Special Session 129: Qualitative and Quantitative Techniques for Differential Equations arising in Economics, Finance and Natural Sciences
Contents
The group invariant solutions for the evolution of a two dimensional fracture with non-zero initial length in permeable rock and driven by a laminar incompressible non-Newtonian fluid of power-law rheology are derived. With the aid of lubrication theory and the PKN approximation a nonlinear diffusion equation for the fracture half-width is derived. Since the fluid-rock interface is permeable the nonlinear diffusion equation contains a leak-off velocity sink term. A condition, in the form of a first order partial differential equation for the leak-off velocity, is obtained for the nonlinear diffusion equation to possess Lie point symmetries. The general form of the leak-off velocity is derived and the effect of power law rheology on the hydraulic fracturing is investigated.