Parallelizable Methods for Modeling Flow and Transport in Heterogeneous Porous Media

  • Myron B. Allen
  • Mark C. Curran
Part of the ISNM International Series of Numerical Mathematics book series (ISNM, volume 114)


Groundwater contaminant modeling presents several challenges to the mathematician. Among these are the need to compute accurate water velocities and difficulties arising from fine-scale heterogeneities and sharp concentration fronts. This paper presents parallelizable numerical methods that address these challenges.

For groundwater flow, mixed finite-element models yield velocities comparable in accuracy to computed heads. However, large variations in hydraulic conductivity can cause iterative matrix solvers to converge slowly. The fine grids needed to resolve heterogeneities aggravate the poor conditioning. A parallelizable, multigrid-based iterative scheme for the lowest-order mixed method largely overcomes both sources of poor behavior.

For contaminant transport, finite-element collocation yields high-order spatial accuracy. The timestepping scheme combines a modified method of characteristics, which reduces temporal errors when advection dominates, with an alternating-direction formulation, which is “embarrassingly parallel” and has a favorable operation count.


Outer Iteration Contaminant Transport Heterogeneous Porous Medium Hydrodynamic Dispersion Miscible Displacement 
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  1. [1]
    Allen M. B., Ewing R. E., Lu P. Well conditioned iterative schemes for mixed finite-element models of porous-media flow. SIAM Jour. Sci. Stat. Comp., 13:794–814, 1992.CrossRefGoogle Scholar
  2. [2]
    Allen M. B., Khosravani A. Solute transport via alternating-direction collocation using the modified method of characteristics. Advances in Water Resources, 15:125–132, 1992.CrossRefGoogle Scholar
  3. [3]
    Celia M. A. Collocation on deformed finite elements and alternating direction collocation methods. PhD thesis, Princeton University, Princeton, 1983.Google Scholar
  4. [4]
    Curran M. C., Allen M. B. Parallel computing for solute transport models via alternating-direction collocation. Adv. Water Resour., 13(2):70–75, 1990.CrossRefGoogle Scholar
  5. [5]
    Douglas J., Ewing R. E., Wheeler M. F. The approximation of the pressure by a mixed method in the simulation of miscible displacement. R.A.I.R.O. Analyse Numerique, 17:17–33, 1983.Google Scholar
  6. [6]
    Dupont T. Galerkin methods for first-order hyperbolics: An example. SIAM J. Numer. Anal., 10:890–899, 1973.CrossRefGoogle Scholar
  7. [7]
    Krishnamachari S. V., Hayes L. J., Russell T. F. A finite element alternating-direction method combined with a modified method of characteristics for convection-diffusion problems. SIAM J. Numer. Anal., 26(6): 1462–1473, 1989.CrossRefGoogle Scholar
  8. [8]
    Percell P., Wheeler M. F. A C 1 finite element collocation method for elliptic equations. SIAM J. Numer. Anal., 17(5):605–622, 1980.CrossRefGoogle Scholar
  9. [9]
    Prenter P. M. Splines and Variational Methods. Wiley, New York, 1975.Google Scholar
  10. [10]
    Raviart P. A., Thomas J. M. A mixed finite element method for second order elliptic problems. In Mathematical Aspects of the Finite Element Method, volume 606 of Lecture Notes in Mathematics, pages 292–315. Springer-Verlag, Berlin and New York, 1977. I. Galligani and E. Magenes, eds.CrossRefGoogle Scholar
  11. [11]
    Russell T. F. An incompletely iterated characteristic finite element method for a miscible displacement problem. PhD thesis, University of Chicago, Chicago, 1980.Google Scholar
  12. [12]
    Shen J. Mixed finite element methods: analysis and computational aspects. PhD thesis, University of Wyoming, Laramie, 1992.Google Scholar
  13. [13]
    Tuminaro R. S., Womble D. E. Analysis of the multigrid FMV cycle on large-scale parallel machines. SIAM Jour. Sci. Stat. Comp. To appear.Google Scholar

Copyright information

© Springer Basel AG 1993

Authors and Affiliations

  • Myron B. Allen
    • 1
  • Mark C. Curran
    • 2
  1. 1.Department of MathematicsUniversity of WyomingLaramieUSA
  2. 2.Applied and Numerical Mathematics DivisionSandia National LaboratoriesAlbuquerqueUSA

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