Journal of Applied Mathematics

Variational analysis for simulating free-surface flows in a porous medium

Shabbir Ahmed and Charles Collins

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A variational formulation has been developed to solve a parabolic partial differential equation describing free-surface flows in a porous medium. The variational finite element method is used to obtain a discrete form of equations for a two-dimensional domain. The matrix characteristics and the stability criteria have been investigated to develop a stable numerical algorithm for solving the governing equation. A computer programme has been written to solve a symmetric positive definite system obtained from the variational finite element analysis. The system of equations is solved using the conjugate gradient method. The solution generates time-varying hydraulic heads in the subsurface. The interfacing free surface between the unsaturated and saturated zones in the variably saturated domain is located, based on the computed hydraulic heads. Example problems are investigated. The finite element solutions are compared with the exact solutions for the example problems. The numerical characteristics of the finite element solution method are also investigated using the example problems.

Article information

J. Appl. Math., Volume 2003, Number 8 (2003), 377-396.

First available in Project Euclid: 3 July 2003

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 65M60: Finite elements, Rayleigh-Ritz and Galerkin methods, finite methods 65N30
Secondary: 35K20


Ahmed, Shabbir; Collins, Charles. Variational analysis for simulating free-surface flows in a porous medium. J. Appl. Math. 2003 (2003), no. 8, 377--396. doi:10.1155/S1110757X03301147.

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  • S. Ahmed, Variational formulations for a parabolic partial differential equation describing variably saturated flow, Master's thesis, Department of Mathematics, The University of Tennessee, Tennessee, 2001.
  • G. A. Baker, J. H. Bramble, and V. Thomée, Single step Galerkin approximations for parabolic problems, Math. Comp. 31 (1977), no. 140, 818--847.
  • J. Bear, Hydraulics of Groundwater, McGraw-Hill, New York, 1979.
  • K. Eriksson and C. Johnson, Error estimates and automatic time step control for nonlinear parabolic problems. I, SIAM J. Numer. Anal. 24 (1987), no. 1, 12--23.
  • G. H. Golub and C. F. Van Loan, Matrix Computations, Johns Hopkins Studies in the Mathematical Sciences, Johns Hopkins University Press, Maryland, 1996.
  • P. S. Huyakorn, P. F. Anderson, J. W. Mercer, and H. O. White Jr., Saltwater intrusion in aquifers: development and testing of a three-dimensional finite element model, Water. Resour. Res. 23 (1987), no. 2, 293--312.
  • P. K. Jimack, A best approximation property of the moving finite element method, SIAM J. Numer. Anal. 33 (1996), no. 6, 2286--2302.
  • C. Johnson, Numerical Solution of Partial Differential Equations by the Finite Element Method, Cambridge University Press, Cambridge, 1990.
  • T. Kuppusamy, J. Sheng, J. C. Parker, and R. J. Lenhard, Finite element analysis of multiphase immiscible flow through soils, Water Resour. Res. 23 (1987), no. 4, 625--631.
  • M. Luskin and R. Rannacher, On the smoothing property of the Galerkin method for parabolic equations, SIAM J. Numer. Anal. 19 (1981), no. 1, 93--113.
  • S. P. Neumann, Saturated-unsaturated seepage by finite elements, ASCE J. Hydraul. Div. 99 (1973), no. 12, 2233--2250.
  • A. Pandit and J. M. Abi-Aoun, Numerical modeling of axisymmetric flow, Journal of Ground Water 32 (1994), no. 3, 458--464.
  • R. Srivastava and T.-C. J. Yeh, A three-dimensional numerical model for water flow and transport of chemically reactive solute through porous media under variably saturated conditions, Adv. in Water Res. 15 (1992), 275--287.
  • V. Thomée, Negative norm estimates and superconvergence in Galerkin methods for parabolic problems, Math. Comp. 34 (1980), no. 149, 93--113.
  • M. T. van Genuchten, A closed-form equation for predicting the hydraulic conductivity of unsaturated soils, Soil Sci. Soc. Am. J. 44 (1980), 892--898.
  • T.-C. J. Yeh, R. Srivastava, A. Guzman, and T. Harter, A numerical model for water flow and chemical transport in variably saturated porous media, Journal of Ground Water 31 (1993), no. 4, 634--644.