Abstract
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.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Bear, J.: Dynamics of Fluids in Porous Media. Elsevier, New York (1972)
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)
Bigi, B.: Using Kullback–Leibler distance for text categorization. In: European Conference on Information Retrieval, pp. 305–319. Springer (2003)
Bijeljic, B., Blunt, M.J.: Pore-scale modeling and continuous time random walk analysis of dispersion in porous media. Water Resour. Res. W01202 (2006). https://doi.org/10.1029/2005WR004578
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)
Cortis, A., Berkowitz, B.: Anomalous transport in ‘classical’ soil and sand columns. Soil Sci. Soc. Am. J. 68(5), 1539 (2004). https://doi.org/10.2136/sssaj2004.1539
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). https://doi.org/10.1063/1.1328075
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)
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). https://doi.org/10.1103/PhysRevFluids.2.124103
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)
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)
Dentz, M., Icardi, M., Hidalgo, J.J.: Mechanisms of dispersion in a porous medium. J. Fluid Mech. 841, 851–882 (2018). https://doi.org/10.1017/jfm.2018.120
Gardiner, C.: Stochastic Methods. Springer, Berlin (2010)
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)
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). https://doi.org/10.1002/2014GL061475
Koponen, A., Kataja, M., Timonen, J.: Tortuous flow in porous media. Phys. Rev. E 54(1), 406 (1996)
Kullback, S., Leibler, R.A.: On information and sufficiency. Ann. Math. Stat. 22(1), 79–86 (1951)
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)
Liu, Y., Kitanidis, P.K.: Applicability of the dual-domain model to nonaggregated porous media. Ground Water 50(6), 927–934 (2012)
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)
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)
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). https://doi.org/10.1063/1.1328076
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)
Painter, S., Cvetkovic, V.: Upscaling discrete fracture network simulations: an alternative to continuum transport models. Water Resour. Res. 41, W02002 (2005). https://doi.org/10.1029/2004WR003682
Puyguiraud, A., Gouze, P., Dentz, M.: Stochastic dynamics of Lagrangian pore-scale velocities in three-dimensional porous media. Water Resour. Res. (2019a). https://doi.org/10.1029/2018WR023702
Puyguiraud, A., Gouze, P., Dentz, M.: Upscaling of anomalous pore-scale dispersion. Transp. Porous Media 128, 837–855 (2019b)
Robert, R., Sommeria, J.: Statistical equilibrium states for two-dimensional flows. J. Fluid Mech. 229, 291–310 (1991). https://doi.org/10.1017/S0022112091003038
Saffman, P.: A theory of dispersion in a porous medium. J. Fluid Mech. 6(03), 321–349 (1959)
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)
Acknowledgements
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.
Author information
Authors and Affiliations
Corresponding author
Additional information
Publisher's Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Rights and permissions
About this article
Cite this article
Puyguiraud, A., Gouze, P. & Dentz, M. Is There a Representative Elementary Volume for Anomalous Dispersion?. Transp Porous Med 131, 767–778 (2020). https://doi.org/10.1007/s11242-019-01366-z
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s11242-019-01366-z