Self-similar solutions for unsteady-state flow of non-Newtonian fluids in porous media

In this paper we study the one-dimensional flow of compressible non-Newtonian power-law fluids generated by fluid withdrawal at the boundary of an infinite reservoir having plane or radial geometry. The withdrawal is effectuated such that the pumped discharge is a prescribed function of time. The power-law fluid flow model is based on a modified Darcy’s law taking into account the nonlinear rheological effects of the fluid behavior. Coupling the flow law with the continuity equation yields a nonlinear second-order partial differential equation in the fluid pressure. The latter equation, with relevant boundary conditions, is amenable to a similarity transformation which reduce the partial differential equation into a nonlinear ordinary differential equation, provided that the injection flow rate as a function of time takes a particular form, depending on the exponent of the flow law and geometry. Solving the nonlinear differential equation yields the pressure distribution in space and time as a function of fluid properties and withdrawal intensity. The resulting integral can be expressed by analytical functions if the fluid consistency index n is of the form (k+1)/k, where k is a positive integer; otherwise, a single numerical integration is required. Explicit expressions are provided for the cases k = 1 and k = 2, while for higher values of k, results can be obtained via recursive formulae. For a Newtonian fluid (n = 1), the self-similar variable reduces to the Boltzmann transformation; in radial geometry, the variable flow rate reduces to a constant one, and the pressure disturbance with respect to the initial condition takes the form of the Theis integral, albeit with pressure replacing hydraulic head.