A methodology to ensure local mass conservation for porous media models under finite element formulations based on convex optimization

Abstract:

It is well know that the standard finite element methods, in general, do not satisfy element-wise mass/species balance properties. It is, however, desirable to have element-wide mass balance property in subsurface modeling. Several studies over the years have aimed to overcome this drawback of finite element formulations. Currently, a post-processing optimization-based methodology is commonly employed to recover the local mass balance for porous media models. However, such a post-processing technique does not respect the underlying variational structure that the finite element formulation may enjoy. Motivated by this, a consistent methodology to satisfy element-wise local mass balance for porous media models is constructed using convex optimization techniques. The assembled system of global equations is reconstructed into a quadratic programming problem subjected to bounded equality constraints that ensure conservation at the element level. The proposed methodology can be applied to any computational mesh and to any non-locally conservative nodal-based finite element method. Herein, we integrate our proposed methodology into the framework of the classical mixed Galerkin formulation using Taylor-Hood elements and the least-squares finite element formulation. Our numerical studies will include computational cost, numerical convergence, and comparision with popular methods. In particular, it will be shown that the accuracy of the solutions is comparable with that of several popular locally conservative finite element formulations like the lowest order Raviart-Thomas formulation. We believe the proposed optimization-based approach is a viable approach to preserve local mass balance on general computational grids and is amenable for large-scale parallel implementation.