On the porous-continuum modeling of gravity-driven fingers in unsaturated materials: Extension of standard theory with a hold-back-pile-up effect

Abstract

The traditional Richards equation (RE) in combination with standard monotonic properties (constitutive relations and hysteretic equations of state) has been shown to lack critical physics required to model gravity-driven fingering (GDF). We extend the RE with an experimentally observed hold-back-pile-up (HBPU) effect not captured in the standard porous-continuum RE formulation. We postulate the HBPU effect is tied to wetting front sharpness and can be mathematically formulated in a variety of ways to include hypodiffusive, hyperbolic, and mixed spatial-temporal forms involving respectively a Laplacian, a second-order derivative in time, and a Laplacian of a first-order derivative in time of the state variables. For each, we can infer an extended flux relation comprised of the Darcy-Buckingham flux plus an additional component due to the HBPU effect. Extended flux relations that are mathematically similar to each can be found in the single-phase and multiphase flow literature, however, all with very different underlying conceptualizations of the possible physics.