A similarity solution for a pre-existing fluid driven fracture in a permeable rock

Abstract

The problem of a two-dimensional, pre-existing, fluid-driven fracture propagating in a permeable rock is investigated. The fracturing fluid is a viscous, incompressible Newtonian fluid and the flow of fluid inside the fracture is laminar. The elasticity of the rock is modelled using the Cauchy principal value integral derived from linear elastic fracture mechanics. With the aid of lubrication theory, a nonlinear partial integro-differential equation relating the fracture half-width to the pressure and leak-off velocity is derived. Similarity solutions are derived for the fracture half-width, pressure and leak-off velocity and are used to reduce the partial integro-differential equation to an ordinary integro-differential equation. In order to close the problem, a model in which the leak-off velocity is proportional to the fracture half-width and the gradient of the fluid–rock interface was used. Numerical results are obtained for the fracture length, fracture half-width and the net fluid pressure.