Numerical solution of two-dimensional fingering patterns.
Summary
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.