  A model of unsteady filtration (seepage) in a porous medium with capillary
retention is considered. It leads to a free boundary problem for a generalized
porous medium equation where the location of the boundary of the water mound is
determined as part of the solution. The numerical solution of the free boundary
problem is shown to possess self-similar intermediate asymptotics. On the other
hand, the asymptotic solution can be obtained from a non-linear boundary value
problem. Numerical solution of the resulting eigenvalue problem agrees with the
solution of the partial differential equation for intermediate times. In the
second part of the work, we consider the problem of control of the water mound
extension by a forced drainage.
