Numerical analysis for generalized Forchheimer flows of slightly compressible fluids in porous media

In this paper, we will consider the generalized Forchheimer flows for slightly compressible fluids. Using Muskat's and Ward's general form of Forchheimer equations, we describe the fluid dynamics by a nonlinear degenerate parabolic equation for density. The long time numerical approximation of the nonlinear degenerate parabolic equation with time dependent boundary conditions is studied. The stability for all positive time is established in both a continuous time scheme and a discrete backward Euler scheme. A Gronwall's inequality-type is used to study the asymptotic behavior of the solution. Error estimates for the solution are derived for both continuous and discrete time procedures. Numerical experiments confirm the theoretical analysis regarding convergence rates.