An H-Adaptive solution of the Richards' Equation

Abstract --

The Richards’ equation describes mathematically the flow of water into the unsaturated soil zone. The model proposed here consists on applying adaptivity with h-refinement in finite element analysis and obtaining a numerical h-adaptive solution of classical Richards' equation. The Euler explicit method discretized the of term time derivative and to confirm the occurrence of the conservation of mass, this adaptive strategy was applied to different meshes with polynomial interpolation function of degree two. Moreover, the h-refinement and the interpolation ensure the concordance between the proposed model and available solutions. When compared with a numerical approximation obtained on uniform meshes, the simulation using h-adaptive scheme required a reduced computational effort.

Keywords --

Adapted mesh techniques, Numerical solution, h-Refinement,Unsaturated porous medium, Water flow.

References --

[1] J. Bear:Hydraulics of groundwater. New York, Dover Publications, INC. Mineola, 2007. 569p.

[2] T. Bunsri, M. Sivakumar, D. Hagare:Numerical Modelling of Tracer Transport in Unsaturated Porous Media. Journal of Applied Fluid Mechanics, v. 1, n. 1, p. 62-70, 2008a.

[3] T. Bunsri, M. Sivakumar, D. Hagare:Influence of Dispersion on Transport of Tracer through Unsaturated Porous Media.J. of Applied Fluid Mechanics, v. 1, n. 2, p. 37-44, 2008b.

[4] M. Celia, E. T. Bouloutas, R. L. Zarba:A general mass conservative numerical solution for the unsaturated flow equation. Water Resour. Res. 26, p. 1483-1496, 1990.

[5] R. Courant, K. Friedrichs, H. Lewy:On the partial difference Equations of Mathematical Physics. v. 11, n.2, p.215, 1967.

[6] H.Darcy: Les Fountaines Publiques de la Ville de Dijon - Appendice D. Paris: Dalmont, 1856.

[7] P. R. B Devloo: PZ: An object oriented environment for scientific programming. Computer Methods in applied mechanics and Engineering,v.150, p.133-153, 1997.

[8] D. G. Fredlund, N. R.Morgenstern: Stress state variables for unsaturated soils. Journal of Geotechnical Engineering Division,ASCE, New York, v. 103, n. 5, p. 447-466, 1977.

[9] J. D.Istok:Groundwater Modeling by the Finite Element Method. Washington: American Geophysical Union, 1989. 495p.

[10] D. K. Jaiswal, A. Kumar,R. R.Yadav:Analytical solution to the one-dimensional advection-diffusion equation with temporally dependent coefficients. Water Resour. 3, 76-84, 2011.

[11] M. M. Namin, M. R. Boroomand:A time splitting algorithm for numerical solution of Richard`s equation. Journal of Hydrology, v. 444-445, 10-21, 2012.

[12] J. S. Pérez Guerrero, T. H.Skaggs: Analytical solution for one-dimensional advection-diffusion transport equation with distance-dependent coefficients. J. Hydrol. 390, 57-65, 2010.

[13] M. L. P.Pizarro:Simulação de Fluxo e Transporte de Solutos na Zona Não-Saturada do Solo (in Portuguese.)  Ph.D. Thesis, Doutorado em Ciências da Engenharia Ambiental Universidade de São Paulo, Escola de Engenharia de São Carlos, São Carlos, 2009.

[14] K.S.H. Prasad, M. S. M. Kumar, M.Sekhar:Modelling flow through unsaturated zones: Sensitivity to unsaturated soil properties. Sãndhanã, v.26, n. 6, p. 517-528, 2001.

[15] W. H.Press:Numerical recipes:the art of scientific computing.Cambridge: Cambridge University Press, 2007. 1235p.

[16] L.A.Richards:Capillary conduction of liquids through porous medium.Physics,v. 1, p. 318-333, 1931.

[17] D. Sperandio, J. T.Mendes, L. H. M. Silva:Cálculo Numérico: Características Matemáticas e Computacionais dos Métodos Numéricos. São Paulo: Pearson Prentice Hall, 2006. 354p.

[18] H. Taheri Shahraiyni, B.AtaieAshtiani: Comparison of finite difference scheme for water flow in unsaturated soils. World Acad. Sci., Eng. Technol. 40, 21-25, 2008.

[19] M. T.van GENUCHTEN:A closed-form equation for predicting the hydraulic conductivity of unsatured soils.Soil Science SocietyofAmericaJournal, v. 44, n. 3, p. 892-898, 1980.

[20] E. Wendland:Contribuição à simulação de processos em meios porosos. Livre Docência (in Portuguese),  Universidade de São Paulo, Escola de Engenharia de São Carlos, São Carlos, 2004. 270p.

[21] M. Wu: A finite-element algorithm for modeling variably saturated flows. J. Hydrol. 394, 315-323, 2010.

[22] J. H.Zar:Biostatistical Analysis. New Jersey: Prentice – Hall, New Jersey, (5ª. Ed.), 2010.



Click here to Download Full Article

About the Author

Maria de Lourdes P. Pizarro

Maria de Lourdes P. Pizarro, Alessandro Firmiano

Brazilian Air Force Academy, Estrada de Aguaí s/n

13.631-972, Pirassununga-SP, Brazil


Edson Wendland

University of São Paulo, Av. Trabalhador Sancarlense, 400
13566-590,São Carlos-SP, Brazil

Most Viewed - All Categories