-
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: [email protected] or [email protected]
(Baole Wen),[email protected] (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
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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
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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
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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
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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
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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
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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
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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
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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
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13
1 Introduction2 Problem formation and computational methodology3
Results4 ConclusionsAppendix A Variation of gas composition