ELSEVIER Available online at www.sciencedirect.com MATHEMATICAL AND SCIENCE ~__~DIRECTe COMPUTER MODELLING Mathematical and Computer Modelling 42 (2005) 1137-1144 www.elsevier.eom/locate/mcm Proposal for a Probabilistic Model of Dispersion: A First Validation E. CARLIER* AND J. EL KHATTABI Centre de Calcul et de Mod41isation de Lens Facult4 Jean Perrin, Universit4 d'Artois SP 18, rue Jean Souvraz, 62 307 Lens cedex, France carlier~univ-art ois. fr (Received and accepted May 2004) Abstract--The probabilistic approach is used to simulate the particle tracking for two types of porous media. The first one is sand grains with a single intergranular porosity. The particle tracking is carried out by advection and dispersion. The second one is chalk granulates with intergranular and matrix porosities. Sorption can occur with advection and dispersion during particle tracking. The particle tracking is simulated as the sum of elementary steps with independent random vari- ables in the sand medium. An exponential distribution is obtained for each elementary step and shows that the whole process is Markovian. A gamma distribution or probability density function is then deduced. The relationship between dispersivity and the elementary step is given using the central limit theorem. The particle tracking in the chalky medium is a non-Markovian process. The probability density function depends of a power to the distance. Experimental simulation by dye tracer tests on a column have been performed for different distances and discharges. The probabilistic approach computations are in good agreement with the experimental data. The probabilistie computation seems an interesting and complementary approach to simulate transfer phenomena in porous media with respect to the traditional numerical methods. @ 2005 Elsevier Ltd. All rights reserved. Keywords--Particle, Tracking, Groundwater, Dispersion, Modelling. 1. INTRODUCTION The probabilistic approach of particle tracking in natural environment is not new. It was used successfully in sedimentary transport [1,2] in solutes transfer in porous medium without sorp- tion phenomena [3,4] and in solutes transfer in fissured medium [5-7]. The modelling of particle tracking in a porous or fissured medium with sorption capacity is very difficult when the ex- change coefficients between matrix and water are difficult to estimate. The modelling of such a process is carried out by the resolution of the dispersion equation by taking account of the sorption phenomena [8 17]. According to these authors, the models are very sensitive to the exchange coefficient values. The error on these coefficients must be very weak in order to keep the model reliable. Furthermore, particle tracking modelling using the classic solution of the *Author to whom all correspondence should be addressed. 0895-7177/05/$ - see front matter @ 2005 Elsevier Ltd. All rights reserved. doi:10.1016/j.mcm.2004.05.014 Typeset by .AjMS-TEX