Non-linear effects on some unsteady non-Darcian flows through porous media

Abstract

In this paper we study two classes of non-Darcian flows through porous media, namely the turbulent flow of gases of polytropic thermodynamic evolution, and the flow of non-Newtonian power law fluids. We consider the problem of determining the one-dimensional flow generated by injecting gas into a porous medium. It is assumed that the injection is effectuated such that the pressure of injection or the mass velocity of injection is a prescribed function of time. For the power law fluid flow we investigate unconfined radial axisymmetric flow with a free surface by considering the spreading of fluid from a centrally located well where the level of fluid or the flow rate varies in time in a prescribed fashion. In the equations governing the gas flow, the momentum equation is the Darcy-Forchheimer equation for high Reynolds number flow, while the power law fluid flow model is based on a modified Darcy's law to take into account the non-linear rheological effects on the flow behavior. In both cases the governing equations belong to a class of non-linear degenerate parabolic equations with solutions exhibiting traveling wave characteristics for certain fluid and porous medium properties. If the boundary conditions are power law functions of time we present similarity transformations which reduce the governing partial differential equations to non-linear ordinary differential equations which are solved numerically by a shooting method. The numerical results are compared with the exact closed form solutions for certain particular cases. The agreement between the numerical and exact solutions is shown to be excellent.