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PETROLEUM SOCIETY PAPER 2008-086 Onset of Convection of CO2-Sequestration in Deep Inclined Saline Aquifers M. JAVAHERI, J. ABEDI University of Calgary H. HASSANZADEH IOR Research Institute, National Iranian Oil Company This paper is accepted for the Proceedings of the Canadian International Petroleum Conference/SPE Gas Technology Symposium 2008 Joint Conference (the Petroleum Society’s 59th Annual Technical Meeting), Calgary, Alberta, Canada, 17-19 June 2008. This paper will be considered for publication in Petroleum Society journals. Publication rights are reserved. This is a pre-print and subject to correction. Abstract convection that help in selecting suitable candidates for geological CO2 sequestration sites. CO2-sequestration in deep geological formations has been suggested as an option to reduce greenhouse gas emissions. Saline aquifers are one of the most promising options for carbon dioxide storage. It has been investigated that if the layer Introduction of aquifer is deep enough, at depths more than 800 meters, Carbon dioxide sequestration is the capture and safe storage of dissolution of CO2 into brine causes density of the mixture to carbon dioxide that would otherwise emit to the atmosphere. increase. If the corresponding Rayleigh number of the porous Sequestration refers to any storage scheme that can keep CO2 medium is enough to initiate convection currents, the rate of out of the atmosphere [1]. In general, storage sites of carbon dissolution will increase. Early time dissolution of CO2 in brine dioxide can be divided into two categories, geological sites and is mainly dominated by molecular diffusion while the late time marine sites. Carbon dioxide sequestration in deep geological dissolution is predominantly governed by convective mixing formations has been suggested as a way of reducing greenhouse mechanism. In this paper, linear stability analysis of density- gas emissions. Geologic sequestration of CO2 is the capture of driven miscible flow for carbon dioxide sequestration in deep CO2 from major sources, transporting it usually by pipeline, and inclined saline aquifers is presented. The effect of inclination injecting it into underground formations such as oil and gas and its influence on the pattern of convection cells has been reservoirs, saline aquifers, and unmineable coal seams for investigated and the results are compared with the horizontal geologically significant period of time [2, 3]. Unlike coal bed layer. The current analysis provides approximations for initial methane reserves and oil reservoirs, sequestration of CO2 in wavelength of the convective instabilities and onset of deep saline aquifers does not produce value-added by-products, but it has other advantages. While there are uncertainties, the 1 world’s total capacity to store CO2 deep underground is large [4]. significant amount of CO2 dissolves in brine, the risk of leakage They are generally unused and are available in many parts of can be ignored. Therefore predicting the onset time of the world [5]. It has been estimated that deep saline formations convection is very important in the fate of CO2 sequestration. in the United States could potentially store up to 500 billion The solution trapping occurs by diffusion and convective tones of CO2. Most existing large CO2 point sources are within mixing. In the early time, diffusion is the only mechanism of easy access to a saline formation injection point, and therefore dissolution. When CO2 diffuses into brine, density of brine sequestration in saline formations is compatible with a strategy increases slightly. This happen because the density of brine of transforming large portions of the existing energy and saturated with carbon dioxide is approximately one percent industrial assets to near-zero carbon emissions via low-cost higher than original formation brine [11, 16, 17]. At the early time carbon sequestration retrofits [3]. However, it is important to of CO2 dissolution, diffusion causes a thin layer of brine to be investigate the behavior of CO2 injected into aquifers for saturated with CO2. The boundary layer of concentration in the effective and safe use of storage. Geological storage of CO2 as a aquifer column increases by time and at a specific time greenhouse gas mitigation option was proposed in the 1970s [6], convective currents start to happen (due to adverse density but little research was done until the early 1990s, when the idea gradient) which will enhance the dissolution rate of CO2 into gained credibility through the work of individual research brine. groups [7-10]. In this work, a prediction of the beginning of convection along When CO2 is injected into the formation above its critical with the pattern of fluid movement is made by using linear temperature and pressure the density of supercritical carbon stability analysis. The analysis is based on the growth of dioxide is usually less than brine. This density difference causes perturbations in the system. Previous analyses were for a CO2 to migrate upwards to the top of the formation and under horizontal layer whereas in this case, effect of inclination of the an impermeable cap rock. Carbon dioxide then spreads laterally formation on the instability time and pattern of convection cells under the cap rock as a separate phase. During migration, a is investigated and results are compared with the horizontal fraction of the injected carbon dioxide will dissolve into case. formation brine and some continue to migrate laterally. Diffusion is a very slow process and because molecular diffusion coefficient is very small, it will take a long time for Previous Work CO2 to dissolve into brine. Analytical study of the convective mixing in porous media was As CO2 free phase migrates through the formation, it can be first analyzed by Horton and Rogers [18] and Lapwood [19]. The trapped by capillary pressure, which is referred to as residual problem was for a horizontal fluid layer with constant boundary CO2 trapping. Residual trapping may cause a significant amount temperatures under a linear and steady vertical temperature of CO2 to be trapped in the formation [11]. If capillary forces are 2 strong, CO2 even cannot migrate to top of the formation [12]. For gradient. They found the critical Rayleigh number to be 4π , dipping formations, this can cause lateral spread of CO2 to be above which perturbation grew and convection currents started. slower, since the effect of gravity segregation is less than In the case of CO2 diffusing into brine, the condition is like the horizontal layers during upwards migration. When residual heat transfer problem in which equation of heat is replaced by trapping happens, much of the trapped CO2 then will dissolve mass transfer equation. In contrast to the Horton and Rogers and into the formation over time [13]. Lapwood problem where the base state profile was steady and In the interface of free-gas phase and formation brine, CO2 will linear, the base problem considered here is a nonlinear and dissolve in water by molecular diffusion. The water in contact transient. with CO2 will be saturated with CO2 and a concentration gradient of CO2 in brine would establish. This process is very There are some approaches to the problem which all have some slow and may take hundreds of years for CO2 to be completely drawbacks. In early attempts to account the nonlinear dissolved in brine. Lindeberg [14] illustrated that 143 kg of CO2 temperature profile, quasi-steady state approximation (QSSA) will dissolve in an infinitely 1 m2 of water saturated reservoir was used [20]. This assumption is valid if the propagation rate of column under a CO2 cap after 1000 years. the nonlinear profile is much smaller than the growth rate of In the very long-term, mineralogical trapping would occur. In perturbations. This approximation is not valid at the early time this case, a sequence of geochemical interactions between when the boundary layer and the diffusing profile change fast. dissolved CO2 and rock minerals, if appropriate minerals are Another method is to use the amplification theory [21]. This encountered, will form ionic species [15] (ionic trapping). The method gives the critical time of instability and the wavelengths speed of the reaction depends on the formation mineralogy and of the start of instability. The wavelength that produces the usually is in the order of hundreds of years. Decomposition of fastest growth is the wavelength of instability. In amplification these minerals over thousands of years or more will precipitate theory, one important factor is the choice of initial condition to new carbonate minerals, which will trap CO2 in its most secure solve the equation. Another issue is the criteria for determining storage state (mineral trapping). Carbonate minerals are stable onset of instability, which is somehow arbitrary due to the and this mechanism is the most permanent form of storage [9], imprecise definition of critical time. Foster [21, 22] and Jhavery but it is very slow and will happen at the end of all the trapping and Homsy [23] used this method for a thermal-diffusing schemes. boundary layer in an incompressible fluid. Caltagirone [24] and Between the short-term (physical trapping) and long-term Kaviany [25] applied this method for a thermal boundary layer in (mineral trapping) processes, there is also a middle-term period, a porous medium for a step change in temperature. Ennis-King solution trapping, which is a very important part of the storage. and Paterson [13] applied this method in the CO2 sequestration The importance of the solution trapping is the reduction in problem for anisotropic porous media. possibility of leakage of CO2 from storage locations. As long as Another method is the global stability (energy) method CO2 remains as a separate phase in the formation under the cap investigated by several authors. This method does not depend rock, there is possibility of leakage of CO2 to higher on the initial condition and gives criteria of stability. In this permeability zones due to lateral movement. However, when method, the onset time of instability is less than measured 2 laboratory data [26] and so it gives a lower bound for the onset of Stability Analysis instability. It also does not give any information about the wave The reference or base state of the concentration profile is given number of the disturbances. by the diffusive mass transfer equation. For pure diffusion All the three methods have been applied to the convective with v = 0 , concentration distribution satisfies: mixing problem in porous media. The aim of this paper is to investigate the critical time of instability for the convective mixing occurred in the geological CO2 storage process into ∂C D ∂ 2 C D = 2 .................................................................................. (5) aquifers due to increase of brine density upon dissolution of ∂t D ∂z D CO2. In this paper, linear stability analysis based on the amplification theory is used for an inclined homogeneous porous layer with the appropriate boundary and initial where t D = Dt / H 2 is the dimensionless time, z D = z / H , and conditions. Results are compared with the horizontal porous cD = C / CS is the dimensionless concentration. layer and show correspondence with thermally induced density- The initial and boundary conditions are: driven flow in porous media. C D = 0 at t D = 0 , 0 ≤ z D ≤ 1 ……….…………........ (6) CD = 1 at z D = 0, t D ≥ 0 …….……….…………….. (7) Governing Equations C D = 0 at z D = 1, t D ≥ 0 ………………………..…... (8) Consider a simplified-geometry saline aquifer, a cross section of The bottom boundary is supposed to be at zero concentration. which is shown in Figure 1 with the appropriate coordinate However, no-flux (∂C D / ∂z D = 0 ) boundary condition can be system. The aquifer is a homogeneous porous medium confined also considered with the same analysis. Solution of Equation 5 between two inclined, infinite parallel impermeable planes. The is: thickness of the layer is H and the fluid is initially quiescent. Diffusion of CO2 from the top of the formation to the quiescent N 1 fluid increases density of the CO2 saturated brine. The fluid is C D = 1 − z D − 2∑ sin( nπz D ) exp( − n 2 π 2 t D ) ........................ (9) assumed to be incompressible: n =1 nπ ∇ . v = 0 …………………………………………………….. (1) We assume that fluid motion is governed by Darcy’s law and For the stability analysis, velocities and concentration are the transport of CO2 into brine is governed by convection- subjected to small perturbations, u ′, v ′, w′, and C ′ , where u, diffusion equation. Due to the choice of the coordinate axis, v, and w are velocities in the x, y, and z directions, respectively. gravity has two terms in the x and z directions leading to the When perturbed velocities and concentration are substituted in following system of equations: equations 2 and 3, by neglecting second order non-linear terms, k ∂p a system of linear equations will be formed as: v=− − ρg cos θ .∇z − ρg sin θ .∇x ……………..…... (2) µ ∂x ∂C kgH∆ρ cosθ 2 ∂ 2CD ′ φ = Dφ∇ 2 C − v.∇C …………………..………………... (3) ∇ 2 w′ = D ′ ∇ H C D − tan θ ...................... (10) ∂t µφD ∂xD ∂z D where v is the Darcy velocity vector, D is the molecular diffusion coefficient of CO2 into brine at the conditions in the porous medium, φ is the effective porosity, µ is the fluid ′ ∂C D ∂C D viscosity, p is the fluid pressure, k is the permeability tensor, g = ∇ 2 C D − w′ ′ D ............................................................. (11) ∂t D ∂z D is the gravity acceleration and C is the CO2 concentration in brine. The model is assumed to be a homogeneous and isotropic porous medium, therefore the permeability tensor k can be 2 ∂2 ∂2 where ∇ H = + , H is the thickness of the aquifer, replaced by a single-term permeability, k. The coupling between ∂x D ∂y D equations occurs through the effect of CO2 concentration on density in the form of: ∆ρ is the density difference between saturated and fresh brine, ρ = ρ0 (1 + βC ) …………………………………...…............. (4) θ is the dip of the aquifer, and w′D = w′H is the where ρ 0 is the density of unsaturated brine and ρ is the φ .D dimensionless perturbed velocity. Rayleigh number is defined density of the solution being saturated with brine and β is the as: density increase coefficient. The relationship between density and concentration is linear since the concentration of CO2 has kgH∆ρ cosθ very little effect on the partial molar volume [27]. Boussinesq Ra = ...................................................................... (12) approximation is assumed valid where the dependence of µφD density on concentration is only retained in the buoyancy term in equations 2 and 3. Effect of geothermal gradient is neglected By setting: for two reasons; first, it has little effect on the results and second it contributes to the destabilization. w′ wD (t D , z D ) ∗ D ′= ∗ [ ] exp i (a x x D + a y y D ) ............................. (13) C D C D (t D , z D ) 3 Figure 3 relates onset of instability with Rayleigh number, ( ) 1 with a = a y + a x 2 2 2 , and using Galerkin technique for the which is similar to the horizontal case [28, 31]. Near the critical amplitude functions of velocity and concentration perturbations value of the Rayleigh number, 4π 2 , the time of instability based on the boundary conditions as: increases dramatically. It seems that based on our criterion of wD = ∑ Al (t D ) exp[ i (lπ z D )] …………………………….. (14) ∗ N the onset of instability and linearization of equations, the l =1 estimated time of instability is an upper bound for it. Results in Figure 3 reveal that dimensionless instability time vary as the C = ∑ Bl (t D ) exp[ i (lπ z D )] …………….………………. (15) N ∗ D minus 2 power of the Rayleigh number at high Ra. Since the l =1 dimensionless critical time is inversely proportional to the the following equations for the time amplitude of the perturbed porous layer thickness H, to the power of 2, this suggests that, velocity and concentration are given: the onset of convection is independent of the porous layer ( ) ( a 2 + (lπ ) 2 Al = a 2 − lπ .a x . tan θ Ra Bl ………………….. (16) ) thickness. Another remark regarding Figure 3 is that the minimum Rayleigh number approaches 4π2, corresponding to ( ) N dBl = − a 2 + (lπ ) 2 Bl − 2∑ Al Elm …………………....…. (17) the critical Rayleigh number given by Horton and Rogers and dt D m =1 Lapwood for a porous layer with a steady linear temperature where Elm is an N by N matrix and for m=l: gradient. In Figure 4 wave number, a, is plotted versus Rayleigh Elm = − 1 2 [ ( 1 − exp − 4l 2π 2 t D …………………………..…. (18))] number. The wave number is equal to the wave number in the y direction. This figure reveals that the wave number is proportional to the Rayleigh number for Ra larger than a few and for m≠l: hundreds. This proportionality implies that the size of 1 Elm = − exp − (l − m) 2 π 2t D − exp − (l + m) 2 π 2t D .……….. (19) 2 [ ( ) ( )] convection cells is independent of the porous layer thickness. The extrapolation of Figure 4 can be used to compare the wave Equations 16 and 17 can be solved numerically for Al and Bl number with the experimental values reported by Green and and the following definition, as in the work of Foster [22] and Foster [32]. The value of wave number in their experiment (0.18 Caltagirone [24], is used for the amplitude of the velocity cm) is greater than the relationship in Figure 4 (which gives a disturbance: value of about 0.07 cm), for the Rayleigh number of about 105. 1 They mentioned that this value is an overestimate due to the ⎛1 * ⎞2 combination of streamers. ⎜ ∫ wD ( z D , t D )dz D ⎟ Figure 5 shows the onset of instability for the no-flux w (t D ) = ⎜ 01 ⎜ ⎟ …………………………… (20) ⎟ concentration boundary at z D = 1. Figure 6 shows the growth ⎜ ∫ wD ( z D ,0)dz D ⎟ ⎜ ⎟ * ⎝ 0 ⎠ of amplitude factor with time. At high Rayleigh numbers, All the coefficients in the Galerkin expansion of the amplitude factor increases sharply, at much shorter time. concentration disturbance were set to unity for the initial The Rayleigh number in a sloping layer contains a cosθ term, condition, since white noise gives the fastest growth rate [28]. which decreases the driving force required for convection compared to the horizontal layer. If cos θ <<1, the Rayleigh number can be less than the critical value and instability will not Results and Discussion be induced. As the inclination increases, for fixed values of Results of the stability analysis are valid when the thickness of fluid and medium parameters, the onset of instability increases. the diffusive boundary layer is much smaller than the thickness of the aquifer layer, which for typical parameters of storage sites is applicable [13]. Conclusion In equations 16 and 17, wave numbers a and ax are unknown. In this study, stability of a fluid in an inclined enclosed domain They are found by seeking which value gives the fastest growth was studied. It was found that the effective driving force in an of amplitudes in time. Based on the definition of a, 0 ≤ a x ≤ a. inclined layer is less than the horizontal case by cos θ factor, Therefore, for each value of a, ax is varied from zero to a and which retards the onset of instability. The pattern of convection the effect of ax is analyzed on the behavior of the velocity currents was different from the horizontal case. While in the amplitude. horizontal case, convection evolves with specific wave numbers In the analysis, the onset of instability was chosen based on the in the horizontal plane, in the inclined layer the wave number in magnitude of velocity amplitude to become one. It was found the direction of the slope is zero. that if a x = 0 , velocity amplitude increases faster than any other values of ax. As ax increases from 0 to a, the onset of Acknowledgment instability increases, but the interesting feature is that for a The authors would like to thank the Department of Chemical specific Rayleigh number greater than the critical value, no and Petroleum Engineering at the University of Calgary. matter what is the value of ax, the onset of instability occurs at a specific wave number, a. This means that the wave number, a, is independent of ax. However, we know that ax is related to a ( ) 1 based on a = a y + a x 2 2 2 . Again, we find that ax should be zero. These two reasons impose the wave number in the x direction to be zero and in this direction instead of convection cells, the instability manifest itself with convection coils. This is in accordance with experiments done for temperature instability in porous media [29, 30]. Pattern of convection rolls is shown in Figure 2 based on Bories’ experimental work. 4 3. U.S. DOE, Carbon Sequestration R&D; NOMENCLATURE http://www.fe.doe.gov/programs/sequestration/ index.html, 2004. A: Time component of the amplitude function for 4. Gale, J., Geological storage of co2: What do we perturbed velocity know, where are the gaps and what more needs to a: Dimensionless wave number be done; Energy, 29, 1329-1338, 2004. B: Time component of the amplitude function for 5. Seo, J.G., Experimental and simulation studies of perturbed concentration sequestration of supercritical carbon dioxide in C: Concentration depleted gas reservoirs; Ph.D. Dissertation, Texas D: Diffusivity coefficient A&M University, 2004. d: Derivative 6. Marchetti, C., On geoengineering and the CO2 E: Square matrix problem; Climate Change, 1, 59-68, 1977. g: Gravitational acceleration 7. Koide, H.G., Tazaki, Y., Noguchi, Y., Nakayama, H: Aquifer thickness S., Iijima, M., Ito, K., and Shindu, Y., Subterranean containment and long-term storage i: Imaginary number, − 1 of carbon dioxide in unused aquifers and in k: Permeability tensor depleted natural gas reservoirs; Energy Conversion k: Permeability and management, 33(5-8), 619-626, 1992. N: Constant number in the summations 8. Van der Meer, L.G.H., Investigation regarding the p: Pressure storage of carbon dioxide in aquifers in the Ra: Rayleigh number Netherlands; Energy Conversion and management, t: Time 33(5-8), 611-618, 1992. u: Velocity in the x direction 9. Gunter, W.D., Perkins, E.H., and McCann, T.J., v: Velocity vector Aquifer disposal of CO2-rich gases: research v: Velocity in the y direction design for added capacity; Energy Conversion and w: Velocity in the z direction management, 34(9-11), pp. 941-948, 1993. w: Velocity amplitude factor 10. Bachu, S., Gunter, W.D., and Perkins, E.H., x: Coordinate direction parallel to the flow Aquifer disposal of CO2: hydrodynamic and y: Coordinate direction mineral trapping; Energy Conversion and z: Coordinate direction perpendicular to the flow management, 35(4), 269-279, 1994. 11. Kumar, A., Noh, M., Pope, G.A., Sepehrnoori, K., φ: Porosity Bryant, S., and Lake, L.W., Reservoir simulation β: Density increase coefficient of CO2 storage in deep saline aquifers; SPE 89343, μ: Viscosity 2004. ρ: Density 12. Doughty, C., and Pruess, K., Modelling supercritical carbon dioxide injection in θ: Dip angle of the aquifer heterogeneous porous media; Vadose Zone Δρ : Density difference Journal, 3(3), pp. 837-847, 2004. ∇H : 2 2D Laplacian 13. Ennis-King, J.K. and L. Paterson, Role of convective mixing in the long-term storage of carbon dioxide in deep saline formations; SPE Superscripts: paper 84344, ATCE 5-8 October, Denver, Colorado, 2003. ‘ Perturbed quantities 14. Lindeberg, E. and Wessel-Berg, D., Vertical * Amplitude of the perturbations convection in an aquifer column under a gas cap of CO2; Energy Conversion and management, 38, Subscripts: S229-S234, 1997. 0 Base state 15. IPCC, Carbon Dioxide Capture and Storage, D Dimensionless Special Report, Intergovernmental Panel on l, m Summation index Climate Change, 2005. s Equilibrium state 16. Bachu, S. and Adams, J.J., Sequestration of CO2 in x, y x and y directions geological media in response to climate change: Capacity of deep saline aquifer to sequester CO2 in solution; Energy Conversion and management, 44(20), 3151-3175, 2003. REFERENCES 17. Enick, R.M. and Klara, S.M., CO2 solubility in 1. Park, A.A., Carbon dioxide sequestration: water and brine under reservoir conditions; chemical and physical activation of aqueous Chemical Engineering Communications, 90, 23- carbonation of Mg-bearing minerals and pH swing 33, 1990. process; M.A.Sc. Dissertation, Ohio State 18. Horton, C.W., and Rogers, F.T., Convection University, 2005. currents in porous media; J. Appl. Phys., 20, pp. 2. Bachu, S., Sequestration of CO2 in geological 367-369, 1945. media in response to climate change: road map for 19. Lapwood. E.R., Convection of a fluid in a porous site selection using the transform of the geological medium; Proc. Cambridge Phil. Soc., 44, pp. 508– space into the co2 phase space; Energy Conversion 521, 1948. and management, 43(5-8), 87-102, 2002. 5 20. Lick, W., The instability of a fluid layer with time- dependent heating; J. Fluid Mech., 21, 565-576, 1965. 21. Foster, T.D., Stability of a homogeneous fluid cooled uniformly from above; Phys. Fluids, 8, 1249-1257, 1965. 22. Foster, T.D., Effect of boundary conditions on the onset of convection; Phys. Fluids, 11, 1257-1262, 1968. 23. Jhavery, B.S., and Homsy, G.M., The onset of convection in fluid layers heated rapidly in a time- dependent manner; J. Fluid Mech., 114, pp. 251- 260, 1982. 24. Caltagirone, J.P., Stability of a saturated porous layer subject to a sudden rise in surface temperature: comparison between the linear and energy methods; Q. J. Mech. Appl. Maths, 33, 47- 58, 1980. 25. Kaviany, M., Onset of thermal convection in a saturated porous medium: experiment and analysis; Intl. J. Heat Mass Transfer, 27, 2101-2110, 1984. 26. Wankat, P.C., and Homsy, G.M., Lower bounds for the onset of instability in heated layers; Phys. Fluids, 20, pp. 1200-1201, 1977. 27. Barbero, J.A., Helper, L.G., McCurdy, K.G., and Tremaine, P.R., Thermodynamics of aqueous carbon dioxide and sulfur dioxide: heat capacities, volume, and the temperature dependence of ionization; Canadian Journal of Chemistry, 61(11), pp. 2509-2519, 1983. 28. Hassanzadeh, H., Pooladi-Darvish, M., and Keith, D.W., Stability of a fluid in a horizontal saturated porous layer: effect of non-linear concentration profile, initial, and boundary conditions; Transport in Porous Media, 65, pp. 193-211, 2006. 29. Hart, J.E., Stability of the flow in a differentially heated inclined box; J. Fluid Mech., 47, pp. 547- 576, 1971b 30. Bories, S.A., and Combarnous, M.A., Natural convection in a sloping porous layer; J. Fluid Mech., 57, pp. 63-79, 1973. 31. Riaz, A., Hesse, M., Tchelepi, H.A., and Orr, J., Onset of convection in a gravitationally unstable diffusive boundary layer in porous media; J. Fluid Mech., 548, pp. 87-111, 2006. 32. Green, L.L., and Foster, T., Secondary convection in a Hele Shaw cell; J. Fluid Mech., 71, pp. 675- 687, 1975. 6 Figure 1. A schematic of the problem’s domain. Figure 2. Convection coils in a sloped porous layer [30]. 7 0 10 -1 10 tD -2 tD = 130 Ra -2 10 -3 10 -4 10 1 2 3 10 10 10 Ra Figure 3. Instability time versus Rayleigh number for C ( z D = 1) = 0 . 60 50 40 a = 0.052 Ra 30 a 20 10 0 0 100 200 300 400 500 600 700 800 900 1000 Ra Figure 4. Wave number versus Rayleigh number. 8 -1 10 -2 10 tD -3 tD = 142 Ra -2 10 -4 10 1 2 3 10 10 10 Ra Figure 5. Instability time versus Rayleigh number for ∂C / ∂z D ( z D = 1) = 0 . 1.2 1.1 Amplitude factor Ra=30 Ra=40 Ra=100 1 0.9 0.8 0.7 0 200 400 600 800 1000 1200 1400 1600 1800 2000 4 tD *10 Figure 6. Amplitude factor versus dimensionless time. 9

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