Group invariant solution for a fluid-driven permeable fracture with Darcy flow in porous rock medium

Group invariant and numerical solutions for the evolution of a two-dimensional fracture with non-zero initial length in permeable rock and driven by a laminar incompressible Newtonian fluid are obtained. The fluid leak-off into the rock mass is modelled using Darcy law. With the aid of lubrication theory and the PKN approximation, a system of nonlinear partial differential equations for the fracture half-width and the extent of leak-off is derived. Since the fluid-rock interface is permeable the nonlinear diffusion equation contains a leak-off velocity sink term. Using the Lie point symmetries the problem is reduced to a boundary value problem for a system of second order ordinary differential equations.