Second-order theory of shallow free-surface flow in porous media

The general equations of shallow free-surface flow in porous media are formally derived by an expansion similar to Friedrichs' (4) theory of shallow water-waves. The same equations are derived in an alternative way by a process of matching of asymptotic expansions, the shallow-flow approximation being obtained rigorously as an inner expansion. The first term of the expansion satisfies the same equations as the usual Dupuit-Forcheimer equations; however, a second-order term is for the first time presented.