High-accuracy Simulation of Density Driven Flow in Porous Media

  • Peter Bastian
  • Klaus Johannsen
  • Stefan Lang
  • Christian Wieners
  • Volker Reichenberger
  • Gabriel Wittum
Conference paper

Abstract

A three-dimensional test case for the density driven flow equations in porous media recently proposed by Oswald, Scheidegger and Kinzelbach is investigated numerically. It was shown in [14] that the results from the physical experiment can be reproduced if parameters are adjusted correctly and that a mathematical benchmark can be defined as an idealization of the physical experiment. Intensive numerical investigations were carried out, with calculations on up to 16.7 million grid points that required the efficient solution of linear systems with more than 35 million unknowns.

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References

  1. 1.
    P. Ackerer and A. Younes. On the modelling of density driven flow. In International Conference on Calibration and Reliability in Groundwater Modelling, pages 13–21. IAHS/AISH, ETH Zürich, 20-23 Sept. 1999.Google Scholar
  2. 2.
    R. Barrett, M. Berry, T. Chan J. Demmel, J. Donato, J. Dongarra, V. Eijkhout, R. Pozo, Ch. Romine, and H. van der Vorst. Templates for the Solution of Linear Systems: Building Blocks for Iterative Methods, 2nd Edition. SIAM, Philadelphia, PA, 1994.CrossRefGoogle Scholar
  3. 3.
    P. Bastian, K. Birken, K. Johannsen, S. Lang, N. Neuss, H. Rentz-Reichert, and C. Wieners. UG-a flexible software toolbox for solving partial differential equations. Computing and Visualization in Science, 1:27–40, 1997.MATHCrossRefGoogle Scholar
  4. 4.
    J. Bear and Y. Bachmat. Introduction to Modeling of Transport Phenomena in Porous Media. Kluwer Academic Publishers, 1991.Google Scholar
  5. 5.
    E. Fein(ed). d3f-Ein Programmpaket zur Modellierung von Dichteströmungen. GRS, Braunschweig, GRS-139, ISBN 3-923875-97-5, 1998.Google Scholar
  6. 6.
    P. Frolkovic. Consistent velocity approximation for density driven flow and transport. Advanced Computational Methods in Engineering, eds. R. van Keer et. al, pages 603–611, 1998.Google Scholar
  7. 7.
    W. Hackbusch. Multi-Grid Methods and Applications. Springer-Verlag, Berlin, Heidelberg, 1985.MATHCrossRefGoogle Scholar
  8. 8.
    W. Hackbusch. Iterative Lösung grosser schwachbesetzter Gleichungssysteme. Teubner-Studienbücher: Mathematik, 1991.Google Scholar
  9. 9.
    E. Hairer and G. Wanner. Solving ordinary differential equations II. Springer-Verlag, Berlin, 1991.MATHCrossRefGoogle Scholar
  10. 10.
    A.W. Herbert, C.P. Jackson, and D.A. Lever. Coupled groundwater flow and solute transport with fluid density strongly dependent upon concentration. Water Resources Research, 24:1781–1795, 1988.CrossRefGoogle Scholar
  11. 11.
    Ekkehard O. Holzbecher. Modeling Density-Driven Flow in Porous Media. Springer-Verlag, Berlin, Heidelberg, 1998.CrossRefGoogle Scholar
  12. 12.
    K. Johannsen. Modified SSOR-a smoother for non M-matrices. in preparation.Google Scholar
  13. 13.
    K. Johannsen. On the stability of the Crank-Nicolson-scheme for density driven flow in porous media, unpublished.Google Scholar
  14. 14.
    K. Johannsen, W. Kinzelbach, S. Oswald, G. Wittum Numerical Simulation of density driven flow in porous media Advances in Water Resources, submitted.Google Scholar
  15. 15.
    Anton Leijnse. Three-dimensional modeling of coupled flow and transport in porous media. PhD thesis, University of Notre Dame, Indiana, 1992.Google Scholar
  16. 16.
    A. Wierse M. Rumpf. GRAPE, eine objektorientierte Visualisierungs-und Numerikplattform. Informatik Forschung und Entwicklung, 7:145–151, 1992.Google Scholar
  17. 17.
    S. Oswald. Dichteströmungen in porösen Medien: Dreidimensionale Experimente und Modellierung. PhD thesis, Institut für Hydromechanik und Wasserwirtschaft, ETH Zürich, 1998.Google Scholar
  18. 18.
    S.E. Oswald, M. Scheidegger, and W. Kinzelbach. A three-dimensional physical benchmark test for verification of variable-density driven flow in time. Water Resources Research, submitted.Google Scholar
  19. 19.
    R. Rannacher. Numerical analysis of nonstationary fluid flow. Technical Report SFB 123, Preprint 492, University of Heidelberg, Germany, November 1988.Google Scholar
  20. 20.
    A.E. Scheidegger. General theory of dispersion in porous media. Journal of Geophysical Research, 66:3273, 1961.CrossRefGoogle Scholar
  21. 21.
    H. van der Vorst. Bi-CGSTAB: A fast and smoothly converging variant of bieg for the solution of nonsymmetric linear systems. SIAM J. Sci. Stat. Comp., 13:631–644, 1992.MATHCrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2002

Authors and Affiliations

  • Peter Bastian
    • 1
  • Klaus Johannsen
    • 1
  • Stefan Lang
    • 1
  • Christian Wieners
    • 1
  • Volker Reichenberger
    • 1
  • Gabriel Wittum
    • 1
  1. 1.Interdisziplinäres Zentrum für Wissenschaftliches Rechnen Technical Simulation GroupUniversität HeidelbergHeidelbergGermany

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