I'm concern a moving boundary problem in rock mechanics. We consider a problem of unsaturated flow of an in-compressible fluid in a porous medium(rock) like D. Moreover suppose that support of a measure, as the fluid source, is located in D close to the boundary.

I am interested in what happen when the saturated part meet the boundary of D and how we can describe the moving boundary of saturated part in a desired time t.

I really appreciate if you introduce me some papers(theory or numerical approach.)