Hostname: page-component-586b7cd67f-t7fkt Total loading time: 0 Render date: 2024-11-24T10:54:56.877Z Has data issue: false hasContentIssue false
  • English
  • Français

Velocity distributions, dispersion and stretching in three-dimensional porous media

Published online by Cambridge University Press:  23 March 2020

M. Souzy Open the ORCID record for M. Souzy [Opens in a new window] ,
H. Lhuissier Open the ORCID record for H. Lhuissier [Opens in a new window] ,
Y. Méheust Open the ORCID record for Y. Méheust [Opens in a new window] ,
T. Le Borgne Open the ORCID record for T. Le Borgne [Opens in a new window]  and
B. Metzger Open the ORCID record for B. Metzger [Opens in a new window]
M. Souzy
Affiliation:
Geosciences Rennes, UMR 6118, Université de Rennes 1, CNRS, 35042Rennes, France
H. Lhuissier
Affiliation:
Aix Marseille Université, CNRS, IUSTI, UMR 7343, 13453Marseille, France
Y. Méheust
Affiliation:
Geosciences Rennes, UMR 6118, Université de Rennes 1, CNRS, 35042Rennes, France
T. Le Borgne
Affiliation:
Geosciences Rennes, UMR 6118, Université de Rennes 1, CNRS, 35042Rennes, France
B. Metzger*
Affiliation:
Aix Marseille Université, CNRS, IUSTI, UMR 7343, 13453Marseille, France
*
†Email address for correspondence: [email protected]
Get access Rights & Permissions [Opens in a new window]

Abstract

Using index matching and particle tracking, we measure the three-dimensional velocity field in an isotropic porous medium composed of randomly packed solid spheres. This high-resolution experimental dataset provides new insights into the dynamics of dispersion and stretching in porous media. Dynamic-range velocity measurements indicate that the distribution of the velocity magnitude, $U$, is flat at low velocity (probability density function $(U)\propto U^{0}$). While such a distribution should lead to a persistent anomalous dispersion process for advected non-diffusive point particles, we show that the dispersion of non-diffusive tracers nonetheless becomes Fickian beyond a time set by the smallest effective velocity of the tracers. We derive expressions for the onset time of the Fickian regime and the longitudinal and transverse dispersion coefficients as a function of the velocity field properties. The experimental velocity field is also used to study, by numerical advection, the stretching histories of fluid material lines. The mean and the variance of the line elongations are found to grow exponentially in time and the distribution of elongation is log-normal. These results confirm the chaotic nature of advection within three-dimensional porous media. By providing the laws of dispersion and stretching, the present study opens the way to a complete description of mixing in porous media.

JFM classification

Low-Reynolds-number flows: Porous media Mixing: Chaotic advection
Type
JFM Papers
Information
Journal of Fluid Mechanics , Volume 891 , 25 May 2020 , A16
Copyright
© The Author(s), 2020. Published by Cambridge University Press

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

AlAdwani, M. S. K. F.2017 Prediction of velocity distribution from the statistics of pore structure in 3D porous media via high-fidelity pore-scale simulation. Doctoral dissertation, Massachusetts Institute of Technology, Cambridge, MA.Google Scholar
Antonsen, T. M. Jr, Fan, Z., Ott, E. & Garcia-Lopez, E. 1996 The role of chaotic orbits in the determination of power spectra of passive scalars. Phys. Fluids 11, 3094–3104.CrossRefGoogle Scholar
Aref, H., Blake, J. R., Budišić, M., Cardoso, S. S. S., Cartwright, J. H. E., Clercx, H. J. H., El Omari, K., Feudel, U., Golestanian, R., Gouillart, E. et al. 2017 Frontiers of chaotic advection. Rev. Mod. Phys. 89, 025007.CrossRefGoogle Scholar
Berkowitz, B., Cortis, A., Dentz, M. & Scher, H. 2006 Modeling non-Fickian transport in geological formations as a continuous time random walk. Rev. Geophys. 44 (2), RG2003.CrossRefGoogle Scholar
Berkowitz, B. & Scher, H. 1995 On characterization of anomalous dispersion in porous and fractured media. Water Resour. Res. 31, 1461–1466.CrossRefGoogle Scholar
Bijeljic, B., Mostaghimi, P. & Blunt, M. J. 2011 Signature of non-Fickian solute transport in complex heterogeneous porous media. Phys. Rev. Lett. 107, 204502.CrossRefGoogle ScholarPubMed
Bijeljic, B., Raeini, A., Mostaghimi, P. & Blunt, M. J. 2013 Predictions of non-Fickian solute transport in different classes of porous media using direct simulation on pore-scale images. Phys. Rev. E 87, 013011.Google ScholarPubMed
Carrel, M., Morales, V. L., Dentz, M., Derlon, N., Morgenroth, E. & Holzner, M. 2018 Pore-scale hydrodynamics in a progressively bioclogged three-dimensional porous medium: 3D particle tracking experiments and stochastic transport modeling. Water Resour. Res. 54, 2183–2198.CrossRefGoogle Scholar
Datta, S. S., Chiang, H., Ramakrishnan, T. S. & Weitz, D. A. 2013 Spatial fluctuations of fluid velocities in flow through a three-dimensional porous medium. Phys. Rev. Lett. 111, 064501.CrossRefGoogle ScholarPubMed
De Anna, P., Dentz, M., Tartakovsky, A. & Le Borgne, T. 2014 The filamentary structure of mixing fronts and its control on reaction kinetics in porous media flows. Geophys. Res. Lett. 13, 4586–4593.CrossRefGoogle Scholar
Dentz, M., Icardi, M. & Hidalgo, J. J. 2018 Mechanisms of dispersion in a porous medium. J. Fluid Mech. 841, 851–882.CrossRefGoogle Scholar
Dentz, M., Kang, P. K., Comolli, A., Le Borgne, T. & Lester, D. R. 2016 Continuous time random walks for the evolution of Lagrangian velocities. Phys. Rev. Fluids 17, 074004.Google Scholar
Dentz, M., Le Borgne, T., Lester, D. R. & de Barros, F. P. J. 2017 Mixing in groundwater. In The Handbook of Groundwater Engineering (ed. Cushman, J. H. & Tartakovsky, D. M.), chap. 13, pp. 383–412. CRC Press.Google Scholar
Duplat, J., Innocenti, C. & Villermaux, E. 2010 A nonsequential turbulent mixing process. Phys. Fluids 22, 035104.CrossRefGoogle Scholar
Haynes, P. H. & Vanneste, J. 2005 What controls the decay of passive scalars in smooth flows? Phys. Fluids 17, 097103.CrossRefGoogle Scholar
Heyman, J. 2019 TracTrac: a fast multi-object tracking algorithm for motion estimation. Comput. Geosci. 128, 11–18.CrossRefGoogle Scholar
Hinch, E. J. 1999 Mixing: turbulence and chaos an introduction. In Mixing, pp. 37–56. Springer.CrossRefGoogle Scholar
Holzner, M., Morales, V. L., Willmann, M. & Dentz, M. 2015 Intermittent Lagrangian velocities and accelerations in three-dimensional porous medium flow. Phys. Rev. E 92, 013015.Google ScholarPubMed
Jensen, K. H., Berg-Sazrensen, K., Bruus, H., Holbrook, N. M., Liesche, J., Schulz, A. & Bohr, T. 2016 Sap flow and sugar transport in plants. Rev. Mod. Phys. 88, 035007.Google Scholar
Jiménez-Martínez, J., Le Borgne, T., Tabuteau, H. & Méheust, Y. 2017 Impact of saturation on dispersion and mixing in porous media: photobleaching pulse injection experiments and shear enhanced mixing model. Water Resour. Res. 2, 1457–1472.CrossRefGoogle Scholar
Jin, C., Langston, P. A., Pavlovskaya, G. E., Hall, M. R. & Rigby, S. P. 2016 Statistics of highly heterogeneous flow fields confined to three-dimensional random porous media. Phys. Rev. E 1, 013122.Google Scholar
Kang, P. K., Anna, P., Nunes, J. P., Bijeljic, B., Blunt, M. J. & Juanes, R. 2014 Pore-scale intermittent velocity structure underpinning anomalous transport through 3D porous media. Geophys. Res. Lett. 17, 6184–6190.CrossRefGoogle Scholar
Kree, M. & Villermaux, E. 2017 Scalar mixtures in porous media. Phys. Rev. Fluids 10, 104502.Google Scholar
Le Borgne, T., Dentz, M. & Villermaux, E. 2013 Stretching, coalescence, and mixing in porous media. Phys. Rev. Lett. 20, 2045018.Google Scholar
Le Borgne, T., Dentz, M. & Villermaux, E. 2015 The lamellar description of mixing in porous media. J. Fluid Mech. 770, 458–498.CrossRefGoogle Scholar
Lester, D. R., Dentz, M. & Le Borgne, T. 2016 Chaotic mixing in three-dimensional porous media. J. Fluid Mech. 803, 144–174.CrossRefGoogle Scholar
Lester, D. R., Metcalfe, G. & Trefry, M. G. 2013 Is chaotic advection inherent to porous media flow? Phys. Rev. Lett. 111, 174101.CrossRefGoogle ScholarPubMed
Maier, R. S., Kroll, D. M., Kutsovsky, Y. E., Davis, H. T. & Bernard, R. S. 1998 Simulation of flow through bead packs using the lattice Boltzmann method. Phys. Fluids 10, 60–74.CrossRefGoogle Scholar
Martínez-Ruiz, D., Meunier, P., Favier, B., Duchemin, L. & Villermaux, E. 2018 The diffusive sheet method for scalar mixing. J. Fluid Mech. 837, 230–257.CrossRefGoogle Scholar
Meunier, P. & Villermaux, E. 2003 How vortices mix. J. Fluid Mech. 476, 213–222.CrossRefGoogle Scholar
Meunier, P. & Villermaux, E. 2010 The diffusive strip method for scalar mixing in two dimensions. J. Fluid Mech. 662, 134–172.CrossRefGoogle Scholar
Morales, V. L., Dentz, M., Willmann, M. & Holzner, M. 2017 Stochastic dynamics of intermittent pore-scale particle motion in three-dimensional porous media: experiments and theory. Geophys. Res. Lett. 44, 9361–9371.CrossRefGoogle Scholar
Moroni, M. & Cushman, J. H. 2001 Three dimensional particle tracking velocimetry studies of the transition from pore dispersion to Fickian dispersion for homogeneous porous media. Water Resour. Res. 37, 873–884.CrossRefGoogle Scholar
Nicholson, C. 2001 Anomalous diffusion inspires anatomical insights. Biophys. J. 108, 2091–2093.CrossRefGoogle Scholar
Ottino, J. M. 1989 The Kinematics of Mixing: Stretching, Chaos, and Transport. Cambridge University Press.Google Scholar
Peyrounette, M., Davit, Y., Quintard, M. & Lorthois, S. 2018 Multiscale modelling of blood flow in cerebral microcirculation: details at capillary scale control accuracy at the level of the cortex. PLoS ONE 13, e0189474.CrossRefGoogle ScholarPubMed
Pore Aventura2019 Velocity field explorer: an open source software to visualize and investige the kinematic properties of a 3-D velocity field. https://github.com/Nico04/Pore-Aventura.Google Scholar
Ranz, W. E. 1979 Applications of a stretch model diffusion, and reaction in laminar and turbulent flows. AlChE J. 25, 1.Google Scholar
Saffman, P. G. 1959 A theory of dispersion in a porous medium. J. Fluid Mech. 03, 321–349.CrossRefGoogle Scholar
Shaqfeh, E. S. & Koch, D. L. 1992 Polymer stretch in dilute fixed beds of fibres or spheres. J. Fluid Mech. 244, 17–54.CrossRefGoogle Scholar
Souzy, M., Lhuissier, H., Meheust, Y., Le Borgne, T. & Metzger, B.2019 Dataset: experimental 3D velocity field in random sphere packing, Digital Rocks Portal. Available at: http://www.digitalrocksportal.org/projects/175.Google Scholar
Souzy, M., Lhuissier, H., Villermaux, E. & Metzger, B. 2017 Stretching and mixing in sheared particulate suspensions. J. Fluid Mech. 812, 611–635.CrossRefGoogle Scholar
Souzy, M., Zaier, I., Lhuissier, H., Le Borgne, T. & Metzger, B. 2018 Mixing lamellae in a shear flow. J. Fluid Mech. 838, R3.CrossRefGoogle Scholar
Turuban, R., Lester, D. R., Le Borgne, T. & Méheust, Y. 2018 Space-group symmetries generate chaotic fluid advection in crystalline granular media. Phys. Rev. Lett. 120, 024501.CrossRefGoogle ScholarPubMed
Villermaux, E. 2019 Mixing versus stirring. Annu. Rev. Fluid Mech. 51, 245–273.CrossRefGoogle Scholar
Wang, C., Gao, Q., Wei, R., Li, T. & Wang, J. 2017 Weighted divergence correction scheme and its fast implementation. Exp. Fluids 58 (5), 44.CrossRefGoogle Scholar

Souzy et al. supplementary movie 1

Blob of non-diffusive tracers advected in a 3D random bead-pack of spheres.

Download Souzy et al. supplementary movie 1(Video)
Video 6.6 MB

Souzy et al. supplementary movie 2

Point particles advected numerically in the 3D velocity field.

Download Souzy et al. supplementary movie 2(Video)
Video 2.7 MB

Souzy et al. supplementary movie 3

Velocity field explorer (https://github.com/Nico04/Pore-Aventura)

Download Souzy et al. supplementary movie 3(Video)
Video 20.6 MB

Souzy et al. supplementary movie 4

Fluid material line advected numerically in the 3D experimental velocity field.

Download Souzy et al. supplementary movie 4(Video)
Video 12.4 MB