Small- and waiting-time behavior of a darcy flow model with a dynamic pressure saturation relation

SIAM Journal on Applied Mathematics

Volumn 66, Issue 5, 2006, Pages 1482-1511

  King, J R ^a   Cuesta, C M ^b  

^a UNIVERSITY OF NOTTINGHAM   (United Kingdom)

^b UNIVERSITY OF VIENNA   (Austria)

Author keywords

Degenerate pseudoparabolic equation; Small time behavior; Time reversibility; Waiting time phenomena

Indexed keywords

INTERFACES (MATERIALS); POROUS MATERIALS; PRESSURE CONTROL; TIME VARYING SYSTEMS;

FINITE TIME; FLOW MODEL; PSEUDOPARABOLIC EQUATION; TIME EVOLUTION; WAITING TIME BEHAVIOR;

FLOW OF FLUIDS;

EID: 33750170480     PISSN: 00361399     EISSN: None     Source Type: Journal    
DOI: 10.1137/040610969     Document Type: Article

References (32)

1. D. G. ARONSON, The porous medium equation, in Nonlinear Diffusion Problems, Lect. 2nd 1985 Seas. C.I.M.E., Montecatini Terme, Italy, 1985, Lecture Notes in Math. 1224, Springer, Berlin, 1986, pp. 1-46.
2. D. G. ARONSON, L. A. CAFFARELLI, AND S. KAMIN, How an initially stationary interface begins to move in porous medium flow, SIAM J. Math. Anal., 14 (1983), pp. 639-658.
3. G. BARENBLATT, V. ENTOV, AND V. RYZHIK, Theory of Fluid Flows through Natural Rocks, Theory and Applications of Transport in Porous Media 3, Kluwer Academic, Dordrecht, The Netherlands, 1990.
4. G. I. BARENBLATT, M. BERTSCH, R. DAL PASSO, V. M. PROSTOKISHIN, AND M. UGHI, A mathematical model of turbulent heat and mass transfer in stably stratified shear flow, J. Fluid Mech., 253 (1993), pp. 341-358.
5. G. I. BARENBLATT, M. BERTSCH, R. DAL PASSO, AND M. UGHI, A degenerate pseudoparabolic regularization of a nonlinear forward-backward heat equation arising in the theory of heat and mass exchange in stably stratified turbulent shear flow, SIAM J. Math. Anal., 24 (1993), pp. 1414-1439.
6. G. I. BARENBLATT, J. GARCÍA-AZORERO, A. DE PABLO, AND J. L. VÁZQUEZ, Mathematical model of the non-equilibrium water-oil displacement in porous strata, Appl. Anal., 65 (1997), pp. 19-45.
7. A. BOURGEAT AND M. PANFILOV, Effective two-phase flow through highly heterogeneous porous media: Capillary nonequilibrium effects, Comput. Geosci., 2 (1998), pp. 191-215.
8. L. A. CAFFARBLLI AND A. FRIEDMAN, Regularity of the free boundary for the one-dimensional flow of gas in a porous medium, Amer. J. Math., 101 (1979), pp. 1193-1218.
9. S. J. CHAPMAN AND J. R. KING, The selection of Saffman-Taylor fingers by kinetic undercooling, J. Engrg. Math., 46 (2003), pp. 1-32.
10. C. CUESTA, C. J. VAN DUIJN, AND J. HULSHOF, Infiltration in porous media with dynamic capillary pressure: Traveling waves, European J. Appl. Math., 11 (2000), pp. 381-397.
11. C. M. CUESTA, Pseudo-parabolic Equations with Driving Convection Term, Ph.D. thesis, Vrije Universiteit Amsterdam, 2003.
12. L. J. CUMMINGS, S. D. HOWISON, AND J. R. KING, Two-dimensional Stokes and Hele-Shaw flows with free surfaces, European J. Appl. Math., 10 (1999), pp. 635-680.
13. W.-P. DÜLL, Theory of a Pseudoparabolic Partial Differential Equation Modeling Solvent Uptake in Polymeric Solids, Ph.D. thesis, University of Bonn, Germany, 2004.
14. C. M. ELLIOTT AND H. GARCKE, On the Cahn-Hilliard equation with degenerate mobility, SIAM J. Math. Anal. 27 (1996), pp. 404-423.
15. J. D. EVANS AND J. R. KING, Asymptotic results for the Stefan problem with kinetic undercooling, Quart. J. Mech. Appl. Math., 53 (2000), pp. 449-473.
16. R. E. GRUNDY, Local similarity solutions for the initial value problem in nonlinear diffusion, IMA J. Appl. Math., 30 (1983), pp. 209-214.
17. S. M. HASSANIZADEH, Dynamic effects in the capillary pressure saturation relationship, in Proceedings of the 4th Annual International Conference on Civil Engineering, Water Resources and Environmental Engineering, Vol. 4, Tehran, Iran, 1997, Sharif University of Technology, pp. 141-149.
18. S. M. HASSANIZADEH, M. A. CELIA, AND H. K. DAHLE, Dynamic effect in the capillary pressure-saturation relationship and their impacts on unsaturated flow, Vadose Zone Hydrology, 1 (2002), pp. 487-510.
19. S. M. HASSANIZADEH AND W. G. GRAY, Thermodynamic basis of capillary pressure in porous media, Water Resour. Res., 29 (1993), pp. 3389-3405.
20. S. M. HASSANIZADEH, R. J. SCHOTTING, AND A. Y. BELIAEV, A new capillary pressure-saturation relationship including hysteresis and dynamic effects, in Proceedings of the 13th Annual International Conference on Computational Methods in Water Resources, Calgary, Canada, 2000, pp. 245-251.
21. J. HULSHOF AND J. R. KING, Analysis of a Darcy flow model with a dynamic pressure saturation relation, SIAM J. Appl. Math., 59 (1998), pp. 318-346.
22. W. L. KATH AND D. S. COHEN, Waiting-time behavior in a nonlinear diffusion equation, Stud. Appl. Math., 67 (1982), pp. 79-105.
23. J. R. KING, Mathematical Aspects of Semiconductors Process Modelling, Ph.D. thesis, University of Oxford, UK, 1986.
24. J. R. KING AND C. M. CUESTA, Small and Waiting Time Behaviour of a Darcy Flow Model with a Dynamic Pressure Saturation Relation, Technical report, RTN "Front and Singularities," 2005; available online from http://www.iac.rm.cnr.it/rtn/.
25. J. R. KING, A. A. LACEY, AND J. L. VÁZQUEZ, Persistence of corners in free boundaries in Hele-Shaw flow, European J. Appl. Math., 6 (1995), pp. 455-490.
26. J. R. KING AND J. M. OLIVER, Thin-film modelling of poroviscous free surface flows, European J. Appl. Math., 4 (2005), pp. 493-517.
27. A. A. LACEY, Initial motion of the free boundary for a nonlinear diffusion equation, IMA J. Appl. Math., 31 (1983), pp. 113-119.
28. A. A. LACEY, J. R. OCKENDON, AND A. B. TAYLER, "Waiting-time" solutions of a nonlinear diffusion equation, SIAM J. Appl. Math., 42 (1982), pp. 1252-1264.
29. A. NOVICK-COHEN, On the viscous Cahn-Hilliard equation, in Material Instabilities in Continuum Mechanics, Oxford University Press, New York, 1988, pp. 329-342.
30. V. PADRÓN, Sobolev regularization of a nonlinear ill-posed parabolic problem as a model for aggregating populations, Comm. Partial Differential Equations, 23 (1998), pp. 457-486.
31. M. RAUSCHER, A. MÜNCH, B. WAGNER, AND R. BLOSSEY, A thin-film equation for viscoelastic liquids of Jeffreys type, Eur. Phys. J. E, 17 (2005), pp. 373-379.
32. F. STAUFFER, Time dependence of the relations between capillary pressure, water content and conductivity during drainage of porous media, in Proceedings of the LAHR Symposium on Scale Effects in Porous Media, Thessaloniki, Greece, LAHR, Madrid, Spain, 1978.