Transport in Porous Media

, Volume 115, Issue 2, pp 265–289 | Cite as

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

  • Alessandro Comolli
  • Juan J. Hidalgo
  • Charlie Moussey
  • Marco Dentz
Article

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 

References

  1. Abramowitz, M., Stegun, I.A.: Handbook of Mathematical Functions. Dover, New York (1972)Google Scholar
  2. 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
  3. 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
  4. Barkai, E., Garini, Y., Metzler, R.: Strange kinetics of single molecules in living cells. Phys. Today 8(65), 29–35 (2012)CrossRefGoogle Scholar
  5. Barthelemy, P., Bertolotti, J., Wiersma, D.S.: A Lévy flight for light. Nature 453, 495–498 (2008)CrossRefGoogle Scholar
  6. 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
  7. 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
  8. 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
  9. 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
  10. Berkowitz, B., Scher, H.: On characterization of anomalous dispersion in porous and fractured media. Water Resour. Res. 31(6), 1461–1466 (1995)CrossRefGoogle Scholar
  11. Berkowitz, B., Scher, H.: Anomalous transport in random fracture networks. Phys. Rev. Lett. 79(20), 4038–4041 (1997)CrossRefGoogle Scholar
  12. 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
  13. 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
  14. 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
  15. 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
  16. 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
  17. 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
  18. Cortis, A., Berkowitz, B.: anomalous transport in “classical” soil and sand columns. Soil Sci. Soc. Am. J. 68(1539–1548), 2004 (2004)Google Scholar
  19. 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
  20. 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
  21. 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
  22. 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
  23. 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
  24. 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
  25. Dentz, M., Castro, A.: Effective transport dynamics in porous media with heterogeneous retardation properties. Geophys. Res. Lett. 36, L03403 (2009)CrossRefGoogle Scholar
  26. 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
  27. 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
  28. 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
  29. 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
  30. 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
  31. 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
  32. 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
  33. 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
  34. 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
  35. 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
  36. 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
  37. 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
  38. 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
  39. 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
  40. 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
  41. 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
  42. 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
  43. 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
  44. 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
  45. Klafter, J., Sokolov, I.: Anomalous diffusion spreads its wings. Phys. World 18(8), 29–32 (2005)CrossRefGoogle Scholar
  46. Koch, D.L., Brady, J.F.: Anomalous diffusion in heterogeneous porous media. Phys. Fluids A 31, 965–1031 (1988)CrossRefGoogle Scholar
  47. 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
  48. 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
  49. 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
  50. Levy, M., Berkowitz, B.: Measurement and analysis of non-Fickian dispersion in heterogeneous porous media. J. Contam. Hydrol. 64, 203–226 (2003)CrossRefGoogle Scholar
  51. Maloszewski, P., Zuber, A.: On the theory of tracer experiments in fissured rocks with a porous matrix. J. Hydrol. 79, 333 (1985)CrossRefGoogle Scholar
  52. 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
  53. 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
  54. 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
  55. Montroll, E.W., Weiss, G.H.: Random walks on lattices, 2. J. Math. Phys. 6(2), 167 (1965)CrossRefGoogle Scholar
  56. 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
  57. 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
  58. 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
  59. Roth, K., Jury, W.A.: Linear transport models for adsorbing solutes. Water Resour. Res. 29(4), 1195–1203 (1993)CrossRefGoogle Scholar
  60. 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
  61. Scher, H., Lax, M.: Stochastic transport in a disordered solid I. Theory. Phys. Rev. B 7(1), 4491–4502 (1973)CrossRefGoogle Scholar
  62. 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
  63. Schumer, R., Benson, D.A., Meerschaert, M.M., Bauemer, B.: Fractal mobile/immobile solute transport. Water Resour. Res. 39(10), 1296 (2003)CrossRefGoogle Scholar
  64. 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
  65. Weiss, G.H.: Aspects and Applications of the Random Walk. Elsevier, North-Holland (1994)Google Scholar
  66. 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
  67. 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
  68. 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

Copyright information

© Springer Science+Business Media Dordrecht 2016

Authors and Affiliations

  1. 1.Institute of Environmental Assessment and Water Research (IDÆA)Spanish National Research Council (CSIC)BarcelonaSpain
  2. 2.Department of Geotechnical Engineering and GeosciencesTechnical University of Catalonia (UPC)BarcelonaSpain
  3. 3.Hydrogeology Group (CSIC-UPC)BarcelonaSpain
  4. 4.École Nationale des Ponts et ChausséesChamps-sur-MarneFrance

Personalised recommendations