Dual-Family Viscous Shock Waves in n Conservation Laws with Application to Multi-Phase Flow in Porous Media
We consider shock waves satisfying the viscous profile criterion in general systems of n conservation laws. We study Si, j dual-family shock waves, which are associated with a pair of characteristic families i and j. We explicitly introduce defining equations relating states and speeds of Si, j shocks, which include the Rankine–Hugoniot conditions and additional equations resulting from the viscous profile requirement. We then develop a constructive method for finding the general local solution of the defining equations for such shocks and derive formulae for the sensitivity analysis of Si, j shocks under change of problem parameters. All possible structures of solutions to the Riemann problems containing Si, j shocks and classical waves are described. As a physical application, all types of Si, j shocks with i>j are detected and studied in a family of models for multi-phase flow in porous media.
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- 1.Brooks R.H., Corey A.T.: Properties of porous media affecting fluid flow. J. Irrig. Drain. E-ASCE 6, 61–88 (1966)Google Scholar
- 2.Guckenheimer J., Holmes P.: Nonlinear Oscillations, Dynamical Systems, and Bifurcations of Vector Fields. Springer, New York, 1983Google Scholar
- 5.Kulikovskij A.G.: Surfaces of discontinuity separating two perfect media of different properties. Recombination waves in magnetohydrodynamics. Prikl. Mat. Mekh. 32, 1125–1131 (1968), in RussianGoogle Scholar
- 6.Kulikovskii A.G., Pogorelov N.V., Semenov A.Yu.: Mathematical aspects of numerical solution of hyperbolic systems. Chapman & Hall/CRC, Boca Raton, 2001Google Scholar
- 8.Mailybaev A.A., Marchesin D.: Dual-family viscous shock waves in systems of conservation laws: a surprising example. In: Proc. Conf. on Analysis, Modeling and Computation of PDE and Multiphase Flow. Stony Brook NY, 2004Google Scholar
- 10.Plohr B., Marchesin D.: Wave structure in WAG recovery. SPE J. 6, 209–219 (2001)Google Scholar
- 14.Sotomayor J.: Generic bifurcations of dynamical systems. Dynamical Systems (Ed. Peixoto, M.M.) (Proc. Sympos., Univ. Bahia, Salvador, 1971). Academic Press, New York, pp 561–582, 1973Google Scholar
- 15.Stone H.L.: Probability model for estimating three-phase relative permeability. J. Petrol. Technol. 249, 214–218 (1970)Google Scholar