# Non-Fickian Transport Under Heterogeneous Advection and Mobile-Immobile Mass Transfer

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## Abstract

We study the combined impact of heterogeneous advection and mobile–immobile mass transfer on non-Fickian transport using the continuous-time random walk (CTRW) approach. The CTRW models solute transport in heterogeneous media as a random walk in space and time. Our study is based on a *d*-dimensional CTRW model that accounts for both heterogeneous advection and mass transfer between mobile and immobile zones, to which we also refer as solute trapping. The flow heterogeneity is mapped into the distribution of advective transition times over a characteristic heterogeneity scale. Mass transfer into immobile zones is quantified by a trapping rate and the distribution of particle return times. The total particle transition time over a characteristic heterogeneity scale then is given by the advective time and the sum of trapping times over the number of trapping events. We establish explicit integro-partial differential equations for the evolution of the concentration and discuss the relation to the multirate mass transfer approach, specifically the relation between the trapping time distribution and the memory function. We then analyze the signatures of anomalous transport due to advective heterogeneity and trapping in terms of spatial moments and first passage times or breakthrough curves. The behaviors for different disorder scenarios are analyzed analytically and through random walk particle tracking simulations. Assuming that advective mass transfer is faster than diffusive, we identify three regimes of distinct transport behaviors, which are separated by the characteristic trapping rate and trapping times. (1) At early times, we identify a pre-asymptotic time regime that is fully determined by advective heterogeneity and which is characterized by superlinear growth of longitudinal dispersion. (2) For longitudinal dispersion, we identify an intermediate regime of strong superlinear diffusion. This regime is determined by the combined effect of advective heterogeneity and trapping. (3) At larger time, the asymptotic trapping-driven regime shows the signatures of diffusion in immobile zones, which leads to both sub- and superlinear dispersion. These results shed some new light on the mechanism of non-Fickian transport and their manifestation in spatial and temporal solute distributions.

## Keywords

Continuous-time random walk Multirate mass transfer Anomalous transport Stochastic modeling## Notes

### Acknowledgments

The support of the European Research Council (ERC) through the project MHetScale (617511) is gratefully acknowledged.

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