Analytical solution to the Riemann problem of three-phase flow in porous media

Ruben Juanes, Tadeusz W. Patzek*

*Corresponding author for this work

Research output: Contribution to journalArticlepeer-review

42 Scopus citations


In this paper we study one-dimensional three-phase flow through porous media of immiscible, incompressible fluids. The model uses the common multiphase flow extension of Darcy's equation, and does not include gravity and capillarity effects. Under these conditions, the mathematical problem reduces to a 2 × 2 system of conservation laws whose essential features are: (1) the system is strictly hyperbolic; (2) both characteristic fields are nongenuinely nonlinear, with single, connected inflection loci. These properties, which are natural extensions of the two-phase flow model, ensure that the solution is physically sensible. We present the complete analytical solution to the Riemann problem (constant initial and injected states) in detail, and describe the characteristic waves that may arise, concluding that only nine combinations of rarefactions, shocks and rarefaction-shocks are possible. We demonstrate that assuming the saturation paths of the solution are straight lines may result in inaccurate predictions for some realistic systems. Efficient algorithms for computing the exact solution are also given, making the analytical developments presented here readily applicable to interpretation of lab displacement experiments, and implementation of streamline simulators.

Original languageEnglish (US)
Pages (from-to)47-70
Number of pages24
JournalTransport in Porous Media
Issue number1
StatePublished - Apr 2004
Externally publishedYes

Bibliographical note

Funding Information:
The authors wish to express their gratitude to Dr Dmitriy B. Silin for his careful revision of the manuscript. We thank Dr A. R. Kacimov for pointing out the book by Charny (1963). The comments of two other anonymous reviewers lead to an improved presentation of the material, and are also appreciated. This work was supported by the Laboratory Directed Research and Development Program of Lawrence Berkeley National Laboratory under the Department of Energy Contract No. DE-AC03-76SF00098. Funding provided by the Barrié de la Maza, Jane Lewis, and Repsol-YPF fellowships, awarded to the first author, is also gratefully acknowledged.


  • Buckley-Leverett
  • Conservation laws
  • Entropy solution
  • Hyperbolic system
  • Three-phase flow
  • Waves

ASJC Scopus subject areas

  • Catalysis
  • General Chemical Engineering


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