Based on the KGD scheme, this paper investigates, with both analytical and numerical approaches, the propagation of a hydraulic fracture with a fluid lag in permeable rock. On the analytical aspect, the general form of normalized governing equations is first formulated to take into account both fluid lag and leak-off during the process of hydraulic fracturing. Then a new self-similar solution corresponding to the limiting case of zero dimensionless confining stress (T=0) and infinite dimensionless leak-off coefficient (L=) is obtained. A dimensionless parameter R is proposed to indicate the propagation regimes of hydraulic fracture in more general cases, where R is defined as the ratio of the two time-scales related to the dimensionless confining stress T and the dimensionless leak-off coefficient L. In addition, a robust finite element-based KGD model has been developed to simulate the transient process from L=0 to L= under T=0, and the numerical solutions converge and agree well with the self-similar solution at T=0 and L=. More general processes from T=0 and L=0 to T= and L= for three different values of R are also simulated, which proves the effectiveness of the proposed dimensionless parameter R for indicating fracture regimes.

References

References
1.
Lecampion
,
B.
, and
Detournay
,
E.
,
2007
, “
An Implicit Algorithm for the Propagation of a Hydraulic Fracture With a Fluid Lag
,”
Comput. Methods Appl. Mech. Eng.
,
196
(
49–52
), pp.
4863
4880
.
2.
Adachi
,
J. I.
, and
Detournay
,
E.
,
2008
, “
Plane Strain Propagation of a Hydraulic Fracture in a Permeable Rock
,”
Eng. Fract. Mech.
,
75
(
16
), pp.
4666
4694
.
3.
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
.
4.
Khristianovic
,
S. A.
, and
Zheltov
,
Y. P.
,
1955
, “
Formation of Vertical Fractures by Means of Highly Viscous Liquid
,”
Fourth World Petroleum Congress
, Rome, Italy, pp. 579–586.
5.
Geertsma
,
J.
, and
De Klerk
,
F.
,
1969
, “
A Rapid Method of Predicting Width and Extent of Hydraulically Induced Fractures
,”
J. Pet. Technol.
,
21
(
12
), pp.
1571
1581
.
6.
Carbonell
,
R.
,
Desroches
,
J.
, and
Detournay
,
E.
,
1999
, “
A Comparison Between a Semi-Analytical and a Numerical Solution of a Two-Dimensional Hydraulic Fracture
,”
Int. J. Solids Struct.
,
36
(
31–32
), pp.
4869
4888
.
7.
Spence
,
D. A.
, and
Sharp
,
P.
,
1985
, “
Self-Similar Solutions for Elastohydrodynamic Cavity Flow
,”
Proc. R. Soc. A: Math., Phys. Eng. Sci.
,
400
(
1819
), pp.
289
313
.
8.
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
.
9.
Detournay
,
E.
,
2004
, “
Propagation Regimes of Fluid-Driven Fractures in Impermeable Rocks
,”
Int. J. Geomech.
,
4
(
1
), pp.
35
45
.
10.
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
.
11.
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
.
12.
Garagash
,
D. I.
,
2006
, “
Propagation of a Plane-Strain Hydraulic Fracture With a Fluid Lag: Early-Time Solution
,”
Int. J. Solids Struct.
,
43
(
18–19
), pp.
5811
5835
.
13.
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
.
14.
Detournay
,
E.
,
2016
, “
Mechanics of Hydraulic Fractures
,”
Annu. Rev. Fluid Mech.
,
48
(
1
), pp.
311
339
.
15.
Adachi
,
J. I.
,
Detournay
,
E.
,
Garagash
,
D. I.
, and
Savitski
,
A. A.
,
2004
, “
Interpretation and Design of Hydraulic Fracturing Treatments
,” U.S. Patent No.
US 7111681 B2
https://patentimages.storage.googleapis.com/dd/20/2d/088d1149525264/US7111681.pdf
16.
Bunger
,
A. P.
,
Detournay
,
E.
, and
Garagash
,
D. I.
,
2005
, “
Toughness-Dominated Hydraulic Fracture With Leak-Off
,”
Int. J. Fract.
,
134
(
2
), pp.
175
190
.
17.
Dong
,
X. L.
,
Zhang
,
G. Q.
,
Gao
,
D. L.
, and
Duan
,
Z. Y.
,
2017
, “
Toughness-Dominated Hydraulic Fracture in Permeable Rocks
,”
ASME J. Appl. Mech.
,
84
(
7
), p.
071001
.
18.
Garagash
,
D.
,
2000
, “
Hydraulic Fracture Propagation in Elastic Rock With Large Toughness
,”
Fourth North American Rock Mechanics Symposium
(
ARMA
), Seattle, WA, July 31–Aug. 3, pp. 221–228.https://www.onepetro.org/conference-paper/ARMA-2000-0221
19.
Wang
,
T.
,
Liu
,
Z.
,
Zeng
,
Q.
,
Gao
,
Y.
, and
Zhuang
,
Z.
,
2017
, “
XFEM Modeling of Hydraulic Fracture in Porous Rocks With Natural Fractures
,”
Sci. China Phys., Mech. Astron.
,
60
(
8
), p.
084614
.
20.
Wang
,
T.
,
Liu
,
Z.
,
Gao
,
Y.
,
Zeng
,
Q.
, and
Zhuang
,
Z.
,
2017
, “
Theoretical and Numerical Models to Predict Fracking Debonding Zone and Optimize Perforation Cluster Spacing in Layered Shale
,”
ASME J. Appl. Mech.
,
85
(
1
), p.
011001
.
21.
Zeng
,
Q.
,
Liu
,
Z.
,
Wang
,
T.
,
Gao
,
Y.
, and
Zhuang
,
Z.
,
2017
, “
Fully Coupled Simulation of Multiple Hydraulic Fractures to Propagate Simultaneously From a Perforated Horizontal Wellbore
,”
Comput. Mech.
,
61
(
1–2
), pp.
137
155
.
22.
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. Fract.
,
205
(
2
), pp.
221
237
.
23.
Detournay
,
E.
,
Hakobyan
,
Y.
, and
Eve
,
R.
,
2017
, “
Self-Similar Propagation of a Hydraulic Fracture in a Poroelastic Medium
,” Biot Conference on Poromechanics, Paris, France, July 9–13, pp. 1909–1914.
24.
Huynen
,
A.
, and
Detournay
,
E.
, “
Self-Similar Propagation of a Plastic Zone Due to Fluid Injection in a Porous Medium
,” Biot Conference on Poromechanics, Paris, France, July 9–13, pp. 1952–1959.
25.
Desroches
,
J.
, and
Thiercelin
,
M.
,
1993
, “
Modelling the Propagation and Closure of Micro-Hydraulic Fractures
,”
Int. J. Rock Mech. Min. Sci. Geomech. Abstr.
,
30
(
7
), pp.
1231
1234
.
26.
Hunsweck
,
M. J.
,
Shen
,
Y.
, and
Lew
,
A. J.
,
2013
, “
A Finite Element Approach to the Simulation of Hydraulic Fractures With Lag
,”
Int. J. Numer. Anal. Methods Geomech.
,
37
(
9
), pp.
993
1015
.
27.
Walters
,
M. C.
,
Paulino
,
G. H.
, and
Dodds
,
R. H.
,
2005
, “
Interaction Integral Procedures for 3D Curved Cracks Including Surface Tractions
,”
Eng. Fract. Mech.
,
72
(
11
), pp.
1635
1663
.
28.
Chen
,
B.
,
Cen
,
S.
,
Barron
,
A. R.
,
Owen
,
D. R. J.
, and
Li
,
C.
, “
Numerical Investigation of the Fluid Lag During Hydraulic Fracturing
,”
Eng. Comput.
(in press).
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