Galerkin finite element method for generalized Forchheimer equation of slightly compressible fluids in porous media

Abstract

We 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 the density. We study Galerkin finite elements method for the initial boundary value problem. The existence and uniqueness of the approximation are proved. A prior estimates for the solutions in L 1 .0, T; L q . //, q 2, time derivative in L 1 .0, T; L 2 . // and gradient in L 1 .0, T; W 1,2 a . //, with a 2 .0, 1/ are established. Error estimates for the density variable are derived in several norms for both continuous and discrete time procedures. Numerical experiments using backward Euler scheme confirm the theoretical analysis regarding convergence rates. Above, the positive constants a, b, c, d are obtained from experiments. The generalized Forchheimer equation of (1.1) and (1.2) were proposed in [8-10] of the form rp D N X iD0 a i jvj˛i v. (1.3)These equations are analyzed numerically in [11][12][13], theoretically in [8,10,[14][15][16][17] for single phase flows, and also in [18,19] for two phase flows.

Generalized Forchheimer flows in heterogeneous porous media

“…Due to the extra dependence on the spatial variables, it is not clear which space of functions without vanishing trace on the boundary, and which version of Poincaré-Sobolev inequality or trace theorem will be suitable. (b) There have been more work on the numerical analysis of generalized Forchheimer flows for incompressible fluids [4,16,19], and slightly compressible fluids [14,15]; see the cited papers for additional references. However, all of these papers are only for homogeneous porous media.…”