Wetting front instability and fingering in random permeability fields

Numerical simulations are used to investigate wetting front instability and finger evolution in the vertical plane during surface-water infiltration into a randomly heterogeneous soil. As is common in the literature, the natural log permeability of the soils is taken to be a random, multivariate Gaussian function of space. The mean (expectation) of this function is constant and its fluctuations about the mean are statistically homogeneous and isotropic with constant variance and autocorrelation scale. The wetting front is allowed to diffuse in accord with Richards' equation of flow in a partially saturated soil. The infiltrating water is labeled with a passive tracer, the 50% concentration contour of which is taken to indicate the position of the wetting front. The variogram of the contour is introduced to characterize the length and width of the fingers. Results show that whereas mean finger length grows at a nonlinear rate that diminishes monotonically with time, mean finger width remains constant in time. Both mean finger growth rate and mean finger width increase with the variances and correlation scale of permeability.