Numerical Investigation of a Dipole Type Solution for Unsteady   Groundwater Flow with Capillary Retention and Forced Drainage
  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. 
Eugene A. Ingerman and Helen Shvets