Stability analysis of traveling wave solution for gravity-driven flow

A linear stability analysis was performed for three models of flow in unsaturated porous media to determine the conditions for growth of small perturbations. The models considered include the conventional Richards equation (RE), a sharp front Richards equation (SFRE) and an extended Richards equation (RRE). The first two models are based on the use of an equilibrium capillary pressure-saturation function, while the third model is derived using a dynamic capillary pressure-saturation function represented by a relaxation coefficient. A traveling wave solution was formulated for each of the governing equations and used as the basic solution of each model. The stability analysis was based on imposing a small perturbation to the basic solution. The RE model yields only the well-known monotonically decreasing saturation profile toward the wetting front, and the wetting front is unconditionally stable. The SFRE model by its nature has a monotonically increasing saturation profile toward the front and an abrupt drop to the initial saturation. This flow is unconditionally unstable. The RRE model is distinct from the others in that it is the only model that is able to produce truly non-monotonic saturation profiles. The wetting front for the RRE model is conditionally stable, i.e. stable for high frequency perturbations, and unstable otherwise. This leads to the existence of a wave-number for maximum amplification, which should relate to the dimensions of fingers in unstable flow.