Galerkin finite element method for generalized Forchheimer equation of slightly compressible fluid in porous media
This work provides rigorous numerical analysis for a nonlinear degenerate parabolic PDE arising in porous media flow, which is important for reservoir engineering but represents an incremental extension of existing finite element theory to a specific equation.
The paper develops a Galerkin finite element method for the generalized Forchheimer equation describing slightly compressible fluid flow in porous media, proving existence, uniqueness, and error estimates, with numerical experiments confirming theoretical convergence rates.
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. The prior estimates for the solutions in $L^\infty(0,T;L^q(Ω)), q\ge 2$, time derivative in $L^\infty(0,T;L^2(Ω))$ and gradient in $L^\infty(0,T;W^{1,2-a}(Ω)),$ with $a\in (0,1)$ 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.