Abstract

Hydraulic fracturing is an industrial process often applied to enhance oil and gas recovery. Under this process, fractures are generated by the injection of highly pressurized fluids, which often exhibit shear-thinning rheology and yield stress. The global fracture propagation is influenced by various processes occurring near the fracture tip. To gain an insight into fracture propagation, the problem of a semi-infinite hydraulic fracture propagating in a permeable linear elastic rock is solved. To investigate the effect of fluid yield stress, we focus on a fracture driven by Herschel–Bulkley fluid. The mathematical model consists of the elasticity equation, the lubrication equation, and the propagation criterion for the semi-infinite plane strain fracture to obtain the fracture opening. The non-linear system of governing equations is represented in the non-singular form and solved numerically using Newton’s method. The solution is influenced by the competing processes related to rock toughness, fluid properties, and leak-off. The effects of these phenomena prevail at different length scales, and the corresponding limits can be described via analytical solutions. For a Herschel-Bulkley fluid, an additional limiting solution related to the fluid yield stress is obtained, and the regions of the dominance of limiting solutions affected by the yield stress are investigated. Finally, a faster approximate solution for the problem is proposed and its accuracy against a numerical solution is evaluated. The obtained result can be applied in hydraulic fracturing simulators to account for the effect of Herschel–Bulkley fluid rheology on the near-tip region.

References

References
1.
Economides
,
M. J.
, and
Nolte
,
K. G.
, eds.,
2000
,
Reservoir Stimulation
,
3rd ed.
,
John Wiley & Sons
,
Chichester, UK
.
2.
Donaldson
,
E. C.
,
Alam
,
W.
, and
Begum
,
N.
,
2014
,
Hydraulic Fracturing Explained: Evaluation, Implementation, and Challenges
,
Elsevier
,
New York
.
3.
Faroughi
,
S. A.
,
Pruvot
,
A. J. -C. J.
, and
McAndrew
,
J.
,
2018
, “
The Rheological Behavior of Energized Fluids and Foams With Application to Hydraulic Fracturing
,”
J. Petrol. Sci. Eng.
,
163
, pp.
243
263
. 10.1016/j.petrol.2017.12.051
4.
Saintpere
,
S.
,
Herzhaft
,
B.
,
Toure
,
A.
, and
Jollet
,
S.
,
1999
, “
Rheological Properties of Aqueous Foams for Underbalanced Drilling
,”
SPE Annual Technical Conference and Exhibition, Society of Petroleum Engineers
,
Houston, TX
,
Oct. 3–6
,
Society of Petroleum Engineers
.
5.
Khade
,
S. D.
, and
Shah
,
S. N.
,
2004
, “
New Rheological Correlations for Guar Foam Fluids
,”
SPE Prod. Facil.
,
19
(
02
), pp.
77
85
. 10.2118/88032-PA
6.
Osiptsov
,
A. A.
,
2017
, “
Fluid Mechanics of Hydraulic Fracturing: A Review
,”
J. Petrol. Sci. Eng.
,
156
, pp.
513
535
.
7.
Gidley
,
J. L.
,
1989
,
Recent Advances in Hydraulic Fracturing
.
Henry L. Doherty Memorial Fund of AIME, Society of Petroleum Engineers
,
Richardson, TX
.
8.
Stickel
,
J. J.
, and
Powell
,
R. L.
,
2005
, “
Fluid Mechanics and Rheology of Dense Suspensions
,”
Annu. Rev. Fluid Mech.
,
37
, pp.
129
149
. 10.1146/annurev.fluid.36.050802.122132
9.
Cherny
,
S. G.
, and
Lapin
,
V. N.
,
2016
, “
3D Model of Hydraulic Fracture With Herschel-Bulkley Compressible Fluid Pumping
,”
Procedia Struct. Integrity
,
2
, pp.
2479
2486
. 10.1016/j.prostr.2016.06.310
10.
Detournay
,
E.
,
2016
, “
Mechanics of Hydraulic Fractures
,”
Annu. Rev. Fluid Mech.
,
48
, pp.
311
339
. 10.1146/annurev-fluid-010814-014736
11.
Rice
,
J. R.
,
1968
, “
A Path Independent Integral and the Approximate Analysis of Strain Concentration by Notches and Cracks
,”
J. Appl. Mech.
,
35
(
2
), pp.
379
386
. 10.1115/1.3601206
12.
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
13.
Desroches
,
J.
,
Detournay
,
E.
,
Lenoach
,
B.
,
Papanastasiou
,
P.
,
Pearson
,
J. R. A.
,
Thiercelin
,
M.
, and
Cheng
,
A. H.-D.
,
1994
, “
The Crack Tip Region in Hydraulic Fracturing
,”
Proc. Royal Soc. Lond. A Math. Phys. Sci.
,
447
, pp.
39
48
. 10.1098/rspa.1994.0127
14.
Bunger
,
A. P.
, and
Detournay
,
E.
,
2008
, “
Experimental Validation of the Tip Asymptotics for a Fluid-Driven Crack
,”
J. Mech. Phys. Solids
,
56
(
11
), pp.
3101
3115
. 10.1016/j.jmps.2008.08.006
15.
Garagash
,
D.
, and
Detournay
,
E.
,
2000
, “
The Tip Region of a Fluid-Driven Fracture in An Elastic Medium
,”
J. Appl. Mech.
,
67
(
1
), pp.
183
192
. 10.1115/1.321162
16.
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
.
17.
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
18.
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
(
R1
). 10.1017/jfm.2015.451
19.
Gomez
,
D.
,
2016
, “
A Non-Singular Integral Equation Formulation of Permeable Semi-Infinite Hydraulic Fractures Driven by Shear-Thinning Fluids
,” Master’s thesis,
University of British Columbia
.
20.
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
21.
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
22.
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. Meth. Appl. M.
,
313
, pp.
53
84
. 10.1016/j.cma.2016.09.017
23.
Gordeliy
,
E.
, and
Peirce
,
A. P.
,
2013
, “
Implicit Level Set Schemes for Modeling Hydraulic Fractures Using the XFEM
,”
Comp. Meth. Appl. Mech. and Eng.
,
266
, pp.
125
143
. 10.1016/j.cma.2013.07.016
24.
Peirce
,
A.
,
2016
, “
Implicit Level Set Algorithms for Modelling Hydraulic Fracture Propagation
,”
Philos. Trans. Royal Soc. A
,
374
(
2078
), p.
20150423
. 10.1098/rsta.2015.0423
25.
Peirce
,
A.
, and
Detournay
,
E.
,
2008
, “
An Implicit Level Set Method for Modeling Hydraulically Driven Fractures
,”
Comput. Method Appl. M.
,
197
(
33–40
), pp.
2858
2885
. 10.1016/j.cma.2008.01.013
26.
Carter
,
E.
,
1957
, “Optimum fluid characteristics for fracture extension,”
Drilling and Production Practices
,
G.
Howard
and
C.
Fast
, eds., pp.
261
270
.
American Petroleum Institute
.
27.
Spence
,
D. A.
,
Sharp
,
P. W.
, and
Turcotte
,
D. L.
,
1987
, “
Buoyancy-Driven Crack Propagation: A Mechanism for Magma Migration
,”
J. Fluid Mech.
,
174
, pp.
135
153
. 10.1017/S0022112087000077
28.
Roper
,
S. M.
, and
Lister
,
J. R.
,
2007
, “
Buoyancy-Driven Crack Propagation: The Limit of Large Fracture Toughness
,”
J. Fluid Mech.
,
580
, pp.
359
380
. 10.1017/S0022112007005472
29.
Polyanin
,
A. D.
, and
Manzhirov
,
A. V.
,
1998
,
Handbook of Integral Equations
,
CRC Press
,
Boca Raton, FL
.
30.
Batchelor
,
G.
,
1967
,
An Introduction to Fluid Dynamics
,
Cambridge University Press
,
Cambridge, UK
.
You do not currently have access to this content.