A maximum principle for multiphase flow models
Keywords:
maximum principle
multi-phase flow
black oil model
Abstract
Two maximum principles for several multi-phase flow models are formulated and proved. The first one is valid for phase saturations in an incompressible two-phase flow model with constant viscosities. The second one is valid for the global pressure in two- and three-phase flow models with constant viscosities and is also valid for phase pressures in the case of zero capillary pressure.
Section
Section 1. Numerical methods and applications
References
- H. Weinberger, “Invariant Sets for Weakly Coupled Parabolic and Elliptic Systems,” Rend. Mat. 8, 295-310 (1975).
- X. Liu and X. Zhang, “The Weak Maximum Principle for a Class of Strongly Coupled Elliptic Differential Systems,” J. Funct. Anal. 263 (7), 1862-1886 (2012).
- G. Gripenberg, “On the Strong Maximum Principle for Degenerate Parabolic Equations,” J. Differ. Equ. 242 (1), 72-85 (2007).
- L. E. Payne and G. A. Philippin, “On Maximum Principles for a Class of Nonlinear Second-Order Elliptic Equations,” J. Differ. Equ. 1980. 37 (1). 39-48.
- J. I. Diaz and J. Hernández, “Global Bifurcation and Continua of Nonnegative Solutions for Some Nonlinear Elliptic Eigenvalue Type Problems,” in Contribuciones Matemáticas: Homenaje al Profesor Enrique Outerelo Domí nguez} (Complutense Univ., Madrid, 2004), pp. 161-170.
- Z. Chen, “Degenerate Two-Phase Incompressible Flow: I. Existence, Uniqueness and Regularity of a Weak Solution,” J. Differ. Equ. 171 (2), 203-232 (2001).
- Z. Chen, “Formulations and Numerical Methods of the Black Oil Model in Porous Media,” SIAM J. Numer. Anal. 38 (2), 489-514 (2000).
- H. Holden, N. H. Risebro, and A. Tveito, “Maximum Principles for a Class of Conservation Laws,” SIAM J. Appl. Math. 55 (3), 651-661 (1995).
- Z. Chen, G. Huan, and Y. Ma, Computational Methods for Multiphase Flows in Porous Media (SIAM Press, Philadelphia, 2006).
- M. H. Protter and H. F. Weinberger, Maximum Principles in Differential Equations (Springer, New York, 1999).