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

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Abstract

[1] 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.

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