Itô equation model for dispersion of solutes in heterogeneous media
Keywords:
Itō equation, random fields, memory effects, ergodicityAbstract
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.Downloads
References
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, https://doi.org/10.1063/1.870855
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, https://doi.org/10.1029/wr019i004p00938
Bouchaud, J.-P. and Georges, A., Anomalous diffusion in disordered media: Statistical mechanisms, models and physical applications, Phys. Rep., 195, pp. 127-293, 1990, https://doi.org/10.1016/0370-1573(90)90099-n
Brouwers, J. J. H., On diffusion theory in turbulence, J. Eng. Mat., 44, pp. 277-295, 2002, https://doi.org/10.1023/a:1020962403844
Compte, A. and Cáceres, M. O., Fractional dynamics in random velocity fields, Phys. Rev. Lett., 81(15), pp. 3140-3143, 1998, https://doi.org/10.1103/physrevlett.81.3140
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.
Dagan, G., Theory of solute transport by groundwater, Annu. Rev. Fluid Mech., 19, pp. 183-215, 1987, https://doi.org/10.1146/annurev.fl.19.010187.001151
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, https://doi.org/10.1088/1751-8113/40/4/002.
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, https://doi.org/10.1016/s0169-7722(00)00123-6
Gardiner, C. W., Handbook of Stochastic Methods (for Physics, Chemistry and Natural Science), Springer, New York, 1985.
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, https://doi.org/10.1029/97wr00553
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, https://doi.org/10.1029/1999wr900224
Le Doussal, P. and Machta, J., Annealed versus quenched diffusion coefficient in random media, Phys. Rev. B, 40(12), pp. 9427-9430, 1989, https://doi.org/10.1103/physrevb.40.9427
Lumley, J. L., An approach to the Eulerian-Lagrangian problem, J. Math. Phys., 3(2), pp. 309-312, 1962, https://doi.org/10.1063/1.1703805
Lundgren, T. S., Turbulent pair dispersion and scalar diffusion, J. Fluid Mech. 111, pp. 27-57, 1981, https://doi.org/10.1017/s0022112081002280
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, https://doi.org/10.1029/93wr02947
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, https://doi.org/10.1029/2005wr004546
Suciu, N., Vamoş, C. and J. Eberhard, Evaluation of the first-order approximations for transport in heterogeneous media, Water Resour. Res., 42, W11504, 2006, https://doi.org/10.1029/2005wr004714
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, https://doi.org/10.1029/2007wr005946
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, https://doi.org/10.1002/pamm.200700057
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 http://www.am.uni-erlangen.de/de/preprints2000.html),
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, https://doi.org/10.1029/2007wr006740, 2008.
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, https://doi.org/10.1016/s0021-9991(03)00073-1
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.
Zirbel, C. L., Lagrangian observations of homogeneous random environments, Adv. Appl. Prob., 33, pp. 810-835, 2001, https://doi.org/10.1017/s0001867800011216
Zwanzig, R., Memory effects and irreversible thermodynamics, Phys. Rev., 124(4), pp. 983-992, 1961, https://doi.org/10.1103/physrev.124.983
Downloads
Published
How to Cite
Issue
Section
License
Copyright (c) 2015 Journal of Numerical Analysis and Approximation Theory
This work is licensed under a Creative Commons Attribution 4.0 International License.
Open Access. This article is distributed under the terms of the Creative Commons Attribution 4.0 International License, which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.
