DMS Colloquium: Béatrice Rivière

Speaker: Béatrice Rivière, Rice University

Title: High order discontinuous Galerkin methods for solving the miscible displacement problem in porous media

Abstract: The accurate prediction of flow and transport in porous media is essential in optimizing the clean-up of contaminated groundwater or the production of hydrocarbons from oil reservoirs. In the miscible displacement problem, a solvent is injected and mixes with the resident fluid (contaminant or oil). The fluid mixture then propagates through the set of connected pores. At the Darcy scale, the mathematical model is a system of partial differential equations coupling flow and transport. This talk presents high order interior penalty discontinuous Galerkin (IPDG) methods and hybridizable discontinuous Galerkin (HDG) methods for solving the nonlinear system of equations. HDG methods retain the positive features of IPDG, but the number of globally coupled degrees of freedom for high order HDG is significantly smaller. The proposed numerical methods are shown to be accurate on coarse meshes when the polynomial degree increases. The numerical approximations of the solvent concentration exhibit sharp fronts even in highly heterogeneous media. Finally, the discontinuous Galerkin methods in space can accurately model viscous fingering. Viscous fingering in porous media may occur when a fluid with low viscosity is used to displace a fluid with high viscosity. For this type of flow instability, a tiny perturbation can be amplified exponentially, which triggers a finger-like pattern in the fluid concentration profile during the fluid displacement.  Simulations in two and in three dimensions show the growth and propagation of fingers for large mobility ratios and large Peclet numbers. Results are compared with those obtained by using a generic cell-centered finite volume method.

Faculty host: Thi-Thao-Phuong Hoang