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Vanishing capillarity solutions of Buckley-Leverett equation with gravity in two-rocks’ medium. (English) Zbl 1392.76033
Summary: For the hyperbolic conservation laws with discontinuous-flux function, there may exist several consistent notions of entropy solutions; the difference between them lies in the choice of the coupling across the flux discontinuity interface. In the context of Buckley-Leverett equations, each notion of solution is uniquely determined by the choice of a “connection,” which is the unique stationary solution that takes the form of an under-compressive shock at the interface. To select the appropriate connection, following E. F. Kaasschieter [ibid. 3, No. 1, 23–48 (1999; Zbl 0952.76085)], we use the parabolic model with small parameter that accounts for capillary effects. While it has been recognized in [C. Cancès, Netw. Heterog. Media 5, No. 3, 635–647 (2010; Zbl 1262.35163)] that the “optimal” connection and the “barrier” connection may appear at the vanishing capillarity limit, we show that the intermediate connections can be relevant and the right notion of solution depends on the physical configuration. In particular, we stress the fact that the “optimal” entropy condition is not always the appropriate one (contrarily to the erroneous interpretation of Kaasschieter’s results which is sometimes encountered in the literature). We give a simple procedure that permits to determine the appropriate connection in terms of the flux profiles and capillary pressure profiles present in the model. This information is used to construct a finite volume numerical method for the Buckley-Leverett equation with interface coupling that retains information from the vanishing capillarity model. We support the theoretical result with numerical examples that illustrate the high efficiency of the algorithm.

MSC:
76M12 Finite volume methods applied to problems in fluid mechanics
35L02 First-order hyperbolic equations
35L65 Hyperbolic conservation laws
65M08 Finite volume methods for initial value and initial-boundary value problems involving PDEs
76S05 Flows in porous media; filtration; seepage
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