Consistency and local conservation in finite-element simulation of flow in porous media

Computational models of subsurface flow involve the coupling between fluid velocity and transport equations. While solving these problems independently can be challenging, we expose the additional complication associated with coupling them. We focus on numerical methods that can readily accommodate unstructured meshes, with emphasis on the mixed finite-element method for the velocity-pressure formulation. The transport equation is discretized using continuous and discontinuous finite elements, and also finite volumes. We theoretically identify the discrete requirements, in space and time, to be satisfied for the constant-preserving test to pass when coupling these two discrete entities: a unit tracer concentration should be discretely preserved to machine precision under the right conditions. In our work, the constant-preserving test is referred to as the consistency test. In particular, we show that a discrete formulation that relies entirely on continuous finite elements is consistent. The overarching conclusion is that consistency is achieved when the discrete tracer equation reduces to the discrete pressure equation upon setting the tracer concentration to one. Equivalently, we conclude that consistency is achieved when coupling discrete formulations that rely on identical local-conservation properties. In explaining this result, we also make yet another attempt at debunking the myth that continuous finite elements are not locally conservative and cannot be used for flow simulation. We emphasize that continuous finite elements are locally conservative in a way that is derived from the discrete form, but not in a finite-volume sense, that is, not by computing the integral of the normal velocity on any element boundary.