## Abstract

For transport in statistically homogeneous random velocity fields with properties that are routinely assumed in stochastic groundwater models, the one‐particle dispersion (i.e., second central moment of the ensemble average concentration for a point source) is a “memory‐free” quantity independent of initial conditions. Nonergodic behavior of large initial plumes, as manifest in deviations of actual solute dispersion from one‐particle dispersion, is associated with a “memory term” consisting of correlations between initial positions and displacements of solute molecules. Reliable numerical experiments show that increasing the source dimensions has two opposite effects: it reduces the uncertainty related to the randomness of center of mass, but, at the same time, it yields large memory terms. The memory effects increase with the source dimension and depend on its shape and orientation. Large narrow sources oriented transverse to the mean flow direction yield ergodic behavior with respect to the one‐particle dispersion of the longitudinal dispersion and nonergodic behavior of the transverse dispersion, whereas for large longitudinal sources, the longitudinal dispersion behaves nonergodically, and the transverse dispersion behaves ergodically. Such memory effects are significantly large over hundreds of heterogeneity scales and should therefore be considered in practical applications, for instance, calibration of model parameters, forecasting, and identification of the contaminant source.

## Authors

N. **Suciu
**Institute of Applied Mathematics, Friedrich-Alexander University of Erlangen-Nuremberg, Erlangen, Germany

C. **Vamoş
**Tiberiu Popoviciu Institute of Numerical Analysis, Romanian Academy, Cluj-Napoca, Romania

H. **Vereecken
**ICG-IV, Research Center Julich, Julich, Germany

K. **Sabelfeld
**Weierstrass Institute for Applied Analysis and Stochastics, Berlin,Germany

Institute of Computational Mathematics and Mathematical Geophysics,

Siberian Branch of Russian Academy of Sciences, Novosibirsk, Russia

P. **Knabner
**Institute of Applied Mathematics, Friedrich-Alexander University of Erlangen-Nuremberg, Erlangen, Germany

## Keywords

## Cite this paper as:

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

doi: 10.1029/2007WR006740

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[1] Chile`s, J. P., and P. Delfiner (1999), *Geostatistics: Modeling Spatial Uncertainty*, John Wiley, New York.

[2] Dagan, G. (1990), *Transport in heterogeneous porous formations: Spatial moments, ergodicity and effective dispersion*, Water Resour. Res., 26, 1281 – 1290.

[3] Dentz, M., and J. Carrera (2007), *Mixing and spreading in stratified flow*, Phys. Fluids, 19, 017107, doi:10.1063/1.2427089.

[4] Dentz, M., H. Kinzelbach, S. Attinger, and W. Kinzelbach (2000), *Temporal behavior of a solute cloud in a heterogeneous porous medium: 2. Spatially extended injection*, Water Resour. Res., 36, 3605 – 3614.

[5] Eberhard, J., N. Suciu, and C. Vamos¸ (2007), *On the self-averaging of dispersion for transport in quasi-periodic random media*, J. Phys. A Math. Theor., 40, 597 – 610, doi:10.1088/1751-8113/40/4/002.

[6] Fiori, A., and I. Jancovic´ (2005), *Can we determine the transverse macrodispersivity by using the method of moments?*, Adv. Water Resour., 28, 589 – 599, doi:10.1016/j.advwaters.2004.09.909.

[7] Kitanidis, P. K. (1988), *Prediction by the method of moments of transport in a heterogeneous formation*, J. Hydrol., 102, 453 – 473.

[8] Le Doussal, P., and J. Machta (1989), *Annealed versus quenched diffusion coefficient in random media*, Phys. Rev. B, 40(12), 9427 – 9430.

[9] Morales-Casique, E., S. P. Neuman, and A. Gaudagnini (2006), *Nonlocal and localized analyses of nonreactive solute transport in bounded randomly heterogeneous porous media: Theoretical framework*, Adv. Water Resour., 29, 1238 – 1255, doi:10.1016/j.advwatres.2005.10.002. Naff, R. L., D. F.

[10] Haley, and E. A. Sudicky (1998), *High-resolution Monte Carlo simulation of flow and conservative transport in heterogeneous porous media: 2. Transport results*, Water Resour. Res., 34, 679 – 697.

[11] Phythian, R., and W. D. Curtis (1978), *The effective long-time diffusivity for a passive scalar in a Gaussian model fluid flow*, J. Fluid Mech., 89(2), 241 – 260.

[12] Radu, F. A., I. S. Pop, and P. Knabner (2004), *Order of convergence estimates for an Euler implicit, mixed finite element discretization of Richards’ equation*, SIAM J. Numer. Anal., 42(4), 1452 – 1478, doi:10.1137/S0036142902405229

[13] Radu, F. A., I. S. Pop, and S. Attinger (2008), *Analysis of an Euler implicit –mixed finite element scheme for reactive solute transport in porous media*, CASA Rep. 08-06, Tech. Univ. Eindhoven, Eindhoven, Netherlands.

[14] Sposito, G. (2001), *Topological groundwater hydrodynamics*, Adv. Water Resour., 24, 793 – 801.

[15] Sposito, G., and G. Dagan (1994), *Predicting solute plume evolution in heterogeneous porous formations*, Water Resour. Res., 30, 585 – 589.

[16] Suciu, N., and C. Vamos¸ (2007), *Comment on ‘‘Nonstationary flow and nonergodic transport in random porous media*’’ by G. Darvini and P. Salandin, Water Resour. Res., 43, W12601, doi:10.1029/ 2007WR005946.

[17] Suciu, N., C. Vamos¸, J. Vanderborght, H. Hardelauf, and H. Vereecken (2006a), *Numerical investigations on ergodicity of solute transport in heterogeneous aquifers*, Water Resour. Res., 42, W04409, doi:10.1029/ 2005WR004546.

[18] Suciu, N., C. Vamos¸, and J. Eberhard (2006b), *Evaluation of the first-order approximations for transport in heterogeneous media*, Water Resour. Res., 42, W11504, doi:10.1029/2005WR004714.

[19] Suciu, N., C. Vamos¸, and K. Sabelfeld (2007a), *Ergodic simulations of diffusion in random velocity fields*, in Monte Carlo and Quasi-Monte Carlo Methods 2006, edited by A. Keller, S. Heinrich, and H. Niederreiter, pp. 659 – 668, Springer, Heidelberg, Germany.

[20] Suciu, N., C. Vamos¸, K. Sabelfeld, and C. Andronache (2007b), *Memory effects and ergodicity for diffusion in spatially correlated velocity fields*, Proc. Appl. Math. Mech., 7, 2010015, doi:10.1002/pamm.20070057.

[21] Suciu, N., C. Vamos¸, H. Vereecken, K. Sabelfeld, and P. Knabner (2008), *Lagrangian stationarity and memory effects for dispersion in ergodic velocity fields,* paper presented at General Assembly 2008, Eur. Geosci. Union, Vienna, 13 – 18 April.

[22] Vamos, C., N. Suciu, and H. Vereecken (2003), *Generalized random walk algorithm for the numerical modeling of complex diffusion processes*, J. Comput. Phys., 186(2), 527 – 544, doi:10.1016/S0021-9991(03)00073-1.

[23] van Kampen, N. G. (1981), *Stochastic Processes in Physics and Chemistry*, North-Holland, Amsterdam. Yaglom, A. M. (1987), Correlation Theory of Stationary and Related Random Functions, vol. 1, Basic Results, Springer, New York.

[24] Yaglom, A. M. (1987), *Correlation Theory of Stationary and Related Random Functions*, vol. 1, Basic Results, Springer, New York.

[25] Zirbel, C. L. (2001), *Lagrangian observations of homogeneous random environments*, Adv. Appl. Prob., 33, 810 – 835.

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