We consider a simpliﬁed model of a two-phase ﬂow through a heterogeneous porous medium, in which convection is neglected. This leads to a nonlinear degenerate parabolic problem in a domain divided into an arbitrary ﬁnite number of homogeneous porous media. We introduce a new way to connect capillary pressures on the interfaces between the homogeneous domains, which leads to a general notion of solution. We then compare this notion of solution with an existing one, showing that it allows one to deal with a larger class of problems. We prove the existence of such a solution in a general case, and the existence and uniqueness of a regular solution in the one-dimensional case for initial data regular enough.
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Alessio Porretta, Clément Cancès, Thierry Gallouët, Two-phase ﬂows involving capillary barriers in heterogeneous porous media. Interfaces Free Bound. 11 (2009), no. 2, pp. 239–258DOI 10.4171/IFB/210