Wetting front instability in randomly stratified soils

A probabilistic criterion is derived for the onset of wetting front instability during surface water infiltration into a randomly stratified soil. It is based on the common assumption that the natural log hydraulic conductivity of the soil is a random, multivariate Gaussian function of space. Whereas the mean (expectation) of this function may exhibit a drift, its fluctuations about the mean are statistically homogeneous with constant variance and autocorrelation scale. The wetting front is taken to form a sharp boundary. Closed-form expressions for the probability of instability, and for the mean critical wave number, are obtained either directly or via a first-order reliability method. Monte Carlo simulations are used to verify these analytical solutions as well as to determine the mean maximum rate of incipient finger growth and corresponding mean wave number. The effects of applied pressure gradient, capillary pressure head at the wetting front, and statistical parameters of the hydraulic conductivity field on instability and incipient finger growth are investigated for a wide range of these variables.