Discontinuous Galerkin methods for solving the miscible displacement problem in porous media

A class of discontinuous locally conservative finite element methods are investigated both theoretically and computationally for modeling second order elliptic and parabolic equations. A priori error estimates for elliptic, parabolic, linear elasticity, and quasi-static viscoelasticity equations which are optimal in the mesh size are derived. In addition for the elliptic case exponential rates of convergence and a posteriori estimates are exhibited.;Numerical implementation of these schemes for modeling miscible displacement in porous media is considered. These problems are described by a coupled system of equations which include an elliptic equation referred to as the pressure equation and a transport-dominated parabolic equation referred to as the concentration equation of the invading fluid.;The error in the energy norm is studied for the pressure equation. Our results confirm the robustness of the proposed error indicators. In particular, we show the advantage of mesh adaptation for discontinuous coefficients. Realistic flow simulations are performed on unstructured meshes and with a high order approximation.;In modeling the concentration equation, the scheme is slightly modified for treating sharp fronts. The transport term is locally upwinded and the degree of the computed solution is reduced to a linear or constant when necessary to avoid undershoot and overshoot. The resulting concentration approximation remains locally mass conservative.;Our computational results indicate that this discontinuous Galerkin approach for modeling miscible displacement performs favorably when compared with a mixed finite element method for pressure based on the RTO spaces and a higher order Godunov method for concentration. In particular, the resulting algebraic problem does not require the solution of a saddle-point problem. Furthermore, larger time steps are allowed for the fully implicit concentration equation and sharper fronts are observed on coarser meshes. Moreover, our complexity analysis indicates that the discontinuous Galerkin method is computationally less expensive than the mixed method for higher order approximations.