Rayleigh fractionation in high-Rayleigh-number solutal convection

in porous media

Baole Wen, Marc A. Hesse

aInstitute of Computational Engineering and Sciences, The University of Texas at Austin, Austin, TX78712 USA

bDepartment of Geological Sciences, Jackson School of Geosciences, The University of Texas at Austin,Austin, TX 78712 USA

Abstract

We study the fractionation of two components between a well-mixed gas and a saturatedconvecting porous layer. Motivated by geological carbon dioxide (CO2) storage we assumethat convection is driven only by the dissolved concentration of the first component, while thesecond acts as a tracer with increased diffusivity. Direct numerical simulations for convectionat high Rayleigh numbers reveal that the partitioning of the components, in general, does notfollow a Rayleigh fractionation trend, as commonly assumed. Initially, increases in tracerdiffusivity also increase its flux, because the diffusive boundary layer penetrates deeperinto the flow. However, for D2 ≥ 10D1, where D1 and D2 are, respectively, the diffusioncoefficients of CO2 and the tracer in water, the transverse leakage of tracer between up- anddown-welling plumes reduces the tracer flux. Rayleigh fractionation between componentsis only realized in the limit of two gases with very large differences in solubility and initialconcentration in the gas.

Keywords: Porous medium convection; multi-component convection; fractionation;Rayleigh fractionation

1. Introduction

Convection in porous media controls many mass and heat transport processes in natureand industry [1] and Rayleigh-Darcy convection is also a classic example of spatiotemporalpattern formation [2, 3]. This subject has received renewed interest due to its potentialimpact on geological carbon dioxide (CO2) storage. The injection of supercritical CO2 intodeep saline aquifers for long-term storage is the only technology that allows large reductionsof CO2 emissions from fossil fuel-based electricity generation [4–8]. Dissolution of CO2 intothe brine eliminates the risk of upward leakage [9–11], because it increases the density ofthe brine and forms a stable stratification [12].

Email addresses: wenbaole@gmail.com or baole@ices.utexas.edu (Baole Wen),mhesse@jsg.utexas.edu (Marc A. Hesse)

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Once the diffusive boundary layer of dissolved CO2 in the brine has grown thick enoughit becomes unstable and convective mass transfer allows a constant dissolution rate [13–15].The time scale for the onset in typical storage formations is at most a few centuries [14, 16–18], so that convective mass transport determines the rate of CO2 dissolution. Recent workhas therefore focused on determining the convective dissolution rate in numerical simulations[19–29] and laboratory experiments [15, 30–32].

However, most of these studies consider convection in homogeneous porous media, whilegeological formations exhibit extreme heterogeneity at all scales [33, 34]. It is thereforeimportant to complement numerical and experimental work with estimates of convectivedissolution rates in real media that have been inferred from field observations. All suchestimates are based on increases in the abundance of Helium (He) relative to CO2 in theresidual gas, as convection strips the more soluble CO2 [35–40]. These studies interpret theobserved changes in the CO2/He ratio in terms of a zero-dimensional Rayleigh fractionationmodel [41–44].

This interpretation assumes that the fractionation depends only on the solubility of thecomponents, but not on their diffusion coefficients. In the absence of convection, however,mass transfer is controlled by diffusion and this assumption must break down. In a stronglyconvecting fluid, in contrast, advective mass transfer is dominant and differences in diffu-sivity may become negligible. One might therefore expect Rayleigh fractionation betweensolutes in the limit of high-Rayleigh-number convection. Here, we directly test this hy-pothesis using highly resolved direct numerical simulations (DNS) of solutal convection ina porous medium. However, unlike the double-diffusive (or combined thermal and solutal)convection [1], the convection considered here is only driven by the buoyancy force due tothe density change induced by the first solute (CO2). Despite the simplicity of this physicalsystem the emergence of complex behavior is observed.

The manuscript is structured as follows. First, we obtain an expression for the evolutionof the residual gas composition as a function of the convective fluxes of the two componentsin the liquid. These fluxes are then obtained from DNS of high-Rayleigh-number solutalconvection in a porous medium. Finally, we determine the conditions under which theresidual gas composition experiences Rayleigh fractionation.

2. Problem formation and computational methodology

In a binary system, the composition of the gas is characterized by the ratio of molesbetween CO2 and the tracer (i.e. He) in the gas field, r = n1,g/n2,g, where the subscripts ‘1’and ‘2’ denote the solutes CO2 and He, respectively, and ‘g’ the gas phase. This gas is incontact with a convecting fluid that equilibrates instantaneously at the gas-water interfaceand constantly removes the dissolved components and carries new unsaturated water to theinterface (Fig. 1). The change of the i-th component (i = 1, 2 here) in the gas is thereforegiven by

dni,gdt

= −FiD∗1CisH

A, (1)

2

Figure 1: Schematics showing the Rayleigh fractionation process in simple geometries. The assumed physicalmechanism leading to Rayleigh fractionation is convection (advection), because it continuously brings in newbrine that is saturated at the gas-water interface and subsequently removed.

where Fi is the corresponding dimensionless flux defined later in Eq. (7), D∗1 is the dimen-

sional diffusivity for the first solute, Cis is the saturated concentration of the i-th componentin the water, H is the thickness of the water layer and A is the gas-water contact area. Weassume an open system in contact with a liquid reservoir at constant pressure. This impliesthat the pressure in the gas remains constant as dissolution proceeds, but the gas volume de-clines. Further we assume that the gas is ideal and that partitioning is described by Henry’slaw [45]. High Rayleigh-number convection is quasi-stationary so that the convective fluxFi is constant. Following [43] and [46] the fraction of the initial CO2 that has dissolved intothe water is given by

F ≡ 1− n1,g/n01,g = 1− (r/r0)α

α−1 , (2)

where the superscript ‘0’ denotes the initial state. The evolution of the gas composition isgoverned by the fractionation factor,

α =F1K1F2K2

, (3)

where Ki is Henry’s law solubility constant of the i-th component (see the detailed derivationin the Appendix section). In the limit of Rayleigh fractionation the fluxes for different solutesare assumed to be identical, F1 ≡ F2, so the Eq. (3) becomes α = K1/K2.

To determine these convective fluxes we study the Boussinesq, Darcy flow in a dimen-sionless 2D porous layer with horizontal and vertical coordinates x and z, respectively, asshown in Fig. 1. We assume the density-driven flow u = (u,w) through the homogeneousand isotropic porous media is incompressible [1],

u = −∇p− Ra(C1 + βC2)ez, (4)∇ · u = 0, (5)

∂Ci∂t

+ u · ∇Ci = Di∇2Ci, i = 1, 2, (6)

where p is the pressure field, ez is a unit vector in the z direction, Ci and Di are, respectively,the concentration and diffusivity of the i-th solute, β is the weighting factor of buoyancy

3

force for C2, and the Rayleigh number Ra = HKg4ρ1/(µϕD∗1) where K is the mediumpermeability, g is the acceleration of gravity, 4ρ1 is the density difference between the freshwater and the saturated water for the first solute, µ is the dynamic viscosity of the fluid,and ϕ is the porosity. Since D∗1 is used for normalization of time, D1 ≡ D∗1/D∗1 = 1,and D2 ≡ D∗2/D∗1 is the ratio of diffusivities between the two solutes. Here, the secondsolute C2 is a passive tracer which does not change the density of the brine, so β = 0.For boundary conditions, the lower boundary is impenetrable to the fluid and solutes, theupper boundary is saturated (i.e., Ci = 1) and impenetrable to the fluid, and all fields areL-periodic in x. One of the key quantities of interest in solutal convection is the dissolutionflux F representing the rate at which the solutes dissolve from the upper boundary of thelayer, defined as

Fi(t) =DiL

∫ L0

∂Ci∂z

∣∣∣∣z=1

dx for i = 1, 2, (7)

where L is the aspect ratio of the domain.The equations (4)–(6) are solved numerically using a Fourier–Chebyshev-tau pseudospec-

tral algorithm [47]. For temporal discretization, a third-order-accurate semi-implicit Runge–Kutta scheme [48] is utilized for computations of the first three steps, and then a four-stepfourth-order-accurate semi-implicit Adams–Bashforth/Backward–Differentiation scheme [49]is used for computation of the remaining steps, so generally it is fourth-order-accurate intime. We performed computations for a discrete set of Rayleigh number and ratio of dif-fusivities from Ra = 50 to Ra = 5 × 104 and D2 = 1.25 to D2 = 100 in the 2D domainwith aspect ratio L = 105/Ra. 8192 Fourier modes were utilized in the lateral discretizationand as Ra was increased, the number of Chebyshev modes used in the vertical discretizationwas increased from 33 to 513. For each case, an error function was utilized as the initialcondition for the diffusive concentration field

Ci = 1 + erf

(−(1− z)2√Dit

), for 0 ≤ z < 1 (8)

at time t = 25/Ra2 or tad = t × Ra2 = 25 in advection-diffusion scaling [50], and a smallrandom perturbation was added as a noise within the upper diffusive boundary layer toinduce the convective instability. The solver has been verified in many previous investigations[28, 51–54].

3. Results

Figure 2 shows the variation of the dissolution flux with time for D2 = 3 with increasingRa. Initially, the diffusion layer is far from the lower wall, the evolution of the purely diffusiveconcentration profile is universal (independent of Ra) in the advection-diffusion framework[50] and follows Eq. (8) so that Fi ∼

√Di/(πt). The top boundary layer becomes unstable

when it is thick enough, thereby inducing convective fingers and making the flow deviatefrom the pure diffusion state [14, 16, 18, 20, 55]. As the nascent, independent-growing

4

102

103

104

105

Ra2·t

10-2

10-1

F/R

a

Diffusion

dominant

Flux growth

and merging Constant flux

Fi ∼√

Diπt

Ra ↑

Figure 2: Variation of the dissolution flux with time at D2 = 3 for Ra = 100, 200, 500, 1000, 2000, 5000, 104,

2 × 104 and 5 × 104. Both the flux and time are rescaled following the advection-diffusion scaling to moreevidently compare different regimes for different Ra. The solid lines are for C1 and dashed lines for C2. Inthe diffusion dominant regime, the flux for the solutes decays as Fi ∼

√Di/(πt); for 2× 104 . tad . 16Ra,

the flow transitions to the constant-flux regime.

fingers penetrate the front of the diffusion layer, the plumes contact with more fresh waterbelow the layer, leading to an increase of flux. Subsequently, a secondary stability leads tolateral motions of the growing fingers and the flux growth regime ends when the neighboringfingers merge from the root. After a series of plume mergers, which cause coarsening of theconvective pattern, the flow transitions to a quasi-steady, constant-flux convective state withF ∼ Ra, consistent with other high-Ra investigations of solutal convection [20, 21, 50] andthermal convection [22, 51, 54, 56, 57] in porous media. At the late time when the water isapproximately 27% saturated, the convection shuts down and the decay of the flux followsa simple box model [23, 25]. In this study, we only focus on the dynamics quasi-steadyconstant-flux regime.

As shown in Fig. 2, although F2 generally follows the same trend with F1 at D2 = 3, theyare not equivalent regardless of the magnitude of Ra. For Ra . 100, diffusion dominates thedynamics, so F̃ = F2/F1 ∼

√D2/D1 before the diffusion front hits the bottom boundary.

Certainly, Rayleigh fractionation does not apply to the diffusion state. Interestingly, evenas Ra →∞, these two dissolution fluxes are still not equivalent, but the ratio F̃ convergesto a constant value in the constant-flux regime at sufficiently large Ra. Figure 3 showssimulated concentration contours of C2 for different D2 at Ra = 20000. In this case, theconcentration contours of C2 basically retain the finger features for D2 < 5. However, the

5

D2 = 1

0 1

D2 = 3

D2 = 10

D2 = 50

D2 = 100

Figure 3: Concentration contours of C2 at tad = 8Ra for different diffusivities at Ra = 20000. For D2 > 1,the downwelling plumes become much whiter, implying that more saturated solute is advected downward;moreover, as D2 is increased, the lateral concentration field is smoothed by diffusion and becomes nearlyuniform for D2 & 50.

increasing diffusivity gradually smooths the long and thin fingers and at sufficiently largeD2, makes the concentration field almost uniformly distributed in x and just diffuse with anew scaling F2 ∼ t−γ with 0 < γ < 1/2 (see D2 = 100 in Fig. 4). As also shown in Fig. 4, forfixed large Ra and at small D2, F2 generally follows the same variation of F1. Nevertheless,the increasing D2 will postpone the occurrence of the constant-flux regime (see D2 = 10),implying that a larger D2 requires corresponding larger Ra’s to obtain the constant-fluxregime before the convection shuts down (see Fig. 5a).

As discussed above, for each fixed D2, the finger features and constant-flux regime canbe retained at sufficiently large Ra. Figure 5(b) shows the ratio of fluxes between tracerand CO2 in the constant-flux regime as a function of D2. At D2 = 1, the two solutes areequivalently transported so that F̃ ≡ 1; interestingly, for D2 ≤ 2.5, the increase of D2enhances the convective mixing of the solute C2, e.g. the flux F2 is nearly 12% increased

6

103

104

105

Ra2· t

10-2

10-1

F/Ra

D2 = 1

3

10

50

100

104

105

10-2

D2 = 50

100

Figure 4: Variation of the dissolution flux with time for C2 at Ra = 20000 for different diffusivities. Thedashed lines are for diffusion state and the inset shows a magnification of flux variation for D2 = 50 and100. At large D2, C2 becomes horizontally averaged and just diffuses with a new effective diffusivity (seethe dashed-dot line for D2 = 100).

at D2 = 2.5; for D2 > 2.5, however, F̃ decreases as D2 is increased, and for D2 > 10,F̃ < 1, implying that the large diffusivity reduces the mixing efficiency of C2. Since the flowfield is only set by C1, the increase of diffusivity thickens the top diffusion boundary layer(see Fig. 3), so that more saturated brine is advected downward by fingers from the upperlayer. Therefore, moderate increase of the diffusivity could increase the dissolution rate ofthe tracer. Nevertheless, due to the conservation of mass, relatively fresh brine rises to thetop through the upwelling flows. As D2 is increased, the strong lateral diffusion smooths thehigh concentrations to the sides, leads to a leakage from the downwellings into the upwellings(see Fig. 5c), and thereby significantly decreases the dissolution rate.

At large Rayleigh number, the solutal convection in the porous layer appears in the formof narrow fingers with the wavelength Lm shrinking as a power-law scale of Ra; namely,the mass transport is generally performed through these downwelling and upwelling plumes.To a certain extent, this phenomenon is analogous to a Taylor (or Taylor–Aris) dispersionproblem [58, 59], where spread of the solute in a 2D channel is enhanced by the axial flow.In the CO2-tracer ‘dispersion’ problem, the channel has a height 1 and width Lm. Awayfrom the top and bottom boundary layers, the horizontal velocity u is negligible and thevertical (axial) velocity can be approximated using w = W0 cos(kx), where W0 = aRa withthe constant pre-factor a and k = 2π/Lm is the fundamental wavenumber. As the traceris advected downward, it also diffuses to both sides and the amplitude of the concentration

7

100

101

102

103

104

Ra

100

101

102

D2

0.6

0.8

1

1.2

F̃

100

101

102

0

0.5

1

1.5

2

2.5

D2

Components

ofF̃

downward

upward

diffusion

(a)

(b)

(c)

Figure 5: (a): Approximated lower bound of Ra required to obtain the constant-flux regime from thesimulations. (b): Variation of the ratio of flux F̃ with D2 in the constant-flux regime at sufficiently largeRa. (c): Three components of F̃ through z = 0.99 at Ra = 50000. In (a), the existence of the constant-fluxregime requires Ra ∼ O(103) for D2 < 5. In (b), through any horizontal plane, F̃ = (downward advection− upward advection + diffusion)/F1.

fluctuation (i.e., deviations from the horizontal mean) decays as the exponential rate e−D2k2t,

so that the time required by diffusion to well smooth the fluctuation term (down to 1%) overLm is t1 = 2 ln 10/(D2k

2). Moreover, the study by Slim [50] indicates that the fingertipstravel with a constant speed 0.13Ra before hitting the lower boundary. Therefore, the timerequired for C2 to be advected downward across the same length Lm is t2 = 2π/(0.13Rak).Hence, to obtain a horizontally uniform concentration field, it requires at least t1 ≤ t2,i.e. D2 ≥ 0.13 ln 10π Ra/k =

0.13 ln 102π2

RaLm, or D2 ∼ O(RaLm). For instance, at Ra = 20000,Lm . 0.14 before the shut-down regime, so D2 ≥ 42.5, quantitatively consistent with the

8

results shown in Fig. 3. It will be shown below D2 ∼ O(RaLm) actually corresponds to O(1)Péclect number in the dispersion model.

Renormalize the variables t̃ = Rat, w̃ = w/W0, X = x/ε, where ε = Lm ∼ Ra−0.4 is asmall parameter at large Ra [22], so that the time and velocity fields are transformed fromdiffusion scales to convection scales. Finally, Eq. (6) for C2 becomes

Pe ε

(1

a

∂C2

∂t̃+ w̃

∂C2∂z

)=

(∂2

∂X2+ ε2

∂2

∂z2

)C2, (9)

where the constant value a and w̃ = cos(2πX) are of order unity, and the Péclet number

Pe =aRaLmD2

=W0ε

D2≡ ε

2/D2ε/W0

(10)

denotes the ratio between the advective and diffusive (dispersive) time scales. From ourprevious analysis, the horizontally uniform concentration requires D2 ∼ O(RaLm), namely,Pe ∼ O(1). For any D2 ∼ o(RaLm), e.g. D2 ∼ O(1), Pe → ∞ as Ra → ∞, and then theconcentration field appears in the form of apparent fingers at sufficiently large Ra.

4. Conclusions

The fundamental role of diffusion in mass or heat transport has been studied extensivelyin the convection problem. In the ‘ultimate’ high-Ra regime, the analysis based on the as-sumption that the molecular diffusive transport is negligible when Ra = advection/diffusion� 1 [60, 61] generally yields an invalid asymptotic F–Ra scaling [62]. For the CO2-tracer,solutal convection problem, our study indicates that the mass transport also depends on themolecular diffusion, which is in contradiction to the classical Rayleigh fractionation assump-tion that the fractionation of different components is only determined by their solubility.When the solubility constants of the two components are close, i.e. K1/K2 ∼ O(1), thedifference between F1 and F2 might have a first-order effect on the fractionation. However,for the noble gases He, Ne, and Ar which are usually used as tracers to identify CO2 disso-lution in carbon sequestration, the ratio of the solubility constant K1/K2 > 20, so that theO(1) variation of F1/F2 will not affect the approximation F ≈ 1 − r/r0 and the Rayleighfractionation is realized.

Acknowledgement

This work was supported as part of the Center for Frontiers of Subsurface Energy Secu-rity, an Energy Frontier Research Center funded by the U.S. Department of Energy, Officeof Science, Basic Energy Sciences under Award # DE-SC0001114. B.W. acknowledges thePeter O’Donnell, Jr. Postdoctoral Fellowship in Computational Engineering and Sciencesat the University of Texas at Austin.

9

Appendix A. Variation of gas composition

The change of i-th component in the gas field can be expressed as

dni,gdt

= −qFiCis, (A.1)

where q = D∗1A/H. For multicomponent ideal gas,

Pi,gVg = ni,gRT ⇒ PgVg = (∑

ni,g)RT, (A.2)

where Pi,g is the partial pressure of the i-th component, Vg is the total gas volume, R isthe universal gas constant, T is the absolute temperature, and Pg =

∑Pi,g is the total gas

pressure. From Henry’s law,

Pi,g =CisKi

. (A.3)

The equations (A.2) and (A.3) yield

Cis = KiPi,g =KiRT

Vgni,g = KiPg

ni,gn1,g + n2,g

. (A.4)

Substituting (A.4) into (A.1) gives

dni,gdt

= −qFiKiRT

Vgni,g = −qFiKiPg

ni,gn1,g + n2,g

. (A.5)

Then, we have

dn1,gdn2,g

= αn1,gn2,g

, (A.6)

where α = F1K1/(F2K2). For a quasi-steady convective system, the dissolution flux Fi isfixed, so that α is constant. Then

lnn1,gn01,g

= α lnn2,gn02,g

orn1,gn01,g

=

(n2,gn02,g

)α. (A.7)

Namely,

r ≡ n1,gn2,g

=n01,g · nα−12,g

(n02,g)α⇒ n2,g =

(n02,g

α · rn01,g

) 1α−1

. (A.8)

Therefore, the fraction of dissolved CO2 into water is

F ≡ 1− n1,gn01,g

= 1− n1,gn2,g· n2,gn01,g

= 1− (r/r0)α

α−1 . (A.9)

Actually, (A.9) is a generic form which is valid for both constant Pg and constant Vg. Whenα� 1,

F ≈ 1− (r/r0). (A.10)10

References

[1] D. A. Nield, A. Bejan, Convection in Porous Media, Springer, New York, 3rd edition, 2006.[2] Lord Rayleigh O.M. F.R.S. , LIX. On convection currents in a horizontal layer of fluid, when the higher

temperature is on the under side, Philosophical Magazine Series 6 32 (1916) 529–546.[3] M. C. Cross, P. C. Hohenberg, Pattern formation outside of equilibrium, Rev. Mod. Phys. 65 (1993)

851–1112.[4] F. Orr, Onshore geologic storage of CO2, Science 325 (2009) 1656–1658.[5] K. Michael, A. Golab, V. Shulakova, J. Ennis-King, G. Allinson, S. Sharma, T. Aiken, Geological

storage of CO2 in saline aquifers a review of the experience from existing storage operations, Int. J.Greenh. Gas Control 4(4) (2010) 659–667.

[6] M. L. Szulczewski, C. W. MacMinn, H. J. Herzog, R. Juanes, Lifetime of carbon capture and storageas a climate-change mitigation technology, Proc. Natl. Acad. Sci. U.S.A. 109(14) (2012) 5185–5189.

[7] Y. Liang, B. Yuan, A guidebook of carbonate laws in china and kazakhstan: Review, comparison andcase studies, Carbon Management Technology Conference, Houston, USA, July, 2017.

[8] Y. Liang, J. Sheng, J. Hildebrand, Dynamic permeability models in dual-porosity system for uncon-ventional reservoirs: Case studies and sensitivity analysis, in: SPE Reservoir Characterisation andSimulation Conference and Exhibition, Society of Petroleum Engineers, Abu Dhabi, UAE, May, 2017.

[9] S. E. Gasda, S. Bachu, M. A. Celia, Spatial characterization of the location of potentially leaky wellspenetrating mature sedimentary basins, Env. Geology 46 (2004) 707–720.

[10] J. J. Roberts, R. A. Wood, R. S. Haszeldine, Assessing the health risks of natural CO2 seeps in Italy,Proc. Natl. Acad. Sci. U.S.A. 108(40) (2011) 16545–16548.

[11] R. C. Trautz, J. D. Pugh, C. Varadharajan, L. Zheng, M. Bianchi, P. S. Nico, N. F. Spycher, D. L.Newell, R. A. Esposito, Y. Wu, B. Dafflon, S. S. Hubbard, J. T. Birkholzer, Effect of dissolved CO2 ona shallow groundwater system: A controlled release field experiment, Environ. Sci. Tech. 47(1) (2013)298–305.

[12] G. J. Weir, S. P. White, W. M. Kissling, Reservoir storage and containment of greenhouse gases,Transport Porous Med. 23 (1996) 37–60.

[13] J. Ennis-King, L. Paterson, Role of convective mixing in the long-term storage of carbon dioxide indeep saline formations, SPE J. 10(3) (2005) 349–356.

[14] A. Riaz, M. Hesse, H. A. Tchelepi, F. M. O. Jr, Onset of convection in a gravitationally unstablediffusive boundary layer in porous media, J. Fluid Mech. 548 (2006) 87–111.

[15] J. A. Neufeld, M. A. Hesse, A. Riaz, M. A. H. andH. A. Tchelepi, H. E. Huppert, Convective dissolutionof carbon dioxide in saline aquifers, Geophys. Res. Lett. 37 (2010) L22404.

[16] J. Ennis-King, I. Preston, L. Paterson, Onset of convection in anisotropic porous media subject to arapid change in boundary conditions, Phys. Fluids 17 (2005) 084107.

[17] D. Wessel-Berg, On a linear stability problem related to underground CO2 storage, SIAM J. Appl.Math. 70(4) (2009) 1219–1238.

[18] A. C. Slim, T. S. Ramakrishnan, Onset and cessation of time-dependent, dissolution-driven convectionin porous media, Phys. Fluids 22 (2010) 124103.

[19] H. Hassanzadeh, M. PooladiDarvish, D. Keith, Scaling behavior of convective mixing, with applicationto geological storage of CO2, AlChE J. 53(5) (2007) 1121–1131.

[20] G. S. Pau, J. B. Bell, K. Pruess, A. S. Almgren, M. J. Lijewski, K. Zhang, High-resolution simulationand characterization of density-driven flow in CO2 storage in saline aquifers, Adv. Water Resour. 33(2010) 443–455.

[21] J. J. Hidalgo, J. Fe, L. Cueto-Felgueroso, R. Juanes, Scaling of convective mixing in porous media,Phys. Rev. Lett. 109 (2012) 264503.

[22] D. R. Hewitt, J. A. Neufeld, J. R. Lister, Ultimate regime of high Rayleigh number convection in aporous medium, Phys. Rev. Lett. 108 (2012) 224503.

[23] A. C. Slim, M. M. Bandi, J. C. Miller, L. Mahadevan, Dissolution-driven convection in a Hele–Shawcell, Phys. Fluids 25 (2013) 024101.

11

[24] X. Fu, L. Cueto-Felgueroso, R. Juanes, Pattern formation and coarsening dynamics in three-dimensionalconvective mixing in porous media, Phil. Trans. R. Soc. A 371 (2013) 20120355.

[25] D. R. Hewitt, J. A. Neufeld, J. R. Lister, Convective shutdown in a porous medium at high Rayleighnumber, J. Fluid Mech. 719 (2013) 551–586.

[26] B. Wen, N. Dianati, E. Lunasin, G. P. Chini, C. R. Doering, New upper bounds and reduced dynam-ical modeling for Rayleigh-Bénard convection in a fluid saturated porous layer, Communications inNonlinear Science and Numerical Simulation 17 (2012) 2191–2199.

[27] B. Wen, G. P. Chini, N. Dianati, C. R. Doering, Computational approaches to aspect-ratio-dependentupper bounds and heat flux in porous medium convection, Phys. Lett. A 377 (2013) 2931–2938.

[28] Z. Shi, B. Wen, M. Hesse, T. Tsotsis, K. Jessen, Measurement and modeling of CO2 mass transfer inbrine at reservoir conditions, Adv. Water Resour. (2017).

[29] B. Wen, D. Akhbari, L. Zhang, M. Hesse, Dynamics of convective carbon dioxide dissolution in a closedporous media system, in revision for J. Fluid Mech., arXiv:1801.02537 [physics.flu-dyn] (2018).

[30] S. Backhaus, K. Turitsyn, R. E. Ecke, Convective instability and mass transport of diffusion layers ina Hele-Shaw geometry, Phys. Rev. Lett. 106 (2011) 104501.

[31] P. A. Tsai, K. Riesing, H. A. Stone, Density-driven convection enhanced by an inclined boundary:Implications for geological CO2 storage, Phys. Fluids 25 (2013) 024101.

[32] Y. Liang, Scaling of Solutal Convection in Porous Media, Ph.D. thesis, The University of Texas atAustin, 2017.

[33] G. Sposito, Scale Dependence and Scale Invariance in Hydrology, Cambridge University Press, 2008.[34] T. Yeh, R. Khaleel, K. Carroll, Flow Through Heterogeneous Geological Media, Flow Through Hetero-

geneous Geologic Media, Cambridge University Press, 2015.[35] M. M. Cassidy, Occurrence and origin of free carbon dioxide gas deposits in the earth crust, Ph.D.

thesis, University of Houston, Houston, 2005.[36] S. M. Gilfillan, C. J. Ballentine, G. Holland, D. Blagburn, B. S. Lollar, S. Stevens, M. Schoell, M. Cas-

sidy, The noble gas geochemistry of natural CO2 gas reservoirs from the Colorado Plateau and RockyMountain provinces, USA., Geochimica et Cosmochimica Acta. 72 (2008) 1174–1198.

[37] S. M. V. Gilfillan, B. S. Lollar, G. Holland, D. Blagburn, S. Stevens, M. Schoell, M. Cassidy, Z. Ding,Z. Zhou, G. Lacrampe-Couloume, C. J. Ballentine, Solubility trapping in formation water as dominantCO2 sink in natural gas fields, Nature. 458 (2009) 614–618.

[38] K. J. Sathaye, M. A. Hesse, M. Cassidy, D. F. Stockli, Constraints on the magnitude and rate of CO2dissolution at Bravo Dome natural gas field, Proc. Natl. Acad. Sci. U.S.A. 111 (2014) 15332–15337.

[39] K. J. Sathaye, A. J. Smye, J. S. Jordan, M. A. Hesse, Noble gases preserve history of retentivecontinental crustin the Bravo Dome natural CO2 field, New Mexico, Earth Planet. Sci. Lett. 443(2016) 32–40.

[40] K. J. Sathaye, T. E. Larson, M. A. Hesse, Noble gas fractionation during subsurface gas migration,Earth Planet. Sci. Lett. 450 (2016) 1–9.

[41] Lord Rayleigh Sec. R.S., L. theoretical considerations respecting the separation of gases by diffusionand similar processes, Philosophical Magazine Series 5 42 (1896) 493–498.

[42] Lord Rayleigh O.M. F.R.S., Lix. on the distillation of binary mixtures, Philosophical Magazine Series6 4 (1902) 521–537.

[43] W. M. White, Geochemistry, Wiley-Blackwell, 2013.[44] I. Clark, Groundwater Geochemistry and Isotopes, CRC Press, 2015.[45] W. Henry, Experiments on the quantity of gases absorbed by water, at different temperatures, and

under different pressures, Philosophical Transactions of the Royal Society of London 93 (1803) 29–274.[46] R. E. Criss, Principles of Stable Isotope Distribution, Oxford University Press, Oxford, 1999.[47] J. P. Boyd, Chebyshev and Fourier Spectral Methods, Dover, New York, 2nd edition, 2000.[48] N. Nikitin, Third-order-accurate semi-implicit Runge–Kutta scheme for incompressible Navier–Stokes

equations, Int. J. Numer. Meth. Fluids 51 (2006) 221–233.[49] R. Peyret, Spectral Methods for Incompressible Viscous Flow, Springer, New York, 2002.[50] A. C. Slim, Solutal-convection regimes in a two-dimensional porous medium, J. Fluid Mech. 741 (2014)

12

461–491.[51] B. Wen, L. T. Corson, G. P. Chini, Structure and stability of steady porous medium convection at

large Rayleigh number, J. Fluid Mech. 772 (2015) 197–224.[52] B. Wen, Porous medium convection at large Rayleigh number: Studies of coherent structure, transport,

and reduced dynamics, Ph.D. thesis, University of New Hampshire, 2015.[53] B. Wen, G. P. Chini, R. R. Kerswell, C. R. Doering, Time-stepping approach for solving upper-bound

problems: Application to two-dimensional Rayleigh-Bénard convection, Phys. Rev. E 92 (2015) 043012.[54] B. Wen, G. P. Chini, Inclined porous medium convection at large Rayleigh number, J. Fluid Mech.

837 (2018) 670–702.[55] X. Xu, S. Chen, D. Zhang, Convective stability analysis of the long-term storage of carbon dioxide in

deep saline aquifers., Adv. Water Resour. 29 (2006) 397–407.[56] C. R. Doering, P. Constantin, Bounds for heat transport in a porous layer, J. Fluid Mech. 376 (1998)

263–296.[57] J. Otero, L. A. Dontcheva, H. Johnston, R. A. Worthing, A. Kurganov, G. Petrova, C. R. Doering,

High-Rayleigh-number convection in a fluid-saturated porous layer, J. Fluid Mech. 500 (2004) 263–281.[58] G. I. Taylor, Dispersion of soluble matter in solvent flowing slowly through a tube, Proc. Roy. Soc. A

219 (1953) 186–203.[59] R. Aris, On the dispersion of a solute in a fluid flowing through a tube, Proc. Roy. Soc. A 235 (1956)

67–77.[60] E. A. Spiegel, Convection in stars, Ann. Rev. Astron. Astrophys. 9 (1971) 323–352.[61] S. Grossmann, D. Lohse, Scaling in thermal convection: a unifying theory, J. Fluid Mech. 407 (2000)

27–56.[62] J. P. Whitehead, C. R. Doering, Ultimate state of two-dimensional Rayleigh–Bénard convection between

free-slip fixed-temperature boundaries, Phys. Rev. Lett. 106 (2011) 244501.

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1 Introduction2 Problem formation and computational methodology3 Results4 ConclusionsAppendix A Variation of gas composition