A phase field model for partially saturated geomaterials describing fluid–fluid displacements. Part I: The model and one-dimensional analysis

A model describing immiscible fluid-fluid displacements in partially saturated porous media is presented. This is based on a phase field approach that interprets the mixture of wetting (liquid water) and non-wetting (air) fluids within the pore space as a single saturating non-uniform pore fluid characterized by a phase field parameter, which is considered to be the saturation degree of the wetting fluid. While the standard retention curve provides for the retention properties of the pore walls, a Cahn-Hilliard like double-well energy is employed to describe the possible co-existence of the immiscible fluid phases. An enhanced description of the macroscopic surface tension between the fluid phases is obtained naturally within the phase field framework due to a regularization that depends on the spatial gradient of the water content. A generalized Darcy's law is used to describe dissipation due to fluid flow driven by the gradient of a generalized chemical potential. Thus, in the context of soil hydrology this model is interpreted as an extension to the classical Richards equation governing the spatio-temporal evolution of the phase field parameter. Employing a convex-concave flux function it is shown, using one-dimensional analysis, that both imbibition and drainage fronts can be modeled in this phase field framework. The non-monotonicities observed in the resolved solutions are explained using a combination of asymptotic matching techniques and dynamical systems analysis.