A Numerical Method for a Nonlocal Form of Richards' Equation Based on Peridynamic Theory

TL;DR

This paper introduces a spectral numerical method based on Chebyshev transforms for solving a nonlocal form of Richards' equation using peridynamic theory, effectively modeling water content in fractured porous media.

Contribution

It develops a novel spectral numerical scheme that converges for a nonlocal Richards' equation, incorporating peridynamic theory and demonstrating its effectiveness on various soil types.

Findings

The scheme converges to the unique solution in Sobolev space.

Effective modeling of water infiltration in fractured soils.

Inclusion of sink terms for root water uptake.

Abstract

Forecasting water content dynamics in heterogeneous porous media has significant interest in hydrological applications; in particular, the treatment of infiltration when in presence of cracks and fractures can be accomplished resorting to peridynamic theory, which allows a proper modeling of non localities in space. In this framework, we make use of Chebyshev transform on the diffusive component of the equation and then we integrate forward in time using an explicit method. We prove that the proposed spectral numerical scheme provides a solution converging to the unique solution in some appropriate Sobolev space. We finally exemplify on several different soils, also considering a sink term representing the root water uptake.