Derivation of seawater intrusion models by formal asymptotics

SIAM Journal on Applied Mathematics

Volumn 74, Issue 4, 2014, Pages 1152-1173

  Jazar, M ^a   Monneau, R ^b  

^a LEBANESE UNIVERSITY   (Lebanon)

^b UNIVERSITÉ PARIS EST   (France)

Author keywords

Boussinesq equation; Confined and unconfined aquifer; Dupuit Forchheimer; Formal asymptotics; Ghyben Herzberg relation; Groundwater flow; Porous medium; Saltwater and freshwater interface; Seawater intrusion; Shallow water

Indexed keywords

SALT WATER INTRUSION;

BOUSSINESQ EQUATIONS; DUPUIT-FORCHHEIMER; FORMAL ASYMPTOTICS; GHYBEN-HERZBERG RELATION; POROUS MEDIUM; SHALLOW WATERS; UNCONFINED AQUIFERS;

GROUNDWATER FLOW;

EID: 84907494167     PISSN: 00361399     EISSN: None     Source Type: Journal    
DOI: 10.1137/120867561     Document Type: Article

References (36)

1. A. Al Bitar, Modélisation des écoulements en milieu poreux hétérogènes 2D/3D, avec couplages surface/souterrain et densitaires, Thèse de l'Institut National Polytechnique de Toulouse, Toulouse, France, 2007.
2. G. Alduncin, J. Esquivel-Avila, and L. Reyes-Avila, Steady filtration problems with seawater intrusion: Variational analysis, Symposium on Advances in Computational Mechanics, Vol. 3 (Austin, TX, 1997), Comput. Methods Appl. Mech. Engrg., 151 (1998), pp. 13-25.
3. H.W. Alt and C.J. van Duijn, A stationary flow of fresh and salt groundwater in a coastal aquifer, Nonlinear Anal., 14 (1990), pp. 625-656.
4. M. Bakker, Simple groundwater flow models for seawater intrusion, in Proceedings of the 16th Salt Water Intrusion Meeting (SWIM16), Wolin Island, Poland, 2000; available online at http://www.swim-site.nl/pdf/swim16.html.
5. J. Bear, Hydraulics of Groundwater, McGraw-Hill, New York, 1979.
6. J. Bear and A.H.D. Cheng, Modeling Groundwater Flow and Contaminant Transport, Theory Appl. Transp. Porous Media 23, Springer, Dordrecht, The Netherlands, 2010.
7. J. Bear, A.H.D. Cheng, S. Sorek, D. Ouazar, and I. Herrera, eds., Seawater Intrusion in Coastal Aquifers - Concepts, Methods and Practices, Theory Appl. Transp. Porous Media 14, Kluwer Academic Publishers, Dordrecht, The Netherlands, 1999.
8. J. Bear and A. Verruijt, Modelling Groundwater Flow and Pollution, D. Reidel, Dordrecht, The Netherlands, 1987.
9. A. Bonnet and R. Monneau, On the mushy region arising between two fluids in a porous medium, Nonlinear Anal. Real World Appl., 5 (2004), pp. 159-182.
10. J. Boussinesq, Recherches théoriques sur l'écoulement des nappes d'eau infiltrées dans le sol et sur le débit des sources, J. Math. Pur. Appl. Sér. 5, 10 (1904), pp. 5-78.
11. W.C. Burnett, P.K. Aggarwal, A. Aureli, H. Bokuniewicz, J.E. Cable, M.A. Charette, E. Kontar, S. Krupa, K.M. Kulkarni, A. Loveless, W.S. Moore, J.A. Oberdorfer, J. Oliveira, N. Ozyurt, P. Povinec, A.M.G. Privitera, R. Rajar, R.T. Ramessur, J. Scholten, T. Stieglitz, M. Taniguchi, and J.V. Turner, Review: Quantifying submarine groundwater discharge in the coastal zone via multiple methods, Sci. Total Environ., 367 (2006), pp. 498-543.
12. Z. Chen and R. Ewing, Mathematical analysis for reservoir models, SIAM J. Math. Anal., 30 (1999), pp. 431-453.
13. M. Chipot, Variational Inequalities and Flow in Porous Media, Appl. Math. Sci. 52, Springer-Verlag, New York, 1984.
14. H. Darcy, Les fontaines publiques de la ville de Dijon; Exposition et application des principes à employer dans les questions de distribution d'eau, Victor Dalmont, Editeur, Paris, 1856.
15. J. Dupuit, Etudes théoriques et pratiques sur le mouvement des eaux dans les canaux découverts et à travers les terrains perméables, 2nd ed., Dunod, Paris, 1863.
16. M. El Alaoui Talibi, D. Ouazar, and M.H. Tber, Identification of the hydraulic conductivities in a saltwater intrusion problem, J. Inverse Ill-Posed Probl., 15 (2007), pp. 935-954.
17. M. El Alaoui Talibi and M.H. Tber, Existence of solutions for a degenerate seawater intrusion problem, Electron. J. Differential Equations, 72 (2005), pp. 1-14.
18. H.I. Essaid, A multilayered sharp interface model of coupled freshwater and saltwater flow in a coastal system: Model development and application, Water Resources Res., 26 (1990), pp. 1431-1454.
19. R. Eymard, R. Herbin, and A. Michel, Mathematical study of a petroleum-engineering scheme, ESAIM Math. Model. Numer. Anal., 37 (2003), pp. 937-972.
20. C.W. Fetter, Jr., Hydrogeology: A short history, Part 2, Ground Water, 42 (2004), pp. 949-953.
21. J.-F. Gerbeau and B. Perthame, Derivation of viscous Saint-Venant system for laminar shallow water; Numerical validation, Discrete Contin. Dyn. Syst. Ser. B, 1 (2001), pp. 89-102.
22. A.R. Kacimov and Yu.V. Obnosov, Analytical solution for a sharp interface problem in sea water intrusion into a coastal aquifer, Proc. R. Soc. Lond. A, 457 (2001), pp. 3023-3038.
23. A.R. Kacimov, Yu.V. Obnosov, M.M. Sherif, and J.S. Perret, Analytical solution to a sea-water intrusion problem with a fresh water zone tapering to a triple point, J. Engrg. Math., 54 (2006), pp. 197-210.
24. C. Kao, Fonctionnement hydraulique des nappes superficielles de fonds de vallées en interaction avec le réseau hydrographique, Thèse de doctorat en Sciences de l'Eau, Ecole Nationale du Génie Rural, des Eaux et Forêts, Paris, 2002.
25. G. Keady, The Dupuit approximation for the rectangular dam problem, IMA J. Appl. Math., 44 (1990), pp. 243-260.
26. M. Kemblowski, The impact of the Dupuit-Forchheimer approximation on salt-water intrusion simulation, Ground Water, 25 (1987), pp. 331-336.
27. C.M. Marle, Henry Darcy et les écoulements de fluides en milieu poreux, Oil & Gas Sci. Tech. Rev. Inst. Français du Petrol, 61 (2006), pp. 599-609.
28. P.Y. Polubarinova-Kochina, Theory of Groundwater Movement (translated from Russian edition of 1952 by R.J.M. De Wiest), Princeton University Press, Princeton, NJ, 1962.
29. S. Sorek, V. Borisov, and A. Yakirevich, A two-dimensional areal model for density dependent flow regime, Transp. Porous Media, 43 (2001), pp. 87-105.
30. O.D.L. Strack, Groundwater Mechanics, Prentice-Hall, Englewood Cliffs, NJ, 1989.
31. O.D.L. Strack, A Dupuit-Forchheimer model for three-dimensional flow with variable density, Water Resources Res., 31 (1995), pp. 3007-3017.
32. O.D.L. Strack, R.J. Barnes, and A. Verruijt, Vertically integrated flows, discharge potential, and the Dupuit-Forchheimer approximation, Ground Water, 44 (2006), pp. 72-75.
33. M.H. Tber, On a seawater intrusion problem, Appl. Sci., 9 (2007), pp. 163-173.
34. C.J. van Duyn and D. Hilhorst, On a doubly nonlinear diffusion equation in hydrology, Nonlinear Anal., 11 (1987), pp. 305-333.
35. J.L. Vazquez, The Porous Medium Equation: Mathematical Theory, Oxford University Press, Oxford, UK, 2007.
36. E.G. Youngs, An examination of computed steady-state water-table heights in unconfined aquifers: Dupuit-Forchheimer estimates and exact analytical results, J. Hydrol., 119 (1990), pp. 201-214.