Scaling dependence on time and distance in nonlinear fractional diffusion equations and possible applications to the water transport in soils

TL;DR

This paper explores the scaling properties of nonlinear fractional diffusion equations and their potential application to modeling water transport in soils, especially under anomalous diffusion conditions.

Contribution

It introduces a fractional generalization of Richards' equation to better simulate water movement in unsaturated soils with anomalous diffusion behaviors.

Findings

Fractional diffusion equations capture scaling dependence on time and distance.

The generalized Richards' equation can model water transport with anomalous diffusion.

Potential applications in soil hydrology and related fields.

Abstract

Recently, fractional derivatives have been employed to analyze various systems in engineering, physics, finance and hidrology. For instance, they have been used to investigate anomalous diffusion processes which are present in different physical systems like: amorphous semicondutors, polymers, composite heterogeneous films and porous media. They have also been used to calculate the heat load intensity change in blast furnace walls, to solve problems of control theory \ and dynamic problems of linear and nonlinear hereditary mechanics of solids. 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 and distance may be expressed in term of a single variable given by $\lambda_q = x/t^{q}.$ 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.