Hostname: page-component-848d4c4894-nmvwc Total loading time: 0 Render date: 2024-06-26T04:53:56.920Z Has data issue: false hasContentIssue false
  • English
  • Français

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: tanguy.le-borgne@univ-rennes1.fr
Get access Rights & Permissions [Opens in a new window]

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
Check for updates
Copyright
© 2015 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

Abramowitz, M. & Stegun, I. A. 1964 Handbook of Mathematical Functions. Dover.Google Scholar
de Anna, P., Dentz, M., Tartakovsky, A. & Le Borgne, T. 2014a The filamentary structure of mixing fronts and its control on reaction kinetics in porous media flows. Geophys. Res. Lett. 41, 4586–4593.CrossRefGoogle Scholar
de Anna, P., Jimenez-Martinez, J., Tabuteau, H., Turuban, R., Le Borgne, T., Derrien, M. & Méheust, Y. 2014b Mixing and reaction kinetics in porous media: an experimental pore scale quantification. Environ. Sci. Technol. 48, 508–516.CrossRefGoogle Scholar
de Anna, P., Le Borgne, T., Dentz, M., Tartakovsky, A., Bolster, D. & Davy, P. 2013 Flow intermittency, dispersion, and correlated continuous time random walks in porous media. Phys. Rev. Lett. 110, 184502.Google Scholar
Attinger, S., Dentz, M., Kinzelbach, H. & Kinzelbach, W. 1999 Temporal behavior of a solute cloud in a chemically heterogeneous porous medium. J. Fluid Mech. 386, 77–104.Google Scholar
Batchelor, G. K. 1959 Small-scale variation of convected quantities like temperature in a turbulent fluid. Part 1. General discussion and the case of small conductivity. J. Fluid Mech. 5, 113–133.CrossRefGoogle Scholar
Bear, J. 1972 Dynamics of Fluids in Porous Media. Elsevier.Google Scholar
Bellin, A. & Tonina, D. 2007 Probability density function of non-reactive solute concentration in heterogeneous porous formations. J. Contam. Hydrol. 94, 109–125.CrossRefGoogle ScholarPubMed
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, 2005RG000178.Google Scholar
Bolster, D., Valdes-Parada, F. J., Le Borgne, T., Dentz, M. & Carrera, J. 2011 Mixing in confined stratified aquifers. J. Contam. Hydrol. 120–121, 198–212.Google Scholar
Bouchaud, J. P. & Georges, A. 1988 A simple model for hydrodynamic dispersion. C. R. Acad. Sci. Paris II 307, 1431–1436.Google Scholar
Chen, H. D., Chen, S. Y. & Kraichnan, R. H. 1989 Probability distribution of a stochastically advected scalar field. Phys. Rev. Lett. 63 (24), 2657–2660.Google Scholar
Chiogna, G., Hochstetler, D. L., Bellin, A., Kitanidis, P. K. & Rolle, M. 2012 Mixing, entropy and reactive solute transport. Geophys. Res. Lett. 39, doi:10.1029/2012gl053295.Google Scholar
Cirpka, O. A., Schwede, R. L., Luo, J. & Dentz, M. 2008 Concentration statistics for mixing-controlled reactive transport in random heterogeneous media. J. Contam. Hydrol. 98, 61–74.Google Scholar
Cirpka, O. A. & Valocchi, A. J. 2007 Two-dimensional concentration distribution for mixing-controlled bioreactive transport in steady state. Adv. Water Resour. 30, 1668–1679.CrossRefGoogle Scholar
Cushman, J. H., Hu, X. & Ginn, T. R. 1994 Nonequilibrium statistical mechanics of preasymptotic dispersion. J. Stat. Phys. 75, 859–878.Google Scholar
Dagan, G. 1987 Theory of solute transport by groundwater. Annu. Rev. Fluid Mech. 19, 183–215.Google Scholar
Danckwerts, P. V. 1952 The definition and measurement of some characteristics of mixtures. Appl. Sci. Res. A 3, 279–296.Google Scholar
Danckwerts, P. V. 1953 Continuous flow systems. Chem. Engng Sci. 2 (1), 1–13.Google Scholar
De Barros, F., Dentz, M., Koch, J. & Nowak, W. 2012 Flow topology and scalar mixing in spatially heterogeneous flow fields. Geophys. Res. Lett. 39, L08404.CrossRefGoogle Scholar
De Gennes, P. G. 1983 Hydrodynamic dispersion in unsaturated porous media. J. Fluid Mech. 136, 189–200.Google Scholar
Delgado, J. M. P. Q. 2007 Longitudinal and transverse dispersion in porous media. Chem. Engng Res. Des. 85 (A9), 1245–1252.Google Scholar
Dentz, M. 2012 Concentration statistics for transport in heterogeneous media due to stochastic fluctuations of the center of mass velocity. Adv. Water Resour. 36, 11–22.Google Scholar
Dentz, M., Kinzelbach, H., Attinger, S. & Kinzelbach, W. 2000 Temporal behavior of a solute cloud in a heterogeneous porous medium. Part 1. Point-like injection. Water Resour. Res. 36, 3591–3604.Google Scholar
Dentz, M., Le Borgne, T., Englert, A. & Bijeljic, B. 2011 Mixing, spreading and reaction in heterogeneous media: a brief review. J. Contam. Hydrol. 120–121, 1–17.Google Scholar
Dentz, M. & Tartakovsky, D. M. 2010 Probability density functions for passive scalars dispersed in random velocity fields. Geophys. Res. Lett. 37, L24406.CrossRefGoogle Scholar
Duplat, J., Innocenti, C. & Villermaux, E. 2010 A nonsequential turbulent mixing process. Phys. Fluids 22, 035104.Google Scholar
Duplat, J. & Villermaux, E. 2008 Mixing by random stirring in confined mixtures. J. Fluid Mech. 617, 51–86.Google Scholar
Fedotov, S., Ihme, M. & Pitsch, H. 2005 Stochastic mixing model with power law decay of variance. Phys. Rev. E 71 (1), 016310.Google Scholar
Fiori, A. 2001 The Lagrangian concentration approach for determining dilution in aquifer transport: theoretical analysis and comparison with field experiments. Water Resour. Res. 37, 3105–3114.Google Scholar
Fiori, A. & Dagan, G. 1999 Concentration fluctuations in transport by groundwater: comparison between theory and field experiments. Water Resour. Res. 35, 105–112.Google Scholar
Fiori, A. & Dagan, G. 2000 Concentration fluctuations in aquifer transport: a rigorous first-order solution and applications. J. Contam. Hydrol. 45 (1–2), 139–163.Google Scholar
Fiorotto, V. & Caroni, E. 2002 Concentration statistics in heterogeneous aquifers for finite Péclet values. Trans. Porous Med. 48, 331–351.Google Scholar
Fox, R. O. 2004 Computational Models for Turbulent Reacting Flows. Cambridge University Press.Google Scholar
Gelhar, L. W. 1993 Stochastic Subsurface Hydrology. Prentice Hall.Google Scholar
Gelhar, L. W. & Axness, C. L. 1983 Three-dimensional stochastic analysis of macrodispersion in aquifers. Water Resour. Res. 19 (1), 161–180.Google Scholar
Grassberger, P. & Procaccia, I. 1983 Characterization of strange attractors. Phys. Rev. Lett. 50, 346–349.Google Scholar
Kalda, J. 2000 Simple model of intermittent passive scalar turbulence. Phys. Rev. Lett. 84, 471–474.Google Scholar
Kitanidis, P. K. 1994 The concept of the dilution index. Water Resour. Res. 30, 2011–2026.Google Scholar
Koch, D. L., Cox, R. G., Brenner, H. & Brady, J. F. 1989 The effect of order on dispersion in porous media. J. Fluid Mech. 200, 173–188.Google Scholar
Le Borgne, T., Dentz, M., Bolster, D., Carrera, J., de Dreuzy, J. R. & Davy, P. 2010 Non-Fickian mixing: temporal evolution of the scalar dissipation rate in heterogeneous porous media. Adv. Water Resour. 33, 1468–1475.Google Scholar
Le Borgne, T., Dentz, M. & Carrera, J. 2008 Lagrangian statistical model for transport in highly heterogeneous velocity fields. Phys. Rev. Lett. 101 (9), 090601.Google Scholar
Le Borgne, T., Dentz, M., Davy, P., Bolster, D., Carrera, J., de Dreuzy, J. R. & Bour, O. 2011 Persistence of incomplete mixing: a key to anomalous transport. Phys. Rev. E (rapid) 84, 015301.Google Scholar
Le Borgne, T., Dentz, M. & Villermaux, E. 2013 Stretching, coalescence and mixing in porous media. Phys. Rev. Lett. 110, 204501.Google Scholar
Mays, D. & Neupauer, M. 2012 Plume spreading in groundwater by stretching and folding. Water Resour. Res. 48, W07501.Google Scholar
Metcalfe, G., Lester, D., Ord, A., Kulkarni, P., Trefry, M., Hobbs, B. E., Regenaur-Lieb, K. & Morris, J. 2010 A partially open porous media flow with chaotic advection: towards a model of coupled fields. Phil. Trans. R. Soc. Lond. A 368, 217–230.Google Scholar
Meunier, P. & Villermaux, E. 2010 The diffusive strip method for scalar mixing in two dimensions. J. Fluid Mech. 662, 134–172.Google Scholar
Neuman, S. P. & Tartakovsky, D. M. 2009 Perspective on theories of anomalous transport in heterogeneous media. Adv. Water Resour. 32, 670–680.Google Scholar
Ottino, J. M. 1989 The Kinematics of Mixing: Stretching, Chaos, and Transport. Cambridge University Press.Google Scholar
Pope, S. B. 2000 Turbulent Flows. Cambridge University Press.CrossRefGoogle Scholar
Ranz, W. E. 1979 Application of a stretch model to mixing, diffusion and reaction in laminar and turbulent flows. AIChE J. 25 (1), 41–47.Google Scholar
Risken, H. 1996 The Fokker–Planck Equation. Springer.Google Scholar
Rolle, M., Eberhardt, C., Chiogna, G., Cirpka, O. A. & Grathwohl, P. 2009 Enhancement of dilution and transverse reactive mixing in porous media: experiments and model-based interpretation. J. Contam. Hydrol. 110, 130–142.Google Scholar
Tartakovsky, D. M. 2013 Assessment and management of risk in subsurface hydrology: a review and perspective. Adv. Water Resour. 51, 247–260.Google Scholar
Tartakovsky, A. M., Redden, G., Lichtner, P. C., Scheibe, T. D. & Meakin, P. 2008 Mixing-induced precipitation: experimental study and multiscale numerical analysis. Water Resour. Res. 44, W06S04.Google Scholar
Tonina, D. & Bellin, A. 2008 Effects of pore-scale dispersion, degree of heterogeneity, sampling size, and source volume on the concentration moments of conservative solutes in heterogeneous formations. Adv. Water Resour. 31, 339–354.Google Scholar
Valiño, L. & Dopazo, C. 1991 A binomial Langevin model for turbulent mixing. Phys. Fluids 3, 3034–3037.Google Scholar
Villermaux, E. 2012 Mixing by porous media. C. R. Mécanique 340, 933–943.Google Scholar
Villermaux, E. & Duplat, J. 2003 Mixing as an aggregation process. Phys. Rev. Lett. 91, 184501.Google Scholar
Villermaux, E. & Duplat, J. 2006 Coarse grained scale of turbulent mixtures. Phys. Rev. Lett. 97, 144506.CrossRefGoogle ScholarPubMed
Villermaux, E. & Innocenti, C. 1999 On the geometry of turbulent mixing. J. Fluid Mech. 393, 123–145.Google Scholar
Zeldovich, Y. B. 1937 The asymptotic law of heat transfer at small velocities in the finite domain problem. Zh. Eksp. Teor. Fiz. 7 (12), 1466–1468.Google Scholar