A Block-Centered Finite Difference Method for Slightly Compressible Darcy-Forchheimer Flow in Porous Media

A block-centered finite difference method is introduced to solve an initial and boundary value problem for a nonlinear parabolic equation to model the slightly compressible flow in porous media, in which the velocity-pressure relation is described by Darcy-Forchheimer's Law. The method can be thought as the lowest order Raviart-Thomas mixed element method with proper quadrature formulation. By using the method the velocity and pressure can be approximated simultaneously. We established the second-order error estimates for pressure and velocity in proper discrete norms on non-uniform rectangular grid. No time-step restriction is needed for the error estimates. The numerical experiments using the scheme show that the convergence rates of the method are in agreement with the theoretical analysis.