We study foam drainage using the large-Q Potts model extended to include gravity on a three dimensional lattice. Without adding liquid, homogeneously distributed liquid drains to the bottom of the foam until equilibrium between capallary effects and gravity is reached, while in an ordered dry foam, if a fixed amount of liquid is added from the top, a sharp flat interface between the wet and dry foam develops. The wetting front profile forms a downward moving pulse, with a constant volecity. The pulse decays over time while its leading edge for a brief time behaves like a solitary wave. With continuous liquid addition from the top, the pulse does not decay and we observe a soliton front moving with a constant velocity. Continuously adding liquid to an initially wet foam keeps the liquid profile constant. Our simulations agree with both experimental data and simplified mean field analytical results for ordered foams but predict an unstable interface for disordered foams.