  We study the large time behavior of the solutions to a two phase extension of
the porous medium equation, which models the so-called seawater intrusion
problem. The goal is to identify the self-similar solutions that correspond to
steady states of a rescaled version of the problem. We fully characterize the
unique steady states that are identified as minimizers of a convex energy and
shown to be radially symmetric. Moreover, we prove the convergence of the
solution to the time-dependent model towards the unique stationary state as
time goes to infinity. We finally provide numerical illustrations of the
stationary states and we exhibit numerical convergence rates.
