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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: bloen.metzger@univ-amu.fr
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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

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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