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A phase-by-phase upstream scheme that converges to the vanishing capillarity solution for countercurrent two-phase flow in two-rock media

Computational Geosciences volume 18, pages 211–226 (2014)Cite this article

Abstract

We discuss the convergence of the upstream phase-by-phase scheme (or upstream mobility scheme) towards the vanishing capillarity solution for immiscible incompressible two-phase flows in porous media made of several rock types. Troubles in the convergence were recently pointed out by Mishra and Jaffré (Comput. Geosci. 14, 105–124, 2010) and Tveit and Aavatsmark (Comput. Geosci. 16, 809–825, 2012). In this paper, we clarify the notion of vanishing capillarity solution, stressing the fact that the physically relevant notion of solution differs from the one inferred from the results of Kaasschieter (Comput. Geosci. 3, 23–48, 1999). In particular, we point out that the vanishing capillarity solution depends on the formally neglected capillary pressure curves, as it was recently proven in by Andreianov and Cancès (Comput. Geosci. 17, 551–572, 2013). Then, we propose a numerical procedure based on the hybridization of the interfaces that converges towards the vanishing capillarity solution. Numerical illustrations are provided.

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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, Sorbonne Universités, UPMC Univ Paris 06, UMR 7598, 75005, Paris, France

    Clément Cancès

  3. Laboratoire Jacques-Louis Lions, CNRS, UMR 7598, 75005, Paris, France

    Clément Cancès

Authors
  1. Boris Andreianov
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  2. Clément Cancès
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Correspondence to Clément Cancès.

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Andreianov, B., Cancès, C. A phase-by-phase upstream scheme that converges to the vanishing capillarity solution for countercurrent two-phase flow in two-rock media. Comput Geosci 18, 211–226 (2014). https://doi.org/10.1007/s10596-014-9403-5

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  • DOI: https://doi.org/10.1007/s10596-014-9403-5

Keywords

  • Porous media two-phase flows
  • Discontinuous capillarity
  • Scalar conservation laws with discontinuous flux functions
  • Vanishing capillarity solution
  • Finite volume scheme
  • Upstream differencing
  • Convergence

Mathematics Subject Classifications (2010)

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