An Error Estimate for Mmoc-mfem Based on Convolution for Porous Media Flow

Mathematical models used to describe porous medium flow processes in petroleum reservoir simulation, groundwater contaminant transport, and other applications lead to a coupled system of time-dependent nonlinear partial differential equations (PDEs) [1]. Conventional second-order finite difference or finite element methods (FDMs, FEMs) tend to yield solutions with spurious oscillations. In industrial applications, first-order upwind methods are commonly used to stabilize the numerical approximations, but they tend to generate excessive numerical diffusion and grid-orientation effect [1]. An MMOC-MFEM time-stepping procedure was proposed and successfully applied in the numerical simulation of miscible displacement processes in petroleum reservoir simulation [2], in which the MMOC [3] was used to solve the transport equation while an MFEM scheme [4, 5] was used to solve the pressure equation. The MMOC symmetrizes and stabilizes the transport equation, greatly reduces temporal errors, and so allows for large time steps in a simulation without loss of accuracy. The MFEM schemes generate an accurate approximation to the Darcy velocity, which are required for accurate approximation to the transport because advection and diffusion dispersion in the transport equation are governed by Darcy velocity. The MFEMs minimize the numerical difficulties occurring in finite difference or finite element caused by differentiation of the pressure and then multiplication by rough coefficients [6]. Numerical experiments showed that the MMOC-MFEM type of solution techniques is numerically very competitive [2, 7]. A delicate and rigorous mathematical analysis was conducted in [8], in which an optimal-order error estimate was proved for a family of MMOC-MFEM time stepping procedure for miscible displacement processes in two space dimensions.