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The lamellar description of mixing in porous media

Published online by Cambridge University Press:  10 April 2015

T. Le Borgne ,
M. Dentz  and
E. Villermaux
T. Le Borgne*
Affiliation:
Université de Rennes 1, CNRS, Géosciences Rennes UMR 6118, 35042 Rennes, France
M. Dentz
Affiliation:
IDAEA-CSIC, Jordi Girona 18-26, 08034 Barcelona, Spain
E. Villermaux
Affiliation:
Aix Marseille Université, CNRS, Centrale Marseille, IRPHE UMR 7342, 13384 Marseille, France Institut Universitaire de France, 103 Boulevard Saint-Michel, 75005 Paris, France
*
†Email address for correspondence: [email protected]
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Abstract

We develop a general framework for modelling mixing in porous media flows, in which the scalar mixture is represented as an ensemble of lamellae evolving through stretching, diffusion and coalescence. Detailed numerical simulations in Darcy scale heterogeneous permeability fields are used to analyse the lamella deformation process, which controls the local concentration gradients and thus the evolution of the concentration mixture through stretching enhanced diffusion. The corresponding Lagrangian deformation process is shown to be well modelled by a Langevin equation with multiplicative noise, which can be coupled with diffusion to predict the temporal evolution of the concentration probability density function (PDF). At late times, lamella interaction is enforced by confinement of the mixture within the dispersion area. This process is shown to be well represented by a random aggregation model, which quantifies the frequency of lamella coalescence and allows us to predict the temporal evolution of the concentration PDF in this regime. The proposed theoretical framework provides an accurate prediction of the concentration PDFs at all investigated times, heterogeneity levels and Péclet numbers. In particular, it relates the temporal behaviour of mixing, as quantified by concentration moments, scalar dissipation rate or spatial increments of concentration, to the degree of structural heterogeneity.

JFM classification

Geophysical and Geological Flows: Geophysical and Geological Flows Low-Reynolds-number flows: Low-Reynolds-number flows
Type
Papers
Information
Journal of Fluid Mechanics , Volume 770 , 10 May 2015 , pp. 458 - 498
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© 2015 Cambridge University Press 

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