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A semi-infinite hydraulic fracture driven by a shear-thinning fluid

Published online by Cambridge University Press:  25 January 2018

Fatima-Ezzahra Moukhtari  and
Brice Lecampion Open the ORCID record for Brice Lecampion [Opens in a new window]
Fatima-Ezzahra Moukhtari
Affiliation:
Geo-Energy Laboratory - Gaznat Chair on Geo-Energy, Ecole Polytechnique Fédérale de Lausanne, ENAC-IIC-GEL-EPFL, Station 18, CH-1015, Switzerland
Brice Lecampion*
Affiliation:
Geo-Energy Laboratory - Gaznat Chair on Geo-Energy, Ecole Polytechnique Fédérale de Lausanne, ENAC-IIC-GEL-EPFL, Station 18, CH-1015, Switzerland
*
†Email address for correspondence: [email protected]
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Abstract

We use the Carreau rheological model which properly accounts for the shear-thinning behaviour between the low and high shear rate Newtonian limits to investigate the problem of a semi-infinite hydraulic fracture propagating at a constant velocity in an impermeable linearly elastic material. We show that the solution depends on four dimensionless parameters: a dimensionless toughness (function of the fracture velocity, confining stress, material and fluid parameters), a dimensionless transition shear stress (related to both fluid and material behaviour), the fluid shear-thinning index and the ratio between the high and low shear rate viscosities. We solve the complete problem numerically combining a Gauss–Chebyshev method for the discretization of the elasticity equation, the quasi-static fracture propagation condition and a finite difference scheme for the width-averaged lubrication flow. The solution exhibits a complex structure with up to four distinct asymptotic regions as one moves away from the fracture tip: a region governed by the classical linear elastic fracture mechanics behaviour near the tip, a high shear rate viscosity asymptotic and power-law asymptotic region in the intermediate field and a low shear rate viscosity asymptotic far away from the fracture tip. The occurrence and order of magnitude of the extent of these different viscous asymptotic regions are estimated analytically. Our results also quantify how shear thinning drastically reduces the size of the fluid lag compared to a Newtonian fluid. We also investigate simpler rheological models (power law, Ellis) and establish the small domain where they can properly reproduce the response obtained with the complete rheology.

JFM classification

Geophysical and Geological Flows: Geophysical and Geological Flows Low-Reynolds-number flows: Low-Reynolds-number flows Non-Newtonian Flows: Non-Newtonian Flows
Type
JFM Papers
Information
Journal of Fluid Mechanics , Volume 838 , 10 March 2018 , pp. 573 - 605
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Copyright
© 2018 Cambridge University Press 

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References

Adachi, J. & Detournay, E. 2002 Self-similar solution of a plane-strain fracture driven by a power-law fluid. Intl J. Numer. Anal. Meth. Geomech. 26 (6), 579–604.CrossRefGoogle Scholar
Barbati, A., Desroches, J., Robisson, A. & McKinley, G. 2016 Complex fluids and hydraulic fracturing. Annu. Rev. Chem. Biomol. Engng 7, 415–453.CrossRefGoogle ScholarPubMed
Bird, R., Armstrong, R. & Hassager, O. 1987 Dynamics of Polymeric Liquids, Fluid Mechanics, vol. 1. Wiley and Sons Inc.Google Scholar
Brodkey, R. S. 1969 The Phenomena of Fluid Motions. Dover.CrossRefGoogle Scholar
Bunger, A. & Detournay, E. 2008 Experimental validation of the tip asymptotics for a fluid-driven crack. J. Mech. Phys. Solids 56 (11), 3101–3115.CrossRefGoogle Scholar
Caricchi, L., Burlini, L., Ulmer, P., Gerya, T., Vassalli, M. & Papale, P. 2007 Non-Newtonian rheology of crystal-bearing magmas and implications for magma ascent dynamics. Earth Planet. Sci. Lett. 264 (3), 402–419.CrossRefGoogle Scholar
Carreau, P. 1972 Rheological equations from molecular network theories. Trans. Soc. Rheol. 16 (1), 99–127.CrossRefGoogle Scholar
Cross, M. 1965 Rheology of non-Newtonian fluids: a new flow equation for pseudoplastic systems. J. Colloid Sci. 20 (5), 417–437.CrossRefGoogle Scholar
Desroches, J., Detournay, E., Lenoach, B., Papanastasiou, P., Pearson, J., Thiercelin, M. & Cheng, A. 1994 The crack tip region in hydraulic fracturing. Proc. R. Soc. Lond. A 447 (1929), 39–48.Google Scholar
Detournay, E. 2016 Mechanics of hydraulic fractures. Annu. Rev. Fluid Mech. 48, 311–339.CrossRefGoogle Scholar
Detournay, E. & Peirce, A. 2014 On the moving boundary conditions for a hydraulic fracture. Intl J. Engng Sci. 84, 147–155.CrossRefGoogle Scholar
Dontsov, E. & Peirce, A. 2015 A non-singular integral equation formulation to analyse multiscale behaviour in semi-infinite hydraulic fractures. J. Fluid Mech. 781, R1.CrossRefGoogle Scholar
Dontsov, E. & Peirce, A. 2017 A multiscale implicit level set algorithm (ILSA) to model hydraulic fracture propagation incorporating combined viscous, toughness, and leak-off asymptotics. Comput. Meth. Appl. Mech. Engng 313, 53–84.CrossRefGoogle Scholar
Economides, M. & Nolte, K. 2000 Reservoir Stimulation, vol. 18. Wiley.Google Scholar
Garagash, D. 2009 Scaling of physical processes in fluid-driven fracture: perspective from the tip. IUTAM Symposium on Scaling in Solid Mechanics, pp. 91–100. Springer.CrossRefGoogle Scholar
Garagash, D. & Detournay, E. 2000 The tip region of a fluid-driven fracture in an elastic medium. Trans. ASME J. Appl. Mech. 67 (1), 183–192.CrossRefGoogle Scholar
Garagash, D., Detournay, E. & Adachi, J. 2011 Multiscale tip asymptotics in hydraulic fracture with leak-off. J. Fluid Mech. 669, 260–297.CrossRefGoogle Scholar
Guillot, D. & Dunand, A. 1985 Rheological characterization of fracturing fluids by using laser anemometry. Soc. Petrol. Engng J. 25 (01), 39–45.CrossRefGoogle Scholar
Ioakimidis, N. & Theocaris, P. 1980 The practical evaluation of stress intensity factors at semi-infinite crack tips. Engng Fracture Mech. 13 (1), 31–42.CrossRefGoogle Scholar
Kefi, S., Lee, J., Pope, T. L., Sullivan, P., Nelson, E., Nunez Hernanez, A., Olsen, T., Parlar, M., Powers, B., Roy, A. et al. 2004 Expanding applications for viscoelastic surfactants. Oilfield Rev. 16 (4), 10–23.Google Scholar
Lister, J. 1990 Buoyancy-driven fluid fracture: the effects of material toughness and of low-viscosity precursors. J. Fluid Mech. 210, 263–280.CrossRefGoogle Scholar
Matsuhisa, S. & Bird, R. 1965 Analytical and numerical solutions for laminar flow of the non-Newtonian Ellis fluid. AIChE J. 11 (4), 588–595.CrossRefGoogle Scholar
Myers, T. 2005 Application of non-Newtonian models to thin film flow. Phys. Rev. E 72 (6), 066302.Google ScholarPubMed
Peirce, A. 2015 Modeling multi-scale processes in hydraulic fracture propagation using the implicit level set algorithm. Comput. Meth. Appl. Mech. Engng 283, 881–908.CrossRefGoogle Scholar
Peirce, A. P. & Detournay, E. 2008 An implicit level set method for modeling hydraulically driven fractures. Comput. Meth. Appl. Mech. Engng 197 (33–40), 2858–2885.CrossRefGoogle Scholar
Pipe, C., Majmudar, T. & McKinley, G. 2008 High shear rate viscometry. Rheol. Acta 47 (5–6), 621–642.CrossRefGoogle Scholar
Rice, J. 1968 Mathematical analysis in the mechanics of fracture. Fracture: An Advanced Treatise 2, 191–311.Google Scholar
Sochi, T. 2014 Using the Euler-Lagrange variational principle to obtain flow relations for generalized Newtonian fluids. Rheol. Acta 53 (1), 15–22.CrossRefGoogle Scholar
Sochi, T. 2015 Analytical solutions for the flow of Carreau and Cross fluids in circular pipes and thin slits. Rheol. Acta 54 (8), 745–756.CrossRefGoogle Scholar
Sousa, J., Carter, B. & Ingraffea, A. 1993 Numerical simulation of 3d hydraulic fracture using newtonian and power-law fluids. Intl J. Rock Mech. Mining Sci. Geomech. Abstracts 30 (7), 1265–1271.Google Scholar
Spence, D. & Sharp, P. 1985 Self-similar solutions for elastohydrodynamic cavity flow. Proc. R. Soc. Lond. A 400 (1819), 289–313.Google Scholar
Spence, D. & Turcotte, D. 1985 Magma-driven propagation of cracks. J. Geophys. Res. 90 (B1), 575–580.CrossRefGoogle Scholar
Stoeckhert, F., Molenda, M., Brenne, S. & Alber, M. 2015 Fracture propagation in sandstone and slate-Laboratory experiments, acoustic emissions and fracture mechanics. J. Rock Mech. Geotech. Engng 7 (3), 237–249.CrossRefGoogle Scholar
Viesca, R. & Garagash, D. 2015 Ubiquitous weakening of faults due to thermal pressurization. Nat. Geosci. 8 (11), 875–879.CrossRefGoogle Scholar