Abstract
The equation of flow through variably saturated porous media is discretized via the Galerkin finite element formulation. The discretization is coupled with an approach for mesh generation and optimization of the node numbering scheme. Sensitivity analysis showed that the solution behavior is controlled by dimensionless quantities equivalent to Peclet and Courant numbers. For the form of equation investigated, no universal limiting values of Pe and Cr can be established because the values of these parameters depend on both the constitutive relations used and on initial conditions. For more efficient solution of the problem, a deformation scheme of the computational mesh is proposed, which accounts for the limiting Peclet and Courant numbers and for the shape of the deformed elements. Comparisons with other solutions showed that the numerical scheme performs very well.
Keywords
Sensitivity Analysis Porous Medium Information Theory Numerical Scheme Constitutive RelationPreview
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References
- Abriola, L. M. (1986): Finite element solution of the unsaturated flow equation using hierarchical basis functions. Proc. Sixth Int. Conf. Finite Elements in Water Resources, Lisboa, Berlin, Heidelberg, New York: SpringerGoogle Scholar
- Bear, J. (1972). Dynamics of fluids in porous media. New York: ElsevierGoogle Scholar
- Bishop, A. W. (1960). The principle of effective stress. Publ. 32, pp. 1–5, Norwegian Geotech. Inst., Oslo, NorwayGoogle Scholar
- Bishop, A. W.; Blight, G. E. (1963): Some aspects of effective stress in saturated and partially saturated soils. Geotechnique, 13/1, 177–197Google Scholar
- Borja, R. I. (1992): Free boundary, fluid flow, and seepage forces in excavations. J. Geotechnical Eng. ASCE 118/1, 125–146Google Scholar
- Campbell, G. S. (1974): A simple method for determining unsaturated conductivity from moisture retention data. Soil Sci. 117, 311–314Google Scholar
- Celia, M. A.; Bouloutas, E. T.; Zarba, R. L. (1990): A general, mass-conservative numerical solution for the unsaturated flow equation. Water Resour. Res. 26/7, 1483–1496Google Scholar
- Clapp, R. B.; Horberger, G. M. (1978): Empirical equations for some soil hydraulic properties. Water Resour. Res. 14/4, 601–604Google Scholar
- Collins, R. J. (1973): Bandwidth reduction by automatic renumbering. Int. J. Num. Meth. Eng. 6, 345–356Google Scholar
- Cooley, R. L. (1983): Some new procedures for numerical solution of variably saturated flow problems. Water Resour. Res. 19/5, 1271–1285Google Scholar
- Duncan, J. M.; Byrne, P.; Wong, K. S.; Mabry, P. (1980): Strength, stress-strain and bulk modulus parameters for finite element analyses of stresses and movements of soil masses. Rep. No. UCB/GT/80-01, University of California, BerkeleyGoogle Scholar
- Durocher, L. L.; Gasper, A. (1979): A versatile two-dimensional mesh generator with automatic bandwidth reduction, Computers and Structures 10, 561–575Google Scholar
- Elsworth, D.; Bai, M. (1992): Flow-deformation response of dual-porosity media. J. Geotechnical Engl. ASCE 118/1, 107–124Google Scholar
- Gelinas, R. J.; Doss, S. K.; Miller, K. (1981): The moving finite element method: Applications to general partial differential equations with multiple large gradients. J. Comp. Phys. 40, 202–249Google Scholar
- Gottardi, G.; Venutelli, M. (1992): Moving finite element model for one-dimensional infiltration in unsaturated soil. Water Resour. Res. 28, 3259–3267Google Scholar
- Huyakorn, P. S.; Pinder, G. F. (1983): Computational methods in subsurface flow. New York: Academic PressGoogle Scholar
- Huyakorn, P. S.; Thomas, S. D.; Thomson, B. M. (1984): Techniques for making finite elements competitive in modeling flow in variably saturated porous media. Water Resour. Res. 20/8, 1099–1115Google Scholar
- Jensen, O. K.; Finlayson, B. A. (1980): Solution of the transport equations using a moving coordinate system: Advan. Water Resour. 3, 9–18Google Scholar
- Kandil, H.; Miller, C. T.; Skaggs, R. W. (1992): Modeling long-term solute transport in drained unsaturated zones. Water Resour. Res. 28/10, 2799–2809Google Scholar
- Leake, S. A. (1990): Interbed storage and compaction in models of regional groundwater flow. Water Resour. Res. 26/9, 1939–1950Google Scholar
- Lynch, D. R.; O'Neill, K. (1980): Elastic grid deformation for moving boundary problems in two space dimensions. Finite Elem. Water Resour. 3. London: PentechGoogle Scholar
- Lynch, D. R.; O'Neill, K. (1981): Continuously deforming finite elements for the solution of parabolic problems with and without phase change. Int. J. Num. Meth. Eng. 17, 81–96Google Scholar
- Lynch, D. R.; Gary, W. G. (1978): Finite element simulation of shallow water problems with moving boundaries. Finite Elem. Water Resour. 2. London: PentechGoogle Scholar
- Lynch, D. R.; Gary, W. G. (1980): Finite element simulation of flow in deforming regions. J. Comp. Phys. 36, 135–153Google Scholar
- McMurdie, J. L.; Day, P. R. (1960): Slow tests under soil moisture suction. Soil Sci. Soc. Amer. Proc. 24, 441–444Google Scholar
- Morel-Seytoux, H. J. (1987): Multiphase flows in porous media. In: Novak, P. (ed.) Developments in hydraulic engineering-4, London: ElsevierGoogle Scholar
- Narashiman, T. N.; Witherspoon, P. A. (1977): Numerical model for saturated-unsaturated flow in deformable porous media, 1. Theory. Water Resour. Res. 13/3, 657–664.Google Scholar
- Neuman, S. P. (1973): Saturated-unsaturated seepage by finite elements. J. Hydr. Div. ASCE, 99/HY12, 2233–2290Google Scholar
- Neuman, S. P.; Feddes, R. A.; Bresler, E. (1974). Finite element simulation of flow in saturated-unsaturated soils considering water uptake by plants. Rep. Proj. ALOSWC-77. Hafia: Technion UniversityGoogle Scholar
- Neuman, S. P.; Feddes, R. A.; Bresler, E. (1975): Finite element analysis of two dimensional flow in soils considering water uptake by roots, 1. Theory. Soil Sci. Soc. Amer. Proc. 39/2, 224–237Google Scholar
- O'Neill, K. (1981): Highly efficient, oscillation-free solution of the transport equation over long time and large spaces. Water Resour. Res. 17/6, 1665–1675Google Scholar
- Philip, J. R. (1955): Numerical solution of equations of the diffusion type with diffusivity concentration-dependent. Trans. Faraday Soc. 51, 885–892Google Scholar
- Philip, J. R. (1969): Theory of infiltration. Adv. Hydrosci. 5, 216–296Google Scholar
- Ralston, A.; Rabinowitz, P. (1989): A first course in numerical analysis. New York: McGraw-HillGoogle Scholar
- Safai, N. M.; Pinder, G. F. (1979): Vertical and horizontal land deformation in a desaturating porous medium. Adv. Water. Resour. 2, 19–25Google Scholar
- Sehayek, L. (1987): Unsaturated-saturated flow of liquids through deformable soils. Numerical solution and applications to hazardous waste landfills, lagoon leaks and associated spills. Ph.D. Thesis, Rutgers University, New Brunswick, N.J.Google Scholar
- Terzaghi, K. (1925): Principles of soil mechanics, A summary of experimental results of clay and sand. Eng. News Rec. 3–98Google Scholar
- Valliappan, S.; Khalili-Naghadeh, N. (1990): Flow through fissured porous media with deformable matrix. Int. J. Numer. Methods Eng. Sci. 29, 1079–1094Google Scholar
- Vichnevetsky, R. (1991): Computer methods for partial differential equations. New York: Prentice-Hall, Engelwood CliffsGoogle Scholar
- Vreugdenhil, C. (1989): Computational Hydraulics. Berlin, Heidelberg, New York: SpringerGoogle Scholar
- Wu, E. R. (1979): Some findings is using the program “A versatile two-dimensional mesh generator with automatic bandwidth reduction”. Comput. Struct. 12, 181–183Google Scholar