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Multiphase Saturation Equations, Change of Type and Inaccessible Regions

  • Barbara Lee Keyfitz
Part of the ISNM International Series of Numerical Mathematics book series (ISNM, volume 114)

Abstract

We identify a class of flux functions which give rise to conservation laws which are hyperbolic except along a codimension one subspace of state space. We show that a number of systems modelling porous medium flow can be regarded as perturbations of such systems, and describe the phenomenon of change of type for these perturbations. We also discuss a property of solutions of such systems, the existence of inaccessible regions - subsets of state space which appear to be avoided by solutions.

Keywords

Hyperbolic System Riemann Problem Flux Function Polymer Flooding Riemann Solution 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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References

  1. [1]
    Allen M. B., Behie G. A., Trangenstein J. A. Multiphase flow in porous media: mechanics, mathematics, and numerics. Lecture Notes in Engineering 34. Springer-Verlag, New York, 1988.CrossRefGoogle Scholar
  2. [2]
    Bick J. H., Newell G. F. A continuum model for two-directional traffic flow. Quart. Appl. Math., XVIII: 191–204, 1960.Google Scholar
  3. [3]
    Holden H., Holden L. On the Riemann problem for a prototype of a mixed type conservation law, II. In Current Progress in Hyperbolic Systems: Riemann Problems and Computations, Contemporary Mathematics 100, pages 331–367. Amer. Math. Soc, Providence, 1989. B. Lindquist, ed.CrossRefGoogle Scholar
  4. [4]
    Holden H., Holden L., Risebro N. H. Some qualitative properties of 2 × 2 systems of conservation laws of mixed type. In Nonlinear Evolution Equations that Change Type, IMA Volumes in Mathematics and its Applications 27, pages 67–78. Springer, 1990. B. Keyfitz and M. Shearer, eds.Google Scholar
  5. [5]
    Johansen T., Winther R. The solution of the Riemann problem for a hyperbolic system of conservation laws modelling polymer flooding. SIAM J. Math. An., 19:541–566, 1988.CrossRefGoogle Scholar
  6. [6]
    Keyfitz B. L. Some elementary connections among nonstrictly hyperbolic conservation laws. In Nonstrictly Hyperbolic Conservation Laws, Contemporary Mathematics 60, pages 67–77. Amer. Math. Soc, Providence, 1987. B. Keyfitz and H. Kranzer, eds.CrossRefGoogle Scholar
  7. [7]
    Keyfitz B. L. A criterion for certain wave structures in systems that change type. In Current Progress in Hyperbolic Systems: Riemann Problems and Computations, Contemporary Mathematics 100, pages 203–213. Amer. Math. Soc, Providence, 1989. B. Lindquist, ed.CrossRefGoogle Scholar
  8. [8]
    Keyfitz B. L. Shocks near the sonic line: a comparison between steady and unsteady models for change of type. In Nonlinear Evolution Equations that Change Type, IMA Volumes in Mathematics and its Applications 27, pages 89–106. Springer, 1990. B. Keyfitz and M. Shearer, eds.Google Scholar
  9. [9]
    Keyfitz B. L. Change of type in simple models of two-phase flow. In Viscous Profiles and Numerical Approximation of Shock Waves, pages 84–104. SIAM, Philadelphia, 1991. M. Shearer, ed.Google Scholar
  10. [10]
    Keyfitz B. L. Conservation laws that change type and porous medium flow: a review. In Modeling and Analysis of Diffusive and Advective Processes in Geosciences, pages 122–145. SIAM, Philadelphia, 1992. W. E. Fitzgibbon and M. F. Wheeler, eds.Google Scholar
  11. [11]
    Keyfitz B. L. A method for generating nonstrictly hyperbolic fluxes with eigenvalue coincidence along a line. In preparation.Google Scholar
  12. [12]
    Liu T.-P. The Riemann problem for general 2×2 conservation laws. Amer. Math. Soc. Trans., 199:89–112, 1974.Google Scholar
  13. [13]
    Osher S. J. Riemann solvers, the entropy condition, and difference approximations. SIAM Jour. Numer. Anal, 21:217–235, 1984.CrossRefGoogle Scholar
  14. [14]
    Pego R. L., Serre D. Instabilities in Glimm’s scheme for two systems of mixed type. SIAM Jour. Numer. Anal, 25:965–988, 1988.CrossRefGoogle Scholar
  15. [15]
    Rhee H.-K., Aris R., Amundson N. R. First-Order Partial Differential Equations: Volume I, Theory and Application of Single Equations. Prentice-Hall, Englewood Cliffs, 1986.Google Scholar
  16. [16]
    Schaeffer D. G., Shearer M. The classification of 2 × 2 systems of non-strictly hyperbolic conservation laws, with application to oil recovery. Comm. Pure Appl. Math., 40:141–178, 1987.CrossRefGoogle Scholar
  17. [17]
    Shearer M., Trangenstein J. A. Loss of real characteristics for models of three-phase flow in a porous medium. Transport in Porous Media, 4:499–525, 1989.CrossRefGoogle Scholar
  18. [18]
    Stewart H. B., Wendroff B. Two-phase flow: models and methods. Jour. Comp. Physics, 56:363–409, 1984.CrossRefGoogle Scholar
  19. [19]
    Temple J. B. The L1-norm distinguishes the strictly hyperbolic from a nonstrictly hyperbolic theory of the initial value problem for systems of conservation laws. In Nonlinear Hyperbolic Equations — Theory, Computational Methods and Applications, Notes Numer. Fluid Mech. 24, pages 608–616. Aachen, 1988; Vieweg, Braunschweig, 1989.Google Scholar
  20. [20]
    Temple J. B. Systems of conservation laws with coinciding shock and rarefaction curves. In Nonlinear Partial Differential Equations, Contemporary Mathematics 17, pages 143–151. American Mathematical Society, Providence, 1983. J. A. Smoller, ed.CrossRefGoogle Scholar
  21. [21]
    Vinod V. Structural stability of Riemann solutions for a multiphase kinematic conservation law model that changes type. PhD thesis, University of Houston, 1992.Google Scholar

Copyright information

© Springer Basel AG 1993

Authors and Affiliations

  • Barbara Lee Keyfitz
    • 1
  1. 1.Mathematics DepartmentUniversity of HoustonHoustonUSA

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