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High Rayleigh number convection in a porous medium containing a thin low-permeability layer

  • Duncan R. Hewitt (a1), Jerome A. Neufeld (a1) (a2) (a3) and John R. Lister (a1)

Porous geological formations are commonly interspersed with thin, roughly horizontal, low-permeability layers. Statistically steady convection at high Rayleigh number $\def \xmlpi #1{}\def \mathsfbi #1{\boldsymbol {\mathsf {#1}}}\let \le =\leqslant \let \leq =\leqslant \let \ge =\geqslant \let \geq =\geqslant \def \Pr {\mathit {Pr}}\def \Fr {\mathit {Fr}}\def \Rey {\mathit {Re}}\mathit{Ra}$ is investigated numerically in a two-dimensional porous medium that is heated at the lower boundary and cooled at the upper, and contains a thin, horizontal, low-permeability interior layer. In the limit that both the dimensionless thickness $h$ and permeability $\Pi $ of the low-permeability layer are small, the flow is described solely by the impedance of the layer $\Omega = h/\Pi $ and by $\mathit{Ra}$ . In the limit $\Omega \to 0$ (i.e. $h \to 0$ ), the system reduces to a homogeneous Rayleigh–Darcy (porous Rayleigh–Bénard) cell. Two notable features are observed as $\Omega $ is increased: the dominant horizontal length scale of the flow increases; and the heat flux, as measured by the Nusselt number $\mathit{Nu}$ , can increase. For larger values of $\Omega $ , $\mathit{Nu}$ always decreases. The dependence of the flow on $\mathit{Ra}$ is explored, over the range $2500 \leqslant \mathit{Ra} \leqslant 2\times 10^4$ . Simple one-dimensional models are developed to describe some of the observed features of the relationship $\mathit{Nu}(\Omega )$ .

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Bachu S. 2008 inline-graphic ${\mathrm{CO}}_2$ storage in geological media: role, means, status and barriers to deployment. Prog. Energy Combust. Sci. 34, 254273.
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, 104501.
Bear J. 1988 Dynamics of Fluids in Porous Media. Dover.
Bickle M., Chadwick A., Huppert H. E., Hallworth M. A. & Lyle S. 2007 Modelling carbon dioxide accumulation at Sleipner: implications for underground carbon storage. Earth Planet. Sci. Lett. 255, 164176.
Daniel D., Tilton N. & Riaz A. 2013 Optimal perturbations of gravitationally unstable, transient boundary layers in porous media. J. Fluid Mech. 727, 456487.
Ennis-King J. P., Preston I. & Paterson L. 2005 Onset of convection in anisotropic porous media subject to a rapid change in boundary conditions. Phys. Fluids 17, 8410784115.
Friedlingstein P., Houghton R. A., Marland G., Hackler J., Boden T. A., Conway T. J., Canadell J. G., Raupach M. R., Ciais P. & Le Quere C. 2010 Update on inline-graphic ${\mathrm{CO}}_2$ emission. Nat. Geosci. 3, 811812.
Genç G. & Rees D. A. S. 2011 The onset of convection in horizontally partitioned porous layers. Phys. Fluids 23, 064107.
Gilfillan S. M. V., Sherwood Lollar B., Holland G., Blagburn D., Stevens S., Schoell M., Cassidy M., Ding Z., Zhou Z., Lacrampe-Couloume G. & Ballentine C. J. 2009 Solubility trapping in formation water as dominant inline-graphic ${\mathrm{CO}}_2$ sinks in natural gas fields. Nature 458, 614618.
Graham M. D. & Steen P. H. 1994 Plume formation and resonant bifurcations in porous-media convection. J. Fluid Mech. 272, 6789.
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, 224503.
Hewitt D. R., Neufeld J. A. & Lister J. R. 2013 Convective shutdown in a porous medium at high Rayleigh number. J. Fluid Mech. 719, 551586.
Hewitt D. R., Neufeld J. A. & Lister J. R. 2014 High Rayleigh number convection in a three-dimensional porous medium. J. Fluid Mech. 748, 879895.
Huppert H. E. & Neufeld J. A. 2014 The fluid mechanics of carbon dioxide sequestration. Annu. Rev. Fluid Mech. 46, 255272.
Kneafsey T. J. & Pruess K. 2010 Laboratory flow experiments for visualizing carbon dioxide-induced, density-driven brine convection. Transp. Porous. Med. 82, 123139.
Lapwood E. R. 1948 Convection of a fluid in a porous medium. Math. Proc. Camb. Phil. Soc. 44, 508521.
McKibbin R. & O’Sullivan M. J. 1980 Onset of convection in a layered porous medium heated from below. J. Fluid Mech. 96, 375393.
McKibbin R. & O’Sullivan M. J. 1981 Heat transfer in a layered porous medium heated from above. J. Fluid Mech. 111, 141173.
McKibbin R. & Tyvand P. A. 1983 Thermal convection in a porous medium composed of alternating thick and thin layers. Intl J. Heat Mass Transfer 26, 761780.
Monkhouse F. J. 1970 Principles of Physical Geography. University of London Press.
Morris K. A. & Shepperd C. M. 1982 The role of clay minerals in influencing porosity and permeability characteristics in the Bridport sands of Wytch Farm, Dorset. Clay Miner. 17, 4154.
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, L22404.
Neufeld J. A., Vella D., Huppert H. E. & Lister J. R. 2011 Leakage from gravity currents in a porous medium. Part 1. A localized sink. J. Fluid Mech. 666, 391413.
Nield D. A. & Bejan A. 2013 Convection in Porous Media, 4th edn. Springer.
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, 263281.
Pau G. S. H., Bell J. B., Pruess K., Almgren A. S., Lijewski M. J. & Zhang K. 2010 High-resolution simulation and characterization of density-driven flow in inline-graphic ${\mathrm{CO}}_2$ storage in saline aquifers. Adv. Water Resour. 33, 443455.
Phillips O. M. 2009 Geological Fluids Dynamics. Cambridge University Press.
Press W. H., Flannery B. P., Teukolsky S. A. & Vetterling W. T. 1989 Numerical Recipes (Fortran), 1st edn. Cambridge University Press.
Pritchard D. 2007 Gravity currents over fractured substrates in a porous medium. J. Fluid Mech. 584, 415431.
Rapaka S., Pawar R. J., Stauffer P. H., Zhang D. & Chen S. 2009 Onset of convection over a transient base-state in anisotropic and layer porous media. J. Fluid Mech. 641, 227244.
Rees D. A. S. & Riley D. S. 1990 The three-dimensional stability of finite-amplitude convection in a layered porous medium heated from below. J. Fluid Mech. 211, 437461.
Riaz A., Hesse M. A., Tchelepi H. A. & Orr F. M. Jr 2006 Onset of convection in a gravitationally unstable diffusive layer in porous media. J. Fluid Mech. 548, 87111.
Simmons C. T., Fenstemaker T. R. & Sharp J. M. 2001 Variable-density groundwater flow and solute transport in heterogeneous porous media: approaches, resolutions and future challenges. J. Contam. Hydrol. 52, 245275.
Slim A. C. 2014 Solutal convection regimes in a two-dimensional porous medium. J. Fluid Mech. 741, 461491.
Slim A. C., Bandi M. M., Miller J. C. & Mahadevan L. 2013 Dissolution-driven convection in a Hele-Shaw cell. Phys. Fluids 25, 024101.
Sternlof K. R., Karimi-Fard M., Pollard D. D. & Durlofsky L. J. 2006 Flow and transport effects of compaction bands in sandstones at scales relevant to aquifer and reservoir management. Water Resour. Res. 42, W07425.
Szulczewski M. L., Hesse M. A. & Juanes R. 2013 Carbon dioxide dissolution in structural and stratigraphic traps. J. Fluid Mech. 736, 287315.
Tilton N. & Riaz A. 2014 Nonlinear stability of gravitationally unstable, transient, diffusive boundary layers in porous meda. J. Fluid Mech. 745, 251278.
Vella D., Neufeld J. A., Huppert H. E. & Lister J. R. 2011 Leakage from gravity currents in a porous medium. Part 2. A line sink. J. Fluid Mech. 666, 414427.
Zhang W. 2013 Density-driven enhanced dissolution of injected inline-graphic ${\mathrm{CO}}_2$ during long-term inline-graphic ${\mathrm{CO}}_2$ storage. J. Earth Syst. Sci. 122, 13871397.
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Journal of Fluid Mechanics
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