A step function density profile model for the convective stability of CO\(_2\) geological sequestration. (English) Zbl 1444.76115

Summary: The convective stability associated with carbon sequestration is usually investigated by adopting an unsteady diffusive basic profile to account for the space and time development of the carbon-saturated boundary layer instability. The method of normal modes is not applicable due to the time dependence of the nonlinear base profile. Therefore, the instability is quantified either in terms of critical times at which the boundary layer instability sets in or in terms of long-time evolution of initial disturbances. This paper adopts an unstably stratified basic profile having a step function density with top heavy carbon-saturated layer (boundary layer) overlying a lighter carbon-free layer (ambient brine). The resulting configuration resembles that of the Rayleigh-Taylor problem with buoyancy diffusion at the interface separating the two layers. The discontinuous reference state satisfies the governing system of equations and boundary conditions and pertains to an unstably stratified motionless state. Our model accounts for anisotropy in both diffusion and permeability and chemical reaction between the carbon dioxide-rich brine and host mineralogy. We consider two cases for the boundary conditions, namely an impervious lower boundary with either a permeable (one-sided model) or poorly permeable upper boundary. These two cases possess neither steady nor unsteady unstably stratified equilibrium states. We proceed by supposing that the carbon dioxide that has accumulated below the top cap rock forms a layer of carbon-saturated brine of some thickness that overlies a carbon-free brine layer. The resulting stratification remains stable until the thickness, and by the same token, the density, of the carbon-saturated layer is sufficient to induce the fluid to overturn. The existence of a finite threshold value for the thickness is due to the stabilizing influence of buoyancy diffusion at the interface between the two layers. With this formulation for the reference state, the stability calculations will be in terms of critical boundary layer thickness instead of critical times, although the two formulations are homologous. This approach is tractable by the classical normal-mode analysis. Even though it yields only conservative threshold instability conditions, it offers the advantage for an analytically tractable study that puts forth expressions for the carbon concentration convective flux at the interface and explores the flow patterns through both linear and weakly nonlinear analyses.


76S05 Flows in porous media; filtration; seepage
76E15 Absolute and convective instability and stability in hydrodynamic stability
86A60 Geological problems
Full Text: DOI arXiv


[1] Change, Intergovernmental Panel On Climate PCC (2007) Aspectos Regionais e Setoriais da Contribuio do Grupo de Trabalho II ao 4 Relatrio de Avaliao Mudana Climtica 2007 do IPCC · Zbl 0032.09203
[2] Plain CO\(_2\) Reduction (PCOR) Partnership. http://www.undeerc.org/pcor/
[3] Pacala, S; Socolow, R, Wedges: solving the climate problem for the next 50 years with current technologies, Science, 305, 968-972, (2004)
[4] Matter, JM; Stute, M; Snbjrnsdottir, SO; Oelkers, EH; Gislason, SR; Aradottir, ES; Sigfusson, B; Gunnarsson, I; Sigurdardottir, H; Gunnlaugsson, E; Axelsson, G; Alfredsson, HA; Wolff-Boenisch, D; Mesfin, K; Reguera Taya, DF; Hall, J; Dideriksen, K; Broecker, WS, Rapid carbon mineralization for permanent disposal of anthropogenic carbon dioxide emissions, Science, 352, 1312-1314, (2016)
[5] Barba Rossa G, Cliffe KA, Power H (2017) Effects of hydrodynamic dispersion on the stability of buoyancy-driven porous media convection in the presence of first order chemical reaction. J Eng Math. 103:55-76 · Zbl 1391.76763
[6] Paoli, M; Zonta, F; Alfredo Soldati, A, Influence of anisotropic permeability on convection in porous media: implications for geological CO\(_2\) sequestration, Phys. Fluids, 28, 056601, (2016)
[7] Xu, X; Chen, S; Zhang, D, Convective stability analysis of the long term storage of carbon dioxide in deep saline aquifers, Adv Water Resour, 29, 397-407, (2006)
[8] Batchelor, GK; Nitsche, J, Instability of stationary unbounded stratified fluid, J Fluid Mech, 227, 357-391, (1991) · Zbl 0850.76205
[9] Simitev RD, Busse FH (2010) Problems of astrophysical convection: thermal convection in layers without boundaries. Proceedings of the 2010 Summer Program. Center of Turbulence Research, Stanford University, pp 485-492
[10] Hadji L, Shahmurov S, Aljahdaly NH (2016) Thermal convection induced by an infinitesimally thin and unstably stratified layer. J Non-equilib Themodyn 41:279-294 · Zbl 1398.76212
[11] Foster, T, Onset of convection in a layer of fluid cooled from above, Phys. Fluids, 8, 1770-1774, (1965)
[12] Foster, T, Onset of manifest convection in a layer of fluid with a time-dependent surface temperature, Phys. Fluids, 12, 2482-2487, (1969)
[13] Kneafsey, TJ; Karsten, P, Laboratory flow experiments for visualizing carbon dioxide-induced, density- driven brine convection, Transp Porous Media, 82, 123-139, (2010)
[14] Neufeld, JA; Hesse, MA; Riaz, A; Hallaworth, MA; Tchelepi, HA, Convective dissolution of carbon dioxide in saline aquifers, Geophys Res Lett, 37, l22404, (2010)
[15] Hill, AA; Morad, MR, Convective stability of carbon sequestration in anisotropic porous media, Proc R Soc A, 470, 20140373, (2014)
[16] Bhadauria, BS, Double-diffusive convection in a saturated anisotropic porous layer with internal heat source, Transp Porous Media, 92, 299-320, (2012)
[17] Horton, CW; Rogers, FT, Convection currents in a porous media, J Appl Phys, 16, 367-370, (1945) · Zbl 0063.02071
[18] Lapwood, ER, Convection of a fluid in a porous medium, Proc Camb Philos Soc, 44, 508-521, (1948) · Zbl 0032.09203
[19] Torre, JM; Busse, FH, Stability of two-dimensional convection in a fluid saturated porous medium, J Fluid Mech, 292, 305-323, (1995) · Zbl 0859.76025
[20] Barletta, A; Tyvand, PA; Nygard, HS, Onset of thermal convection in a porous layer with mixed boundary conditions, J Eng Math, 91, 105-120, (2015) · Zbl 1398.76212
[21] Riaz, A; Hesse, M; Tchelepi, HA; Orr, FM, Onset of convection in a gravitationally unstable diffusive boundary layer in porous media, J Fluid Mech, 548, 87-111, (2006)
[22] Ennis-King, J; Preston, I; Paterson, L, Onset of convection in anisotropic porous media subject to a rapid change in boundary conditions, Phys Fluids, 17, 084107, (2005) · Zbl 1187.76141
[23] Slim, AC; Ramakrishnan, TS, Onset and cessation of time-dependent, dissolution-driven convection in porous media, Phys Fluids, 22, 124103, (2010)
[24] Slim, AC, Solutal-convection regimes in a two-dimensional porous medium, J Fluid Mech, 741, 461-491, (2014)
[25] Klinkenberg K (1941) The permeability of porous media to liquids and gases. In: Drilling and production practice. American Petroleum Institute, New York · Zbl 0507.76049
[26] Rasenat, S; Busse, FH; Rehberg, I, A theoretical and experimental study of double-layer convection, J Fluid Mech, 199, 519-540, (1989) · Zbl 0659.76115
[27] Busse, FH; Riahi, N, Nonlinear thermal convection with poorly conducting boundaries, J Fluid Mech, 96, 243-256, (1980) · Zbl 0429.76055
[28] Chapman, CJ; Proctor, MRE, Nonlinear Rayleigh-BĂ©nard convection between poorly conducting boundaries, J Fluid Mech, 101, 759-782, (1980) · Zbl 0507.76049
[29] Nayfeh AH (2004) Introduction to perturbation techniques, vol 139. Wiley, New York · Zbl 0449.34001
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