Group invariant solutions for a pre-existing fracture driven by a non-Newtonian fluid in permeable and impermeable rock

Abstract

The aim of the thesis is to derive group invariant, exact, approximate analytical and numerical solutions for a two-dimensional laminar, non-Newtonian pre-existing hydraulic fracture propagating in impermeable and permeable elastic media. The fracture is driven by the injection of an incompressible, viscous non-Newtonian fluid of power law rheology in which the fluid viscosity depends on the magnitude of the shear rate and on the power law index n > 0. By the application of lubrication theory, a nonlinear diffusion equation relating the half-width of the fracture to the fluid pressure is obtained. When the interface is permeable the nonlinear diffusion equation has a leak-off velocity sink term. The half-width of the fracture and the net fluid pressure are linearly related through the PKN approximation. 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 have Lie point symmetries. The general form of the leak-off velocity is derived. Using the Lie point symmetries the problem is reduced to a boundary value problem for a second order ordinary differential equation. The leak-off velocity is further specified by assuming that it is proportional to the fracture half-width. Only fluid injection at the fracture entry is considered. This is the case of practical importance in industry. Two exact analytical solutions are derived. In the first solution there is no fluid injection at the fracture entry while in the second solution the fluid velocity averaged over the width of the fracture is constant along the length of the fracture. For other working conditions at the fracture entry the problem is solved numerically by transforming the boundary value problem to a pair of initial value problems. The numerical solution is matched to the asymptotic solution at the fracture tip. Since the fracture is thin the fluid velocity averaged over the width of the fracture is considered. For the two analytical solutions the ratio of the averaged fluid velocity to the velocity of the fracture tip varies linearly along the fracture. For other working conditions the variation is approximately linear. Using this observation approximate analytical solutions are derived for the fracture half-width. The approximate analytical solutions are compared with the numerical solutions and found to be accurate over a wide range of values of the power-law index n and leak-off parameter β. The conservation laws for the nonlinear diffusion equation are investigated. When there is fluid leak-off conservation laws of two kinds are found which depend in which component of the conserved vector the leak-off term is included. For a Newtonian fluid two conservation laws of each kind are found. For a non-Newtonian fluid the second conservation law does not exist. The behaviour of the solutions for shear thinning, Newtonian and shear thickening fluids are qualitatively similar. The characteristic time depends on the properties of the fluid which gives quantitative differences in the solution for shear thinning, Newtonian and shear thickening fluids.