Is There a Representative Elementary Volume for Anomalous Dispersion?

  • Alexandre Puyguiraud
  • Philippe Gouze
  • Marco DentzEmail author


The concept of the representative elementary volume (REV) is often associated with the notion of hydrodynamic dispersion and Fickian transport. However, it has been frequently observed experimentally and in numerical pore-scale simulations that transport is non-Fickian and cannot be characterized by hydrodynamic dispersion. Does this mean that the concept of the REV is invalid? We investigate this question by a comparative analysis of the advective mechanisms of Fickian and non-Fickian dispersions and their representation in large-scale transport models. Specifically, we focus on the microscopic foundations for the modeling of pore-scale fluctuations of Lagrangian velocity in terms of Brownian dynamics (hydrodynamic dispersion) and in terms of continuous-time random walks, which account for non-Fickian transport through broad distributions of advection times. We find that both approaches require the existence of an REV that, however, is defined in terms of the representativeness of Eulerian flow properties. This is in contrast to classical definitions in terms of medium properties such as porosity, for example.


Representative elementary volume Upscaling Anomalous dispersion Continuous-time random walks Velocity statistics 



The research leading to these results has received funding from the European Research Council under the European Union’s Seventh Framework Programme (FP7/2007-2013)/ERC Grant Agreement No. 617511 (MHetScale). This work was partially funded by the CNRS-PICS project CROSSCALE, Project Number 280090.


  1. Bear, J.: Dynamics of Fluids in Porous Media. Elsevier, New York (1972)Google Scholar
  2. Berkowitz, B., Cortis, A., Dentz, M., Scher, H.: Modeling non-Fickian transport in geological formations as a continuous time random walk. Rev. Geophys. 44(2), RG2003 (2006)Google Scholar
  3. Bigi, B.: Using Kullback–Leibler distance for text categorization. In: European Conference on Information Retrieval, pp. 305–319. Springer (2003)Google Scholar
  4. Bijeljic, B., Blunt, M.J.: Pore-scale modeling and continuous time random walk analysis of dispersion in porous media. Water Resour. Res. W01202 (2006). CrossRefGoogle Scholar
  5. Bijeljic, B., Mostaghimi, P., Blunt, M.J.: Signature of non-Fickian solute transport in complex heterogeneous porous media. Phys. Rev. Lett. 107(20), 204502 (2011)CrossRefGoogle Scholar
  6. Cortis, A., Berkowitz, B.: Anomalous transport in ‘classical’ soil and sand columns. Soil Sci. Soc. Am. J. 68(5), 1539 (2004). CrossRefGoogle Scholar
  7. Cushman, J.H., Moroni, M.: Statistical mechanics with three-dimensional particle tracking velocimetry experiments in the study of anomalous dispersion. I. Theory. Phys. Fluids 13(1), 75–80 (2001). CrossRefGoogle Scholar
  8. 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(18), 184502 (2013)CrossRefGoogle Scholar
  9. De Anna, P., Quaife, B., Biros, G., Juanes, R.: Prediction of velocity distribution from pore structure in simple porous media. Phys. Rev. Fluids 2, 124103 (2017). CrossRefGoogle Scholar
  10. 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
  11. Dentz, M., Kang, P.K., Comolli, A., Le Borgne, T., Lester, D.R.: Continuous time random walks for the evolution of lagrangian velocities. Phys. Rev. Fluids 1(7), 074004 (2016)CrossRefGoogle Scholar
  12. Dentz, M., Icardi, M., Hidalgo, J.J.: Mechanisms of dispersion in a porous medium. J. Fluid Mech. 841, 851–882 (2018). CrossRefGoogle Scholar
  13. Gardiner, C.: Stochastic Methods. Springer, Berlin (2010)Google Scholar
  14. 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
  15. 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
  16. Koponen, A., Kataja, M., Timonen, J.: Tortuous flow in porous media. Phys. Rev. E 54(1), 406 (1996)CrossRefGoogle Scholar
  17. Kullback, S., Leibler, R.A.: On information and sufficiency. Ann. Math. Stat. 22(1), 79–86 (1951)CrossRefGoogle Scholar
  18. Lindgren, B., Johansson, A.V., Tsuji, Y.: Universality of probability density distributions in the overlap region in high reynolds number turbulent boundary layers. Phys. Fluids 16(7), 2587–2591 (2004)CrossRefGoogle Scholar
  19. Liu, Y., Kitanidis, P.K.: Applicability of the dual-domain model to nonaggregated porous media. Ground Water 50(6), 927–934 (2012)CrossRefGoogle Scholar
  20. Meyer, D.W., Bijeljic, B.: Pore-scale dispersion: bridging the gap between microscopic pore structure and the emerging macroscopic transport behavior. Phys. Rev. E 94(1), 013107 (2016)CrossRefGoogle Scholar
  21. Morales, V.L., Dentz, M., Willmann, M., Holzner, M.: Stochastic dynamics of intermittent pore-scale particle motion in three-dimensional porous media: experiments and theory. Geophys. Res. Lett. 44(18), 9361–9371 (2017)CrossRefGoogle Scholar
  22. 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(1), 81–91 (2001). CrossRefGoogle Scholar
  23. Noetinger, B., Roubinet, D., Russian, A., Le Borgne, T., Delay, F., Dentz, M., De Dreuzy, J.R., Gouze, P.: Random walk methods for modeling hydrodynamic transport in porous and fractured media from pore to reservoir scale. Transp. Porous Media 115, 1–41 (2016)CrossRefGoogle Scholar
  24. Painter, S., Cvetkovic, V.: Upscaling discrete fracture network simulations: an alternative to continuum transport models. Water Resour. Res. 41, W02002 (2005). CrossRefGoogle Scholar
  25. Puyguiraud, A., Gouze, P., Dentz, M.: Stochastic dynamics of Lagrangian pore-scale velocities in three-dimensional porous media. Water Resour. Res. (2019a). CrossRefGoogle Scholar
  26. Puyguiraud, A., Gouze, P., Dentz, M.: Upscaling of anomalous pore-scale dispersion. Transp. Porous Media 128, 837–855 (2019b)CrossRefGoogle Scholar
  27. Robert, R., Sommeria, J.: Statistical equilibrium states for two-dimensional flows. J. Fluid Mech. 229, 291–310 (1991). CrossRefGoogle Scholar
  28. Saffman, P.: A theory of dispersion in a porous medium. J. Fluid Mech. 6(03), 321–349 (1959)CrossRefGoogle Scholar
  29. Taylor, G.: Dispersion of soluble matter in solvent flowing slowly through a tube. Proc. R. Soc. Lond. Ser. A Math. Phys. Sci. 219(1137), 186–203 (1953)CrossRefGoogle Scholar

Copyright information

© Springer Nature B.V. 2019

Authors and Affiliations

  1. 1.Spanish National Research Council (IDAEA-CSIC)BarcelonaSpain
  2. 2.Géosciences MontpellierCNRS-Université de MontpellierMontpellierFrance

Personalised recommendations