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

- 479 Downloads
- 7 Citations

## 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.

## References

- Abramowitz, M., Stegun, I.A.: Handbook of Mathematical Functions. Dover, New York (1972)Google Scholar
- Adams, E.E., Gelhar, L.W.: Field study of dispersion in a heterogeneous aquifer, 2. Spatial moment analysis. Water Resour. Res.
**28**(12), 3293–3308 (1992)CrossRefGoogle Scholar - Barenblatt, G.I., Zheltov, I.P., Kochina, I.N.: Basic concepts in the theory of seepage of homogeneous liquids in fissured rocks [strata]. PMM
**24**(5), 825–864 (1960)Google Scholar - Barkai, E., Garini, Y., Metzler, R.: Strange kinetics of single molecules in living cells. Phys. Today
**8**(65), 29–35 (2012)CrossRefGoogle Scholar - Barthelemy, P., Bertolotti, J., Wiersma, D.S.: A Lévy flight for light. Nature
**453**, 495–498 (2008)CrossRefGoogle Scholar - Becker, M.W., Shapiro, A.M.: Interpreting tracer breakthrough tailing from different forced-gradient tracer experiment configurations in fractured bedrock. Water Resour. Res.
**39**, 1024 (2003)CrossRefGoogle Scholar - Benke, R., Painter, S.: Modeling conservative tracer transport in fracture networks with a hybrid approach based on the Boltzmann transport equation. Water Resour. Res.
**39**, 1324 (2003)CrossRefGoogle Scholar - Benson, D.A., Meerschaert, M.M.: A simple and efficient random walk solution of multi-rate mobile/immobile mass transport equations. Adv. Water Resour.
**32**(4), 532–539 (2009)CrossRefGoogle Scholar - Benson, D.A., Wheatcrat, S.W., Meerschaert, M.M.: Application of a fractional advection-dispersion equation. Water Resour. Res.
**36**, 1403–1421 (2000)CrossRefGoogle Scholar - Berkowitz, B., Scher, H.: On characterization of anomalous dispersion in porous and fractured media. Water Resour. Res.
**31**(6), 1461–1466 (1995)CrossRefGoogle Scholar - Berkowitz, B., Scher, H.: Anomalous transport in random fracture networks. Phys. Rev. Lett.
**79**(20), 4038–4041 (1997)CrossRefGoogle Scholar - Berkowitz, B., Kosakowski, G., Margolin, G., Scher, H.: Application of continuous time random walk theory to tracer test measurements in fractured and heterogeneous porous media. Groundwater
**39**(4), 593–604 (2001)CrossRefGoogle Scholar - Berkowitz, B., Cortis, A., Dentz, M., Scher, H.: Modeling non-fickian transport in geological formations as a continuous time random walk. Rev. Geophys.
**44**, RG2003 (2006)CrossRefGoogle Scholar - Berkowitz, B., Emmanuel, S., Scher, H.: Non-fickian transport and multiple-rate mass transfer in porous media. Water Resour. Res.
**44**, W03402 (2008). doi: 10.1029/2007WR005906 CrossRefGoogle Scholar - Bijeljic, B., Blunt, M.J.: Pore-scale modeling and continuous time random walk analysis of dispersion in porous media. Water Resour. Res.
**42**, W01202 (2006)CrossRefGoogle Scholar - Bouchaud, J.P., Georges, A.: Anomalous diffusion in disordered media: statistical mechanisms, models and physical applications. Phys. Rep.
**195**(4,5), 127–293 (1990)CrossRefGoogle Scholar - Carrera, J., Sánchez-Vila, X., Benet, I., Medina, A., Galarza, G., Guimerà, J.: On matrix diffusion: formulations, solution methods, and qualitative effects. Hydrol. J.
**6**, 178–190 (1998)Google Scholar - Cortis, A., Berkowitz, B.: anomalous transport in “classical” soil and sand columns. Soil Sci. Soc. Am. J.
**68**(1539–1548), 2004 (2004)Google Scholar - Cortis, A., Chen, Y., Scher, H., Berkowitz, B.: Quantitative characterization of pore-scale disorder effects on transport in “homogeneous” granular media. Phys. Rev. E
**70**, 041108 (2004)CrossRefGoogle Scholar - Cushman, J.H., Ginn, T.R.: Fractional advection-dispersion equation: a classical mass balance with convolution-Fickian flux. Water Resour. Res.
**36**, 3763–3766 (2000)CrossRefGoogle Scholar - Cushman, J.H., Hu, X., Ginn, T.R.: Nonequilibrium statistical mechanics of preasymptotic dispersion. J. Stat. Phys.
**75**(5/6), 859–878 (1994)CrossRefGoogle Scholar - Cvetkovic, V., Fiori, A., Dagan, G.: Solute transport in aquifers of arbitrary variability: a time-domain random walk formulation. Water Resour Res.
**50**, WR015449 (2014)Google Scholar - de Anna, P., Le Borgne, T., Dentz, M., Tartakovsky, A.M., Bolster, D., Davy, P.: Flow intermittency, dispersion and correlated continuous time random walks in porous media. Phys. Rev. Lett.
**110**, 184502 (2013)CrossRefGoogle Scholar - Dentz, M., Berkowitz, B.: Transport behavior of a passive solute in continuous time random walks and multirate mass transfer. Water Resour. Res.
**39**(5), 1111 (2003)CrossRefGoogle Scholar - Dentz, M., Castro, A.: Effective transport dynamics in porous media with heterogeneous retardation properties. Geophys. Res. Lett.
**36**, L03403 (2009)CrossRefGoogle Scholar - Dentz, M., Cortis, A., Scher, H., Berkowitz, B.: Time behavior of solute transport in heterogeneous media: transition from anomalous to normal transport. Adv. Water Resour.
**27**(2), 155–173 (2004)CrossRefGoogle Scholar - Dentz, M., Gouze, P., Carrera, J.: Effective non-local reaction kinetics for transport in physically and chemically heterogeneous media. J. Contam. Hydrol.
**120**, 222–236 (2011)CrossRefGoogle Scholar - Dentz, M., Gouze, P., Russian, A., Dweik, J., Delay, F.: Diffusion and trapping in heterogeneous media: an inhomogeneous continuous time random walk approach. Adv. Water Resour.
**49**, 13–22 (2012)CrossRefGoogle Scholar - Dentz, M., Kang, P.K., Le Borgne, T.: Continuous time random walks for non-local radial solute transport. Adv. Water Resour.
**82**, 16–26 (2015)CrossRefGoogle Scholar - Edery, Y., Guadagnini, A., Scher, H., Berkowitz, B.: Origins of anomalous transport in heterogeneous media: structural and dynamic control. Water Resour. Res.
**50**(2), 1490–1505 (2014)CrossRefGoogle Scholar - Fiori, A., Jankovic, I., Dagan, G., Cvetkovic, V.: Ergodic transport trough aquifers of non-gaussian log conductivity distribution and occurence of anomalous behavior. Water Resour. Res.
**43**, W09407 (2007)Google Scholar - Gjetvaij, F., Russian, A., Gouze, P., Dentz, M.: Dual control of flow field heterogeneity and immobile porosity on non-fickian transport in berea sandstone. Water Resour. Res. (2015). doi: 10.1002/2015WR017645 Google Scholar
- Gouze, P., Le Borgne, T., Leprovost, R., Lods, G., Poidras, T., Pezard, P.: Non-Fickian dispersion in porous media: 1. multi-scale measurements using single well injection withdrawal tracer tests at the ses sitjoles/aliance test site (Spain). Water Resour. Res.
**44**, W06426 (2008a)Google Scholar - Gouze, P., Melean, Z., Le Borgne, T., Dentz, M., Carrera, J.: Non-fickian dispersion in porous media explained by heterogeneous microscale matrix diffusion. Water Resour. Res.
**44**, W11416 (2008b)Google Scholar - Haggerty, R., Gorelick, S.M.: Multiple-rate mass transfer for modeling diffusion and surface reactions in media with pore-scale heterogeneity. Water Resour. Res.
**31**(10), 2383–2400 (1995)CrossRefGoogle Scholar - Haggerty, R., McKenna, S.A., Meigs, L.C.: On the late time behavior of tracer test breakthrough curves. Water Resour. Res.
**36**(12), 3467–3479 (2000)CrossRefGoogle Scholar - Haggerty, R.S., Fleming, S.W., Meigs, L.C., McKenna, S.A.: Tracer tests in a fractured dolomite: 2. Analysis of mass transfer in single-well injection-withdrawal tests. Water Resour. Res.
**37**, 1129–1142 (2001)CrossRefGoogle Scholar - Harvey, C.F., Gorelick, S.M.: Temporal moment-generating equations: modeling transport and mass transfer in heterogeneous aquifers. Water Resour. Res.
**31**(8), 1895–1911 (1995)CrossRefGoogle Scholar - Hatano, Y., Hatano, N.: Dispersive transport of ions in column experiments: an explanation of long-tailed profiles. Water Resour. Res.
**34**(5), 1027–1033 (1998)CrossRefGoogle Scholar - Holzner, M., Morales, V.L., Willmann, M., Dentz, M.: Intermittent lagrangian velocities and accelerations in three-dimensional porous medium flow. Phys. Rev. E
**92**, 013015 (2015)CrossRefGoogle Scholar - Kang, P.K., Dentz, M., Le Borgne, T., Juanes, R.: Spatial markov model of anomalous transport through random lattice networks. Phys. Rev. Lett.
**107**, 180602 (2011)CrossRefGoogle Scholar - Kang, P.K., de Anna, P., Nunes, J.P., Bijeljic, B., Blunt, M.J., Juanes, R.: Pore-scale intermittent velocity structure underpinning anomalous transport through 3-D porous media. Geophys. Res. Lett.
**41**(17), 6184–6190 (2014)CrossRefGoogle Scholar - Kang, P.K., Le Borgne, T., Dentz, M., Bour, O., Juanes, R.: Impact of velocity correlation and distribution on transport in fractured media: field evidence and theoretical model. Water Resour. Res.
**51**, 940–959 (2015)CrossRefGoogle Scholar - Kenkre, V.M., Montroll, E.W., Shlesinger, M.F.: Generalized master equations for continuous-time random walks. J. Stat. Phys.
**9**(1), 45–50 (1973)CrossRefGoogle Scholar - Klafter, J., Sokolov, I.: Anomalous diffusion spreads its wings. Phys. World
**18**(8), 29–32 (2005)CrossRefGoogle Scholar - Koch, D.L., Brady, J.F.: Anomalous diffusion in heterogeneous porous media. Phys. Fluids A
**31**, 965–1031 (1988)CrossRefGoogle Scholar - Le Borgne, T., de Dreuzy, J.R., Davy, P., Bour, O.: Characterization of the velocity field organization in heterogeneous media by conditional correlation. Water Resour. Res.
**43**, 2006WR004875 (2007)CrossRefGoogle Scholar - Le Borgne, T., Dentz, M., Carrera, J.: Spatial Markov processes for modeling Lagrangian particle dynamics in heterogeneous porous media. Phys. Rev. E
**78**, 041110 (2008a)CrossRefGoogle Scholar - Le Borgne, T., Dentz, M., Carrera, J.: A Lagrangian statistical model for transport in highly heterogeneous velocity fields. Phys. Rev. Lett.
**101**, 090601 (2008b)CrossRefGoogle Scholar - Levy, M., Berkowitz, B.: Measurement and analysis of non-Fickian dispersion in heterogeneous porous media. J. Contam. Hydrol.
**64**, 203–226 (2003)CrossRefGoogle Scholar - Maloszewski, P., Zuber, A.: On the theory of tracer experiments in fissured rocks with a porous matrix. J. Hydrol.
**79**, 333 (1985)CrossRefGoogle Scholar - Margolin, G., Dentz, M., Berkowitz, B.: Continuous time random walk and multirate mass transfer modeling of sorption. Chem. Phys.
**295**, 71–80 (2003)CrossRefGoogle Scholar - Metzler, R., Jeon, J.H., Cherstvy, A.G., Barkai, E.: Anomalous diffusion models and their properties: non-stationary, non-ergodicity, and ageing at the centenary of single particle tracking. Phys. Chem. Chem. Phys.
**16**, 24128 (2014)CrossRefGoogle Scholar - Michalak, A.M., Kitanidis, P.K.: Macroscopic behaviour and random-walk particle tracking of kinetically sorbing solutes. Water Resour. Res.
**36**, 2133–2146 (2000)CrossRefGoogle Scholar - Montroll, E.W., Weiss, G.H.: Random walks on lattices, 2. J. Math. Phys.
**6**(2), 167 (1965)CrossRefGoogle Scholar - Morales-Casique, E., Neuman, S.P., Guadagnini, A.: Nonlocal and localized analyses of nonreactive solute transport in bounded randomly heterogeneous porous media: theoretical framework. Adv. Water Resour.
**29**(8), 1238–1255 (2006)CrossRefGoogle Scholar - Moroni, M., Cushman, J.H.: Statistical mechanics with three-dimensional particle tracking velocimetry experiments in the study of anomalous dispersion. II. Experiments. Phys. Fluids
**13**, 81–91 (2001)CrossRefGoogle Scholar - Neuman, S., Zhang, Y.: A quasi-linear theory of non-fickian and fickian subsurface dispersion 1. Theoretical analysis with application to isotropic media. Water Resour. Res.
**26**(5), 887–902 (1990)Google Scholar - Roth, K., Jury, W.A.: Linear transport models for adsorbing solutes. Water Resour. Res.
**29**(4), 1195–1203 (1993)CrossRefGoogle Scholar - Salamon, P., Fernàndez-Garcia, D., Gómez-Hernández, J.J.: Modeling mass transfer processes using random walk particle tracking. Water Resour. Res.
**42**, W11417 (2006)CrossRefGoogle Scholar - Scher, H., Lax, M.: Stochastic transport in a disordered solid I. Theory. Phys. Rev. B
**7**(1), 4491–4502 (1973)CrossRefGoogle Scholar - Scher, H., Margolin, G., Metzler, R., Klafter, J., Berkowitz, B.: The dynamical foundation of fractal stream chemistry: the origin of extremely long retention times. Geophys. Res. Lett.
**29**, 1061 (2002)CrossRefGoogle Scholar - Schumer, R., Benson, D.A., Meerschaert, M.M., Bauemer, B.: Fractal mobile/immobile solute transport. Water Resour. Res.
**39**(10), 1296 (2003)CrossRefGoogle Scholar - Villermaux, J.: Chemical engineering approach to dynamic modeling of linear chromatography, a simple method for representing complex phenomena from simple concepts. J. Chromatogr.
**406**, 11–26 (1987)CrossRefGoogle Scholar - Weiss, G.H.: Aspects and Applications of the Random Walk. Elsevier, North-Holland (1994)Google Scholar
- Willmann, M., Carrera, J., Sanchez-Vila, X.: Transport upscaling in heterogeneous aquifers: what physical parameters control memory functions? Water Resour. Res.
**44**, W12437 (2008)CrossRefGoogle Scholar - Zhang, Y., Green, C.T., Baeumer, B.: Linking aquifer spatial properties and non-fickian transport in alluvial settings. J. Hydrol.
**512**, 315–331 (2014)CrossRefGoogle Scholar - Zinn, B., Meigs, L.C., Harvey, C.F., Haggerty, R., Peplinski, W.J., von Schwerin, C.F.: Experimental visualization of solute transport and mass transfer processes in two-dimensional conductivity fields with connected regions of high conductivity. Environ. Sci. Technol.
**38**, 3916–3926 (2004). doi: 10.1021/es034958g CrossRefGoogle Scholar