  We study the long-time behavior of an exterior Hele-Shaw problem in random
media with a free boundary velocity that depends on position in dimensions $n
\geq 2$. A natural rescaling of solutions that is compatible with the evolution
of the free boundary leads to homogenization of the free boundary velocity. By
studying a limit obstacle problem for a Hele-Shaw system with a point source,
we are able to show uniform convergence of the rescaled solution to a
self-similar limit profile and we deduce that the rescaled free boundary
uniformly approaches a sphere.
