Asymptotic expansions of the pressure and the free boundary for flows through porous media

The free boundary value problem for the pressure head u of a compressible fluid flowing through a homogeneous porous medium is studied. This process is governed by the partial differential equation epsilon delta t u - delta squared x u = 0, where epsilon is proportional to the compressibility of the fluid. It is shown that the pressure as well as the free boundary converge to the corresponding stationary solutions when epsilon tends to zero. The error is estimated in terms of powers of epsilon. In the case of water, for example, if compressibility is neglected, the error can be estimated.