A finite element formulation of the outlet gradient boundary condition for convective-diffusive transport problems

TL;DR

This paper introduces a finite element method for accurately implementing the outlet gradient boundary condition in convective-diffusive transport problems, improving modeling of contaminant transport in porous media.

Contribution

It presents a novel finite element formulation that evaluates the outlet gradient boundary condition using an upstream approach with covariant bases and metric tensors.

Findings

The method effectively handles mixed-dimensional finite elements.

It corrects the common zero-gradient assumption at outlets.

Illustrative examples demonstrate improved accuracy in contaminant transport modeling.

Abstract

A simple finite element formulation of the outlet gradient boundary condition is presented in the general context of convective-diffusive transport processes. Basically, the method is based on an upstream evaluation of the dependent variable gradient along open boundaries. Boundary normal unit vectors and gradient operators are evaluated using covariant bases and metric tensors, which allow handling finite elements of mixed dimensions. Even though the presented method has implications for many fields where diffusion processes are involved, discussion and illustrative examples address more particularly the framework of contaminant transport in porous media, in which the outlet gradient concentration is classically, but wrongly assumed to be zero.