
Carroll, Jr
,
H. B.
,
1983
, “
Numerical Solution for Height and Elongated Hydraulic Fractures
,”
SPE/DOE Low Permeability Gas Reservoirs
,
Denver, CO
, Symposium Society of Petroleum Engineers.
37.
Adachi
,
J. I.
,
Detournay
,
E.
, and
Peirce
,
A. P.
,
2010
, “
Analysis of the Classical Pseudo-3D Model for Hydraulic Fracture With Equilibrium Height Growth Across Stress Barriers
,”
Int. J. Rock Mech. Mining Sci.
,
47
(
4
), pp.
625
639
. 10.1016/j.ijrmms.2010.03.008
38.
Palmer
,
I. D.
,
1983
, “
Three-Dimensional Hydraulic Fracture Propagation in the Presence of Stress Variations
,”
Soc. Petroleum Eng. J.
,
23
(
6
), pp.
870
878
. 10.2118/10849-PA
39.
Palmer
,
I. D.
, and
Carroll
,
H. B.
,
1983
, “
Numerical Solution for Height and Elongated Hydraulic Fractures
,”
SPE/DOE Low Permeability Gas Reservoirs Symposium
,
Denver, CO
,
March
, Society of Petroleum Engineers of AIME.
40.
Weng
,
X.
,
1992
, “
Incorporation of 2D Fluid Flow Into a Pseudo-3D Hydraulic Fracturing Simulator
,”
SPE Production Eng.
,
7
(
4
), pp.
331
337
. 10.2118/21849-PA
41.
Xiujuan
,
Y.
,
Tongtao
,
W.
,
Xiangzhen
,
Y.
, and
Xin
,
W.
,
2010
, “
A Pseudo-3D Model with 2D Flow of Hydraulic Fracture Propagation
,”
ISRM International Symposium on In-Situ Rock Stress
,
Beijing, China
,
August
, International Society for Rock Mechanics, pp.
429
433
.
42.
Adachi
,
J. I.
, and
Peirce
,
A. P.
,
2008
, “
Asymptotic Analysis of An Elasticity Equation for a Finger-like Hydraulic Fracture
,”
J. Elasticity
,
90
(
1
), pp.
43
69
. 10.1007/s10659-007-9122-4
43.
Carter
,
E.
,
1957
, “Optimum Fluid Characteristics: Appendix I,”
Drilling and Production Practice
,
G. C.
Howard
, and
C. R.
Fast
, eds.,
American Petroleum Institute
,
New York
, pp.
261
270
.
44.
Garagash
,
D.
, and
Detournay
,
E.
,
2000
, “
The Tip Region of a Fluid-Driven Fracture in An Elastic Medium
,”
ASME J. Appl. Mech.
,
67
(
1
), pp.
183
192
. 10.1115/1.321162
45.
Detournay
,
E.
, and
Garagash
,
D. I.
,
2003
, “
The Near-Tip Region of a Fluid-Driven Fracture Propagating in a Permeable Elastic Solid
,”
J. Fluid. Mech.
,
494
, pp.
1
32
. 10.1017/S0022112003005275
46.
Detournay
,
E.
, and
Peirce
,
A.
,
2014
, “
On the Moving Boundary Conditions for a Hydraulic Fracture
,”
Int. J. Eng. Sci.
,
84
, pp.
147
155
. 10.1016/j.ijengsci.2014.06.010
47.
Crouch
,
S. L.
, and
Starfield
,
A. M.
,
1983
,
Boundary Element Methods in Solid Mechanics
,
George Allen and Unwin
,
London
.
48.
Hills
,
D. A.
,
Kelly
,
P. A.
,
Dai
,
D. N.
, and
Korsunsky
,
A. M.
,
1996
,
Solution of Crack Problems: The Distributed Dislocation Technique
, Vol.
44
,
Kluwer Academic Publisher
,
Dordrecht
.
49.
Cramer
,
D. D.
,
1987
, “
The Application of Limited-Entry Techniques in Massive Hydraulic Fracturing Treatments
,”
SPE Production Operations Symposium
,
Oklahoma City, OK
, Society of Petroleum Engineers.
50.
Crump
,
J. B.
, and
Conway
,
M. W.
,
1988
, “
Effects of Perforation-Entry Friction on Bottomhole Treating Analysis
,”
J. Petroleum Technol.
,
40
(
8
), pp.
1
41
. 10.2118/15474-PA
51.
Lord
,
D. L.
,
1994
, “
Study of Perforation Friction Pressure Employing a Large-scale Fracturing Flow Simulator
,”
SPE Annual Technical Conference and Exhibition
,
New Orleans, LA
, Society of Petroleum Engineers.
52.
Romero
,
J.
,
Mack
,
M. G.
, and
Elbel
,
J. L.
,
1995
, “
Theoretical Model and Numerical Investigation of Near-Wellbore Effects in Hydraulic Fracturing
,”
SPE Annual Technical Conference and Exhibition
,
Dallas, TX
, Society of Petroleum Engineers.
53.
El-Rabaa
,
A. M.
,
Shah
,
S. N.
, and
Lord
,
D. L.
,
1997
, “
New Perforation Pressure Loss Correlations for Limited Entry Fracturing Treatments
,”
SPE Rocky Mountain Regional Meeting
,
Casper, WY
, Society of Petroleum Engineers.
54.
Rice
,
J. R.
,
1968
, “Mathematical Analysis in the Mechanics of Fracture,”
Fracture: An Advanced Treatise
, Vol.
II
,
Liebowitz
,
H.
, ed.,
Academic Press
,
New York, NY
, pp.
191
311
. ch. 3.
55.
Tada
,
H.
,
Paris
,
P. C.
, and
Irwin
,
G. R.
,
2000
,
The Stress Analysis of Cracks Handbook: Third Edition
,
ASME Press
,
New York
.
56.
Erdogan
,
F.
, and
Sih
,
G. C.
,
1963
, “
On the Crack Extension in Plates Under Plane Loading and Transverse Shear
,”
J. Fluid. Eng.
,
85
(
4
), pp.
519
525
.
57.
Dontsov
,
E. V.
,
2016
, “
An Approximate Solution for a Penny-Shaped Hydraulic Fracture that Accounts for Fracture Toughness, Fluid Viscosity and Leak-Off
,”
R. Soc. Open Sci.
,
3
(
12
), p.
160737
. 10.1098/rsos.160737
58.
Dontsov
,
E. V.
, and
Peirce
,
A. P.
,
2015
, “
A Non-Singular Integral Equation Formulation to Analyse Multiscale Behaviour in Semi-Infinite Hydraulic Fractures
,”
J. Fluid. Mech.
,
781
, p.
R1
. 10.1017/jfm.2015.451
59.
Dontsov
,
E. V.
, and
Peirce
,
A. P.
,
2015
, “
Incorporating Viscous, Toughness, and Intermediate Regimes of Propagation Into Enhanced Pseudo-3D Model
,”
49th US Rock Mechanics/Geomechanics
,
San Francisco, CA
, American Rock Mechanics Association.
60.
Adachi
,
J. I.
, and
Detournay
,
E.
,
2002
, “
Self-Similar Solution of a Plane-Strain Fracture Driven by a Power-Law Fluid
,”
Int. J. Numer. Anal. Methods Geomech.
,
26
(
6
), pp.
579
604
. 10.1002/nag.213
61.
Garagash
,
D. I.
, and
Detournay
,
E.
,
2005
, “
Plane-Strain Propagation of a Fluid-Driven Fracture: Small Toughness Solution
,”
ASME J. Appl. Mech.
,
72
(
6
), pp.
916
928
. 10.1115/1.2047596
62.
Garagash
,
D. I.
,
2006
, “
Plane-Strain Propagation of a Fluid-Driven Fracture During Injection and Shut-In: Asymptotics of Large Toughness
,”
Eng. Fract. Mech.
,
73
(
4
), pp.
456
481
. 10.1016/j.engfracmech.2005.07.012
63.
Adachi
,
J. I.
, and
Peirce
,
A. P.
,
2008
, “
Asymptotic Analysis of An Elasticity Equation for a Finger-Like Hydraulic Fracture
,”
J. Elasticity
,
90
, pp.
43
69
. 10.1007/s10659-007-9122-4
64.
Dontsov
,
E. V.
,
2017
, “
An Approximate Solution for a Plane Strain Hydraulic Fracture That Accounts for Fracture Toughness, Fluid Viscosity, and Leak-Off
,”
Int. J. Fracture
,
205
(
2
), pp.
221
237
. 10.1007/s10704-017-0192-4
65.
Hu
,
J.
, and
Garagash
,
D. I.
,
2010
, “
Plane-Strain Propagation of a Fluid-Driven Crack in a Permeable Rock With Fracture Toughness
,”
J. Eng. Mech.
,
136
(
9
), pp.
1152
1166
. 10.1061/(ASCE)EM.1943-7889.0000169
66.
Savitski
,
A. A.
, and
Detournay
,
E.
,
2002
, “
Propagation of a Penny-Shaped Fluid-Driven Fracture in An Impermeable Rock: Asymptotic Solutions
,”
Int. J. Solids. Struct.
,
39
(
26
), pp.
6311
6337
. 10.1016/S0020-7683(02)00492-4
67.
Bunger
,
A. P.
,
Detournay
,
E.
, and
Garagash
,
D. I.
,
2005
, “
Toughness-Dominated Hydraulic Fracture With Leak-Off
,”
Int. J. Fracture
,
134
(
2
), pp.
175
190
. 10.1007/s10704-005-0154-0
68.
Bunger
,
A. P.
, and
Detournay
,
E.
,
2007
, “
Early Time Solution for a Radial Hydraulic Fracture
,”
J. Eng. Mech.
,
133
(
5
), pp.
175
190
. 10.1061/(ASCE)0733-9399(2007)133:5(534)
69.
Madyarova
,
M. V.
,
2003
, “
Fluid-Driven Penny-Shaped Fracture in Elastic Medium
,” Ph.D. thesis,
University of Minnesota
.
70.
Cotterell
,
B.
, and
Rice
,
J. R.
,
1980
, “
Slightly Curved Or Kinked Cracks
,”
Int. J. Fracture
,
16
(
2
), pp.
155
169
. 10.1007/BF00012619
71.
Dontsov
,
E. V.
, and
Suarez-Rivera
,
R.
,
2020
, “
Propagation of Multiple Hydraulic Fractures in Different Regimes
,”
Int. J. Rock Mech. Mining Sci.
,
128
, p.
104270
. 10.1016/j.ijrmms.2020.104270
72.
Lenoach
,
B.
,
1995
, “
The Crack Tip Solution for Hydraulic Fracturing in a Permeable Solid
,”
J. Mech. Phys. Solids.
,
43
(
7
), pp.
1025
1043
. 10.1016/0022-5096(95)00026-F
73.
Desroches
,
J.
,
Detournay
,
E.
,
Lenoach
,
B.
,
Papanastasiou
,
P.
,
Pearson
,
J. R. A.
,
Thiercelin
,
M.
, and
Cheng
,
A.
,
1994
, “
The Crack Tip Region in Hydraulic Fracturing
,”
Proc. R. Soc. London., A.
,
447
(
1929
), pp.
39
48
. 10.1098/rspa.1994.0127
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