On the performance of the variational multiscale formulation for subsurface flow and transport in heterogeneous porous media

TL;DR

This paper compares the performance of Raviart-Thomas and variational multiscale finite elements in modeling heterogeneous subsurface flow, highlighting the variational multiscale element's limitations in mass conservation under certain conditions.

Contribution

It provides a comparative analysis of two finite element methods for subsurface flow, emphasizing the conditions under which the variational multiscale element exhibits mass conservation issues.

Findings

Variational multiscale element shows mass loss or gain in certain flow conditions.

Raviart-Thomas element maintains better mass conservation in heterogeneous media.

Mass conservation violations are more pronounced with flow tangential to permeability layers.

Abstract

The following work compares two popular mixed finite elements used to model subsurface flow and transport in heterogeneous porous media; the lowest order Raviart-Thomas element and the variational multiscale stabilized element. Comparison is made based on performance for several problems of engineering relevance that involve highly heterogenous material properties (permeability ratios of up to $1\times10^5$), open flow boundary conditions (pressure driven flows), and large scale domains in two dimensions. Numerical experiments are performed to show the degree to which mass conservation is violated when a flow field computed using either element is used as the advection velocity in a transport model. The results reveal that the variational multiscale element shows considerable mass production or loss for problems that involve flow tangential to layers of differing permeability, but marginal violation of local mass balance for problems of less orthogonality in the permeability. The results are useful in establishing rudimentary estimates of the error produced by using the variational mutliscale element for several different types of problems.