Asymptotic Analysis of an Elasticity Equation for a Finger-Like Hydraulic Fracture |
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Authors: | José I Adachi Anthony P Peirce |
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Institution: | (1) Schlumberger Data and Consulting Services, 1325 S. Dairy Ashford Rd., Houston, TX 77077, USA;(2) Dept. of Mathematics, The University of British Columbia, Vancouver, BC V6T 1Z2, Canada |
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Abstract: | We derive a novel integral equation relating the fluid pressure in a finger-like hydraulic fracture to the fracture width.
By means of an asymptotic analysis in the small height to length ratio limit we are able to establish the action of the integral
operator for receiving points that lie within three distinct regions: (1) an outer expansion region in which the dimensionless
pressure is shown to be equal to the dimensionless width plus a small correction term that involves the second derivative
of the width, which accounts for the nonlocal effects of the integral operator. The leading order term in this expansion is
the classic local elasticity equation in the PKN model that is widely used in the oil and gas industry; (2) an inner expansion
region close to the fracture tip within which the action of the elastic integral operator is shown to be the same as that
of a finite Hilbert transform associated with a state of plane strain. This result will enable pressure singularities and
stress intensity factors to be incorporated into analytic models of these finger-like fractures in order to model the effect
of material toughness; (3) an intermediate region within which the action of the Fredholm integral operator of the first kind
is reduced to a second kind operator in which the integral term appears as a small perturbation which is associated with a
convergent Neumann series. These results are important for deriving analytic models of finger-like hydraulic fractures that
are consistent with linear elastic fracture mechanics.
Submitted to Journal of Elasticity on February 5, 2007. Re-submitted with revisions on May 30, 2007. |
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Keywords: | Hydraulic fracturing Fracture mechanics Integral equations Asymptotic expansions |
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