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Chaotic mixing in three-dimensional porous media

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

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Corresponding author
Email address for correspondence: daniel.lester@rmit.edu.au
References
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AbramowitzM. & StegunI. A. 1972 Handbook of Mathematical Functions. Dover.
de AnnaP., Jimenez-MartinezJ., TabuteauH., TurubanR., Le BorgneT., DerrienM. & MéheustY. 2014 Mixing and reaction kinetics in porous media: an experimental pore scale quantification. Environ. Sci. Technol. 48, 508516.
de AnnaP., Le BorgneT., DentzM., TartakovskyA. M., BolsterD. & DavyP. 2013 Flow intermittency, dispersion, and correlated continuous time random walks in porous media. Phys. Rev. Lett. 110, 184502.
BajerK. 1994 Hamiltonian formulation of the equations of streamlines in three-dimensional steady flows. Chaos, Solitons Fractals 4 (6), 895911.
BajerK. & MoffattH. K. 1990 On a class of steady confined Stokes flows with chaotic streamlines. J. Fluid Mech. 212, 337363.
de BarrosF., DentzM., KochJ. & NowakW. 2012 Flow topology and scalar mixing in spatially heterogeneous flow fields. Geophys. Res. Lett. 39, L08404.
BattiatoI., TartakovskyD. M., TartakovskyA. M. & ScheibeT. 2009 On breakdown of macroscopic models of mixing-controlled heterogeneous reactions in porous media. Adv. Water Resour. 32, 16641673.
BerkowitzB., CortisA., DentzM. & ScherH. 2006 Modeling non-Fickian transport in geological formations as a continuous time random walk. Rev. Geophys. 44, RG2003.
BijeljicB., MostaghimiP. & BluntM. J. 2011 Signature of non-fickian solute transport in complex heterogeneous porous media. Phys. Rev. Lett. 107, 204502.
BijeljicB., MuggeridgeA. H. & BluntM. J. 2003 Pore-scale modeling of longitudinal dispersion. Water Resour. Res. 40, W11501.
ChiognaG., HochstetlerD., BellinA., KitanidisP. & RolleM. 2012 Mixing, entropy and reactive solute transport. Geophys. Res. Lett. 39, L20405.
ChongM. S., MontyJ. P., ChinC. & MarusicI. 2012 The topology of skin friction and surface velocity fields in wall-bounded flows. J. Turbul. 13 (6), 110.
DentzM., Le BorgneT., LesterD. R. & de BarrosF. P. J. 2015 Scaling forms of particle densities for Lévy walks and strong anomalous diffusion. Phys. Rev. E 92, 032128.
DentzM., LeborgneT., EnglertA. & BijeljicB. 2011 Mixing, spreading and reaction in heterogeneous media: a brief review. J. Contam. Hydrol. 120–121, 117.
DuplatJ., InnocentiC. & VillermauxE. 2010 A nonsequential turbulent mixing process. Phys. Fluids 22, 035104.
DuplatJ. & VillermauxE. 2008 Mixing by random stirring in confined mixtures. J. Fluid Mech. 617, 5186.
ErdélyiA. 1956 Asymptotic Expansions. Dover.
GramlingC. M., HarveyC. F. & MeigsL. C. 2002 Reactive transport in porous media: a comparison of model prediction with laboratory visualization. Environ. Sci. Technol. 36, 25082514.
HolznerM., MoralesV. L., WillmannM. & DentzM. 2015 Intermittent Lagrangian velocities and accelerations in three-dimensional porous medium flow. Phys. Rev. E 92, 013015.
JonesS. W., ThomasO. M. & ArefH. 1989 Chaotic advection by laminar flow in a twisted pipe. J. Fluid Mech. 209, 335357.
KangP. K., de AnnaP., NunesJ. P., BijeljicB., BluntM. & JuanesR. 2014 Pore-scale intermittent velocity structure underpinning anomalous transport through 3d porous media. Geophys. Res. Lett. 41, 61846190.
Le BorgneT., BolsterD., DentzM., de AnnaP. & TartakovskyA. 2011 Effective pore-scale dispersion upscaling with a correlated continuous time random walk approach. Water Resour. Res. 47, W12538.
Le BorgneT., DentzM. & VillermauxE. 2013 Stretching, coalescence, and mixing in porous media. Phys. Rev. Lett. 110 (20), 204501.
Le BorgneT., DentzM. & VillermauxE. 2015 The lamellar description of mixing in porous media. J. Fluid Mech. 770, 458498.
LesterD. R., MetcalfeG. & TrefryM. G. 2013 Is chaotic advection inherent to porous media flow? Phys. Rev. Lett. 111, 174101.
LesterD. R., MetcalfeG. & TrefryM. G. 2014 Anomalous transport and chaotic advection in homogeneous porous media. Phys. Rev. E 90, 063012.
MacKayR. S. 1994 Transport in 3D volume-preserving flows. J. Nonlinear Sci. 4, 329354.
MacKayR. S. 2008 A steady mixing flow with non-slip boundaries. In Chaos, Complexity and Transport (ed. Chandre C., Leoncini X. & Zaslavsky G. M.), pp. 5568. World Scientific.
MetcalfeG., SpeetjensM., LesterD. & ClercxH. 2012 Beyond passive: chaotic transport in stirred fluids. In Advances in Applied Mechanics (ed. van der Giessen E. & Aref H.), vol. 45, pp. 109188. Elsevier.
MeunierP. & VillermauxE. 2010 The diffusive strip method for scalar mixing in two dimensions. J. Fluid Mech. 662, 134172.
MezićI. & WigginsS. 1994 On the integrability and perturbations of three-dimensional fluid flows with symmetry. J. Nonlinear Sci. 4, 157194.
MoffattH. K. 1964 Viscous and resistive eddies near a sharp corner. J. Fluid Mech. 18, 118.
MoroniM. & CushmanJ. 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.
OttinoJ. M. 1989 The Kinematics of Mixing: Stretching, Chaos, and Transport. Cambridge University Press.
OttinoJ. M. & WigginsS. 2004 Introduction: mixing in microfluidics. Phil. Trans. R. Soc. Lond. A 362 (1818), 923935.
RanzW. E. 1979 Application of a stretch model to mixing, diffusion and reaction in laminar and turbulent flows. AIChE J. 25 (1), 4147.
ScholzC., WirnerF., GötzJ., RüdeU., Schröder-TurkG. E., MeckeK. & BechingerC. 2012 Permeability of porous materials determined from the Euler characteristic. Phys. Rev. Lett. 109, 264504.
SienaM., RivaM., HymanJ., WinterC. & GuadagniniA. 2014 Relationship between pore size and velocity probability distributions in stochastically generated porous media. Phys. Rev. E 89 (1), 013018.
SuranaA., GrunbergO. & HallerG. 2006 Exact theory of three-dimensional flow separation. Part 1. Steady separation. J. Fluid Mech. 564, 57103.
TartakovskyA. M., ReddenG., LichtnerP. C., ScheibeT. D. & MeakinP. 2008a Mixing-induced precipitation: experimental study and multiscale numerical analysis. Water Resour. Res. 44, W06S04.
TartakovskyA. M., TartakovskyD. M. & MeakinP. 2008b Stochastic Langevin model for flow and transport in porous media. Phys. Rev. Lett. 101, 044502.
TartakovskyA. M., TartakovskyD. M., ScheibeT. D. & MeakinP. 2008c Hybrid simulations of reaction–diffusion systems in porous media. SIAM J. Sci. Comput. 30 (6), 27992816.
TartakovskyA. M., TartakovskyG. D. & ScheibeT. D. 2009 Effects of incomplete mixing on multicomponent reactive transport. Adv. Water Resour. 32, 16741679.
UchaikinV. V. & ZolotarevM. Z. 1999 Chance and Stability, Stable Distributions and Their Applications. Walter de Gruyter.
VillermauxE. 2012 Mixing by porous media. C. R. Mécanique 340, 933943.
VillermauxE. & DuplatJ. 2003 Mixing as an aggregation process. Phys. Rev. Lett. 91, 18.
VogelH. J. 2002 Topological characterization of porous media. In Morphology of Condensed Matter (ed. Mecke K. & Stoyan D.), Lecture Notes in Physics, vol. 600, pp. 7592. Springer.
WigginsS. 2010 Coherent structures and chaotic advection in three dimensions. J. Fluid Mech. 654, 14.
de WinkelE. & BakkerP. 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.
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Journal of Fluid Mechanics
  • ISSN: 0022-1120
  • EISSN: 1469-7645
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