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A non-singular integral equation formulation to analyse multiscale behaviour in semi-infinite hydraulic fractures

Published online by Cambridge University Press:  16 September 2015

E. V. Dontsov  and
A. P. Peirce
E. V. Dontsov
Affiliation:
Department of Mathematics, University of British Columbia, Vancouver, BCV6T 1Z2, Canada
A. P. Peirce*
Affiliation:
Department of Mathematics, University of British Columbia, Vancouver, BCV6T 1Z2, Canada
*
†Email address for correspondence: peirce@math.ubc.ca
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Abstract

This study revisits the problem of a steadily propagating semi-infinite hydraulic fracture in which the processes of toughness-related energy release, viscous dissipation and leak-off compete on multiple length scales. This problem typically requires the solution of a system of integro-differential equations with a singular kernel, which is complicated by the need to capture extremely disparate length scales. In this study the governing equations are rewritten in the form of one non-singular integral equation. This reformulation enables the use of standard numerical techniques to capture the complete multiscale behaviour accurately and efficiently. This formulation also makes it possible to approximate the problem by a separable ordinary differential equation, whose closed-form solution captures the multiscale behaviour sufficiently accurately to be used in practical applications. We also consider a similar reformulation of the equations governing the propagation of a buoyancy-driven semi-infinite hydraulic fracture. The resulting numerical solution is able to capture the near-tip multiscale behaviour efficiently and agrees well with published solutions calculated in the large-toughness limit.

JFM classification

Boundary Layers: Boundary layer structure Geophysical and Geological Flows: Magma and lava flow Low-Reynolds-number flows: Porous media
Type
Rapids
Information
Journal of Fluid Mechanics , Volume 781 , 25 October 2015 , R1
Copyright
© 2015 Cambridge University Press 

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Footnotes

‡

Department of Civil and Environmental Engineering, University of Houston, Houston, TX 77204, USA.

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