Compositional modeling by the combined discontinuous Galerkin and mixed methods

In this work, we present a numerical procedure that combines the mixed finite-element (MFE) and the discontinuous Galerkin (DG) methods. This numerical scheme is used to solve the highly nonlinear coupled equations that describe the flow processes in homogeneous and heterogeneous media with mass transfer between the phases. The MFE method is used to approximate the phase velocity based on the pressure (more precisely average pressure) at the interface between the nodes. This approach conserves the mass locally at the element level and guarantees the continuity of the total flux across the interfaces. The DG method is used to solve the mass-balance equations, which are generally convection-dominated. The DG method associated with suitable slope limiters can capture sharp gradients in the solution without creating spurious oscillations. We present several numerical examples in homogeneous and heterogeneous media that demonstrate the superiority of our method to the finite-difference (FD) approach. Our proposed MFE-DG method becomes orders of magnitude faster than the FD method for a desired accuracy in 2D.