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Vanishing capillarity solutions of Buckley–Leverett equation with gravity in two-rocks’ medium

Computational Geosciences volume 17, pages 551–572 (2013)Cite this article

Abstract

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 Kaasschieter (Comput Geosci 3(1):23–48, 1999), we use the parabolic model with small parameter that accounts for capillary effects. While it has been recognized in Cancès (Networks Het Media 5(3):635–647, 2010) 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.

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Authors and Affiliations

  1. Laboratoire de Mathématiques de Besançon CNRS UMR 6623, Université de Franche-Comté, 16 route de Gray, 25030, Besançon, Cedex, France

    Boris Andreianov

  2. Laboratoire Jacques-Louis Lions, UPMC Univ Paris 06 & CNRS, UMR 7598, 75005, Paris, France

    Clément Cancès

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  1. Boris Andreianov
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  2. Clément Cancès
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Andreianov, B., Cancès, C. Vanishing capillarity solutions of Buckley–Leverett equation with gravity in two-rocks’ medium. Comput Geosci 17, 551–572 (2013). https://doi.org/10.1007/s10596-012-9329-8

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  • DOI: https://doi.org/10.1007/s10596-012-9329-8

Keywords

  • Scalar conservation laws with discontinuous-flux functions
  • Discontinuous capillarity
  • Two-phase flows in heterogeneous porous media
  • Finite volume schemes

Mathematics Subject Classifications (2010)

  • 35L02
  • 35L65
  • 65M08
  • 76S05