Mathematical Geology

, Volume 28, Issue 1, pp 45–71 | Cite as

Transport in a 2-D saturated porous medium: A new method for particle tracking

  • Frédérick Delay
  • Hélène Housset-Resche
  • Gilles Porel
  • Ghislain de Marsily


A new method for solving the transport equation based on the management of a large numbe of particles in a discretized 2-D domain is presented. The method uses numerical variables to represent the number of particles in a given mesh and is more complex than the 1-D problem. The first part of the paper focuses on the specific management of particles in a 2-D problem. The method also would be valid for three dimensions as long as the medium can be modeled similar to a layered system. As the particles are no longer tracked individually, the algorithm is fast and does not depend on the number of particles present. The numerical tests show that the method is nearly numerical dispersion free and permits accurate calculations even for simulations of low-concentration transport. Because each mesh is considered as a closed system between two successive time steps, it is easy to add adsorption phenomenon without any problem of numerical stability. The model is tested under conditions that are extremely demanding for its operating mode and gives a good fit to analytical solutions. The conditions in which it can be used to best advantage are discussed.

Key words

mass transport 2-D advection-dispersion adsorption particle tracking 


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  1. Ackerer, P., 1988, Random-walk method to simulate pollutant transport in alluvial aquifers or fractured rocks,in Custodio, C., and others, eds., Groundwater flow and quality modeling, Spec. Vol: D. Reidel Publ. Co., Dordrecht, The Netherlands, p. 475–486.Google Scholar
  2. Ackerer, P., and Kinzelbach, W., 1985, Modélisation du transport de contaminants par la méthode de marche au hasard. Influence des variations du champ d'écoulement au cours du temps sur la dispersion: Proc. Intern. Symp. Stochastic approach to subsurface flow (Montvillagenne, France), Intern. Assoc. Hydraul. Res., 12 p.Google Scholar
  3. Ackerer. P., Mose, R., and Sernba, K., 1990, Natural tracer test simulation by stochastic particle tracking method,in Moltyaner, G., ed., Proc. Intern. Conf., Transport and mass exchange processes in sand and gravel aquifers, Ottawa, Canada, p. 595–604.Google Scholar
  4. Ackerer, P., Magnico, P., and Mose, R., 1994, About pollutants transport modelling in groundwater: Rev. Sci. Eau, v. 7, no. 2, p. 201–212.Google Scholar
  5. Bear, J., 1979, Hydraulics of groundwater: McGraw-Hill Book Co., New York, 567 p.Google Scholar
  6. Bobba, A. G., 1990, Numerical model of contaminant transport through conduit-porous matrix system: Math. Geology, v. 21, no. 8, p. 861–890.CrossRefGoogle Scholar
  7. Cameron, D. R., and Klute, A., 1977, Convective dispersive solute transport with a combined equilibrium and kinetics adsorption model: Water Resources Res., v. 13, no. 1, p. 183–188.Google Scholar
  8. Coats, K. H., and Smith, B. D., 1964, Dead-end pore volume and dispersion in porous media: Soc. Petrol. Eng. Jour., v. 4, p. 78–84.Google Scholar
  9. Delay, F., Dzikowsky, M., and de Marsily. G., 1993, A new algorithm for representing transport in porous media in one dimension, including convection, dispersion and interaction with the immobile phase with first-order kinetics: Math. Geology, v. 25, no. 6, p. 689–712.CrossRefGoogle Scholar
  10. Delay, F., de Marsily, G., and Carlier, E., 1994, One-dimensional solution of the transport equation in porous media in transient state by a new numerical method for the management of particle track: Computers & Geosciences, v. 20, no. 7/8, p. 1169–1200.Google Scholar
  11. Dzikowsky, M., 1992, L'analyse des systémes traçages à débit Variable et volume constant. Possibilités d'application en milieu karstique. unpubl. Thèse Univ. Lille, (France), 182 p.Google Scholar
  12. Euvrard, D., 1990, Résolution numérique des équations aux dérivées partielles, différences finies, éléments finis, méthode des singularités: Masson ed., Paris. 341 p.Google Scholar
  13. Goblet, P., 1981, Modélisation des transferts de masse et d'énergie en aquifère: unpubl. Thèse Docteur Ingénieur, Ecole Mines Paris, 199 p.Google Scholar
  14. Grisak, G. E., and Pickens, J. F., 1980a, Solute transport through fractured media. 1 the effect of matrix diffusion: Water Resources Res., v. 16, no. 4, p. 719–730.Google Scholar
  15. Grisak, G. E., and Pickens, J. F., 1980b, Solute transport through fractured media. 2 column study of fractured till: Water Resources Res., v. 16, no. 4, p. 731–739.Google Scholar
  16. Kinzelbach, W., 1986, Groundwater modelling: an introduction with sample programs in BASIC: Elsevier ed., Amsterdam, 333 p.Google Scholar
  17. Legrand-Marc, C., and Laudelout, H., 1985, Longitudinal dispersion in a forest stream: Jour. Hydrology, v. 78, no. 3–4, p. 317–324.CrossRefGoogle Scholar
  18. Maloszewsky, P., and Zuber, A., 1991, Influence of matrix diffusion and exchange reactions on radiocarbon ages in fissured carbonate aquifers: Water Resources Res., v. 27, no. 8, p. 1937–1945.CrossRefGoogle Scholar
  19. de Marsily, G., 1986, Quantitative hydrogeology: groundwater hydrology for engineers: Academic Press, San Diego, California, 440 p.Google Scholar
  20. Pironneau, O., 1988, Finite element methods for fluids: Applied Mathematics Research Collection: Masson ed., Paris, 200 p.Google Scholar
  21. Prickett, T. A., Naymik, T. G., and Lonnquist, P. P., 1981, A random-walk solute transport model for selected groundwater quality evaluations: Illinois State Water Survey, v. 65, p. 26–34.Google Scholar
  22. Saiers, J. E., Hornberger, M., and Liang, L., 1994, First- and second-order kinetics approaches for modeling the transport of colloidal particles in porous media: Water Resources Res., v. 30, no. 9, p. 2499–2506.CrossRefGoogle Scholar
  23. Uffink, G. J. M., 1985, A random-walk method for the simulation of macrodispersion in a stratified Aquifer: I.U.G.G., 18th Gen. Assembly Proc. (Hamburg), Symposium, I.A.S.H. Publ., v. 146, p. 103–114.Google Scholar
  24. Uffink, G. J. M., 1988, Modelling of solute transport with the random-walk method,in Custodio, M., and others, eds., groundwater flow and quality modeling, Spec. Vol: D. Reidel Publ. Co., Dordrecht, The Netherlands, p. 247–265.Google Scholar
  25. Uffink, G. J. M., 1990, Analysis of dispersion by the random-walk Method: unpubl. doctoral dissertation, Univ. Delft, The Netherlands, 147 p.Google Scholar
  26. Valocchi, A. J., 1985, Validity of the local equilibrium assumption for modeling sorbing solute transport through homogeneous soils: Water Resources Res, v. 21, no. 6, p. 808–820.Google Scholar
  27. Van der Lee, J., Ledoux, E., and de Marsily, G., 1992, Modelling of colloidal transport in a fractured medium: Jour. Hydrology, v. 139, no. 1–4, p. 135–158.CrossRefGoogle Scholar
  28. Zuber, A., 1986, On the interpretation of tracer data in variable flow systems: Jour. Hydrology, v. 86, no. 1–2, p. 45–57.CrossRefGoogle Scholar
  29. Zwater, P. C., 1993, Groundwater Flow Contaminant Transport Software, Zace Services Limited Software Engineering. PO Box 2, CH 1015, Lausanne, Switzerland.Google Scholar

Copyright information

© International Association for Mathematical Geology 1996

Authors and Affiliations

  • Frédérick Delay
    • 1
  • Hélène Housset-Resche
    • 1
  • Gilles Porel
    • 2
  • Ghislain de Marsily
    • 1
  1. 1.Laboratoire de Géologie AppliquéeUniversité Paris VIParis, Cedex 05France
  2. 2.Laboratoire HydrogéologieUniversité de PoitiersPoitiersFrance

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