Advertisement

Block Iterative Strategies for Multiaquifer Flow Models

  • G. Gambolati
  • P. Teatini
Part of the International Centre for Mechanical Sciences book series (CISM, volume 364)

Abstract

When groundwater flow takes place in aquifer-aquitard systems characterized by a high contrast of hydrogeological parameters and non-linear aquitard behavior, quasi — three-dimensional models of flow must be solved by a fully numerical approach. The numerical implementation with finite difference or finite element methods involves large systems whose solution requires much CPU time and computer storage. A new solution strategy of the resulting algebraic equations is suggested to overcome non-linearity in the aquitards. The global nonlinear system is decoupled into a number of smaller subsystems, which are consistent with the geologic structure of the multiaquifer system and are ideally suited for updating the nonlinear hydrologic parameters in the aquitards. The aquifer and the aquitard equations are solved separately with the modified conjugate gradient (MCG) and the Thomas algorithms, respectively, while the final coupled solution is obtained with an iterative procedure. The procedure, as is naturally suggested by the special block tridiagonal pattern of the coefficient matrix, can be shown to be equivalent to a block Gauss-Seidel strategy and can therefore be generalized into a block SOR (successive over-relaxation) strategy, the blocks corresponding to the aquifer and aquitard equations. The convergence properties of the new SOR scheme are initially analyzed with linear porous systems for which the optimum over-relaxation factor ω opt can be theoretically computed, and the asymptotic convergence rate is studied in relation to the geometry, the hydrogeological parameters of the multiaquifer system and the size of the numerical model. The results obtained with steady state sample problems show that if flow in the various aquifers is weakly interconnected (i.e. the hydraulic exchange between the aquifer units is quite limited) ω opt is close to 1, while in strongly coupled systems ω opt falls into the upper range (1.5 <ω opt <2) and the SOR asymptotic convergence rate can be as much as one order of magnitude larger than the Gauss-Seidel rate. Subsequently the block iterative method has been extended to non-linear porous media and early results indicate that the relaxation procedure with ω = ω opt provides a significant acceleration of convergence as well. The linear theory, however, does no more apply and ω opt must be assessed empirically.

Keywords

Hydraulic Head Iteration Matrix Asymptotic Rate Hydrogeological Parameter Computer Storage 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Herrera, I. and G. E. Figueroa Vega, A correspondence principle for the theory of leaky aquifers, Water Resour. Res. 5, 900–904, 1969.CrossRefGoogle Scholar
  2. 2.
    Herrera, I., Theory of multiple leaky aquifers, Water Resour. Res. 6, 185–193, 1970.CrossRefGoogle Scholar
  3. 3.
    Herrera, I. and R. Yates, Integrodifferential equations for systems of leaky aquifers and applications, 3, A numerical method of unlimited applicability, Water Resour. Res. 13 (4), 725–732, 1977.CrossRefGoogle Scholar
  4. 4.
    de Marsily, G., E. Ledoux, A. Levassor, D. Poitrinal and A. Salem, Modeling of large multiaquifer systems: Theory and application, J. Hydrol. 36, 1–33, 1978.CrossRefGoogle Scholar
  5. 5.
    Hennart, J. P., R. Yates and I. Herrera, Extension of the integrodifferential approach to inhomogeneous multiaquifer systems, Water Resour. Res. 17, 1044–1050, 1981.CrossRefGoogle Scholar
  6. 6.
    Premchitt, J. A., A technique in using integrodifferential equations for model simulation of multiaquifer systems, Water Resour. Res. 17, 162–168, 1981.CrossRefGoogle Scholar
  7. 7.
    Gambolati, G., F. Sartoretto and F. Uliana, A conjugate gradient finite element model of flow for large multiaquifer systems, Water Resour. Res. 22 (7), 1003–1015, 1986.CrossRefGoogle Scholar
  8. 8.
    Fujinawa, K., Finite element analysis of groundwater flow in multiaquifer systems, 2, A quasi three dimensional flow model, J. Hydrol. 33, 349–362, 1977.CrossRefGoogle Scholar
  9. 9.
    Chorley, D. W. and E. O. Frind, An iterative quasi three-dimensional finite element model for heterogeneous multiaquifer systems, Water Resour. Res. 14 (5), 943–952, 1978.CrossRefGoogle Scholar
  10. 10.
    Neuman, S. P., C. Preller and T. N. Narashiman, Adaptive explicit-implicit quasi three-dimensional finite element model of flow and subsidence in multiaquifer systems, Water Resour. Res. 18 (5), 1551–1561, 1982.CrossRefGoogle Scholar
  11. 11.
    Rivera, A., Modèle hydrogéologique quasi-tridimensionnel non-linéaire pour simuler la subsidence dans les systèmes aquifères multicouches. Cas de Mexico. PhD thesis, Ecole des Mines de Paris-CIG, 1990.Google Scholar
  12. 12.
    Hantush, M. S., Modification of the theory of leaky aquifers, J. Geophys. Res. 65, 3713–3725,1960.Google Scholar
  13. 13.
    Westlake, J. R., Numerical Matrix Inversion and Solution of Linear Equations. John Wiley, New York, 1968.zbMATHGoogle Scholar
  14. 14.
    Kershaw, D. S., The incomplete Cholesky-conjugate gradient method for the iterative solution of systems of linear equations, J. Comp. Phys. 26, 43–65, 1978.MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    Gambolati, G., Fast solution to finite element flow equations by Newton iteration and modified conjugate gradient method, Int. J. Numer. Methods Eng. 15, 661–675, 1980.CrossRefzbMATHGoogle Scholar
  16. 16.
    Gambolati, G., Perspective on a modified conjugate gradient method for the solution of linear sets of subsurface equations. In: Wang, S. Y. and et al. (eds.) 3rd Int. Conf. Finite Elements in Water Resources. Missisipi University Press, pp 2.15–2. 30, 1980.Google Scholar
  17. 17.
    Gambolati, G. and A. M. Perdon, The conjugate gradients in flow and land subsidence modeling. In: Bear, J. and Y. Corapcioglu (eds.) Fundamentals of Transport Phenomena in Porous Media. NATO-ASI Series, Applied Sciences 82, Martinus Nijoff B.V., The Hague, pp 953–984, 1984.Google Scholar
  18. 18.
    Varga, R. S., Matrix Iterative Analysis. Prentice-Hall, Englewood Cliffs, New Jersey, 1962.Google Scholar
  19. 19.
    Young, D., Iterative methods for solving partial differential equations of the elliptic type, Trans. Amer. Math. Soc. 76, 92–111, 1954.MathSciNetCrossRefzbMATHGoogle Scholar
  20. 20.
    Rudolph, D. and E. O. Frind, Hydraulic response of highly compressible aquitards during consolidation, Water Resour. Res. 27 (1), 17–30, 1991.CrossRefGoogle Scholar

Copyright information

© Springer-Verlag Wien 1995

Authors and Affiliations

  • G. Gambolati
    • 1
  • P. Teatini
    • 1
  1. 1.University of PaduaPaduaItaly

Personalised recommendations