In this talk, we propose to use discontinuous Galerkin (DG) methods for simulating the viscous fingering phenomena in porous media. DG methods are specialized finite element methods that utilize discontinuous spaces to approximate solutions, and employ bilinear forms to weakly enforce boundary conditions and interelement continuities. The methods are locally mass conservative by construction, and have small numerical diffusion and little oscillation. In addition, due to the discontinuous approximation spaces, they handle sharp fronts and discontinuities in the solution very well. Moreover, the treatment of full-tensor permeability (for flow) and diffusion-dispersion (for transport) tensors is flexible and efficient in primal DG methods.
A family of four primal DG methods with dynamic adaptivity are applied and investigated for simulating the viscous fingering phenomena. The four primal DG include Symmetric Interior Penalty Galerkin (SIPG), Oden-Baumann-Babuska DG formulation (OBB-DG), Nonsymmetric Interior Penalty Galerkin (NIPG) and Incomplete Interior Penalty Galerkin (IIPG) methods. We study the effect of anisotropic dispersion on nonlinear viscous fingering in miscible displacements in both two-dimensional Hele–Shaw cells and full three-dimensional heterogenous porous media, and we show that DG algorithms effectively treat dispersion coefficient-velocity field couplings (i.e., mechanical dispersivities). We address efficient implementation issues with an emphasis on dynamic mesh adaptation strategies. A number of numerical examples are presented to illustrate various features of DG methods for viscous fingering simulations.