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Asymptotic behavior of two-phase flows in heterogeneous porous media for capillarity depending only on space. II: Nonclassical shocks to model oil-trapping. (English) Zbl 1219.35139

The author considers a one-dimensional problem modeling two-phase flow in heterogeneous porous media made of two homogeneous subdomains, with discontinuous capillarity at the interface between them. One supposes that the capillary forces vanish inside the domains, but not on the interface. Under the assumption that the gravity forces and the capillary forces are oriented in opposite directions, one shows that the limit, for vanishing diffusion, is not in general the optimal entropy solution of the hyperbolic scalar conservation law as in the first paper of the series [SIAM J. Math. Anal. 42, No. 2, 946–971 (2010; Zbl 1219.35136)]. A nonclassical shock can occur at the interface, modeling oil-trapping.
In this problem the lack of entropy comes only from the discontinuity of the porous medium. It is also shown that only a finite quantity of oil can be definitely trapped. This quantity is determined only by the capillary pressure curves and the difference between the volume mass of both phases, and does not depend on the initial data \(u_{0}\).

MSC:

35L67 Shocks and singularities for hyperbolic equations
35L65 Hyperbolic conservation laws
76S05 Flows in porous media; filtration; seepage
35B40 Asymptotic behavior of solutions to PDEs

Citations:

Zbl 1219.35136
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