A physically-based entropy production rate method to simulate sharp-front transport problems in porous medium systems

摘要

Sharp-front problems arising from equations that tend toward a hyperbolic limit occur routinely in modeling transport phenomena in porous medium systems. Numerical methods to approximate hyperbolic conservation equations are mature for finite volume methods. However, irregularly shaped domains make finite element methods (FEMs) an attractive approach for many subsurface problems. For conforming FEM approaches, sharp fronts continue to pose a computational challenge. We build on recent work to develop and evaluate an entropy viscosity approach. Recent theoretical advancements have resulted in the formulation of entropy inequality equations for many porous medium problems, which we use to compute a physically-based entropy production rate for both linear and nonlinear species transport problems. The entropy production rate is in turn used to add dissipation to the approximation of the hyperbolic operator using an entropy-viscosity formulation. We demonstrate how existing, problem-specific theoretical results can be leveraged in a generic and efficient numerical method to create problem-specific dissipation functions.