Abstract
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.
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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.
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.
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.
Ackerer, P., Magnico, P., and Mose, R., 1994, About pollutants transport modelling in groundwater: Rev. Sci. Eau, v. 7, no. 2, p. 201–212.
Bear, J., 1979, Hydraulics of groundwater: McGraw-Hill Book Co., New York, 567 p.
Bobba, A. G., 1990, Numerical model of contaminant transport through conduit-porous matrix system: Math. Geology, v. 21, no. 8, p. 861–890.
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.
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.
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.
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.
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.
Euvrard, D., 1990, Résolution numérique des équations aux dérivées partielles, différences finies, éléments finis, méthode des singularités: Masson ed., Paris. 341 p.
Goblet, P., 1981, Modélisation des transferts de masse et d'énergie en aquifère: unpubl. Thèse Docteur Ingénieur, Ecole Mines Paris, 199 p.
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.
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.
Kinzelbach, W., 1986, Groundwater modelling: an introduction with sample programs in BASIC: Elsevier ed., Amsterdam, 333 p.
Legrand-Marc, C., and Laudelout, H., 1985, Longitudinal dispersion in a forest stream: Jour. Hydrology, v. 78, no. 3–4, p. 317–324.
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.
de Marsily, G., 1986, Quantitative hydrogeology: groundwater hydrology for engineers: Academic Press, San Diego, California, 440 p.
Pironneau, O., 1988, Finite element methods for fluids: Applied Mathematics Research Collection: Masson ed., Paris, 200 p.
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.
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.
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.
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.
Uffink, G. J. M., 1990, Analysis of dispersion by the random-walk Method: unpubl. doctoral dissertation, Univ. Delft, The Netherlands, 147 p.
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.
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.
Zuber, A., 1986, On the interpretation of tracer data in variable flow systems: Jour. Hydrology, v. 86, no. 1–2, p. 45–57.
Zwater, P. C., 1993, Groundwater Flow Contaminant Transport Software, Zace Services Limited Software Engineering. PO Box 2, CH 1015, Lausanne, Switzerland.
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Delay, F., Housset-Resche, H., Porel, G. et al. Transport in a 2-D saturated porous medium: A new method for particle tracking. Math Geol 28, 45–71 (1996). https://doi.org/10.1007/BF02273523
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DOI: https://doi.org/10.1007/BF02273523