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Solution of the equations for one-dimensional, two-phase, immiscible flow by geometric methods

An Erratum to this article was published on 28 February 2017

This article has been updated


Buckley–Leverett equations describe non viscous, immiscible, two-phase filtration, which is often of interest in modelling of oil production. For many parameters and initial conditions, the solutions of these equations exhibit non-smooth behaviour, namely discontinuities in form of shock waves. In this paper we obtain a novel method for the solution of Buckley–Leverett equations, which is based on geometry of differential equations. This method is fast, accurate, stable, and describes non-smooth phenomena. The main idea of the method is that classic discontinuous solutions correspond to the continuous surfaces in the space of jets - the so-called multi-valued solutions (Bocharov et al., Symmetries and conservation laws for differential equations of mathematical physics. American Mathematical Society, Providence, 1998). A mapping of multi-valued solutions from the jet space onto the plane of the independent variables is constructed. This mapping is not one-to-one, and its singular points form a curve on the plane of the independent variables, which is called the caustic. The real shock occurs at the points close to the caustic and is determined by the Rankine–Hugoniot conditions.

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  • 28 February 2017

    An erratum to this article has been published.


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Corresponding author

Correspondence to Andrey Shevlyakov.

Additional information

This work is supported by Russian Scientific Foundation (project No. 15-19-00275).

In the original publication of this article the given name and family name of the authors have been published incorrectly; this error has now been corrected.

An erratum to this article is available at

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Boronin, I., Shevlyakov, A. Solution of the equations for one-dimensional, two-phase, immiscible flow by geometric methods. Anal.Math.Phys. 8, 1–8 (2018).

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  • Filtration
  • Buckley–Leverett equations
  • Shock waves
  • Geometric methods

Mathematics Subject Classification

  • 53Z05
  • 76S05