Itô equation model for dispersion of solutes in heterogeneous media


  • N. Suciu Friedrich-Alexander University of Erlangen-Nuremberg, Germany
  • C. Vamoş Tiberiu Popoviciu Institute of Numerical Analysis, Romania
  • H. Vereecken Research Center Jülich, Germany
  • K. Sabelfeld Weierstrass Institute for Applied Analysis and Stochastics, Berlin, Germany
  • P. Knabner Friedrich-Alexander University of Erlangen-Nuremberg, Germany



Itō equation, random fields, memory effects, ergodicity
Abstract views: 272


Transport processes in heterogeneous media such as ionized plasmas, natural porous media, and turbulent atmosphere are often modeled as diffusion processes in random velocity fields. Using the Itô formalism, we decompose the second spatial moments of the concentration and the equivalent effective dispersion coefficients in terms corresponding to various physical factors which influence the transport. We explicitly define "ergodic'' dispersion coefficients, independent of the initial conditions and completely determined by local dispersion coefficients and velocity correlations. Ergodic coefficients govern an upscaled process which describes the transport at large tine-space scales. The non-ergodic behavior at finite times shown by numerical experiments for large initial plumes is explained by "memory terms'' accounting for correlations between initial positions and velocity fluctuations on the trajectories of the solute molecules.


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Balescu, R., Wang, H.-D. and Misguich, J. H., Langevin equation versus kinetic equation: Subdiffusive behavior of charged particles in a stochastic magnetic field, Phys. Plasmas 1(12), pp. 3826-3842, 1994, DOI:

Bhattacharya, R. N. and Gupta, V. K., A Theoretical Explanation of Solute Dispersion in Saturated Porous Media at the Darcy Scale, Water Resour. Res., 19, pp. 934-944, 1983, DOI:

Bouchaud, J.-P. and Georges, A., Anomalous diffusion in disordered media: Statistical mechanisms, models and physical applications, Phys. Rep., 195, pp. 127-293, 1990, DOI:

Brouwers, J. J. H., On diffusion theory in turbulence, J. Eng. Mat., 44, pp. 277-295, 2002, DOI:

Compte, A. and Cáceres, M. O., Fractional dynamics in random velocity fields, Phys. Rev. Lett., 81(15), pp. 3140-3143, 1998, DOI:

Conlon, J. G. and Naddaf, A., Green's functions for elliptic and parabolic equations with random coefficients, New York Journal of Mathematics, 6, pp. 153-225, 2000. DOI:

Dagan, G., Theory of solute transport by groundwater, Annu. Rev. Fluid Mech., 19, pp. 183-215, 1987, DOI:

Doob, J. L., Stochastic Processes, John Wiley & Sons, London, 1953.

Eberhard J., Suciu, N. and C. Vamoş, On the self-averaging of dispersion for transport in quasi-periodic random media, J. Phys. A: Math. Theor., 40, pp. 597-610, 2007, DOI:

Fiori, A. and Dagan, G., Concentration fluctuations in aquifer transport: a rigorous first-order solution and applications, J. Contam. Hydrol., 45, pp. 139-163, 2000, DOI:

Gardiner, C. W., Handbook of Stochastic Methods (for Physics, Chemistry and Natural Science), Springer, New York, 1985. DOI:

Kloeden, P. E. and Platten, E, Numerical solutions of stochastic differential equations, Springer, Berlin, 1999.

Jaekel, U. and Vereecken, H., Renormalization group analysis of macrodispersion in a directed random flow, Water Resour. Res., 33, pp. 2287-2299, 1997, DOI:

LaBolle, E. M., Quastel, J., Fogg, G. E. and Gravner, J., Diffusion processes in composite media and their numerical integration by random walks: Generalized stochastic differential equations with discontinuous coefficients, Water Resour. Res., 36(3), pp. 651-662, 2000, DOI:

Le Doussal, P. and Machta, J., Annealed versus quenched diffusion coefficient in random media, Phys. Rev. B, 40(12), pp. 9427-9430, 1989, DOI:

Lumley, J. L., An approach to the Eulerian-Lagrangian problem, J. Math. Phys., 3(2), pp. 309-312, 1962, DOI:

Lundgren, T. S., Turbulent pair dispersion and scalar diffusion, J. Fluid Mech. 111, pp. 27-57, 1981, DOI:

Monin, A. S. and Yaglom, A. M., Statistical Fluid Mechanics: Mechanics of Turbulence, MIT Press, Cambridge, M A, 1971.

Sposito, G. and Dagan, G., Predicting solute plume evolution in heterogeneous porous formations, Water Resour. Res., 30(2), pp. 585-589, 1994, DOI:

Suciu, N., Some Relations Between Microscopic and Macroscopic Modelling of Thermodynamic Processes (in Romanian), Ed. Univ. Piteşti, Appl. and Ind. Math. Series, No. 5, 2001.

Suciu, N. and Vamoş, C., Effective diffusion in heterogeneous media, Internal Report ICG-IV.00303, Research Center Jülich, 2003.

Suciu, N., Vamoş, C., Vanderborght, J., H. Hardelauf and Vereecken, H., Numerical investigations on ergodicity of solute transport in heterogeneous aquifers, Water Resour. Res., 42, W04409, 2006, DOI:

Suciu, N., Vamoş, C. and J. Eberhard, Evaluation of the first-order approximations for transport in heterogeneous media, Water Resour. Res., 42, W11504, 2006, DOI:

Suciu, N. and Vamoş, C., Comment on "Nonstationary flow and nonergodic transport in random porous media" by G. Darvini and P. Salandin, Water Resour. Res., 43, W12601, 2007, DOI:

Suciu N., Vamoş, C., Sabelfeld, K. and Andronache, C., Memory effects and ergodicity for diffusion in spatially correlated velocity fields, Proc. Appl. Math. Mech., 7, 2010015-2010016, 2007, DOI:

Suciu N., Vamos, C., Vereecken, H., Sabelfeld, K. and Knabner, P., Dependence on initial conditions, memory effects, and ergodicity of transport in heterogeneous media, Preprint No. 324, Institute of Applied Mathematics, Friedrich-Alexander University Erlangen-Nuremberg, 2008, (available online at,

Suciu, N., Vamos, C., Vereecken, H., Sabelfeld, K. and Knabner, P., Memory effects induced by dependence on initial conditions and ergodicity of transport in heterogeneous media, Water Resour. Res., 44, W08501,, 2008. DOI:

Vamoş, C., Suciu, N., Vereecken, H., Vanderborht, J. and Nitzsche, O., Path decomposition of discrete effective diffusion coefficient, Internal Report ICG-IV.00501, Research Center Jülich, 2001.

Vamoş, C., Suciu, N. and Vereecken, H., Generalized random walk algorithm for the numerical modeling of complex diffusion processes, J. Comp. Phys., 186(2), pp. 527-544, 2003, DOI:

van Kampen, N. G., Stochastic Processes in Physics and Chemistry, North-Holland, Amsterdam, 1981.

Yaglom, A. M., Correlation Theory of Stationary and Related Random Functions, Volume I: Basic Results, Springer-Verlag, New York, 1987. DOI:

Zirbel, C. L., Lagrangian observations of homogeneous random environments, Adv. Appl. Prob., 33, pp. 810-835, 2001, DOI:

Zwanzig, R., Memory effects and irreversible thermodynamics, Phys. Rev., 124(4), pp. 983-992, 1961, DOI:




How to Cite

Suciu, N., Vamoş, C., Vereecken, H., Sabelfeld, K., & Knabner, P. (2008). Itô equation model for dispersion of solutes in heterogeneous media. Rev. Anal. Numér. Théor. Approx., 37(2), 221–238.