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On the porous continuum‐scale modeling of gravity‐driven fingers in unsaturated materials: Numerical solution of a hypodiffusive governing equation that incorporates a hold‐back‐pile‐up effect

We consider the use of a hypodiffusive governing equation (HDE) for the porous‐continuum modeling of gravity‐driven fingers (GDF) as occur in initially dry, highly nonlinear, and hysteretic porous media. In addition to the capillary and gravity terms within the traditional Richards equation, the HDE contains a hypodiffusive term that models an experimentally observed hold‐back‐pile‐up (HBPU) effect and thus imparts nonmonotonicity at the wetting front. In its dimensionless form the HDE contains the dimensionless hypodiffusion number, NHD. As NHD increases, one‐dimensional (1D) numerical solutions transition from monotonic to nonmonotonic. Considering the experimentally observed controls on GDF occurrence, as either the initial moisture content and applied flux increase or the material nonlinearity decreases, solutions undergo the required transition back to monotonic. Additional tests for horizontal imbibition and capillary rise show the HDE to yield the required monotonic response but display sharper fronts for NHD > 0. Finally, two‐dimensional (2D) numerical solutions illustrate that in parameter space where the 1D HDE yields nonmonotonicity, in 2D it forms nonmonotonic GDF.