Mathematical analysis of the influence of power-law fluid rheology on a capillary diffusion zone

Non-Newtonian fluids are used in current oil recovery processes. These fluids do not satisfy the linear Darcy Law for flow through porous media. A generalization is needed to model the flow processes involved. Furthermore, when two immiscible fluids are present in a porous medium, capillary pressure will cause a transition zone to develop between them. This transition zone may lead to early breakthrough of water into an oil well. In this paper, we study the effect of the non-Newtonian behaviour of fluids on a capillary transition zone. A general framework is set up for modelling processes involving two-phase flow of non-Newtonian, immiscible and incompressible fluids in a porous medium. The equations are applied to the one-dimensional diffusion process of power-law fluids. The model allows for general capillary pressure and relative permeability functions. The mathematical model consists of a degenerate diffusion equation, giving rise to a free boundary formulation. The free boundaries represent the endpoint of the diffusion zone. Qualitative properties, and some analytical solutions, can be obtained for the saturation profile. Two numerical methods are presented. One is applicable if the rheology of both fluids is modelled with equal powers. The other is applicable to the general situation. Both methods make use of the qualitative, analytical properties of the solutions, which clearly improved the results obtained by standard methods. One-dimensional results can be used to interpret the general multi-dimensional flow behaviour of non-Newtonian fluids when capillarity is considered. They can also be used to test numerical algorithms developed for multi-dimensional displacement processes.