An efficient lumped mixed hybrid finite element formulation for variably saturated groundwater flow

Vadose Zone Journal

Volumn 8, Issue 2, 2009, Pages 352-362

  Belfort, Benjamin ^a   Ramasomanana, Fanilo ^a   Younes, Anis ^a   Lehmann, François ^a  

^a UNIVERSITÉ DE STRASBOURG   (France)

Author keywords

[No Author keywords available]

Indexed keywords

FINITE ELEMENT METHOD; GROUNDWATER FLOW; INFILTRATION; NUMERICAL MODEL; PERMEABILITY; RICHARDS EQUATION; TRIANGULATION; UNSATURATED FLOW; VADOSE ZONE;

EID: 67149108357     PISSN: None     EISSN: 15391663     Source Type: Journal    
DOI: 10.2136/vzj2008.0108     Document Type: Article

References (43)

1. Ackerer, Ph., A. Younes, and R. Mosé. 1999. Modeling variable density flow and solute transport in porous medium: 1. Numerical model and verification. Transp. Porous Media 35:345-373.
2. Arbogast, T., C.N. Dawson, P.T. Keenan, M.F. Wheeler, and I. Yotov. 1998. Enhanced cell-centered finite differences for elliptic equations on general geometry. SIAM J. Sci. Comput. 19:404-425.
3. Baranger, J., J.F. Maitre, and F. Oudin. 1994. Application de la théorie des éléments finis mixtes à létude d'une classe de schémas aux volumesdifférences finis pour les problèmes elliptiques. C.R. Acad. Sci. I. Math. 319:401-404.
4. Belfort, B., and F. Lehmann. 2005. Comparison of equivalent conductivities for numerical simulation of one-dimensional unsaturated flow. Vadose Zone J. 4:1191-1200.
5. Bergamaschi, L., S. Mantica, and F. Saleri. 1998. A mixed finite element-finite volume formulation of the black-oil model. SIAM J. Sci. Comput. 20:970-997.
6. Bergamaschi, L., and M. Putti. 1999. Mixed finite elements and Newton-type linearizations for the solution of Richards' equation. Int. J. Numer. Methods Eng. 45:1025-1046.
7. Brezzi, F., and M. Fortin. 1991. Mixed and hybrid finite element methods. Springer-Verlag, Berlin.
8. Celia, M.A., E.T. Bouloutras, and R.L. Zarba. 1990. A general mass-conservative numerical solution for the unsaturated flow equation. Water Resour. Res. 26:1483-1496.
9. Chavent, G., and J. Jaffré. 1986. Mathematical models and finite elements for reservoir simulation. North-Holland, Amsterdam.
10. Chavent, G., and J.E. Roberts. 1991. A unified physical presentation of mixed, mixed hybrid finite elements and standard finite difference approximations for the determination of velocities in waterflow problems. Adv. Water Resour. 14:329-348.
11. Chavent, G., A. Younes, and Ph. Ackerer. 2003. On the finite volume reformulation of the mixed finite element method for elliptic and parabolic PDE on triangles. Comput. Methods Appl. Mech. Eng. 192:655-682.
12. Farthing, M.W., C.E. Kees, and C.T. Miller. 2003. Mixed finite element methods and higher order temporal approximations for variably saturated groundwater flow. Adv. Water Resour. 26:373-394.
13. Freeze, R.A. 1971. Th ree-dimensional, transient, saturated-unsaturated flow in a groundwater basin. Water Resour. Res. 7:347-366.
14. Ippisch, O., H.-J. Vogel, and P. Bastian. 2006. Validity limits for the van Genuchten- Mualem model and implications for parameter estimation and numerical simulation. Adv. Water Resour. 29:1780-1789.
15. Ju, S.-H., and K-.J.S. Kung. 1997. Mass types, element orders and solution schemes for the Richards equation. Comput. Geosci. 23:175-187.
16. Karthikeyan, M., T.-S. Tan, and K.K. Phoon. 2001. Numerical oscillation in seepage analysis of unsaturated soils. Can. Geotech. J. 38:639-651.
17. Kavetski, D., P. Binning, and S.W. Sloan. 2001. Adaptive time stepping and error control in a mass conservative numerical solution of the mixed form of Richards equation. Adv. Water Resour. 24:595-605.
18. Mansell, R.S., L. Ma, L.R. Ahuja, and S.A. Bloom. 2002. Adaptive grid refinement in numerical models for water flow and chemical transport in soil: A review. Vadose Zone J. 1:222-238.
19. Marini, L.D., and P. Pietra. 1990. New mixed finite element schemes for current continuity equations. COMPEL 9:257-268.
20. Mazzia, A. 2008. An analysis of monotonicity conditions in the mixed hybrid finite element method on unstructured triangulations. Int. J. Numer. Methods Eng. 76:351-375.
21. Micheletti, S., and R. Sacco. 1999. Stabilized mixed finite elements for fluid models in semiconductors. Comput. Visual. Sci. 2:139-147.
22. Micheletti, S., R. Sacco, and F. Saleri. 2001. On some mixed finite element methods with numerical integration. SIAM J. Sci. Comput. 23:245-270.
23. Milly, P.C.D. 1985. A mass-conservative procedure for time-stepping in models of unsaturated flow. Adv. Water Resour. 8:32-36.
24. Mosé, R., P. Siegel, P. Ackerer, and G. Chavent. 1994. Application of the mixed hybrid finite element approximation in a groundwater flow model: Luxury or necessity? Water Resour. Res. 30:3001-3012.
25. Mualem, Y. 1976. A new model for predicting the hydraulic conductivity of unsaturated porous media. Water Resour. Res. 12:513-522.
26. Narasimhan, T.N. 2004. Darcy's law and unsaturated flow. Vadose Zone J. 3:1059.
27. Narasimhan, T.N. 2006. On storage coefficient and vertical strain. Ground Water 44:488-491.
28. Pan, L., A.W. Warrick, and P.J. Wierenga. 1996. Finite element methods for modeling water flow in variably saturated porous media: Numerical oscillation and mass-distributed schemes. Water Resour. Res. 32:1883-1889.
29. Philip, J.R. 1957. Th eory of infiltration: 1. The infiltration equation and its solution. Soil Sci. 83:435-448.
30. Rathfelder, K., and L.M. Abriola. 1994. Mass conservative numerical solutions of the head-based Richards equation. Water Resour. Res. 30:2579-2586.
31. Raviart, P.A., and J.M. Th omas. 1977. A mixed finite element method for second order elliptic problems. p. 292-315. In I. Galligani and E. Magenes (ed.) Mathematical aspects of finite element methods. Proc. Conf., Rome. 10-12 Dec. 1975. Lect. Notes Math. 606. Springer, New York.
32. Roberts, J.E., and J.M. Th omas. 1990. Mixed and hybrid methods. P. 523-640. In P.G. Ciarlet and J.L. Lyons (ed.) Handbook of numerical analysis. Vol. 2. Finite element methods (Part 1). North-Holland, Amsterdam.
33. Sacco, R., and F. Saleri. 1997. Stabilized mixed finite volume methods for convection- diffusion problems. East West J. Numer. Math. 5:291-311.
34. Schaap, M.G., and M.Th. van Genuchten. 2006. A modified Mualem-van Genuchten formulation for improved description of the hydraulic conductivity near saturation. Vadose Zone J. 5:27-34.
35. Segerlind, L.J. 1984. Applied finite element analysis. 2nd ed. John Wiley & Sons. Simunek, J., and S.A. Bradford. 2008. Vadose zone modeling: Introduction and importance. Vadose Zone J. 7:581-586.
36. Th omas, J.M. 1977. Sur l'Analyse Numérique des Méthodes d'Eléments Finis Hybrides et Mixtes. Ph.D. diss. Université Pierre et Marie Curie, Paris.
37. van Genuchten, M.Th. 1980. A closed-form equation for predicting the hydraulic conductivity of unsaturated soils. Soil Sci. Soc. Am. J. 44:892-898.
38. Vogel, T., M.Th. van Genuchten, and M. Cislerova. 2001. Effect of the shape of the soil hydraulic functions near saturation on variably-saturated flow predictions. Adv. Water Resour. 24:133-144.
39. Weiser, A., and M.F. Wheeler. 1988. On convergence of block-centered finite differences for elliptic problems. SIAM J. Numer. Anal. 25:351-375.
40. Wheeler, M.F., and M. Peszynska. 2002. Computational engineering and science methodologies for modeling and simulation of subsurface applications. Adv. Water Resour. 25:1147-1173.
41. Younes, A., P. Ackerer, and G. Chavent. 2004. From mixed finite elements to fi- nite volumes for elliptic PDEs in two and three dimensions. Int. J. Numer. Methods Eng. 59:365-388.
42. Younes, A., P. Ackerer, and F. Lehmann. 2006. A new mass lumping scheme for the mixed hybrid finite element method. Int. J. Numer. Methods Eng. 67:89-107.
43. Younes, A., Ph. Ackerer, R. Mosé, and G. Chavent. 1999. A new formulation of the mixed finite element method for solving elliptic and parabolic PDE with triangular elements. J. Comput. Phys. 149:148-167.