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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Fitt, A.D., Mason, D.P. & Moss, E.A. Group invariant solution for a pre-existing fluid-driven fracture in impermeable rock. Z. angew. Math. Phys. 58, 1049–1067 (2007). https://doi.org/10.1007/s00033-007-7038-2
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DOI: https://doi.org/10.1007/s00033-007-7038-2