  We provide rigorous asymptotic models for the free boundary Darcy and
Forchheimer problem under the assumption of weak nonlinear interaction, in a
regime in which the steepness parameter of the interface is considered to be
very small. The models we derive capture the nonlinear interaction of the
original free boundary Darcy and Forchheimer problem up to quadratic terms.
Furthermore, we provide models that consider both the two-dimensional and
three-dimensional cases, with and without bottom topography.
