Physics Preserving Finite Element Methods for Coupled Multi-Physical Subsurface Applications
In recent years, the main challenges in subsurface energy systems (e.g., enhanced geothermal systems and CO2 sequestration) have included issues arising from the multi-physical and multi-scale nature of the problem, as well as uncertainty quantification. Multi-physics involves coupling solid mechanics, fluid mechanics, thermal energy, and chemical reactions, while multi-scale considerations involve relating pore-scale problems to field-scale problems. These problems are complex and require interdisciplinary efforts to achieve meaningful outcomes in research. In this talk, we will focus on the challenges of the multi-physical formulations for solving subsurface applications, and discuss new enriched Galerkin (EG) finite element methods for coupled flow and transport and poro-elasticity systems. The primary goal of the study is to develop computationally efficient and robust numerical methods that are free from oscillations due to a lack of local conservation, maximum principle violations, int-sup issues, and locking effects. The locally conservative enriched Galerkin (LC-EG) method, which will be used to solve the flow problem, is constructed by adding a constant function to each element based on the classical continuous Galerkin methods (CG). The locking-free enriched Galerkin (LF-EG) method adds a piecewise linear vector to the displacement space. These EG methods incorporate well-known discontinuous Galerkin (DG) techniques but use approximation spaces with fewer degrees of freedom than typical DG methods, offering an efficient alternative. We will present a priori error estimates of optimal order and demonstrate, through numerical examples, that the new method is free from oscillations and locking.
