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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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