On the effects of discontinuous capillarities for immiscible two-phase flows in porous media made of several rock-types

Networks and Heterogeneous Media

Volumn 5, Issue 3, 2010, Pages 635-647

  Cancès, Clément ^a  

^a UNIVERSITÉ PIERRE ET MARIE CURIE   (France)

Author keywords

Discontinuous flux; Non classical shock; Oiltrapping; Scalar conservation laws

Indexed keywords

DISCONTINUOUS CAPILLARITIES; DISCONTINUOUS FLUXES; EXISTENCE/UNIQUENESS; HETEROGENEOUS POROUS MEDIA; IMMISCIBLE TWO-PHASE FLOWS; NON-CLASSICAL SHOCK; OILTRAPPING; SCALAR CONSERVATION LAWS;

ENTROPY; PHYSICAL PROPERTIES; POROUS MATERIALS; ROCKS; TWO PHASE FLOW;

CAPILLARITY;

EID: 77955024616     PISSN: 15561801     EISSN: 1556181X     Source Type: Journal    
DOI: 10.3934/nhm.2010.5.635     Document Type: Article

References (26)

1. Adimurthi, J. Jaffré and G. D. Veerappa Gowda, Godunov-type methods for conservation laws with a flux function discontinuous in space, SIAM J. Numer. Anal., 42 (2004), 179-208.
2. Adimurthi, S. Mishra and G. D. Veerapa Gowda, Optimal entropy solutions for onservation laws with discontinuous flux-functions, J. Hyperbolic Differ. Equ., 2 (2005), 783-837.
3. B. Andreianov, P. Goatin and N. Seguin, Finite volume schemes for locally constrained conservation laws, Numer. Math., 115 (2010), 609-645.
4. S. N. Antontsev, A. V. Kazhikhov and V. N. Monakhov, "Boundary Value Problems in Mechanics of Nonhomogeneous Fluids," Studies in Mathematics and its Applications, 22, North-Holland Publishing Co., Amsterdam, 1990.
5. T. Arbogast, The existence of weak solutions to single porosity and simple dual-porosity models of two-phase incompressible flow, Nonlinear Anal., 19 (1992), 1009-1031.
6. C. Bardos, A. Y. le Roux and J. C. Nédélec, First order quasilinear equations with boundary conditions, Comm. Partial Differential Equations, 4 (1979), 1017-1034.
7. J. Bear, "Dynamic of Fluids in Porous Media," American Elsevier, New-York, 1972.
8. M. Bertsch, R. Dal Passo and C.J. van Duijn, Analysis of oil trapping in porous media flow, SIAM J. Math. Anal., 35 (2003), 245-267 (electronic).
9. R. Bürger, K. H. Karlsen. and J. D. Towers, An Engquist-Osher-type scheme for conservation laws with discontinuous flux adapted to flux connections, SIAM J. Numer. Anal., 47 (2009), 1684-1712.
10. F. Buzzi, M. Lenzinger and B. Schweizer, Interface conditions for degenerate two-phase flow equations in one space dimension, Analysis, 29 (2009), 299-316.
11. C. Cancès, Finite volume scheme for two-phase flow in heterogeneous porous media involving capillary pressure discontinuities, M2AN Math. Model. Numer. Anal., 43 (2009), 973-1001.
12. C. Cancès, Two-phase flows involving discontinuities on the capillary pressure, in "Finite Volumes for Complex Applications V: Problems and Perspectives," Papers from the 5th Symposium held in Aussois, June 08-13, 2008 (eds. R. Eymard and J.-M. Hérard), Hermes Science Publications, Paris, (2008), 249-256.
13. C. Cancès, Asymptotic behavior of two-phase flows in heterogeneous porous media for capillarity depending only of the space. I. Convergence to the optimal entropy solution, SIAM J. Math. Anal., 42 (2010), 946-971.
14. C. Cancès, Asymptotic behavior of two-phase flows in heterogeneous porous media for capillarity depending only of the space. II. Non-classical shocks to model oil-trapping, SIAM J. Math. Anal., 42 (2010), 972-995.
15. C. Cancès, T. Gallouët and A. Porretta, Two-phase flows involving capillary barriers in heterogeneous porous media, Interfaces Free Bound., 11 (2009), 239-258.
16. G. Chavent and J. Jaffré, "Mathematical Models and Finite Elements for Reservoir Simulation," Studies in Mathematics and its Applications, 17, North-Holland Publishing Co., Amsterdam, 1986.
17. Z. Chen, Degenerate two-phase incompressible flow. I. Existence, uniqueness and regularity of a weak solution J. Differential Equations, 171 (2001), 203-232.
18. R. M. Colombo and P. Goatin, A well posed conservation law with a variable unilateral constraint J. Differential Equations, 234 (2007), 654-675.
19. G. Enchéry, R. Eymard and A. Michel, Numerical approximation of a two-phase flow in a porous medium with discontinuous capillary forces, SIAM J. Numer. Anal., 43 (2006), 2402-2422.
20. A. Ern, I. Mozolevski and L. Schuh, Discontinuous Galerkin approximation of two-phase flows in heterogeneous porous media with discontinuous capillary pressures, Comp. Meth. Appl. Mech. Eng., 199 (2010), 1491-1501.
21. B. G. Ersland, M. S. Espedal and R. Nybø., Numerical methods for flow in a porous medium with internal boundaries, Comput. Geosci., 2 (1998), 217-240.
22. G. Gagneux and M. Madaune-Tort, "Analyse Mathématique de Modèles non Linéaires de L'ingénierie Pétrolière," Mathématiques & Applications (Berlin) [Mathematics & Applications], 22, Springer-Verlag, Berlin, 1996.
23. E. F. Kaasschieter, Solving the Buckley-Leverett equation with gravity in a heterogeneous porous medium, Comput. Geosci., 3 (1999), 23-48.
24. J. Málek, J. Nečas, M. Rokyta and M. Ru{ring operator}žička, "Weak and Measure-valued Solutions to Evolutionary PDEs," Applied Mathematics and Mathematical Computation, 13, Chapman & Hall, London, 1996.
25. F. Otto, Initial-boundary value problem for a scalar conservation law, C. R. Acad. Sci. Paris Sér. I Math., 322, (1996), 729-734.
26. E. Yu. Panov, Existence of strong traces for quasi-solutions of multidimensional conservation laws, J. Hyperbolic Differ. Equ., 4 (2007), 729-770.