Skip to main content
    • Aa
    • Aa

Chaotic mixing in three-dimensional porous media

  • Daniel R. Lester (a1), Marco Dentz (a2) and Tanguy Le Borgne (a3)

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.

Corresponding author
Email address for correspondence:
Linked references
Hide All

This list contains references from the content that can be linked to their source. For a full set of references and notes please see the PDF or HTML where available.

P. de Anna , J. Jimenez-Martinez , H. Tabuteau , R. Turuban , T. Le Borgne , M. Derrien  & Y. Méheust 2014 Mixing and reaction kinetics in porous media: an experimental pore scale quantification. Environ. Sci. Technol. 48, 508516.

P. de Anna , T. Le Borgne , M. Dentz , A. M. Tartakovsky , D. Bolster  & P. Davy 2013 Flow intermittency, dispersion, and correlated continuous time random walks in porous media. Phys. Rev. Lett. 110, 184502.

K. Bajer 1994 Hamiltonian formulation of the equations of streamlines in three-dimensional steady flows. Chaos, Solitons Fractals 4 (6), 895911.

F. de Barros , M. Dentz , J. Koch  & W. Nowak 2012 Flow topology and scalar mixing in spatially heterogeneous flow fields. Geophys. Res. Lett. 39, L08404.

I. Battiato , D. M. Tartakovsky , A. M. Tartakovsky  & T. Scheibe 2009 On breakdown of macroscopic models of mixing-controlled heterogeneous reactions in porous media. Adv. Water Resour. 32, 16641673.

B. Berkowitz , A. Cortis , M. Dentz  & H. Scher 2006 Modeling non-Fickian transport in geological formations as a continuous time random walk. Rev. Geophys. 44, RG2003.

B. Bijeljic , P. Mostaghimi  & M. J. Blunt 2011 Signature of non-fickian solute transport in complex heterogeneous porous media. Phys. Rev. Lett. 107, 204502.

G. Chiogna , D. Hochstetler , A. Bellin , P. Kitanidis  & M. Rolle 2012 Mixing, entropy and reactive solute transport. Geophys. Res. Lett. 39, L20405.

M. S. Chong , J. P. Monty , C. Chin  & I. Marusic 2012 The topology of skin friction and surface velocity fields in wall-bounded flows. J. Turbul. 13 (6), 110.

J. Duplat , C. Innocenti  & E. Villermaux 2010 A nonsequential turbulent mixing process. Phys. Fluids 22, 035104.

C. M. Gramling , C. F. Harvey  & L. C. Meigs 2002 Reactive transport in porous media: a comparison of model prediction with laboratory visualization. Environ. Sci. Technol. 36, 25082514.

P. K. Kang , P. de Anna , J. P. Nunes , B. Bijeljic , M. Blunt  & R. Juanes 2014 Pore-scale intermittent velocity structure underpinning anomalous transport through 3d porous media. Geophys. Res. Lett. 41, 61846190.

T. Le Borgne , D. Bolster , M. Dentz , P. de Anna  & A. Tartakovsky 2011 Effective pore-scale dispersion upscaling with a correlated continuous time random walk approach. Water Resour. Res. 47, W12538.

T. Le Borgne , M. Dentz  & E. Villermaux 2013 Stretching, coalescence, and mixing in porous media. Phys. Rev. Lett. 110 (20), 204501.

D. R. Lester , G. Metcalfe  & M. G. Trefry 2013 Is chaotic advection inherent to porous media flow? Phys. Rev. Lett. 111, 174101.

R. S. MacKay 1994 Transport in 3D volume-preserving flows. J. Nonlinear Sci. 4, 329354.

R. S. MacKay 2008 A steady mixing flow with non-slip boundaries. In Chaos, Complexity and Transport (ed. C. Chandre , X. Leoncini  & G. M. Zaslavsky ), pp. 5568. World Scientific.

G. Metcalfe , M. Speetjens , D. Lester  & H. Clercx 2012 Beyond passive: chaotic transport in stirred fluids. In Advances in Applied Mechanics (ed. E. van der Giessen  & H. Aref ), vol. 45, pp. 109188. Elsevier.

I. Mezić  & S. Wiggins 1994 On the integrability and perturbations of three-dimensional fluid flows with symmetry. J. Nonlinear Sci. 4, 157194.

M. Moroni  & J. Cushman 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), 873884.

J. M. Ottino  & S. Wiggins 2004 Introduction: mixing in microfluidics. Phil. Trans. R. Soc. Lond. A 362 (1818), 923935.

W. E. Ranz 1979 Application of a stretch model to mixing, diffusion and reaction in laminar and turbulent flows. AIChE J. 25 (1), 4147.

C. Scholz , F. Wirner , J. Götz , U. Rüde , G. E. Schröder-Turk , K. Mecke  & C. Bechinger 2012 Permeability of porous materials determined from the Euler characteristic. Phys. Rev. Lett. 109, 264504.

A. M. Tartakovsky , G. Redden , P. C. Lichtner , T. D. Scheibe  & P. Meakin 2008a Mixing-induced precipitation: experimental study and multiscale numerical analysis. Water Resour. Res. 44, W06S04.

A. M. Tartakovsky , D. M. Tartakovsky  & P. Meakin 2008b Stochastic Langevin model for flow and transport in porous media. Phys. Rev. Lett. 101, 044502.

A. M. Tartakovsky , D. M. Tartakovsky , T. D. Scheibe  & P. Meakin 2008c Hybrid simulations of reaction–diffusion systems in porous media. SIAM J. Sci. Comput. 30 (6), 27992816.

A. M. Tartakovsky , G. D. Tartakovsky  & T. D. Scheibe 2009 Effects of incomplete mixing on multicomponent reactive transport. Adv. Water Resour. 32, 16741679.

V. V. Uchaikin  & M. Z. Zolotarev 1999 Chance and Stability, Stable Distributions and Their Applications. Walter de Gruyter.

E. Villermaux 2012 Mixing by porous media. C. R. Mécanique 340, 933943.

E. Villermaux  & J. Duplat 2003 Mixing as an aggregation process. Phys. Rev. Lett. 91, 18.

H. J. Vogel 2002 Topological characterization of porous media. In Morphology of Condensed Matter (ed. K. Mecke  & D. Stoyan ), Lecture Notes in Physics, vol. 600, pp. 7592. Springer.

Recommend this journal

Email your librarian or administrator to recommend adding this journal to your organisation's collection.

Journal of Fluid Mechanics
  • ISSN: 0022-1120
  • EISSN: 1469-7645
  • URL: /core/journals/journal-of-fluid-mechanics
Please enter your name
Please enter a valid email address
Who would you like to send this to? *



Full text views

Total number of HTML views: 10
Total number of PDF views: 136 *
Loading metrics...

Abstract views

Total abstract views: 255 *
Loading metrics...

* Views captured on Cambridge Core between September 2016 - 18th August 2017. This data will be updated every 24 hours.