# Memory effects induced by dependence on initial conditions and ergodicity of transport in heterogeneous media

## 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

## 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

### References

see the expansion block below.

## PDF

https://agupubs.onlinelibrary.wiley.com/doi/pdf/10.1029/2007WR006740

soon

##### Journal

Water Resour. Res.

##### Print ISSN

Not available yet.

##### Online ISSN

Not available yet.