  We consider the homogenization of the Hele-Shaw problem in periodic media
that are inhomogeneous both in space and time. After extending the theory of
viscosity solutions into this context, we show that the solutions of the
inhomogeneous problem converge in the homogenization limit to the solution of a
homogeneous Hele-Shaw-type problem with a general, possibly nonlinear
dependence of the free boundary velocity on the gradient. Moreover, the free
boundaries converge locally uniformly in Hausdorff distance.
