Speed of Propagation for Some Models of Two-Phase Flow in Porous Media

Flow of two immiscible, incompressible fluids in a porous medium is typically described by a nonlinear advection-diffusion equation for one of the fluid saturations. The diffusion coefficient, which represents the effect of capillary forces on the fluids, is zero when the medium is locally saturated by either fluid since in these limiting cases the effects of capillary forces tend to zero. This degeneracy in the second-order term usually gives rise to the qualitative property that perturbations in saturation propagate with finite speed through regions that are fully saturated by either fluid. This qualitative property is physically realistic. In this work we show that, under certain choices of constitutive relations and modeling approximations, the finite speed of propagation property is lost, despite the fact that the diffusion coefficient is degenerate. The loss of finite speed of propagation is due to unbounded derivatives in the closure relations as the medium becomes saturated by wetting phase. We present analytical and numerical solutions, compare solution dynamics that display finite and infinite speed of propagation, and provide a brief account of numerical difficulties related to the degenerate coefficients.