Stochastic Foundation to Solving Transient Unsaturated Flow Problems Using a Fractional Dispersion Term

Abstract

Mathematical modeling of unsaturated water seepage through soils uses the core concepts of continuum mechanics. More precisely, it combines the continuum mass balance equation and Darcy–Buckingham Law into the Richards equation for water flow in an unsaturated porous medium. This paper proposes the possibility of an interpretation following another perspective: using the concepts of statistical mechanics and the movement of quasi-molecules of water in soils. It resumes an underexplored approach using the Langevin equation to describe the movement of the quasi-molecules. By replacing the term that accounts for the Gaussian white noise for a term that follows a stable distribution, the mathematical manipulations render a fractional partial differential equation that describes the water flow into unsaturated soils. A constructed analytical solution is proposed for the new equation. A set of experimental data for an unsaturated flow on a soil column is fitted using the new model. Its results are compared to a fit using the integer-order linearized Richards equation solution.