Numerical simulation of two-phase immiscible incompressible flows in heterogeneous porous media with capillary barriers

We present a new version of the sequential discontinuous Galerkin method introduced in [25] for two-phase immiscible incompressible flows in heterogeneous porous media with a discontinuous capillary field. Here, a new implementation of the extended interface condition, that does not use the threshold saturation value at the interface and permits treatment of different residual saturations in different rocks, is considered. Another novel ingredient is the implicit treatment of the diffusion term and the non-linear interface conditions in the saturation equation. The proposed method is validated in two-dimensional test cases and confirms theoretically known optimal orders of convergence. The numerical experiments demonstrate that for two-dimensional interface problems the method exhibits better performance on moderately refined meshes, which is particularly important for multidimensional heterogeneous problems related to realistic field studies. We consider also two heterogeneous five-spot benchmark problems to assess the potential of the proposed method in simulations of reservoirs with discontinuous permeability and capillary pressure fields in different types of rock.