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TECHNICAL PAPERS

# An Asymptotic Framework for the Analysis of Hydraulic Fractures: The Impermeable Case

[+] Author and Article Information
S. L. Mitchell1

Department of Mathematics, University of British Columbia, Vancouver, BC, V6T 1Z2, Canadasarah@iam.ubc.ca

R. Kuske, A. P. Peirce

Department of Mathematics, University of British Columbia, Vancouver, BC, V6T 1Z2, Canada

1

Author to whom correspondence should be addressed.

J. Appl. Mech 74(2), 365-372 (Mar 13, 2006) (8 pages) doi:10.1115/1.2200653 History: Received October 31, 2005; Revised March 13, 2006

## Abstract

This paper presents a novel asymptotic framework to obtain detailed solutions describing the propagation of hydraulic fractures in an elastic material. The problem consists of a system of nonlinear integro-differential equations and a free boundary problem. This combination of local and nonlocal effects leads to transitions on a small scale near the crack tip, which control the behavior across the whole fracture profile. These transitions depend upon the dominant physical process(es) and are identified by simultaneously scaling the associated parameters with the distance from the tip. A smooth analytic solution incorporating several physical processes in the crucial tip region can be constructed using this new framework. In order to clarify the exposition of the new methodology, this paper is confined to considering the impermeable case in which only the two physical processes of viscous dissipation and structure energy release compete.

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## Figures

Figure 1

The KGD profile and its cross-section

Figure 2

Diagram of the solution Ω versus ξ near the fracture tip with transition region at 1−ξ=O(Pkm2)

Figure 3

Solution profiles of Ω versus ξ in the viscosity dominated regime with Gm=1 and Gk=0.45,0.35,0.15 (and so Pkm=0.091,0.043,0.003)

Figure 4

Plot of logΩ versus log(1−ξ) in the viscosity dominated regime with Gm=1, Gk=0.35 (and so Pkm=0.043). The solid line denotes the leading order power law solutions, as indicated above.

Figure 5

Solution profiles of Π versus ξ corresponding to Ω in Fig. 3

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