Department of Mathematics, Florida State University
"Enriched Galerkin Approximations for Coupled Flow and Transport in Porous Media"
We present and analyze enriched Galerkin ﬁnite element methods (EG) to solve coupled ﬂow and transport system with jump coeﬃcients referred as miscible displacement problems. Miscible displacement of one fluid by another in a porous medium has attracted considerable attention in subsurface modeling with emphasis on enhanced oil recovery applications. Here flow instabilities arising when a fluid with higher mobility displaces another fluid with lower mobility is referred to as viscous fingering. The latter has been the topic of major physical and mathematical studies for over half a century. The EG is formulated by enriching the conforming continuous Galerkin ﬁnite element method (CG) with piecewise constant functions. This approach is shown to be locally and globally conservative while keeping fewer degrees of freedom in comparison with discontinuous Galerkin ﬁnite element methods (DG). The brief introduction to the finite element method will be presented before discussing EG. Dynamic mesh adaptivity using entropy residual and hanging nodes is applied to save computational cost for large-scale physical problems. Some numerical tests in two and three dimensions are presented to conﬁrm our theoretical results as well as to demonstrate the advantages of the EG. This work was done in collaboration with Mary F. Wheeler and Young-Ju Lee.