Dissolution-driven porous-medium convection in the

Under consideration for publication in J. Fluid Mech. 1

Dissolution-driven porous-mediumconvection in the presence of chemical

reaction

T.J. Ward1, K.A. Cliffe1 O.E. Jensen2†, & H. Power3

1School of Mathematical Sciences, University of Nottingham, University Park, NottinghamNG7 2RD, UK

2School of Mathematics, University of Manchester, Oxford Road, Manchester M13 9PL, UK3Faculty of Engineering, University of Nottingham, University Park, Nottingham NG7 2RD,

UK

(Received ?; revised ?; accepted ?. - To be entered by editorial office)

Motivated by processes occurring during CO2 sequestration in an underground salineaquifer, we examine two-dimensional convection in a finite-depth porous medium inducedby a solute introduced at the upper boundary. Once dissolved, the solute concentrationis assumed to decay via a first-order chemical reaction, restricting the depth over whichsolute can penetrate the domain. Using spectral and asymptotic methods, we explorethe resulting convective mixing using linear stability analysis, computation of nonlin-ear steady solution branches and time-dependent simulations, as a function of Rayleighnumber, Damkohler number and domain size. Long-wave eigenmodes show how deeprecirculation can be driven by a shallow solute field while explicit approximations arederived for the growth of short-wave eigenmodes. Steady solution branches undergo nu-merous secondary bifurcations, forming an intricate network of mixed states. Althoughmany of these states are unstable, some play an important role in organising the phasespace of time-dependent states, providing approximate bounds for time-averaged mixingrates.

Key words:

1. IntroductionThe concentration of CO2 in the atmosphere has increased by approximately a third

since the 1850s, and is currently increasing by 0.46% per year (Lal 2008), making it a focusof international concern. The principal reason for this increase is the burning of fossil fuels,with the power and industry sectors combined accounting for approximately 60% of totalCO2 emissions (IPCC 2005). Storage of CO2 in underground geological formations is apotential means of limiting greenhouse gas emissions to the atmosphere while continuingthe use of fossil fuels (IPCC 2005). Storage takes place through structural, residual andsolubility trapping over short and intermediate timescales, and through the reaction ofCO2 with minerals in the storage site over geological time periods. However a numberof factors are delaying the large-scale deployment of carbon sequestration, includinguncertainties about surface leakage over long periods (Lemieux 2011). There is therefore

† Email address for correspondence: [email protected]

2 T.J. Ward, K.A. Cliffe, O.E. Jensen & H. Power

considerable current interest in understanding the fundamental physical and chemicalmechanisms involved in long-term CO2 storage.

When supercritical CO2 is injected into an underground aquifer, dissolution in brinecauses an increase in the density of the CO2-rich brine. The resulting convection en-hances mixing (Ennis-King & Paterson 2005; Hassanzadeh et al. 2007; Neufeld et al.2010), reducing reservoir mixing times from thousands to hundreds of years (Backhauset al. 2011). To investigate this process, researchers have drawn on, and significantlyextended, the substantial literature on Rayleigh–Benard convection in porous media.Recent attention has focussed on the onset time of convection following the introductionof solute from an upper surface (Riaz et al. 2005; Ghesmat et al. 2009; Slim & Ramakr-ishnan 2010; Andres & Cardoso 2011; Kim & Choi 2012) and the asymptotic transportproperties of high-Rayleigh-number convection (Otero et al. 2004; Neufeld et al. 2010;Hidalgo et al. 2012; Hewitt et al. 2012, 2013). Associated experiments have exploitedthe analogy between porous-medium flow (described most simply by the Darcy equationcoupled to a solute transport equation) and flow in a Hele–Shaw cell (Hartline & Lister1977; Backhaus et al. 2011; Kneafsey & Pruess 2010; Neufeld et al. 2010; Slim et al.2013).

The chemical interaction between CO2 and brine is complex, with a sequence of re-actions taking place leading to acidification and ultimately to a reaction with the hostrock. The reaction timescales span many orders of magnitude (Mitchell et al. 2010),which makes great demands of computational studies that attempt to integrate flow andchemistry. This has motivated fundamental studies where chemistry is represented in ahighly simplified manner, either as a first-order reaction (where the dissolved CO2 reactswith an abundant substrate; Andres & Cardoso (2011)) or a second-order reaction (wherethe substrate may be depleted at the leading edge of the advancing solute field; Ennis-King & Paterson (2007); Ghesmat et al. (2009)). Related studies, addressing the effectof chemical reaction on Rayleigh–Taylor instability, have employed nonlinear reactionterms that allow for chemical front propagation (De Wit 2001, 2004). It is well estab-lished that consumption of dissolved CO2 prevents the solute from penetrating deep intothe underlying fluid and delays the onset of convection (Ennis-King & Paterson 2007).However the resulting convective flows remain poorly characterised.

In the present study, we consider two-dimensional porous-medium convection in thepresence of a first-order reaction. This is typically described by three parameters (aRayleigh number, a Damkohler number and the domain aspect ratio), although two pa-rameters suffice for sufficiently deep layers (Andres & Cardoso 2011) or for sufficientlyshallow ones (as we show below). We keep the description of the chemistry deliberatelysimple in order to understand how a linear reaction term influences the structure ofthe resulting dynamical system. We regard this as a valuable precursor to studies in-volving multiple interacting chemical species and we deliberately address a canonicalproblem with minimal parameterization. We focus first on computing steady solutionsand their stability, comparing the structure of primary and secondary solution branchesto the reaction-free case in a box of finite depth (Riley & Winters 1990, 1991). We thenshow how the pattern of steady solutions provides a framework for understanding long-term mixing processes at high Rayleigh number when the system is inherently unsteady.Numerous physical features of real porous media are neglected, including heterogene-ity and disorder (Bestehorn & Firoozabadi 2012; Ranganathan et al. 2012), anisotropyof the medium (Ennis-King et al. 2005), hydrodynamic dispersion of the solute, capil-lary trapping (Hesse et al. 2008), chemically-induced permeability or porosity changes(either through dissolution or precipitation reactions) (Luquot & Gouze 2009) and three-dimensionality (Pau et al. 2010).

Dissolution-driven convection 3

We consider a planar domain with the solute concentration held constant at the uppersurface. Previous studies have focused on the stability of the thickening boundary layer(Andres & Cardoso 2011; Ghesmat et al. 2009); here we examine the stability of thehorizontally uniform no-flow steady state for a wide range of Damkohler numbers. Themodel and the methods that we use to analyse the problem are outlined in §2. We thenpresent results of a linear stability analysis (§ 3) and derive asymptotic approximationsof the short-wave and long-wave cut-off eigenmodes arising at high Rayleigh number.For linear and nonlinear simulations we use a numerical method with spectral accuracyin space; arc-length continuation and numerical bifurcation tracking demonstrate howmode interactions lead to an intricate network of mixed-mode states (§ 4). Finally (§ 5)we consider the time-evolution of the system, showing that the unsteady flux of solventinto the system fluctuates between values associated with a subset of the steady states.

2. Model and MethodsWe consider the dissolution of a solute in a two dimensional homogeneous porous

medium. We assume that the solute undergoes a first-order A→B reaction, where speciesA (but not B) increases solution density, and that the substrate of the reaction remainsabundant (i.e. its concentration is sufficiently large for any changes to have negligibleeffect on reaction rates, an approximation that can be expected to fail at very largetimes). The equations governing this system are the continuity equation, Darcy’s lawand the convection-diffusion-reaction equation:

∇∗ · u∗ = 0, u∗ =− K

µ(∇∗p∗ −∆gC∗z) , (2.1a)

ϕC∗t∗ + u∗ · ∇∗C∗ =ϕD∇∗2C∗ − αϕC∗. (2.1b)

C ∗, p∗, u∗ ≡ (u∗, w∗) and ∇∗ ≡ (∂/∂x∗, ∂/∂z∗) are the concentration of the speciesA dissolved in the solution, pressure, velocity and gradient operator respectively. Theparameters K, µ, D, ϕ, α, g and ∆ are respectively the medium permeability, solventviscosity, solute diffusion coefficient, medium porosity, reaction rate of the solute, accel-eration due to gravity and the density increase per unit concentration of solute. (Withsuitable adjustments to parameters, we can also use (2.1) to describe a reacting solute ina vertical Hele–Shaw cell.) We assume the solution density is linear in C∗ and that ∆ issufficiently small to apply the Boussinesq approximation. Equations (2.1) are defined ona domain with horizontal length L and vertical depth 2H, i.e. x ∈ [0, L] and z ∈ [−H,H].The initial solute distribution is assumed known, with C∗ = C∗ and u∗ = u∗ at timet∗ = 0. Equations (2.1) are subject to the boundary and initial conditions

C∗(x∗,−H, t∗) = C∗, C∗z∗(x∗, H, t∗) = 0, (2.2a)

w∗(x∗,−H, t∗) = 0, w∗(x∗, H, t∗) = 0, (2.2b)C∗x∗(0, z

∗, t∗) = C∗x∗(L, z∗, t∗) = 0, u∗(0, z∗, t∗) = u∗(L, z∗, t∗) = 0, (2.2c)

C∗(x∗, z∗, 0) = C∗, u∗(x∗, z∗, 0) = u∗. (2.2d)

According to (2.2), the solute concentration is held at a constant value C∗ at the upperboundary and there is no flux of solute across the lower and lateral boundaries.

Equations (2.1) are non-dimensionalized using (x, z) = (x∗, z∗)/H, t = t∗∆KgC∗/(Hϕµ),(C, C) = (C∗, C∗)/C∗, p = p∗/(Hg∆C∗) and (u, u) = (u∗, u∗)µ/(KgC∗∆), yielding

∇ · u = 0, u =−∇p+ Cz, (2.3a)

Ct + u · ∇C =Ra−1∇2C −DaC, (2.3b)


subject to the boundary and initial conditions

C(x,−1, t) = 1, Cz(x, 1, t) = 0, (2.4a)w(x,−1, t) = 0, w(x, 1, t) = 0, (2.4b)

Cx(0, z, t) = Cx(L, z, t) = 0, u(0, z, t) = u(L, z, t) = 0, (2.4c)

C(x, z, 0) = C, u(x, z, 0) = u, (2.4d)

where L = L/H is the aspect ratio of the box. The Rayleigh and Damkohler numbersare defined here as

Ra =HKgC ∗∆ϕµD

, Da =αϕHµ

KgC ∗∆. (2.5)

As a result, (2.3, 2.4) produce a system dependent on Ra, Da, and L.When RaDa = αH2/D = O(1), the time taken for the solute to diffuse across the

box is comparable to the reaction time. For sufficiently large RaDa, however, we mayexpect the domain depth to have negligible influence on the convection. To describe thisdeep-domain case, we introduce the H -independent parameters

β2 =RaDa

=1

αDϕ2

(KgC∗∆

µ

)2

, Λ =√

RaDaL, (2.6)

with which we define the new variables (x′, z′) =√

RaDa (x, (z + 1)), (C ′, C ′) = β(C, C),C ′ = βC, ∇′ = (∂/∂x′, ∂/∂z′), t′ = tRa/β2, (u′, u′) = β(u, u), p′ = pRa. Expressed inthese variables, (2.3, 2.4) become

∇′ · u′ = 0, u′ =−∇′p′ + C ′z, (2.7a)

C ′t + u′ · ∇C ′ =∇′2C ′ − C ′, (2.7b)

subject to

C ′(x′, 0, t′) = β, C ′z′ → 0 as z′ →∞, (2.8a)w′(x′, 0, t′) = 0, w′ → 0 as z′ →∞, (2.8b)

C ′x′(0, z′, t′) = C ′x′(Λ, z

′, t′) = 0, u′(0, z′, t′) = u′(Λ, z′, t′) = 0 (2.8c)

C ′(x′, z′, 0) = C ′ u′(x′, z′, 0) = u′. (2.8d)

Equations (2.7, 2.8) depend only upon the parameters β and Λ; a similar reduction wasemployed by Andres & Cardoso (2011).

Alternatively, for RaDa� 1, solute diffuses rapidly across the domain. Defining

γ = DaRa2 =H3KgC∗∆φµD2

, (2.9)

and considering the limit RaDa � 1 � Ra with γ = O(1), we can set (C − 1, C − 1) =

RaDa(C, ˜C), (u, u) = RaDa(u, ˜u), p = z + RaDa p, t = Ra t, so that (2.3, 2.4) become,

to leading order in RaDa,

∇ · u = 0, u =−∇p+ Cz, (2.10a)

Ct + γu · ∇C =∇2C − 1, (2.10b)


subject to

C(x,−1, t) = 0, Cz(x, 1, t) = 0, (2.11a)w(x,−1, t) = 0, w(x, 1, t) = 0, (2.11b)

Cx(0, z, t) = Cx(L, z, t) = 0, u(0, z, t) = u(L, z, t) = 0, (2.11c)

C(x, z, 0) = ˆC, u(x, z, 0) = ˆu. (2.11d)

This nonlinear shallow-layer version of the problem is parameterized by γ and L; becausethe concentration field is nearly uniform, the chemical reaction term in (2.10b) followszeroth-order, rather than first-order, kinetics.

We define the Sherwood number, Sh, a dimensionless measure of the solute flux perunit length across the upper boundary, by

Sh =1L

∫ L0

−Cz

∣∣∣∣∣z=−1

dx. (2.12)

The corresponding dimensional flux per unit length is (ϕDC∗/H) Sh; when Sh isproportional to Ra, the dimensional net flux is independent of molecular diffusivity D.The total mass of dissolved solute, M(t) =

∫ 1

−1

∫ L0C dxdz, satisfies

Mt = (LSh/Ra)−DaM. (2.13)

The mean square concentration, obtained by multiplying (2.3a) by C and integratingover the domain, satisfies

12M2t = (LSh/Ra)−DaM2 −N2, (2.14)

where M2 ≡∫ 1

−1

∫ L0C2dxdz and N2 ≡

∫ 1

−1

∫ L0| ∇C |2 dxdz. Multiplying (2.3a) by u

and integrating over the domain implies∫ 1

−1

∫ L0

u · u dxdz =∫ 1

−1

∫ L0

wC dx dz. (2.15)

Identities (2.13–2.15) provide useful computational validation. In the deep-domain andshallow-layer cases we define

Sh′ =1Λ

∫ Λ

0

−C ′z′∣∣∣∣∣z′=0

dx′, Sh =1L

∫ L0

−Cz

∣∣∣∣∣z=−1

dx, (2.16)

where Sh = Sh′√

RaDa/β = RaDaSh.

2.1. Linear stability of the no-flow steady solutionEquations (2.3, 2.4) admit the no-flow steady solution

C0(z) =cosh

(√RaDa(1− z)

)cosh(2

√RaDa)

, u0 = 0, (2.17)

for which Sh =√

RaDa tanh(2√

RaDa). We investigate small disturbances to this stateof the form

C(x, z, t) =C0(z) + Re [C1(z) cos (kx) exp (σt)] , (2.18a)u(x, z, t) =u0(z) + Re [u1(z) sin (kx) exp (σt)] , (2.18b)


where σ and k are the growth rate and wavenumber of perturbations respectively; weassume k = nπ/L for n = 1, 2, 3, . . . to ensure disturbances fit within the domain, andallow σ to be complex. Writing (2.3a) in terms of the stream function ψ, such that ψz = uand ψx = −w, eliminating the pressure and linearising about the base state, gives

σC1 − kψ1C0z =Ra−1C1zz − k2Ra−1C1 −DaC1, (2.19a)

ψ1zz − k2ψ1 − kC1 =0, (2.19b)

subject to the boundary conditions

C1(−1) = 0 C1z(1) = 0, ψ1(−1) = 0 ψ1(1) = 0. (2.20)

This is an eigenvalue problem for σ = σ(k,Ra,Da). We define Rac(Da) as the smallestRa satisfying σ(k,Ra,Da) = 0 for any k.

In the deep-domain case (RaDa � 1, see (2.7, 2.8)), for which we might expect thesolution to be confined to a boundary layer at the top of the domain, the base state plusperturbations are

C ′ =β exp (−z′) + Re [βC ′1(z′) cos (k′x′) exp (σ′t′)] , (2.21a)(u′, w′) =Re [(ψ′1z′ ,−ψ′1x′) sin (k′x′) exp (σ′t′)] , (2.21b)

with σ=σ′Ra/β2, ψ′1 = βψ1 and k=k′√

RaDa. Linearising in C ′1 and ψ′1, we obtain from(2.7, 2.8)

σ′C ′1 + k′ψ′1 exp(−z′) =C ′1z′z′ −(

1 + k′2)C ′1, (2.22a)

ψ′1z′z′ − k′2ψ′1 − k′βC ′1 =0, (2.22b)

subject to

C ′1(0) = ψ′1(0) = 0; C ′1z′ → 0 and ψ′1 → 0 as z′ →∞. (2.22c)

This gives an eigenvalue problem for σ′=σ′(k′, β). We define βc as the smallest β for anyk′ for which σ′(k′, β) = 0.

When RaDa� 1 with γ = O(1) (see (2.9)), vertical diffusion dominates reaction andthe solute can penetrate to the base of the domain, the problem reducing to (2.10, 2.11).The steady state is almost uniform with a small parabolic deviation due to reaction, withC0 = 1

2 (z − 3)(z + 1). Then, setting σ = σ/Ra, (2.19) becomes, to leading order,

σC1 + γk(1− z)ψ1 =C1zz − k2C1, (2.23a)

ψ1zz − k2ψ1 − kC1 =0, (2.23b)

with boundary conditions analogous to (2.20). The reaction term affects the base statebut not the perturbations in (2.23). We define γc as the smallest γ for any k for whichσ(k, γ) = 0.

2.2. Numerical MethodsThe linear stability problem (2.19, 2.20) was solved numerically using a spectral method.C and ψ were approximated in z as a sum of N Chebyshev polynomials and the governingequations were evaluated at the Chebyshev collocation points kj = cos ((2j − 1)π/(2N)),for j =1...N, yielding an eigenvalue problem of the form

σB

(Cψ

)= A

(Cψ

), (2.24)


for some 2N×2N matrices A and B. The spectral accuracy of the approximation wasverified by comparison with the exact solution for k = 0, namely σ = −π2/(16Ra)−Da.The problems (2.22, 2.23) were addressed in a similar manner.

The nonlinear equations (2.3) and (2.7) subject to relevant boundary conditions werealso solved using a spectral method (Trefethen 2000; Boyd 2001), using N Chebyshevpolynomials (satisfying the vertical boundary conditions) in z and a truncated cosineand sine series (using M modes) for C and ψ respectively in x. The problem was for-mulated as follows. Equations (2.3a) imply that ∇2p = Cz, with boundary conditions(2.4b) becoming pz(x,±1, t) = C(x,±1, t). To specify a unique solution, we imposed theadditional constraint

∫ 1

−1

∫ L0p dxdz = 0, allowing the pressure to be considered as a

functional of concentration, p = p[C]. Thus (2.3a) yields u = u[C], making (2.3b) a non-linear evolution equation for C. Steady solutions (Cs, ψs) were obtained using Newton’smethod, yielding the concentration at each of the N×M collocation points with spectralaccuracy; the Crank–Nicholson method was employed to evolve the solution in time.

The linear stability of a steady solution (Cs, ψs) was calculated by forming an eigen-value problem using Chebyshev differentiation operators in the z -direction and cosineand sine series differentiation operators in the x -direction. In particular, the steady state(2.17) in a box of width L yields eigenmodes C(j)

1 which can be labelled by j = 1, 2, 3, . . .according to the number of half-wavelengths of disturbance in the box. As we illustratefor example in figure 10 below, it is helpful to project nonlinear solutions C onto theseeigenmodes (evaluated at a chosen Ra and Da in the range of parameters being consid-ered) using

C(j) = 〈C,C(j)1 (z) cos(jπx/L)〉, where 〈f, g〉 ≡

∫ 1

−1

∫ L0

fg dxdz. (2.25)

Such projections help reveal the symmetries of nonlinear solution branches. Project-ing nonlinear solutions onto eigenmodes evaluated at nearby parameter values givesmarginally different curves but does not change the topological relationships or sym-metries exhibited by the projections.

We used arc-length continuation to determine how nonlinear steady solutions changewith a parameter, allowing for solution branches with turning points (Cliffe et al. 2000).In addition to solving (2.3) and (2.4) at each parameter value using Newton’s method,we introduced an arc-length parameter, λ, satisfying

(Ci+1 − Ci)Ci + (λi+1 − λi)λi = δs, (2.26)

where (Ci, λi) is the tangent vector to the solution branch with respect to the steppingparameter at the previous step (Ci, λi), and δs is the step size. This approach accountsfor the curvature of the solution branch and allows continuation around turning points.

Computing the number of unstable eigenvalues along each step of the solution branchalso allowed us to locate secondary bifurcations and turning points with respect to abifurcation parameter. The principle of exchange of stabilities between primary and sec-ondary solution branches, a characteristic feature of transcritical and pitchfork bifurca-tions, then suggested whether the bifurcation diagram is consistent, and helped determinewhether all secondary bifurcations had been found for a given parameter range. At bifur-cation points, linear eigenvectors were used to perturb the solution, providing a meansof locating the bifurcating solution branch. We verified these numerical techniques byreproducing the bifurcation diagram of the Rayleigh-Benard problem in a square cavitypresented by Riley & Winters (1990).


Figure 1. Growth rates of the perturbations about the no-flow steady state for Da = 0.1 andindicated k and Ra.

3. Results: linear stability of the no-flow steady solutionThe eigenvalue problem (2.19, 2.20) was solved numerically to obtain the growth rates

of perturbations, which were found to satisfy Im(σ) = 0. Figure 1 shows how, for fixedDa, increasing Ra leads to a band of wavenumbers for which perturbations grow. ForDa = 0.1, for example, the critical Ra for the onset of instability is Rac ≈ 41.6.

Neutral stability curves in figure 2(a) illustrate the values of Ra and Da for fixed kat which the onset of linear instability occurs. For parameters above a neutral stabilitycurve, perturbations of that wavenumber are linearly unstable. The neutral stabilityenvelope bounding all such curves, Ra = Rac(Da), increases like Da−1/2 for Da � 1and like Da for large Da (as demonstrated in figure 2a). The corresponding wavenumberkc(Da) of neutral modes, satisfying σ(kc,Rac,Da) = 0, tends to a constant for Da � 1and rises like Da for large Da (figure 2b). The inset to figure 2(b), showing the shape ofthe concentration profile in the no-flow state, explains this: for low Da, solute gradientsextend to the base of the layer and neutral perturbations have wavelength comparable tothe layer depth; for large Da, solute gradients are confined to a narrow boundary layerand perturbations correspondingly shrink in wavelength. In the former case, reducing Damakes the concentration profile more uniform, hence increasing Rac; in the latter case,increasing Da draws solute out of the domain more rapidly, again increasing Rac. Wefind that Rac is minimized at Da ≈ 0.038, where Rac ≈ 30.7.

As figure 1 illustrates, for given Da and Ra > Rac(Da), there are two wavenumbersfor which σ = 0 bounding the range of unstable eigenmodes, corresponding to a short-wave and a long-wave neutral mode; these cut-off wavenumbers are represented by theintersections of curves k = constant in the (Da,Ra)-plane (figure 2a). For given k, we findthat there is a maximum Da above which longer-wavelength modes are stable; thus largerDa favours modes of shorter wavelength, corresponding to shrinkage of the underlyingconcentration boundary layer.

Figure 2(a) gives stability boundaries for the problem with no-flux boundary conditionsin a box of width L for k = nπ/L, for mode number n = 1, 2, 3, . . . (where n denotes the


(a)

(b)

Figure 2. (a) Neutral stability envelope (bold curve) in (Da,Ra)-space, plus asymptotic limits(dashed, showing (3.9) for Da�1 and (3.1) for RaDa�1), and neutral stability curves for givenwavenumbers. Points below the neutral stability envelope are linearly stable. Triangles indicatescaling behaviour. Asterisks show the turning point of the neutral curve for given k with thesolid straight line giving the asymptotic approximation (3.3). (b) Critical wavenumber kc(Da)at which linear instability first occurs (solid), showing the asymptotic limits for Da � 1 (3.9)and for RaDa �1 (3.1). The inset shows the profiles of the no-flow steady solutions for givenDa (circular symbols in the main figure) and Ra = Rac(Da).

number of convection cells in the box). Figure 3 shows the corresponding effect of varyingL for fixed n and Da = 0.1. For small L, any mode becomes unstable for sufficiently largeRa, whereas for sufficiently large L lower-order modes (with flatter convection cells)remain linearly stable. For L = π, for example, the box is sufficiently long for mode-1(single-roll) solutions always to be stable, but higher-order modes bifurcate from theno-flow state in the sequence shown in figure 3; low-order modes restabilise when Ra is


Figure 3. Neutral stability curves showing Rac for the onset of linear instability as L varies forDa = 0.1 and indicated mode number. Solid circular symbols denote the modal onset points forincreasing Ra. The horizontal line shows the approximate stability envelope (3.1) with approx-imate modal minima (3.2) shown with solid squares. The turning points for given k (asterisks)are approximated by (3.4) (open circles). The asymptotic approximation (3.7) for the onset ofconvection for small L and large Ra is illustrated by the dotted line.

sufficiently large (because of shrinkage of the underlying concentration boundary layer).Modes j and j + 1 bifurcate simultaneously when L ≈ 0.595π (j = 1) and L ≈ 1.065π(j = 2); below, we will explore how finite-amplitude solution branches interact in theneighbourhood of these crossing points. As expected from figure 1, the lowest value atwhich any mode becomes unstable is Rac ≈ 41.6 when Da = 0.1; this threshold isindicated by the solid circular symbols in figure 3.

3.1. Deep domain: RaDa� 1

To understand the structure of the solutions arising at high Ra, we turn to the stabilityproblem (2.22), governed by the single parameter β, applying in the limit RaDa � 1.Figure 4(a) shows how the growth rate σ′ depends on wavenumber k′ in this case: anunstable band of wavenumbers exists for β > βc ≈ 19.9, and the neutral wavenumber atonset (satisfying σ′(k′c, βc) = 0) is k′c ≈ 1.02. Thus the envelope of neutral curves andcritical wavenumbers in figure 2(a,b) asymptote respectively to

Ra = β2cDa, k = βck

′cDa (RaDa� 1). (3.1)

This turns out to provide a good estimate for the minimum Ra for stability even whenDa = 0.1, as shown in figure 3. Likewise, we can estimate the domain lengths at whichmode n becomes unstable at this minimum Ra, namely

Lπ

=n

βck′cDa(n = 1, 2, 3, . . . ), (3.2)


Figure 4. (a) Growth rates obtained from (2.22), showing the values of k′ resulting in instabili-ties for values of β as indicated. (b) Neutral stability envelope obtained from (2.22) showing thecritical value of β resulting in the onset of linear instability, along with the asymptotic limitsfor k′ � 1 and k′ � 1 from (B 7) and (B 29) respectively. The dot illustrates β−1 and k′−1 atwhich the gradient is -1.

points that are indicated with solid squares in figure 3. Thus longer domains or an increasein Da favour a greater number of rolls at onset (as reported previously by Ritchie &Pritchard (2011)).

In contrast to classical porous medium convection, the neutral curves for fixed wavenum-ber (figure 2(a)) and fixed mode number (figure 3) exhibit turning points with respect toDa and L respectively (marked with asterisks in both cases). As explained in Appendix A,when RaDa � 1 these turning points are defined by the point d log β/d log k′ = −1 onσ′(k′, β) = 0. At this point (marked on figure 4(b)), β = β−1 ≈ 34.4 and k′ = k′−1 ≈ 0.38.Thus the turning points in figure 2(a) lie on the asymptote

Ra = β2−1Da, (3.3)

sitting at Da = k/(β−1k′−1). Correspondingly, the turning points of the neutral curves in

figure 3 are well approximated by (3.3) with

L =nπ

Daβ−1k−1. (3.4)

This provides an explicit prediction of the maximum domain length in which mode n islinearly unstable, and confirms that the turning point arises because of shrinkage of theunderlying concentration boundary layer.

For large β, figure 4(b) shows the existence of long- and short-wave neutral modes forwhich k′ = O(β−1/2) and k′ = O(β1/2) respectively, corresponding to k = O((RaDa3)1/4)and k = O((Ra3Da)1/4) respectively. Thus sufficiently far above Rac in the (log Da, log Ra)-plane (figure 2a), with RaDa� 1, curves of constant k intersect with slope -3 and -1/3(as indicated on figure 2a), representing the long- and short-wave cut-offs respectively.The corresponding eigenmodes for β = 104 are illustrated in figure 5; recall from (2.21)that the base state is proportional to exp(−z′). The long-wave neutral mode (figure 5a, c)has its concentration perturbation confined to a depth comparable to the base state, butit drives a much deeper recirculating flow which penetrates a distance comparable tothe wavelength of the perturbation. In contrast, the short-wave mode (figure 5b, d) hasconcentration perturbation and streamfunction that are almost identical, but which are


Figure 5. Concentration perturbation and streamfunction profiles of neutrally stable eigen-modes for β = 104 and (a, c) k′c = 0.013 and (b, d) k′c = 93.1. In (a,b), solutions to (2.22)are shown with curves and asymptotic approximations (B 9), (B 30) with dots. The maximumconcentration perturbation is set to unity in each case. In (c,d), the concentration perturbationand stream function are shown as contour plots. Note differing vertical scales in (c).

confined to a thin boundary layer that is much shorter than the penetration depth of thebase state. These modes illustrate how distinct physical balances can drive instability,which we characterise below.

Further analysis (for details see Appendix B) reveals that σ′(k′, β) has four distin-guished asymptotic limits when β � 1, identified as regions Ia, Ib, II and III in Table 1,two of which capture the neutral modes. In each distinguished limit, the 2-parametereigenvalue problem (2.22) can be reduced to a distinct 1-parameter problem. In the over-laps between these regions, σ′ can be captured by simpler zero-parameter approximations(after appropriate rescaling). The predictions of σ′ in each distinguished limit are com-pared to a numerical prediction of the growth rate in figure 6, showing good agreementfor large β. Each case captures a distinct physical interaction leading (in cases Ib, II andIII) to instability.

In regions Ia and Ib, where perturbations are long compared to the penetration depthof the base state, the eigenmodes have the boundary-layer structure illustrated in fig-ure 5(a, c). In both cases, horizontal flow redistributes solute within the base-state bound-ary layer (see (B 1, B 6)); in region Ia, the concentration perturbations then diffuse ver-tically to drive a recirculating flow at depth (see (B 2)); in region Ib, the concentra-


k’ σ’ z’ ψ’

Ia O(β−1) −1− 4βk′3 + 4k′4β2 1, β β/k′

β−1 � k′ � β−1/2 −1 + 4k′4β2 1, 1/βk′2, 1/k′ 1/(βk′3)

Ib O(β−1/2) σIb

“k′β1/2

”1, β1/2 β1/2

β−1/2 � k′ � 1 σIbβk′2 1, 1/k′ βk′

II O(1) βσII(k′) 1 β

1� k′ � β1/2 β“

1− η0k′−2/3”

k′−2/3 β/k′

III O(β1/2) β − k′2 − η0β2/3“k′2+βk′2

”1/3

β−1/3 β1/2

Table 1. Summary of the asymptotic approximation of ( 2.22) for β � 1 for varying orders of k′.Four regions (Ia, Ib, II, III) are identified along with intermediate limits, where σIb ≈ 0.6917, andη0 ≈ 2.338. σIb and σII are solutions to the eigenvalue problems (B 6) and (B 14) respectively.The table shows the approximation of σ′, the length scales of the concentration and streamfunction profiles, and the magnitude of ψ′1, given that maxC′1 = 1.

Figure 6. Growth rates obtained from (2.22) (solid) compared with the asymptotic approxi-mations for β � 1 shown in table 1 for β = 104; the constant c = 1.1 is used to offset thegrowth-rate so that regions Ia and Ib (with σ′ < 0) may be plotted on a log scale.

tion perturbations drive the flow within the base-state boundary layer (see (B 6)), whilethe flow recirculates passively at depth (via B 8). The neutral eigenmode illustrated infigure 5(a) is well captured by the region-Ib approximation (B 9). The correspondingcriticality condition (B 7) provides the long-wave asymptote shown in figure 4(b). Thecondition for the recirculation depth of the neutrally stable long-wave mode to be lessthan the depth of the domain is Ra � Da, consistent with the underlying assumptionof large β. For a domain of fixed width L, with constant Da, a mode of given number nultimately becomes neutrally stable as Ra increases due to shrinkage of the underlying


boundary layer relative to the fixed box width, the threshold being at

Ra = Da−3

(nπ

LkIbc

)4

, (3.5)

which provides the upper asymptote of each curve n = constant in figure 3.In Region II, with k′ = O(1), which represents rapidly growing modes, the stream

function and concentration vary over the same vertical lengthscale, with the flow re-distributing solute and solute gradients reinforcing the flow (see (B 13)). Diffusion andreaction of the solute perturbation do not arise at leading order, so that the eigen-value problem reduces to second order (B 14) and the predicted asymptotic growth rateβσII(k′) increases monotonically with k′. However, once k′ = O(β1/2) (Region III), solutediffusion (both horizontal and vertical) becomes important (see (B 17)). We can recoverthe short-wave neutral mode asymptotically using a WKB method, as illustrated in fig-ure 5(b). The eigenmode forms shallow and slender recirculation cells (figure 5(d)) withdepth-to-width aspect ratio scaling like β1/6. The prediction (B 29) for the wavenumberof the short-wave neutral mode, which can be written

k′c = β1/2 − η0β1/6

41/3+ . . . (β � 1), (3.6)

where −η0 ≈ −2.338 is the largest zero of Ai(η), is shown in Figure 4(b). Equation (3.6)also captures the onset condition for convection in a narrow domain at high Ra, given by

nπ

L= (RaDa)1/2k′c, (3.7)

which is illustrated in figure 3 for n = 1. Region III also contains the most rapidlygrowing modes, for which (from (B 28)) k′ = (η0β/3)3/8, with eigenmodes resemblingthose in figure 5(b, d). From this we can estimate that, for RaDa� 1, the number ofrecirculation cells nf initially arising in a large domain is approximately

nf ≈Lπ

(η0

3

)3/8

Ra11/16Da5/16. (3.8)

3.2. Shallow domain: RaDa� 1� RaReturning to figure 2, we see very different behaviour at high Ra when Da� 1. This canbe captured by taking the limit while holding γ = O(1) (see (2.9)). Solving the eigenvalueproblem (2.23, 2.20) for σ(k, γ), we find that increasing γ destabilizes the linear system(figure 7a). Instability arises for γ > γc ≈ 7.733, with onset wavenumber kc ≈ 1.22. Thusthe envelope of neutral curves and critical wavenumbers asymptote respectively to

Ra2 =γcDa

, k = kc (3.9)

as Da→ 0, as illustrated in figure 2(a,b).For γ = DaRa2 � 1, which corresponds to the special case Da � 1 with Da−1/2 �

Ra � Da−1, we see from figure 7(b) how the short- and long-wave neutral modes havek ∝ γ−1/2 = (RaDa1/2)−1 and k ∝ γ1/2 = (RaDa1/2) respectively, implying that curvesσ = 0 for constant k all have slope − 1

2 in the (log Da, log Ra)-plane (figure 2a). Thelong-wave neutral mode occupies the full depth of the domain (figure 8a, c), whereas theshort-wave mode is confined to a boundary layer near the upper surface (figure 8a) withrolls resembling figure 5(d). An analysis of (2.23) for γ � 1, outlined in Appendix C,reveals that σ(k, γ) has three distinct asymptotic limits, labelled I–III in table 2, withregions I and III capturing the neutral modes. The mathematical structure of regions II


Figure 7. (a) Growth rates for the asymptotic limit Da�1, 1� Ra� 1/Da, showing unstablewavenumbers for given γ. (b) Neutral stability envelope obtained from (2.23,2.20) showing thecritical value of β resulting in the onset of linear instability, with the asymptotic limits (C 2) fork � 1 and (C 10) k � 1.

k σ z ψ

I O(γ−1/2) σI

“kγ1/2

”1 γ1/2

γ−1/2 � k � 1 σIk2 1 1/k

II O(1) γσII (k) 1 1

1� k � γ1/2 γ(2− 21/3η0/k2/3) k−2/3 k

III O(γ1/2) 2γ − k2 − η0γ2/3“k2+2γk2

”1/3

γ−1/3 γ1/2

Table 2. Summary of the asymptotic approximation of (2.23,2.20) for neutral modes withγ � 1 where σI ≈ 0.422 and η0 ≈ 2.338. σI and σII are solutions to the eigenvalue problemsdefined in (C 1) and (C 5) respectively. The table shows the approximation of σ, the length scalesof the concentration and stream function profiles, and the magnitude of ψ1 relative to C1.

and III corresponds closely to that identified in the deep-layer case, with diffusion termsbeing subdominant in region II (see (C 5)) but both horizontal and vertical diffusionappearing in region III (see (C 7)); however the boundary-layer behaviour seen at smallerwavenumbers (regions Ia and Ib in table 1) is replaced by a single-layer structure in whichvertical diffusion of solute and streamfunction appear at leading order (C 1). The lowerboundary has an influence only in regions I and II; the modes in region III are identical(up to rescaling) with those in the deep-layer case.

4. Results: nonlinear steady solutionsFigure 9 shows the Sherwood number (2.12) for steady solutions of (2.3, 2.4) for 40 6

Ra 6 95, arising in a box of width L = π for Da = 0.1. At the anticipated bifurcationpoints , corresponding to the values of Ra for which linear eigenmodes are neutrally stable(see figure 3 for L/π = 1), mode-2,3,4,. . . solutions bifurcate supercritically from the basestate. Here the mode number corresponds to the number of rolls in the streamfunction.For every mode, there exist two potential solutions, each with the same Sh, for which


Figure 8. (a) Concentration and stream function profiles of neutrally stable eigenmodes, show-ing solutions to (2.23,2.20) (curves) and asymptotic approximations (C 1, C 9) (dots) for γ = 104

and k = 0.011 and k = 440.04. The long-wave mode occupies the whole domain, as illustratedin (c), while the short-wave mode is confined to a boundary layer near the upper surface. (b)Growth rates obtained from (2.23,2.20) with the asymptotic approximations for γ � 1 (dashed,see table 2) for γ = 104 and c = 1.1.

flow is upward (downward) adjacent to the right-hand wall of the box. Figure 9 revealsadditional mixed-mode steady solutions arising via secondary bifurcations, and turningpoints in the primary solution branches. For example, the mode-2 branch does not extendbeyond Ra ≈ 67. Secondary bifurcations of the mode-5,6,7,. . . branches are not shownin figure 9. Equation (2.14) shows that, for steady states, LSh/Ra is a positive-definitefunctional of C and ∇C; it is straightforward to show that solutions bifurcating fromthe no-flow state must increase Sh. Recirculation enhances mixing, although we foundno clear relationship between Sh and mode number or Ra, at least over the range ofparameters investigated. We note however that for fixed Ra, Sh is not monotonic inmode number; thus for Ra = 95, say, mode 4 provides the most effective mixing. Thefigure highlights the complexity of the system even at moderate Ra. We now examine inmore detail the stability and structure of the primary and mixed states.

To explore the bifurcation structure of the system, we project the solution branchesonto the underlying eigenmodes using (2.25). Figure 10 shows the projections of thesteady solutions illustrated in figure 9, with thick curves denoting stable states. Of thetwo primary mode-2 solutions, that for which fingers descend adjacent to the walls of thedomain is stable as far as the turning point at Ra ≈ 67, whereas the solution with a fingerdescending in the middle of the domain loses stability to a mixed mode at Ra ≈ 52.5.This is because the no-flux boundary conditions suppress some classes of disturbances to


Figure 9. Change in the Sherwood number for steady states with different modes as Ra varies.The steady states are calculated for Da = 0.1 and L = π using the no-flux boundary conditions.This shows steady states with indicated modes bifurcating from the no-flow state along withsecondary bifurcation branches.

the wall-bound fingers. A mixed mode (illustrated by example A−) arises subcriticallyand coexists with an unstable solution of opposite parity (A+); the latter has a turningpoint, regaining stability at Ra ≈ 46.6 before turning into a pure mode-3 state. Thismode-3 branch in turn loses stability at Ra ≈ 67.4 to a mixed state that has a mode-4 component, which in turn loses stability at a turning point. Again there are strongdifferences between states with fingers attached to the walls of the domain (C(2) > 0,C(4) > 0; example C), those with fingers not adjacent to boundaries (example B) andthose with mixed properties (C(2) < 0, C(4) > 0, example D). A pure mode-4 stateis stable for 62.4 < Ra < 90, which in turn loses stability to a new mixed state asRa increases beyond 90 (example E). This coexists with a number of other mixed states,including example F. For large Ra we were able to identify some isolated solution branches(example D).

The projections used in figure 10 show a notable difference from the bifurcation struc-ture of the Rayleigh–Benard problem (Riley & Winters 1990), for which secondary solu-tion branches extend to large values of Ra and all solutions are symmetric about z = 0,so that modes with C(i) < 0 are equivalent to those with C(i) > 0 for all integers i.Replacing a fixed-concentration boundary condition with the no-flux condition (2.4a)at z = −1 breaks this vertical symmetry. While odd modes remain symmetric (in thatsolutions with C(3) < 0, for example, resemble those with C(3) > 0 after reflection aboutx = L/2), even modes are not (so that solutions with C(2) > 0 or C(4) > 0 have de-scending fingers attached to no-flux boundaries, unlike the cases C(2) < 0 or C(4) < 0).


Furthermore, consistent with the folding back of neutral curves in figures 2 and 3 atlarge Ra, inclusion of the reaction term truncates solution branches bifurcating from thebase state as Ra increases; for L = π we also find that the reaction term prevents theexistence of the mode-1 state that would be present in the Rayleigh–Benard problem.

Much of this bifurcation structure can be understood by considering how secondarybranches emerge from mode interactions as the domain length changes. For example,when Da = 0.1 the mode-1 and mode-2 solutions bifurcate simultaneously from the no-flow state when L ≈ 0.59π (figure 3). Taking values of L near this critical value revealshow the bifurcation diagram evolves as L changes (figure 11). For L = 0.55π (figure11a), the mode-1 state exists only as a pair of mixed modes: one which connects withthe pure mode-2 state (see point II) having fingers attached to the walls of the domain,and which is predominantly stable; and the other which connects to the mode-2 state(between points I and III) with a finger in the centre of the domain, which is unstable(example A). As L increases through 0.59π (figures 11b, c), the former mixed mode shrinks(example B) and the latter grows, until ultimately the pure mode-2 state with the fingerdescending in the middle of the domain provides the primary stable solution branch fromthe no-flow state (figure 11d). Neutral stability curves of the mode-1 and mode-2 states(figure 11e) confirm the critical L ≈ 0.59π at which the two steady solutions bifurcatesimultaneously. Bifurcation points of the mode-2 steady states and limit points of themode-1 steady state show how the domains of parameter space in which stable mode-1and stable mode-2 solutions overlap (figure 11e). These results can be interpreted asan unfolding of one of the canonical normal forms describing 1:2 mode interactions, asclassified by Armbruster & Dangelmayr (1986, 1987).

Increasing L further notably alters the bifurcation structure of the system with theintroduction of secondary bifurcations not present at smaller L. To illustrate, the bifur-cation digram for L = 1.1π (figure 12) shows the coexistence for a narrow range of Raof four locally stable solutions: mode-2 with the fingers attached to the domain walls;mode-3; and two mode-4 solutions. Of the latter, the mode with fingers attached to thedomain walls is stable over a greater range of Ra. Despite a modest change in parametervalues, the pattern of secondary states differs substantially from figure 10. Examples A–Fillustrate the diversity of steady states that arise in this case. These and other bifurca-tion diagrams that we obtained for nearby parameter values suggest that steady solutionswith fingers growing down the walls tend to be stable over a greater range of Ra thansolutions having fingers growing down the centre of the domain.

5. Results: time-dependent solutionsThe existence of multiple steady states at higher Ra, many of which are unstable,

suggests complex time-dependent behaviour. This is confirmed by simulations runningfrom different initial conditions.

For example, with Da = 0.1, L = π (see figure 10), the stable steady states at lowerRa are attractors from a range of different initial conditions. However for larger Ra,persistent time-dependent behaviour emerges. Figure 13 shows solutions computed forRa = 84.15, 87 and 92. After an initial transient, periodic oscillations arise in each case,increasing in amplitude and period as Ra falls. For Ra = 92, the system cycles betweenstates in which two widely spaced fingers cling to the wall of each domain; the left-rightreflection symmetry is broken with opposite parity in each half-phase of the oscillation.The orbit cycles close to the steady states A and B, each of which has one pair of complexconjugate unstable eigenvalues (σ ≈ 0.03 ± 0.07i for Ra = 87). For Ra = 87, there areperiods during each cycle when one or other of the lateral fingers detaches from the


Figure 10. Bifurcation diagram showing steady states for Da = 0.1, L = π at low Ra. Solutionbranches are projected onto the mode 2, 3 and 4 eigenvectors respectively computed for Ra = 50,Da = 0.1, with stable branches denoted by bold curves. The left insets show concentration

fields of the eigenmodes C(k)1 used in (2.25), illustrating the difference between positive and

negative mode components; solutions with C(3) < 0 are identical (up to reflection) with those

with C(3) > 0; bifurcations from the no-flow state are indicated with circles; squares indicateexamples A-F shown in insets (right).


Figure 11. Crossing of the mode 1 and mode 2 branches for Da = 0.1 as the box lengthincreases for L/π=0.55 (a), 0.58 (b), 0.59 (c) and 0.595 (d). (a-d) show projections of steadysolutions onto mode-1 and 2 eigenvectors computed for Ra = 50. In (e), neutral stability curvesfor these parameter values are labelled by n; bifurcations from mode 2 branches are solid curves(including labels I, II and III corresponding to points in (a)); limit points of mode 1 branchesare dashed curves (one bearing an asterisk corresponding to the point shown in (a), the othera cross corresponding to that in (c)). Squares in (a,c) indicate examples A and B shown in theinsets.


Figure 12. Bifurcation diagram showing steady states for Da = 0.1, L = 1.1π at low Ra.Solution branches are projected onto the mode-2, 3 and 4 eigenvectors with stable branchesdenoted by bold curves. The value of Ra of bifurcations from the no-flow state (circles) and ofthe insets (squares) are as indicated.

wall. This is more exaggerated when Ra = 84.15, when the cycle spends long periodsclose to a nearby unstable steady mode-3 solution (labelled C,D in figure 13). Thesesteady states have one real unstable eigenvalue (σ ≈ 0.01 for Ra = 87) correspondingto a symmetry-breaking eigenmode. The rapid increase in period, and the evident close


approach to saddle-points in phase space, provide strong evidence of a nearby heteroclinicbifurcation. In these examples, all solutions began as small perturbations from the steadysolution C. The phase portrait in figure 13(c) shows how the steady states A, B,C, Dorganise the structure of the periodic orbits.

More disordered behaviour arises at higher Ra, as an increasing number of modes losestability in a domain of a given size. Figure 14 shows LSh/Ra for two time-dependent so-lutions for Ra = 200, Da = 0.1, and L = π, starting from two different initial conditions.The no-flow steady state has LSh/Ra ≈ 0.070. For t < 100, the solution arising from aperturbed steady-state immediately produces fluctuating Sh whereas the solution involv-ing growth of a boundary layer from the upper surface has a long initial transient as theboundary layer diffuses into the box. However, for t > 100, both solutions are disorderedand apparently unpredictable. Insets A–H illustrate how the penetration depth of fingersof the steady states decreases as the number of fingers increases (a feature reflected bythe lengthscales reported in table 1; the longer-wave modes drive recirculation that fillsthe full domain). Modes 6 and 7 have the largest Sh/Ra, lower modes having fewer fingersand higher modes a penetration depth less than that of the base state. Consequently thevalues of Sh/Ra for the time-dependent solutions are approximately bounded by thoseof steady modes 3 and 7 in this instance, with the system spending long intervals closeto the mode 4 steady state. We anticipate that steady-state solutions — even if they areunstable — are likely to provide useful and practical estimates of dissolution rates overlong time periods.

Finally, to illustrate the effect of finite domain depth and of a reduction in Da, figure 15shows a simulation of the reduced model (2.10, 2.11) arising in the limit RaDa� 1� Ra.As in figure 14, the flow is characterised by isolated plumes descending from a thinboundary layer near the upper surface, with plumes at lateral walls again being stabilisedby the no-flux conditions. In this example, the plumes penetrate to the base of the layer.Peaks in mass transfer are associated with eruptions of new plumes from the boundarylayer, as illustrated in example C in figure 15.

6. DiscussionWe have examined the effect of a first-order chemical reaction on the linear and nonlin-

ear features of buoyancy-induced solutal convection in a porous medium. In comparisonto classical thermal convection, the problem that we formulated with chemical reactionand mixed boundary conditions lacks the up-down reflection symmetry of the classi-cal problem; in particular, chemical reaction prevents solute introduced at the upperboundary from penetrating deep into the domain. Thus at high Rayleigh and Damkohlernumbers, solute is confined to a thin boundary layer near the top of the domain, sothat short-wave modes become unstable and convective motions are not sensitive to thedomain depth. However longer-wave modes can drive recirculation that is substantiallydeeper than the boundary layer at small amplitudes (figure 5(a)), and which can developslender vertical fingers at large amplitudes (figure 14).

The dissolution of CO2 in an aquifer involves multiple coupled chemical reactions oc-curring over widely disparate timescales, including reactions with the host rock that canlead to changes in its physical structure. As dissolution and acidification proceed, thedominant reactions change character, exhibiting a variety of linear and nonlinear inter-actions at different times (Mitchell et al. 2010); correspondingly, the effective Damkohlernumbers for the reactions can be expected to fall from high values in early stages whenrapid reactions dominate, to low values when slower reactions persist. Instead of address-ing the full complexity of this problem, we have addressed only a first-order reaction of


Figure 13. Possible existence of a heteroclinic orbit for Da = 0.1, L=π. (a-c) show how themode-2 and mode-3 components change in time for these simulations at Ra = 84.15, 87 and 92.The insets show the steady states A, B, C and D calculated for Ra = 85; the corresponding C2

and C3 projected onto eigenvectors computed for Ra = 50 are indicated with curves in (a,b)and dots in (c).


Figure 14. LSh/Ra computed from two time-dependent solutions, one starting from a perturbedmode-2 steady-state (thick curve) and the other starting with C = 0 in the box (thin curve)for Ra = 200, Da = 0.1 and L = π . The mode-2 to 8 steady states are illustrated by insetsA,B,C,D and E with the respective values of Sh labelled. Inset H illustrates the concentrationprofile of the no-flow base state (2.17).

a single species in the presence of an abundant substrate. However we have describedbehaviour across the full range of Damkohler number, which gives a crude indication ofpossible behaviour across different timescales in geophysical applications. For example,we identified Rac ≈ 30.7 as the minimum value below which convection will not takeplace for any Da, and showed that Ra ≈ β2

cDa (Da� 1) and Ra ≈ (γc/Da)1/2 (Da� 1)provide limiting thresholds for fast and slow reactions respectively. At sufficiently lowDa, the solute penetrates to the base of the domain and solute gradients diminish (seethe inset to figure 2(b)), to the extent that the reaction effectively becomes zeroth order(see (2.10b)). It is notable that for both small and large Da, fully developed convectionis characterised by vertical plumes emerging from, and sustained by, a thin dynamicboundary layer at the upper surface (see figures 14, 15).

We have chosen to focus on a specific class of chemical reaction and therefore thepredictions of our study are necessarily of limited validity. For example, previous studies(Ennis-King & Paterson 2007; Ghesmat et al. 2009) have addressed a two-species modelin which the reaction rate α in (2.1b) is replaced by αw∗, where C∗ represents dissolvedCO2 and w∗ is the concentration of a second species (a mineral fixed in the host rock)that is depleted through its reaction with C∗ at rate −αw∗ϕC∗. Suppose that w∗ isof initially uniform concentration w∗. Then on nondimensionalisation the reaction termin (2.3b) becomes −DawC where Da = Daw∗/C

∗and Da = αφµH/(∆Kg), while the


Figure 15. Unsteady evolution of Sh for γ = 103, L = π. A–E show snapshots of theconcentration field at different times.

evolution of w follows wt = −DawC. Thus, setting w = 1 + w, a necessary conditionfor perturbations in w to be neglected (and for the present model to be valid, with Dareplaced by Da) is Da � 1, requiring the reaction time between mineral and soluteto be much greater than the convective timescale. However because the ratio w∗/C

∗

can be large when the solute is dilute relative to the substrate in the host rock, thismodel still accommodates a wide range of Da; nevertheless over very long timescales thecapacity of the host rock to react with the solute will be diminished and the effectiveDamkohler number Da will fall. A similar argument could apply to a chemical reactionin a Hele–Shaw cell.

In addition to identifying regimes of linear instability of the spanwise uniform steadystate numerically (figures 2 and 3), we used asymptotic approximations to characterisethe onset of disturbances across different parts of parameter space. At large Ra and Da(table 1, figure 6), the short-wave neutral mode, for which the concentration and stream-function perturbations are confined to a narrow boundary layer (figure 5(b)), provides


a new analytic threshold for the onset of convection in a narrow domain at large Ra(see (3.7) and figure 3). Determination of the asymptotic structure of linear eigenmodesreveals dominant lengthscales that may persist at finite amplitudes (Tables 1 and 2). Asimilar asymptotic approach could be used to investigate disturbances to a time-evolvingbase state, or of the boundary layer arising when convection is fully developed, but wehave not pursued this here.

At finite amplitudes, the bifurcation diagrams obtained for constant Da and L andvarying Ra reveal a complex web of steady solution branches. As Ra increases, the no-flow state loses stability to a primary mode which then loses stability to a variety of mixedstates, which in turn lead to disordered time-varying solutions (figure 14). The brokenvertical symmetry influences the stability properties of primary solution branches, pro-moting (on average) the stability of fingers falling adjacent to no-flux lateral boundariesrelative to those in the interior of the domain (figures 10, 11, 12). We paid particularattention to domain lengths close to critical values at which there is a switch in theprimary mode bifurcating from the base state, as such points provide organising centresin parameter space from which curves of bifurcations emerge. For example, the interac-tion between modes 1 and 2 is illustrated in detail in figure 11, and that between modes2 and 3 in figure 12. While many of the steady solution branches emerging from suchpoints are unstable, they are nevertheless useful as they in turn appear to organise thehigh-dimensional phase space inhabited by time-dependent solutions.

For the parameter values that we investigated, we did not find clear evidence of Hopfbifurcations. Nevertheless we did observe periodic limit cycles, which show evidence ofappearing through a heteroclinic bifurcation (figure 13). For larger Ra, solutions becomemore disordered (figure 14), yet the steady states provide practical bounds for the fluc-tuating Sherwood number. In particular, our numerical evidence suggests that the time-dependent solutions rarely exceed the largest Sh of any individual steady mode, and arebounded below by the lowest-order unstable steady state. Further studies are required tocharacterise the statistical and scaling properties of Sh at higher Ra, particularly at val-ues more characteristic of CO2 sequestration. Inclusion of additional chemical reactions,taking place over a range of time scales, can be expected to add further complexity tothis system.

AcknowledgementsThis paper is dedicated to the memory of Professor Andrew Cliffe, who died on 5th

January 2014.We are grateful to Dr C.A. Rochelle and Dr. D. Noy for valuable discussions. This

project was funded by the Nottingham Centre for Carbon Capture and Storage and theEU FP7 PANACEA project.

Appendix A. Turning points of neutral curvesWe estimate the location of turning points on neutral curves as follows. On a curve

σ(k,Ra,Da) = 0 for k = constant,

0 =∂σ

∂Ra

∣∣∣∣∣k

dRa +∂σ

∂Da

∣∣∣∣∣k

dDa. (A 1)


Thus at a turning point where dDa = 0, ∂σ/∂Ra|k,Da = 0. Now since σ ≈ Daσ′(k′, β)when RaDa� 1, it follows that

∂σ

∂Ra

∣∣∣∣∣k,Da

≈ Da∂σ′

∂Ra

∣∣∣∣∣k,Da

= Da∂σ′

∂k′

∣∣∣∣∣β

∂k′

∂Ra

∣∣∣∣∣k,Da

+ Da∂σ′

∂β

∣∣∣∣∣k′

∂β

∂Ra

∣∣∣∣∣k,Da

. (A 2)

But ∂k′/∂Ra|k,Da = −k′/(2Ra) and ∂β/∂Ra|k,Da = β/(2Ra), so at a turning point

0 = −∂σ′

∂k′

∣∣∣∣∣β

k′

2Ra+∂σ′

∂β

∣∣∣∣∣k′

β

2Ra. (A 3)

But on σ′ = 0,

0 = −∂σ′

∂k′

∣∣∣∣∣β

dk′ +∂σ′

∂β

∣∣∣∣∣k′

dβ, (A 4)

Combining (A 3, A 4), it follows that that turning points of the neutral stability curvescorrespond to the points where d log β/d log k′ = −1.

Appendix B. Deep-layer high-Ra asymptoticsWhen β � 1, the deep-layer eigenvalue problem (2.22) has four distinguished limits, as

summarised in Table 1. We derive below the distinct one-parameter approximations forthe growth rate σ′(k′, β′) in each limit. We use Roman labels I–III to denote variablesand parameters that remain O(1) within each region as β → ∞. Where an analyticapproximation to σ′ does not exist, the growth rates are evaluated numerically to obtainthe results shown in figure 6. We evaluate the limiting forms of σ′ at the periphery of eachregion, where the distinguished limits overlap, to ensure complete coverage of parameterspace.

B.1. Region Ia: k′ = O(β−1)For k′ = O(β−1), the eigenfunction has distinct inner and outer approximations. Theinner zone has depth comparable to the decay length of the base state; the outer zone hasdepth comparable to the perturbation wavelength. In the inner zone, we set k′ = kIa/β,ψ′1 = ψIa(z′)/k′, C ′1 = CIa(z′), σ′ = σIa0 + . . . . To leading order, (2.22) becomes

σIa0CIa + ψIa exp(−z′) = CIa

z′z′ − CIa, ψIaz′z′ = 0 (B 1)

with CIa(0) = ψIa(0) = 0. Thus ψIa = αz′ for some α. We seek a solution for whichCIaz′ → 0 for large z′, requiring σIa0 = −1. Thus CIa = α(2 + z′)e−z

′ − 2α. In the outerzone, we set z′ = βz, σ′ = −1 + β−2σIa1 + . . . , C ′ = CIa(z) and ψ′

1 = β2ψIa(z). Then(2.22) becomes

σIa1CIa = CIa

zz − k2IaC

Ia, ψIazz − k2

IaψIa − kIaC

Ia = 0, (B 2)

with CIa → 0, ψIa → 0 as z → ∞. Matching with the outer limit of the inner zonerequires CIa ∼ −2α and ψIa ∼ αz/kIa as z → 0. We find after some algebra that

CIa = −2α exp(−√k2

Ia + σ1z

), ψIa =

2αkIa

σ1

(exp(−kIaz)− exp

(−√k2

Ia + σ1z

))(B 3)

where σIa1 = 4k4Ia − 4k3

Ia. This gives the region-Ia approximation

σ′ ≈ −1− 4βk′3 + 4β2k′4, (B 4)


which is plotted in figure 6. Thus σ < −1 for 0 < k′ < β−1, a feature suggested by thecurve β = 10 in figure 4. However for β−1 � k′ � β−1/2, (B 4) reduces to

σ′ ≈ −1 + 4β2k′4, (B 5)

as shown in Table 1, implying that σ′ increases with k′ in the overlap between regionsIa and Ib. In this intermediate regime, as suggested by (B 3), three spatial zones can beidentified: C ′ rises to its maximum for z′ = O(1); ψ′ rises to its maximum and C ′ decaysto zero for z′ = O(1/βk′2); and ψ′ decays to zero for z′ = O(1/k′).

B.2. Region Ib: k′ = O(β−1/2)

In Region Ib there is again a two-zone structure: an inner zone with z′ = O(1), and adeeper recirculation zone for which z′ = O(β−1/2). In the inner zone, we set k′ = kIb/β

1/2,C ′1 = CIb(z′) and ψ′1 = ψIb(z′)β1/2 with σ′ = σIb(kIb). To leading order, (2.22) becomes

σIbCIb + kIbψ

Ibe−z′

= CIbz′z′ − CIb, ψIb

z′z′ − kIbCIb = 0 (B 6)

subject to CIb(0) = ψIb(0) = 0 and CIb → 0, ψIbz′ → 0 for z′ � 1. This eigenvalue problem

for the inner zone determines the growth rate σIb(kIb), which is plotted in figure 6. Wefind numerically that σIb(kIbc) = 0 for kIbc ≈ 1.34, i.e.

k′c = kIbcβ−1/2. (B 7)

Numerical solution of (B 6) also determines the limiting value ψIb → ψIbL(kIb) for largez′. The outer zone has a length scale z′ = zβ1/2 and, setting ψ′ = β1/2ψIb(z), (2.22)reduces to

ψIbzz − kIb

2ψIb = 0 (B 8)

to leading order. Matching to the inner problem gives the solution ψIb = ψIbL exp (−kIbz).The uniformly valid approximation to (2.22) in region Ib, namely

C = CIb ψ = ψIb + ψIb − ψIbL, (B 9)

is shown in figure 5(a) for kIb = kIbc. This illustrates how the concentration perturbationin the inner zone drives recirculation in the outer zone.

For large kIb (specifically β−1/2 � k′ � 1) with σ = O(k2Ib) and z′ = O(1), the

dominant balance of terms in (B 6) is

σIbCIb + kIbψ

Ibe−z′

= 0, ψIbz′z′ − kIbC

Ib = 0, (B 10)

reducing the leading-order eigenvalue problem to second order as

σIbψIbz′z′ + ψIbe−z

′= 0. (B 11)

Applying ψIb(0) = 0, a power-series expansion for z′ � 1 is

ψIb = a

(z′ − z′3

6σIb+

z′4

12σIb+ . . .

), (B 12)

for some normalisation constant a. For large z′, we impose the condition ψ1z′ → 0 in

order to match with the outer zone as before, which makes σIb an eigenvalue. We findnumerically that σIb ≈ 0.6917. Thus in the overlap between regions Ib and II we haveσ′ ≈ σIbβk

′2, as indicated in Table 1. As k′ increases further, the outer zone shrinks tobecome the same size as the inner zone, and we enter region II.


B.3. Region II: k′ = O(1)Taking k′ = O(1) and writing σ = βσII, C ′1 = CII(z′) and ψ′1 = βψII(z′), (2.22) reducesto

σIICII + k′ψIIe−z

′= 0, ψII

z′z′ − k′2ψII − k′CII = 0 (B 13)to leading order. Rearranging gives

ψIIz′z′ + k′2ψII

(e−z

′

σII− 1

)= 0, (B 14)

with ψII ∼ z′ for z → 0 and ψ → 0 for z′ →∞. This is another second-order eigenvalueproblem for σII(k′), the solution for which is shown in figure 6. For k′ � 1, it is evidentthat (B 14) reduces to the simpler problem (B 11) by setting σII = σIIk

′2.For large k′, it is evident from figure 6 that σII increases towards unity. We therefore

set z′ = k′−2/3z and σII = 1 − δk′−2/3 for some δ to be determined. Then to leadingorder, (B 14) becomes

ψIIzz = ψII (z − δ) , (B 15)

with ψII(0) = 0 and ψII → 0 as z →∞. Thus ψII can be written as a linear combinationof Airy functions Ai(η) and Bi(η) where η = z−δ. We discard Bi to ensure ψII is boundedfor large z. Now the largest zero of Ai, satisfying Ai(−η0) = 0, has η0 ≈ 2.338. Thus wechoose δ = η0. Thus for 1� k′ � β1/2,

ψII ≈ Ai(z′k′2/3 − η0

), CII ≈ −k′e−z

′Ai(z′k′2/3 − η0

)(B 16)

and σ′ ≈ β(1− η0k′−2/3), as reported in Table 1.

B.4. Region III: k′ = O(β1/2)Setting k′ = β1/2kIII, ψ′1 = β1/2ψIII, σ′ = βσIII, (2.22) becomes to leading order

σIIICIII + kIIIψ

IIIe−z′

=β−1CIIIz′z′ − k2

IIICIII +O(1/β), (B 17a)

β−1ψIIIz′z′ − k2

IIIψIII − kIIIC

III =0, (B 17b)

with boundary conditions (2.22c). We find that there is no longer a straightforwardseparation of scales and the stabilising diffusive terms cannot be discounted, despiteapparently being subdominant. Thus we pursue a WKB-style analysis. We first seeksolutions of the form

CIII = g(X) exp(−αβ1/2z′3/2

), ψIII = f(X) exp

(−αβ1/2z′3/2

), (B 18)

where X = z′β1/3, for some α > 0. Then (B 17) becomes

β−1/3Πg =k2IIIg + σg + kIIIf

(1−X/β1/3 +O(β−2/3)

), (B 19a)

β−1/3Πf =k2IIIf + kIIIg, (B 19b)

whereΠ ≡ ∂2

X − 3αX1/2∂X + 94α

2X − 34αX

−1/2. (B 20)Expanding in the form

g = g0 +β−1/3g1 + . . . , f = f0 +β−1/3f1 + . . . , σIII = σIII0 +β−1/3σIII1 + . . . , (B 21)

to leading order (B 19) becomes

k2IIIg0 + σIII0g0 + kIIIf0 = 0, k2

IIIf0 + kIIIg0 = 0. (B 22)


Thus σIII0 = 1−k2III and f0 = −g0/kIII, implying a short-wave cut-off wavenumber near

kIII = 1 and suggesting that the streamfunction and concentration have very similarshapes. At O(β−1/3), (B 19) gives

Πg0 = k2IIIg1 + (1− k2

III)g1 + σIII1g0 + kIIIf1 − kIIIXf0, Πf0 = k2IIIf1 + kIIIg1. (B 23)

Eliminating kIIIf1 + g1, we obtain on ODE for g0:

(1 + k2III)Πg0 = k2

III(σIII1g0 +Xg0). (B 24)

For X �1, Π ≈ 9α2X/4, so to leading order (B 24) requires that α = 23kIII(1 + k2

III)−1/2.

Then defining the new variablesX = (k2III+1)1/3X/k

2/3III and σIII1 = (k2

III+1)1/3σIII1/k2/3III ,

(B 24) becomesg0XX − 2X1/2g0X −

12X−1/2g0 = σIII1g0. (B 25)

Imposing g0(0) = 0, a power series expansion for X �1 takes the form

g0 = a(X + 2

3X5/2 + 1

6 σIII1X3 + . . .

), (B 26)

for some constant a. The normalisation constant a can be set to unity, but σIII1 remainsa free parameter. In the far field, g0 takes the form

g0 ≈ b1 exp(

43X3/2 + . . .

)+ b2 exp

(−σIII1X

1/2 + . . .), (B 27)

for some constants b1, b2. In order for CIII and ψIII1 in (B 26) to decay for large z ’, werequire b1=0, which determines σIII1.

Using (B 26), (B 25) is solved numerically using a shooting method to find σIII1 ≈−2.34. This allows us to construct the asymptotic approximations

σIII ≈ 1− k2III +

(1 + k2

III

βk2III

)1/3

σIII1 (B 28)

For kIII � 1, the approximation (B 28a) reduces to σ′ ≈ β(1 − σIII1k′−2/3). This inner

limit of the region III solution matches the outer limit of the region III solution forσIII1 = −η0, consistent with our numerical solution. Thus the estimate of the cut-offwavenumber for short-wave neutral modes is

k′ = β1/2kIIIc, kIIIc = 1− η0

(4β)1/3+ . . . . (B 29)

The first-order asymptotic approximation to the concentration and stream function pro-files are constructed via

C ′1 = g0(z′) exp(−αβ1/2z′3/2

), ψ′1 = −β

1/2C ′1k′

. (B 30)

and are plotted in figure 5(b).

Appendix C. Shallow layer asymptoticsWhen γ � 1, we can identify three distinct asymptotic descriptions of σ(k, γ).

C.1. Region I: k = O(γ−1/2)Writing k = kIγ

−1/2, ψ1 = ψIγ−1/2, C1 = CI(kI) with σ = σI(kI), (2.23) reduces to

σICI + kIψ

I(1− z) = CIzz, ψI

zz − kICI = 0, (C 1)


with boundary conditions (2.20). The solution of this eigenvalue problem σI(kI) is shownin figure 8(b); the growth rate rises monotonically from −π2/16 as k increases. The criticalwavenumber satisfying σI(kIc) = 0 is kIc =1.53, providing the asymptote

k = kIcγ−1/2 (C 2)

shown in figure 7(b). The neutrally stable long-wave eigenmode is illustrated in fig-ure 8(a). For large kI, the solute diffusion term in (C 1) becomes subdominant, σI ≈ σIk

2I

and the problem reduces to

ψIzz −

ψI(z − 1)σI

= 0, (C 3)

with ψI(−1) = ψI(1) = 0. Thus

ψI ≈ Ai

(z − 1

σ1/3I

)− bBi

(z − 1

σ1/3I

), CI ≈ (z − 1)

kIσIψI, (C 4)

where b = Ai(0)/Bi(0) and σI ≈ 0.422, as indicated in Table 2. With σ ≈ σIγk2, once k

grows to order unity we expect σ to be O(γ) and we enter region II.

C.2. Region II: k = O(1)Writing σ = γσII(k), ψ1 = ψII, (2.23) reduces to

ψIIzz + k2ψII

(1− zσII

− 1)

= 0, (C 5)

subject to (2.20). The solution of this eigenvalue problem for σII(k) is shown in figure 8(b).In the absence of solute diffusion, the growth rate grows monotonically with k. For largek, the penetration depth of the eigenmode shrinks and σII rises towards 2. Thus settingz = −1 + z/k2/3, σII = 2 − σII/k

2/3, (C 5) reduces to ψIIzz = ψII(z − σII) and, following

the argument used for (B 15), we deduce that

ψ ≈ Ai(2−1/3z − η0), σII ≈ 2− 21/3η0k−2/3. (C 6)

However once k = O(γ1/2) the stabilising effect of solute diffusion kicks in and we enterregion III.

C.3. Region III: k = O(γ1/2)Setting k = (2γ)1/2k, z = −1 + 2z, ψ1 = (2γ)−1/2ψIII(z), C1 = CIII(z) σ = 2γσ andγ = γ/8, (2.23) becomes

σCIII + kψIII(1− z) =1γCIIIzz − k2CIII,

1γψIIIzz − k2ψIII − kCIII = 0 (C 7)

with boundary conditions

CIII(0) = ψIII(0) = 0, CIIIz (1) = ψIII(1) = 0. (C 8)

We can deduce immediately by comparison with (B 17) that (via B 28)

σ = 2γ − k2 − γ2/3η0

(2γ + k2

k2

)1/3

+ . . . . (C 9)

The neutral mode is shown in figure 8(a). It arises for

k = (2γ)1/2 − η0

( γ27

)1/6

. (C 10)


This asymptote is shown in figure 7(b). Setting γ = β/8, k = k′/2 and σ = σ′/4 recoversthe short-wave, deep-domain approximation for β � 1 given by (3.6).

REFERENCES

Andres, J.T.H. & Cardoso, S.S.S. 2011 Onset of convection in a porous medium in thepresence of chemical reaction. Phys. Rev. E 83, 046312.

Armbruster, D. & Dangelmayr, G 1986 Corank–two bifurcations for the Brusselator withno-flux boundary conditions. Dynamics and Stability of Systems 1 (3), 187–200.

Armbruster, D. & Dangelmayr, G 1987 Coupled stationary bifurcations in non-flux bound-ary value problems. Mech. Proc. Camb. Phil. Soc. 101, 167–192.

Backhaus, S., Turitsyn, K. & Ecke, R.E. 2011 Convective instability and mass transportof diffusion layers in a Hele–Shaw geometry. Phys. Rev. Lett. 106, 104501.

Bestehorn, M. & Firoozabadi, A. 2012 Effect of fluctuations on the onset of density-drivenconvection in porous media. Phys. Fluids 24, 114102.

Boyd, J.P. 2001 Chebyshev and Fourier spectral methods. Dover publications.Cliffe, K.A., Spence, A. & Tavener, S.J. 2000 The numerical analysis of bifurcation prob-

lems with application to fluid mechanics. Acta Numerica pp. 39–131.De Wit, A. 2001 Fingering of chemical fronts in porous media. Phys. Rev. Lett. 87, 054502.De Wit, A. 2004 Miscible density fingering of chemical fronts in porous media: Nonlinear

simulations. Phys. Fluids 16, 163–175.Ennis-King, J. & Paterson, L. 2005 Role of convective mixing in the long-term storage of

carbon dioxide in deep saline formations. SPE Journal 10, 349–356.Ennis-King, J. & Paterson, L. 2007 Coupling of geochemical reactions and convective mixing

in the long-term geological storage of carbon dioxide. Int. J. Greenhouse Gas Cont. 1, 86–93.

Ennis-King, J., Preston, I. & Paterson, L. 2005 Onset of convection in anisotropic porousmedia subject to a rapid change in boundary conditions. Phys. Fluids 17, 084107.

Ghesmat, K., Hassanzadeh, H. & Abedi, J. 2009 The impact of geochemistry on convectivemixing in a gravitationally unstable diffusive boundary layer in porous media : CO2 storagein saline aquifers. J. Fluid Mech. 673, 480–512.

Hartline, B.K. & Lister, C.R.B. 1977 Thermal convection in a Hele–Shaw cell. J. FluidMech. 79, 379–389.

Hassanzadeh, H., Pooladi-Darvish, M. & Keith, D.W. 2007 Scaling behavior of convectivemixing, with application to geological storage of CO2. AIChE J. 53 (5), 1121–1131.

Hesse, M.A., Orr, F.M. & Tchelepi, H.A. 2008 Gravity currents with residual trapping. J.Fluid Mech. 611, 35–60.

Hewitt, D.R., Neufeld, J.A. & Lister, J.R. 2012 Ultimate regime of high Rayleigh numberconvection in a porous medium. Phys. Rev. Lett. 108 (22), 224503.

Hewitt, D.R., Neufeld, J.A. & Lister, J.R. 2013 Convective shutdown in a porous mediumat high Rayleigh number. J. Fluid Mech. 719, 551–586.

Hidalgo, J.J., Fe, J., Cueto-Felgueroso, L. & Juanes, R. 2012 Scaling of convectivemixing in porous media. Phys. Rev. Lett. 109, 264503.

IPCC 2005 IPCC special report on carbon dioxide capture and storage. Tech. Rep..Kim, M.C. & Choi, C.K. 2012 Linear stability analysis on the onset of buoyancy-driven con-

vection in liquid-saturated porous medium. Phys. Fluids 24, 044102.Kneafsey, T.J. & Pruess, K. 2010 Laboratory flow experiments for visualizing carbon dioxide-

induced, density-driven brine convection. Trans. Por. Med. 82, 123–139.Lal, R. 2008 Carbon sequestration. Phil. Trans R. Soc B 363, 815–830.Lemieux, J-M 2011 Review : The potential impact of underground geological storage of carbon

dioxide in deep saline aquifers on shallow groundwater resources. Hydrology J. 19, 757–778.Luquot, L. & Gouze, P. 2009 Experimental determination of porosity and permeability

changes induced by injection of CO2 into carbonate rocks. Chem. Geol. 265 (1), 148–159.Mitchell, M.J., Jensen, O.E., Cliffe, K.A. & Maroto-Valer, M.M. 2010 A model of

carbon dioxide dissolution and mineral carbonation kinetics. Proc. R. Soc. A 466, 1265–1290.


Neufeld, J.A., Hesse, M.A., Riaz, A., Hallworth, M.A., Tchelepi & Huppert, H.E.2010 Convective dissolution of carbon dioxide in saline aquifers. Geophys. Res. Lett. 37.

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 porouslayer. J. Fluid Mech. 500, 263–281.

Pau, G.S.H., Bell, J.B., Pruess, K., Almgren, A.S., Lijewski, M.J. & Zhang, K. 2010High-resolution simulation and characterization of density-driven flow in CO2 storage insaline aquifers. Advances in Water Resources 33, 443–455.

Ranganathan, P., Farajzadeh, R., Bruining, H. & Zitha, P.L.J. 2012 Numerical sim-ulation of natural convection in heterogeneous porous media for CO2 geological storage.Trans. Porous Media 95, 25–54.

Riaz, A., Hesse, M., Tchelepi, H.A. & Orr, F.M. 2005 Onset of convection in a gravita-tionally unstable diffusive boundary layer in porous media. J. Fluid Mech. 548, 87–111.

Riley, D.S. & Winters, K.H. 1990 A numerical bifurcation study of natural convection in atilted two-dimensional porous cavity. J. Fluid Mech. 215, 309–329.

Riley, D.S. & Winters, K.H. 1991 Time-periodic convection in porous media : the evolutionof hopf bifurcations with aspect ratio. J. Fluid Mech. 223, 457–474.

Ritchie, L. T. & Pritchard, D. 2011 Natural convection and the evolution of a reactiveporous medium. J. Fluid Mech. 673, 286–317.

Slim, A.C., Bandi, M.M., Miller, J.C. & Mahadevan, L. 2013 Dissolution-driven convectionin a Hele–Shaw cell. Phys. Fluids 25, 024101.

Slim, A.C. & Ramakrishnan, T.S. 2010 Onset and cessation of time-dependent, dissolution-driven convection in porous media. Phys. Fluids 22, 124103.

Trefethen, L.N. 2000 Spectral Methods in MATLAB . Society for Industrial and Applied Math-ematics.

Documents

Dissolution-driven porous-medium convection in the