Group invariant solution for a pre-existing fluid-driven fracture in impermeable rock |
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Authors: | A. D. Fitt D. P. Mason E. A. Moss |
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Affiliation: | (1) Vattenfall Power Consultant AB, Timmermansgatan 25, 971 77 Lule?, Sweden;(2) GeoForschungZentrum Potsdam, Potsdam, Germany;(3) Institut de Physique du Globe de Strasbourg, Strasbourg, France;(4) Lule? University of Technology, Lule?, Sweden |
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Abstract: | The propagation of a two-dimensional fluid-driven fracture in impermeable rock is considered. The fluid flow in the fracture is laminar. By applying lubrication theory a partial differential equation relating the half-width of the fracture to the fluid pressure is derived. To close the model the PKN formulation is adopted in which the fluid pressure is proportional to the half-width of the fracture. By considering a linear combination of the Lie point symmetries of the resulting non-linear diffusion equation the boundary value problem is expressed in a form appropriate for a similarity solution. The boundary value problem is reformulated as two initial value problems which are readily solved numerically. The similarity solution describes a preexisting fracture since both the total volume and length of the fracture are initially finite and non-zero. Applications in which the rate of fluid injection into the fracture and the pressure at the fracture entry are independent of time are considered. |
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