Generalized Richards equation to simulate water transport in unsaturated soils

Simulations of water transport in soil are ubiquitous, and the Richardsʹ equation introduced in 1931 is the main tool for that purpose. For experiments on water transport in soil horizontal columns, Richardsʹ equation predicts that volumetric water contents should depend solely on the ratio (distance)/(time)q where q=0.5. Substantial experimental evidence shows that value of q is significantly less than 0.5 in some cases. Donald Nielsen and colleagues in 1962 related values of q<0.5 to `jerky movementsʹ of the wetting front, i.e. occurrences of rare large movements. The physical model of such transport is the transport of particles being randomly trapped and having a power law distribution of waiting periods. The corresponding mathematical model is a generalized Richardsʹ equation in which the derivative of water content on time is a fractional one with the order equal or less than one. We solved this equation numerically and fitted the solution to data on horizontal water transport. The classical Richardsʹ equation predicted a decrease of the soil water diffusivity for the same water content as infiltration progressed whereas the generalized Richardsʹ equation described all observations well with a single diffusivity function. Validity of the generalized Richardsʹ equation indicates presence of memory effects in soil water transport phenomena and may help to explain scale-dependence and variability in soil hydraulic conductivity encountered by researchers who applied classical Richardsʹ equation.