Some remarks on a mathematical model for water flow in porous media with competition between transport and diffusion

TL;DR

This paper analyzes a nonlinear PDE model for water flow in porous media, incorporating gravity effects, and presents a numerical solution approach using finite element methods in Matlab to study the competition between transport and diffusion.

Contribution

It introduces a novel way to include gravity in a nonlinear PDE model for water flow in soils and develops a numerical solution method for complex cases.

Findings

Numerical simulations demonstrate the model's behavior under various conditions.

The finite element method effectively solves the nonlinear PDE in Matlab.

Gravity effects significantly influence water transport and diffusion in the model.

Abstract

The contribution deals with the mathematical modelling of fluid flow in porous media, in particular water flow in soils, with the aim of describing the competition between transport and diffusion. The analysis is based on a mathematical model developed by B. Detmann, C. Gavioli, and P. Krej\v{c}\'i, in which the effects of gravity are included in a novel way. The model consists of a nonlinear partial differential equation describing both the diffusion and the gravitational transport of water. Although analytical solutions can be obtained for some special cases, only numerical solutions are available in more general situations. The solving algorithm is based on a time discretisation and the finite element method, and is written in Matlab. The results of the numerical simulations are shown and the behaviour of the model is discussed.