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Group invariant solution for a pre-existing fluid-driven fracture in impermeable rock

Group invariant solution for a pre-existing fluid-driven fracture in impermeable rock
Group invariant solution for a pre-existing fluid-driven fracture in impermeable rock
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.
lie point symmetries, similarity solution, fluid solid interaction, fracture, lubrication theory, nonlinear diffusion, PKN fracture theory
0044-2275
1049-1067
Fitt, A.D.
51b348d7-b553-43ac-83f2-3adbea3d69ab
Mason, D.P.
fbe5c473-ae44-4a2d-a7ef-1c2c72f44b8d
Moss, E.A.
f199b0b4-dd61-4fe1-be22-99dcb585f290
Fitt, A.D.
51b348d7-b553-43ac-83f2-3adbea3d69ab
Mason, D.P.
fbe5c473-ae44-4a2d-a7ef-1c2c72f44b8d
Moss, E.A.
f199b0b4-dd61-4fe1-be22-99dcb585f290

Fitt, A.D., Mason, D.P. and Moss, E.A. (2007) Group invariant solution for a pre-existing fluid-driven fracture in impermeable rock. Zeitschrift für Angewandte Mathematik und Physik (ZAMP), 58 (6), 1049-1067. (doi:10.1007/s00033-007-7038-2).

Record type: Article

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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More information

Published date: November 2007
Keywords: lie point symmetries, similarity solution, fluid solid interaction, fracture, lubrication theory, nonlinear diffusion, PKN fracture theory

Identifiers

Local EPrints ID: 50316
URI: http://eprints.soton.ac.uk/id/eprint/50316
ISSN: 0044-2275
PURE UUID: 829380f3-59e7-4a31-a0bf-f288945c7610

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Date deposited: 14 Feb 2008
Last modified: 07 Jan 2022 22:28

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Contributors

Author: A.D. Fitt
Author: D.P. Mason
Author: E.A. Moss

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