Numerical solution of two-dimensional fingering patterns.
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In this report we numerically simulate fingering patters in unsaturated porous media using a combination of implicit and explicit finite difference methods. These fingering patterns can be modeled using the nonequilibrium Richards Equation (NERE). The NERE is an extended version of the Richards Equation (RE), the conventional equation for modeling flow in porous media. Theoretical research in the past has shown that a propagating water front that is uniform in the lateral direction is conditionally unstable to finite perturbations to the flow field. This instability causes the ?ngering patterns. Our numerical experiments reproduce the theoretical results: small perturbations start to grow if the wave number of the perturbations is small enough and the parameter T is large enough, where T is a measure for the nonequilibrium contribution in the NERE. Initially our numerical experiments were carried out on a fixed grid. To allow for a more accurate numerical solution of the fingering patterns we solve the NERE on an adaptive grid as well.