TWO FLUID FLOW IN POROUS MEDIA

The Gray-Hassanizadeh model, for flow in a porous medium of two immiscible fluids such as oil and water, is a scalar equation for the evolution of the saturation of one of the fluids. The model is based on Darcy's law, coupled with a constitutive equation for the capillary pressure that incorporates a rate-dependence to capture the relaxation of interfacial energy towards equilibrium. The model has interesting properties, including the structure of traveling waves explored in this paper. In particular, we find that for certain forms of relative permeability, there are undercompressive shocks that are degenerate in that the corresponding smooth traveling wave drops to zero saturation in finite time, due to the singularity in the PDE at zero saturation. In the second half of the paper, we report on a two-dimensional stability result for the shock wave, regarded as a plane wave in two dimensions. We find a criterion for linearized stability that predicts that some Lax shocks are stable, while others are unstable. This analysis relates to the Saffman-Taylor instability [9] and is a version of a result of Yortsos and Hickernell [15] for stability of smooth traveling waves of the full system including capillary pressure. Whereas matched asymptotics are used in [15], at the hyperbolic level of this paper the analysis of stability and instability is much more transparent.