  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 $L^\infty(0,T;L^q(\Omega)), q\ge 2$,
time derivative in $L^\infty(0,T;L^2(\Omega))$ and gradient in
$L^\infty(0,T;W^{1,2-a}(\Omega)),$ 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.
