Approximate analytical solutions of the Forchheimer equation

In this paper we derive approximate analytical solutions for non-steady-state, non-linear flows through porous media, described by the Forchheimer equation. We demonstrate that one has to distinguish between two characteristic regimes. In early times, the hydraulic gradient is steep, and subsequently the inertial terms are dominant. One obtains the leading hydraulic behavior by neglecting linear terms describing the viscous dissipation mechanisms. In moderate times, as the disturbance introduced upstream propagates through the entire medium, the hydraulic gradient, and subsequently the inertial effects become less important: the leading behavior corresponds to the Darcy solution. The influence of the inertia mechanisms in this regime is taken into account by computing higher order correction terms, by means of perturbation analysis. The verification of the analytical solution is done by comparison with numerical results of a finite volume code. As application of the developed theory the water volume accumulated in an aquifer due to river flooding is presented.