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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.)

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Some "quick" references concerning incompressible porous flow:

1) For the theory: "The confined Muskat problem: differences with the deep water regime" (D. Córdoba, R. Granero and R. Orive). Preprint arXiv:1209.1575.

2) For the numerics: “A note on the interface dynamics for convection in porous media” (D. Córdoba, F. Gancedo & R. Orive), Physica D 237 (2008), 1488-1497.

3) More theory: Steady-state fingering patterns for a periodic Muskat problem (Mats Ehrnstrom, Joachim Escher, Bogdan-Vasile Matioc) Preprint arXiv:1303.6724

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