[1] Stability analysis of gravity-driven unsaturated flow is examined for the general case of Darcian flow with a generalized nonequilibrium capillary pressure-saturation relation. With this nonequilibrium relation the governing equation is referred to as the nonequilibrium Richards equation (NERE). For the special case where the nonequilibrium vanishes, the NERE reduces to the Richards equation (RE), the conventional governing equation for describing unsaturated flow. A generalized linear stability analysis of the RE shows that this equation is unconditionally stable and therefore not able to produce gravity-driven unstable flows for infinitesimal perturbations to the flow field. A much stronger result of unconditional stability for the RE is derived using a nonlinear stability analysis applicable to the general case of heterogeneous porous media. For the general case of the NERE model, results of a linear stability analysis show that the NERE model is conditionally stable, with lower-frequency perturbations being unstable. A result of this analysis is that the nonmonotonicity of the pressure and saturation profile is a requisite condition for flow instability.

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Several alternative mathematical models for describing water flow in unsaturated porous media are presented. These models are based on an equation for conservation of mass of water, and a generalized linear law for water flux (Darcy's law) containing a term called the dynamic capillary pressure. The distinct form of each alternative model is based on the specific form of expression used to describe the dynamic capillary pressure. The conventional representation arises when this pressure is set equal to the equilibrium pressure given by the capillary pressure -saturation function for unsaturated porous media, and this conventional approach leads to the Richards equation. Other models are derived by representing the dynamic capillary pressure by a rheological relationship stating that the pressure is not given directly by the capillary pressure -saturation function. Two forms of rheological relationship are considered in this manuscript, a very general non-equilibrium relation, and a more specific relation expressed by a first-order kinetic equation referred to as a relaxation relation. For the general non-equilibrium relation the system of governing equations is called the general Non-Equilibrium Richards Equation (NERE), and for the case of the relaxation relation the system is called the Relaxation Non-Equilibrium Richards Equation (RNERE). Each of the alternative models was analyzed for flow characteristics under gravity-dominant conditions by using a traveling wave transformation for the model equations, and more importantly the flow described by each model was analyzed for linear stability. It is shown that when a flow field is perturbed by infinitesimal disturbances, the RE is unconditionally stable, while both the NERE and the RNERE are conditionally stable. The stability analysis for the NERE was limited to disturbances in the very low frequency range because of the general form of the NERE model. This analysis resulted in what we call a low-frequency criterion (LFC) for stability.This LFC is also shown to apply to the stability of the RE and the RNERE. The LFC is applied to stability analysis of the RNERE model for conditions of initial saturation less than residual.