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Let us consider a vertical column Ω:— (0, L) of height L which is filled with a homogeneous porous medium. Initially it is saturated with an incompressible fluid up to some level b. Above level b there is an unsaturated region. The medium is allowed to drain under the influence of gravity. At the top of the column there is no flow of moisture and at the bottom the pressure u of the fluid is constant. The physical behaviour of the fluid in the porous medium is described by Darcy's law (see Bear [5], p. 487)

Parabolic regularization and behaviour of the free boundary for unsaturated flow in a porous medium.