Surface permeability, capillary transport and the Laplace-Beltrami problem

TL;DR

This paper links surface permeability in porous media to the solution of the Laplace-Beltrami problem, enhancing the modeling of liquid spreading at low saturation levels through surface finite element methods.

Contribution

It introduces a method to determine effective surface permeability via the Laplace-Beltrami Dirichlet problem using surface finite element techniques.

Findings

Effective surface permeability can be accurately computed.

The approach improves predictive modeling of liquid spreading.

Mathematical results align with experimental data.

Abstract

We have established previously, in a lead-in study, that the spreading of liquids in particulate porous media at low saturation levels, characteristically less than 10% of the void space, has very distinctive features in comparison to that at higher saturation levels. In particular, we have found that the dispersion process can be accurately described by a special class of partial differential equations, the super-fast non-linear diffusion equation. The results of mathematical modelling have demonstrated very good agreement with experimental observations. However, any enhancement of the accuracy and predictive power of the model, keeping in mind practical applications, requires the knowledge of the effective surface permeability of the constituent particles, which defines the global, macroscopic permeability of the particulate media. In the paper, we demonstrate how this quantity can be determined through the solution of the Laplace-Beltrami Dirichlet problem, we study this using the well-developed surface finite element method.