Scaling Properties in the Non-Linear Diffusion Equation and Its Application to the Problem of Water Transport in Water Unsaturated Soils

Abstract

In this work, we investigate the scaling properties related to the nonlinear fractional diffusion equations and indicate the possibilities to the applications of these equations to simulate the water transport in unsaturated soils. Usually, the water transport in soils with anomalous diffusion, the dependence of concentration on time (t) q and distance (x) may be expressed in term of a single variable given by λ_q = x / t^q . In particular, for q = 1/2 the systems obey Fick’s law and Richards’ equation for water transport. We show that a generalization of Richards’ equation via fractional approach can incorporate the above property.