Modeling finger development and persistence in initially dry porous media

The mechanism for the growth and persistence of gravity-driven fingered flow of water in initially dry porous media is described. A Galerkin finite element solution of the two-dimensional Richards equation with the associated parameter equations for capillary hysteresis in the water retention function is presented. A scheme for upstream weighting of internodal unsaturated hydraulic conductivities is applied to limit smearing of steep wetting fronts. The growth and persistence of a single finger in an initially dry porous media is simulated using this numerical solution scheme. To adequately simulate fingered flow, it was found that the upstream weighting factor had to be negative, meaning that the internodal unsaturated hydraulic conductivities were weighted more by the downstream node. It is shown that the growth and persistence of a finger is sensitive to the character of the porous media water retention functions. For porous media where the water-entry capillary pressure on the main wetting function is less than the air-entry capillary pressure on the main drainage function, a small perturbation will grow into a finger, and during sequential drainage and wetting the finger will persist. In contrast, for porous media where the water-entry capillary pressure on the main wetting function is greater than the air-entry capillary pressure on the main drainage function, the same small perturbation will dissipate by capillary diffusion. The finger widths derived from the numerical simulation are similar to those predicted by analytical theory.