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Chaotic mixing in three-dimensional porous media

Published online by Cambridge University Press:  17 August 2016

Daniel R. Lester ,
Marco Dentz  and
Tanguy Le Borgne
Daniel R. Lester*
Affiliation:
School of Chemical and Environmental Engineering, RMIT University, 3000 Melbourne, Victoria, Australia
Marco Dentz
Affiliation:
Spanish National Research Council (IDAEA-CSIC), 08034 Barcelona, Spain
Tanguy Le Borgne
Affiliation:
Geosciences Rennes, UMR 6118, Université de Rennes 1, CNRS, 35042 Rennes, France
*
†Email address for correspondence: [email protected]
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Abstract

Under steady flow conditions, the topological complexity inherent to all random three-dimensional (3D) porous media imparts complicated flow and transport dynamics. It has been established that this complexity generates persistent chaotic advection via a 3D fluid mechanical analogue of the baker’s map which rapidly accelerates scalar mixing in the presence of molecular diffusion. Hence, pore-scale fluid mixing is governed by the interplay between chaotic advection, molecular diffusion and the broad (power-law) distribution of fluid particle travel times which arise from the non-slip condition at pore walls. To understand and quantify mixing in 3D porous media, we consider these processes in a model 3D open porous network and develop a novel stretching continuous time random walk (CTRW), which provides analytic estimates of pore-scale mixing which compare well with direct numerical simulations. We find that the chaotic advection inherent to 3D porous media imparts scalar mixing which scales exponentially with the longitudinal advection, whereas the topological constraints associated with two-dimensional porous media limit the mixing to scale algebraically. These results decipher the role of wide transit time distributions and complex topologies on porous media mixing dynamics, and provide the building blocks for macroscopic models of dilution and mixing which resolve these mechanisms.

JFM classification

Mixing: Chaotic advection Low-Reynolds-number flows: Porous media
Type
Papers
Information
Journal of Fluid Mechanics , Volume 803 , 25 September 2016 , pp. 144 - 174
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Copyright
© 2016 Cambridge University Press 

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References

Abramowitz, M. & Stegun, I. A. 1972 Handbook of Mathematical Functions. Dover.Google Scholar
de Anna, P., Jimenez-Martinez, J., Tabuteau, H., Turuban, R., Le Borgne, T., Derrien, M. & Méheust, Y. 2014 Mixing and reaction kinetics in porous media: an experimental pore scale quantification. Environ. Sci. Technol. 48, 508–516.Google Scholar
de Anna, P., Le Borgne, T., Dentz, M., Tartakovsky, A. M., Bolster, D. & Davy, P. 2013 Flow intermittency, dispersion, and correlated continuous time random walks in porous media. Phys. Rev. Lett. 110, 184502.Google Scholar
Bajer, K. 1994 Hamiltonian formulation of the equations of streamlines in three-dimensional steady flows. Chaos, Solitons Fractals 4 (6), 895–911.Google Scholar
Bajer, K. & Moffatt, H. K. 1990 On a class of steady confined Stokes flows with chaotic streamlines. J. Fluid Mech. 212, 337–363.CrossRefGoogle 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.Google Scholar
Battiato, I., Tartakovsky, D. M., Tartakovsky, A. M. & Scheibe, T. 2009 On breakdown of macroscopic models of mixing-controlled heterogeneous reactions in porous media. Adv. Water Resour. 32, 1664–1673.Google 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, RG2003.Google 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.Google Scholar
Bijeljic, B., Muggeridge, A. H. & Blunt, M. J. 2003 Pore-scale modeling of longitudinal dispersion. Water Resour. Res. 40, W11501.Google Scholar
Chiogna, G., Hochstetler, D., Bellin, A., Kitanidis, P. & Rolle, M. 2012 Mixing, entropy and reactive solute transport. Geophys. Res. Lett. 39, L20405.Google Scholar
Chong, M. S., Monty, J. P., Chin, C. & Marusic, I. 2012 The topology of skin friction and surface velocity fields in wall-bounded flows. J. Turbul. 13 (6), 1–10.Google Scholar
Dentz, M., Le Borgne, T., Lester, D. R. & de Barros, F. P. J. 2015 Scaling forms of particle densities for Lévy walks and strong anomalous diffusion. Phys. Rev. E 92, 032128.Google Scholar
Dentz, M., Leborgne, 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
Duplat, J., Innocenti, C. & Villermaux, E. 2010 A nonsequential turbulent mixing process. Phys. Fluids 22, 035104.CrossRefGoogle Scholar
Duplat, J. & Villermaux, E. 2008 Mixing by random stirring in confined mixtures. J. Fluid Mech. 617, 51–86.Google Scholar
Erdélyi, A. 1956 Asymptotic Expansions. Dover.Google Scholar
Gramling, C. M., Harvey, C. F. & Meigs, L. C. 2002 Reactive transport in porous media: a comparison of model prediction with laboratory visualization. Environ. Sci. Technol. 36, 2508–2514.Google 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
Jones, S. W., Thomas, O. M. & Aref, H. 1989 Chaotic advection by laminar flow in a twisted pipe. J. Fluid Mech. 209, 335–357.CrossRefGoogle Scholar
Kang, P. K., de Anna, P., Nunes, J. P., Bijeljic, B., Blunt, M. & Juanes, R. 2014 Pore-scale intermittent velocity structure underpinning anomalous transport through 3d porous media. Geophys. Res. Lett. 41, 6184–6190.Google Scholar
Le Borgne, T., Bolster, D., Dentz, M., de Anna, P. & Tartakovsky, A. 2011 Effective pore-scale dispersion upscaling with a correlated continuous time random walk approach. Water Resour. Res. 47, W12538.Google Scholar
Le Borgne, T., Dentz, M. & Villermaux, E. 2013 Stretching, coalescence, and mixing in porous media. Phys. Rev. Lett. 110 (20), 204501.CrossRefGoogle ScholarPubMed
Le Borgne, T., Dentz, M. & Villermaux, E. 2015 The lamellar description of mixing in porous media. J. Fluid Mech. 770, 458–498.Google Scholar
Lester, D. R., Metcalfe, G. & Trefry, M. G. 2013 Is chaotic advection inherent to porous media flow? Phys. Rev. Lett. 111, 174101.Google Scholar
Lester, D. R., Metcalfe, G. & Trefry, M. G. 2014 Anomalous transport and chaotic advection in homogeneous porous media. Phys. Rev. E 90, 063012.Google ScholarPubMed
MacKay, R. S. 1994 Transport in 3D volume-preserving flows. J. Nonlinear Sci. 4, 329–354.Google Scholar
MacKay, R. S. 2008 A steady mixing flow with non-slip boundaries. In Chaos, Complexity and Transport (ed. Chandre, C., Leoncini, X. & Zaslavsky, G. M.), pp. 55–68. World Scientific.Google Scholar
Metcalfe, G., Speetjens, M., Lester, D. & Clercx, H. 2012 Beyond passive: chaotic transport in stirred fluids. In Advances in Applied Mechanics (ed. van der Giessen, E. & Aref, H.), vol. 45, pp. 109–188. Elsevier.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
Mezić, I. & Wiggins, S. 1994 On the integrability and perturbations of three-dimensional fluid flows with symmetry. J. Nonlinear Sci. 4, 157–194.Google Scholar
Moffatt, H. K. 1964 Viscous and resistive eddies near a sharp corner. J. Fluid Mech. 18, 1–18.Google Scholar
Moroni, M. & Cushman, J. 2001 Three-dimensional particle tracking velocimetry studies of the transition from pore dispersion to Fickian dispersion for homogeneous porous media. Water Resour. Res. 37 (4), 873–884.CrossRefGoogle Scholar
Ottino, J. M. 1989 The Kinematics of Mixing: Stretching, Chaos, and Transport. Cambridge University Press.Google Scholar
Ottino, J. M. & Wiggins, S. 2004 Introduction: mixing in microfluidics. Phil. Trans. R. Soc. Lond. A 362 (1818), 923–935.Google 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.CrossRefGoogle Scholar
Scholz, C., Wirner, F., Götz, J., Rüde, U., Schröder-Turk, G. E., Mecke, K. & Bechinger, C. 2012 Permeability of porous materials determined from the Euler characteristic. Phys. Rev. Lett. 109, 264504.Google Scholar
Siena, M., Riva, M., Hyman, J., Winter, C. & Guadagnini, A. 2014 Relationship between pore size and velocity probability distributions in stochastically generated porous media. Phys. Rev. E 89 (1), 013018.Google Scholar
Surana, A., Grunberg, O. & Haller, G. 2006 Exact theory of three-dimensional flow separation. Part 1. Steady separation. J. Fluid Mech. 564, 57–103.Google Scholar
Tartakovsky, A. M., Redden, G., Lichtner, P. C., Scheibe, T. D. & Meakin, P. 2008a Mixing-induced precipitation: experimental study and multiscale numerical analysis. Water Resour. Res. 44, W06S04.CrossRefGoogle Scholar
Tartakovsky, A. M., Tartakovsky, D. M. & Meakin, P. 2008b Stochastic Langevin model for flow and transport in porous media. Phys. Rev. Lett. 101, 044502.CrossRefGoogle ScholarPubMed
Tartakovsky, A. M., Tartakovsky, D. M., Scheibe, T. D. & Meakin, P. 2008c Hybrid simulations of reaction–diffusion systems in porous media. SIAM J. Sci. Comput. 30 (6), 2799–2816.Google Scholar
Tartakovsky, A. M., Tartakovsky, G. D. & Scheibe, T. D. 2009 Effects of incomplete mixing on multicomponent reactive transport. Adv. Water Resour. 32, 1674–1679.Google Scholar
Uchaikin, V. V. & Zolotarev, M. Z. 1999 Chance and Stability, Stable Distributions and Their Applications. Walter de Gruyter.Google Scholar
Villermaux, E. 2012 Mixing by porous media. C. R. Mécanique 340, 933–943.CrossRefGoogle Scholar
Villermaux, E. & Duplat, J. 2003 Mixing as an aggregation process. Phys. Rev. Lett. 91, 18.CrossRefGoogle ScholarPubMed
Vogel, H. J. 2002 Topological characterization of porous media. In Morphology of Condensed Matter (ed. Mecke, K. & Stoyan, D.), Lecture Notes in Physics, vol. 600, pp. 75–92. Springer.CrossRefGoogle Scholar
Wiggins, S. 2010 Coherent structures and chaotic advection in three dimensions. J. Fluid Mech. 654, 1–4.Google Scholar
de Winkel, E. & Bakker, P. 1988 On the Topology of Three-dimensional Viscous Flow Structures Near a Plane Wall: A Classification of Hyperbolic and Non-hyperbolic Singularities on the Wall. Delft University of Technology, Faculty of Aerospace Engineering.Google Scholar