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Mixing across fluid interfaces compressed by convective flow in porous media

Published online by Cambridge University Press:  10 January 2018

Juan J. Hidalgo Open the ORCID record for Juan J. Hidalgo [Opens in a new window]  and
Marco Dentz Open the ORCID record for Marco Dentz [Opens in a new window]
Juan J. Hidalgo*
Affiliation:
IDAEA-CSIC, Barcelona, 08034, Spain Associated Unit, Hydrogeology Group (UPC-CSIC), Barcelona, Spain
Marco Dentz
Affiliation:
IDAEA-CSIC, Barcelona, 08034, Spain Associated Unit, Hydrogeology Group (UPC-CSIC), Barcelona, Spain
*
†Email address for correspondence: juanj.hidalgo@idaea.csic.es
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Abstract

We study mixing in the presence of convective flow in a porous medium. Convection is characterized by the formation of vortices and stagnation points, where the fluid interface is stretched and compressed enhancing mixing. We analyse the behaviour of the mixing dynamics in different scenarios using an interface deformation model. We show that the scalar dissipation rate, which is related to the dissolution fluxes, is controlled by interfacial processes, specifically the equilibrium between interface compression and diffusion, which depends on the flow field configuration. We consider different scenarios of increasing complexity. First, we analyse a double-gyre synthetic velocity field. Second, a Rayleigh–Bénard instability (the Horton–Rogers–Lapwood problem), in which stagnation points are located at a fixed interface. This system experiences a transition from a diffusion controlled mixing to a chaotic convection as the Rayleigh number increases. Finally, a Rayleigh–Taylor instability with a moving interface, in which mixing undergoes three different regimes: diffusive, convection dominated and convection shutdown. The interface compression model correctly predicts the behaviour of the systems. It shows how the dependency of the compression rate on diffusion explains the change in the scaling behaviour of the scalar dissipation rate. The model indicates that the interaction between stagnation points and the correlation structure of the velocity field is also responsible for the transition between regimes. We also show the difference in behaviour between the dissolution fluxes and the mixing state of the systems. We observe that while the dissolution flux decreases with the Rayleigh number, the system becomes more homogeneous. That is, mixing is enhanced by reducing diffusion. This observation is explained by the effect of the instability patterns.

JFM classification

Convection: Convection in porous media Geophysical and Geological Flows: Geophysical and Geological Flows Mixing: Mixing
Type
JFM Papers
Information
Journal of Fluid Mechanics , Volume 838 , 10 March 2018 , pp. 105 - 128
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Copyright
© 2018 Cambridge University Press 

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References

Abarca, E., Carrera, J., Sánchez-Vila, X. & Voss, C. I. 2007 Quasi-horizontal circulation cells in 3d seawater intrusion. J. Hydrol. 339 (3–4), 118–129.Google Scholar
Backhaus, S., Turitsyn, K. & Ecke, R. E. 2011 Convective instability and mass transport of diffusion layers in a Hele-Shaw geometry. Phys. Rev. Lett. 106 (10), 104501.Google Scholar
Batchelor, G. K. 1959 Small-scale variation of convected quantities like temperature in turbulent fluid part 1. General discussion and the case of small conductivity. J. Fluid Mech. 5, 113–133.Google Scholar
Cheng, P. 1979 Heat transfer in geothermal systems. Adv. Heat Transfer 14, 1–105.CrossRefGoogle Scholar
Ching, E. S. C. & Lo, K. F. 2001 Heat transport by fluid flows with prescribed velocity fields. Phys. Rev. E 64 (4), 046302.Google ScholarPubMed
Cooper, H. H. 1964 Sea Water in Coastal Aquifers, USGS Numbered Series 1613. USGS.Google Scholar
De Simoni, M., Carrera, J., Sánchez-Vila, X. & Guadagnini, A. 2005 A procedure for the solution of multicomponent reactive transport problems. Water Resour. Res. 41 (11), W11410.Google Scholar
Dentz, M., Le Borgne, T., Englert, A. & Bijeljic, B. 2011 Mixing, spreading and reaction in heterogeenous media: a brief review. J. Contam. Hydrol. 120–121, 1–17.Google Scholar
Dow Chemical2011 Propylene glycols – density values. https://dowac.custhelp.com/app/answers/detail/a_id/7471/∼/propylene-glycols—density-values.Google Scholar
Dyga, R. & Troniewski, L. 2015 Convective heat transfer for fluids passing through aluminum foams. Arch. Thermodyn. 36 (1), 139–156.Google Scholar
Elder, J. W. 1968 The unstable thermal interface. J. Fluid Mech. 32 (01), 69–96.Google Scholar
Ennis-King, J. P. & Paterson, L. 2005 Role of convective mixing in the long-term storage of carbon dioxide in deep saline formations. SPE J. 10 (03), 349–356.CrossRefGoogle Scholar
Graham, M. D. & Steen, P. H. 1994 Plume formation and resonant bifurcations in porous-media convection. J. Fluid Mech. 272, 67–90.CrossRefGoogle Scholar
Hamadouche, A., Nebbali, R., Benahmed, H., Kouidri, A. & Bousri, A. 2016 Experimental investigation of convective heat transfer in an open-cell aluminum foams. Exp. Therm. Fluid Sci. 71, 86–94.Google Scholar
Hewitt, D. R., Neufeld, J. A. & Lister, J. R. 2012 Ultimate regime of high Rayleigh number convection in a porous medium. Phys. Rev. Lett. 108 (22), 224503.Google Scholar
Hewitt, D. R., Neufeld, J. A. & Lister, J. R. 2013a Convective shutdown in a porous medium at high Rayleigh number. J. Fluid Mech. 719, 551–586.Google Scholar
Hewitt, D. R., Neufeld, J. A. & Lister, J. R. 2013b Stability of columnar convection in a porous medium. J. Fluid Mech. 737, 205–231.Google Scholar
Hidalgo, J. J. & Carrera, J. 2009 Effect of dispersion on the onset of convection during co2 sequestration. J. Fluid Mech. 640, 441–452.Google Scholar
Hidalgo, J. J., Dentz, M., Cabeza, Y. & Carrera, J. 2015 Dissolution patterns and mixing dynamics in unstable reactive flow. Geophys. Res. Lett. 42 (15), 6357–6364.Google Scholar
Hidalgo, J. J., Fe, J., Cueto-Felgueroso, L. & Juanes, R. 2012 Scaling of convective mixing in porous media. Phys. Rev. Lett. 109 (26), 264503.Google Scholar
Hidalgo, J. J., MacMinn, C. W. & Juanes, R. 2013 Dynamics of convective dissolution from a migrating current of carbon dioxide. Adv. Water Resour. 62, 511–519.Google Scholar
Horton, C. W. & Rogers, F. T. 1945 Convection currents in a porous medium. J. Appl. Phys. 16 (6), 367–370.Google Scholar
Howard, L. N. 1966 Convection at High Rayleigh Number, pp. 1109–1115. Springer.Google Scholar
Kimura, S., Schubert, G. & Straus, J. M. 1986 Route to chaos in porous-medium thermal convection. J. Fluid Mech. 166 (-1), 305–324.Google Scholar
Kitanidis, P. K. 1994 The concept of the dilution index. Water Resour. Res. 30 (7), 2011–2026.Google Scholar
Kueper, B. H. & Frind, E. O. 1991 Two-phase flow in heterogeneous porous media: 1. Model development. Water Resour. Res. 27 (6), 1049–1057.Google Scholar
Lapwood, E. R. 1948 Convection of a fluid in a porous medium. Math. Proc. Camb. Phil. Soc. 44 (04), 508–521.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 (12), 1468–1475.Google Scholar
Le Borgne, T., Dentz, M. & Villermaux, E. 2013 Stretching, coalescence, and mixing in porous media. Phys. Rev. Lett. 110 (20), 204501.Google Scholar
Le Borgne, T., Dentz, M. & Villermaux, E. 2015 The lamellar description of mixing in porous media. J. Fluid Mech. 770, 458–498.Google Scholar
Martin, D., Griffiths, R. W. & Campbell, I. H. 1987 Compositional and thermal convection in magma chambers. Contrib. Mineral. Petrol. 96 (4), 465–475.Google Scholar
Musgrave, D. L. 1985 A numerical study of the roles of subgyre-scale mixing and the western boundary current on homogenization of a passive tracer. J. Geophys. Res. 90 (C4), 7037–7043.Google Scholar
Neufeld, J. A., Hesse, M. A., Riaz, A., Hallworth, M. A., Tchelepi, H. A. & Huppert, H. E. 2010 Convective dissolution of carbon dioxide in saline aquifers. Geophys. Res. Lett. 37 (22), L22404.Google Scholar
Otero, J., Dontcheva, L. A., Johnston, H., Worthing, R. A., Kurganov, A., Petrova, G. & Doering, C. R. 2004 High-Rayleigh-number convection in a fluid-saturated porous layer. J. Fluid Mech. 500, 263–281.Google Scholar
Ottino, J. M. 1989 The Kinematics of Mixing: Stretching, Chaos, and Transport. Cambridge Texts in Applied Mathematics. Cambridge University Press.Google Scholar
Ranz, W. E. 1979 Applications of a stretch model to mixing, diffusion, and reaction in laminar and turbulent flows. AIChE J. 25 (1), 41–47.CrossRefGoogle Scholar
Rees, D. A. S., Selim, A. & Ennis-King, J. P. 2008 The instability of unsteady boundary layers in porous media. In Emerging Topics in Heat and Mass Transfer in Porous Media. From Bioengineering and Microelectronics to Nanotechnology (ed. Vadasz, P.), Theory and Applications of Transport in Porous Media, vol. 22, pp. 85–110. Springer.Google Scholar
Riaz, A., Hesse, M., Tchelepi, H. A. & Orr, F. M. Jr. 2006 Onset of convection in a gravitationally unstable diffusive boundary layer in porous media. J. Fluid Mech. 548, 87–111.Google Scholar
Sanford, W. E., Whitaker, F. F., Smart, P. L. & Jones, G. 1998 Numerical analysis of seawater circulation in carbonate platforms: I, geothermal convection. Am. J. Sci. 298 (10), 801–828.Google Scholar
Shadden, S. C., Lekien, F. & Marsden, J. E. 2005 Definition and properties of lagrangian coherent structures from finite-time Lyapunov exponents in two-dimensional aperiodic flows. Physica D 212 (3–4), 271–304.Google Scholar
Slim, A. C. 2014 Solutal-convection regimes in a two-dimensional porous medium. J. Fluid Mech. 741, 461–491.Google Scholar
Slim, A. C. & Ramakrishnan, T. S. 2010 Onset and cessation of time-dependent, dissolution-driven convection in porous media. Phys. Fluids 22 (12), 124103.Google Scholar
Szulczewski, M. L., Hesse, M. A. & Juanes, R. 2013 Carbon dioxide dissolution in structural and stratigraphic traps. J. Fluid Mech. 736, 287–315.Google Scholar
Tait, S. & Jaupart, C. 1989 Compositional convection in viscous melts. Nature 338 (6216), 571–574.Google Scholar
Villermaux, E. 2012 Mixing by porous media. C. R. Méc 340 (11–12), 933–943.Google Scholar
Villermaux, E. & Duplat, J. 2006 Coarse grained scale of turbulent mixtures. Phys. Rev. Lett. 97, 144506.Google Scholar
Wells, A. J., Wettlaufer, J. S. & Orszag, S. A. 2011 Brine fluxes from growing sea ice. Geophys. Res. Lett. 38 (4), l04501.Google Scholar