PANACEA. Predicting and monitoring the long-term behavior of CO2 injected in deep geological formations Contract #282290

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1 PANACEA Predicting and monitoring the long-term behavior of CO injected in deep geological formations Contract #89 Deliverable Number D3.4 Title D34: Coupled dissolution and fingering processes Work-Package WP3 Lead Participant Contributors CSIC CSIC, UNOTT Version: Revision level: Due Date: month 8 Reviewed by: Tobias S. Rötting Status: Final Dissemination level PU Monday, September, 3

2 PANACEA Executive summary During geologic storage of carbon dioxide (CO), trapping of the buoyant CO after injection is essential in order to minimize the risk of leakage into shallower formations through a pre-existing well or fracture, or via the activation of a fault. Accurate tools to predict and to model the behavior of the CO plume are essential for the design, implementation and long term monitoring of injection sites. However, traditional reservoir-simulation tools are currently unable to resolve the impact of small scale trapping processes on fluid flow at the scale of a geologic basin. CSIC has studied the impact of dissolution on the up-dip migration of a CO gravity current in a sloping aquifer by means of high-resolution numerical simulations. During migration CO is trapped via convective dissolution, where dense fingers of CO-rich groundwater carry CO away from the buoyant plume as it dissolves. We analyze how the slope and system parameters affect the dissolution flux as dissolved CO accumulates beneath the buoyant plume. We show how the main features of the migrating plume, i.e., thickness and position of the leading edge, can be reproduced by using upscaled parameters in a one dimensional sharp interface model of practical applicability. UNOTT has examined two-dimensional convection in a finite-depth porous medium induced by a solute introduced at the upper boundary. The solute concentration is assumed to decay via a first-order chemical reaction, restricting the depth over which solute can penetrate the domain. Using spectral and asymptotic methods, UNOTT explored the resulting convective mixing using linear stability analysis, computation of nonlinear steady solution branches and time-dependent simulations, as a function of Rayleigh number, Damköhler number and domain size. Long-wave eigenmodes show how deep recirculation can be driven by a shallow solute field while explicit approximations are derived for the growth of shirt-wave eigenmodes. Steady solution branches undergo numerous secondary bifurcations, forming to an intricate network of mixed states. Although many of these states are unstable, some play an important role in organising the phase space of time-dependent states, providing approximate bounds for time-averaged mixing rates. Keywords CO sequestration; gravity current; convective mixing; fingering. Monday, September, 3

3 Table of Contents: Table of Contents Table of Contents Dynamics of Convective Dissolution from a Migrating Current of Carbon Dioxide (CSIC) Introduction Analogue fluid system Governing equations Effect of dissolution on CO migration Conclusions References CSIC Dissolution-driven porous-medium convection in the presence of chemical reaction (UNOTT).... Introduction.... Models and Methodologies..... Conceptual model..... Linear stability of the equilibrium state Numerical Methods Results Numerical solution of the linear stability of the equilibrium state Asymptotic limits: deep domain Asymptotic limits: shallow domain Numerical solution of the nonlinear steady states Numerical solution of time-dependent problems Conclusions Appendixes: Deep-layer high-ra asymptotics Shallow layer asymptotics References UNOTT PANACEA - 89 Deliverable D3.4 - Version. Page 3 of 35

4 . Dynamics of Convective Dissolution from a Migrating Current of Carbon Dioxide (CSIC). Introduction After injection, the buoyant CO will spread and migrate laterally as a gravity current relative to the denser ambient brine, increasing the risk of leakage into shallower formations through fractures, outcrops, or abandoned wells. One mechanism that acts to arrest and securely trap the migrating CO is dissolution of CO into the brine. Dissolved CO is considered trapped because brine with dissolved CO is denser than the ambient brine, and sinks to the bottom of the aquifer. In addition to providing storage security by hindering the return of the CO to the atmosphere, this sinking fluid triggers a hydrodynamic fingering instability that drives convection in the brine and greatly enhances the rate of CO dissolution [-4]. The interaction of convective dissolution with a migrating gravity current remains poorly understood. This is due primarily to the disparity in scales between the long, thin gravity current and the details of the fingering instability. Resolving these simultaneously has proven challenging for traditional reservoir simulation tools [5]. Upscaled theoretical models [6,7] and laboratory experiments [8,9] have recently provided some macroscopic insights, but by design these capture only the averaged dynamics of the dissolution process. Here, we study the impact of convective dissolution on the migration of a buoyant gravity current in a sloping aquifer by conducting high-resolution numerical simulations of a pair of miscible analogue fluids. Our simulations fully resolve the small-scale features of the convective dissolution process. We define an average dissolution flux and use it to study the dynamic interactions of the fingering instability with the migrating current.. Analogue fluid system For simplicity, and to focus on the role of convective dissolution, we neglect capillarity and assume that the two fluids are perfectly miscible. We adopt constitutive laws for density and viscosity that are inspired by a pair of miscible analogue fluids that have been used to study this problem experimentally [,, 8, 9]. This system captures three key features of the CO-brine system: () a density contrast that stratifies the pure fluids and drives the migration of the gravity current, () an intermediate density maximum that triggers and drives convective dissolution (discussed below), and (3) a viscosity contrast between the pure fluids that influences the shape and propagation speed of the gravity current. To trigger convective dissolution, the essential feature of the density law is that it must be a nonmonotonic function of concentration with an intermediate maximum (Fig..). This shape introduces a neutral concentration c=cm for which the density of the mixture is equal to the density of the ambient fluid. Fluid with concentration c>cn (i.e., to the right of cn) is less dense than the ambient and tends to float, whereas fluid with concentration c<cn (i.e., to the left of cn) is denser than the ambient and tends to sink. The contour of neutral concentration within the transition zone therefore emerges as a natural interface between buoyant and sinking fluids: the fluid above is buoyant and stably stratified (density decreasing as concentration increases from c=cn to c=), the fluid below is dense and unstably stratified (density decreasing as concentration decreases from c=cn to c=), and diffusion continuously transfers fluid from the stable region to the unstable region. PANACEA - 89 Deliverable D3.4 - Version. Page 4 of 35

5 Figure.: Non-monotonic density law (dimensional) inspired by miscible analogue fluids [, ]. The density has a maximum at c=cm. The contour of neutral concentration c=cn (red line) acts as an interface: mixtures with c<cn (left of the red line) are denser than the ambient brine and will sink, whereas those with c>cn (right of the red line) are buoyant relative to the ambient brine and will rise. Δρm is the characteristic density difference that drives convective dissolution and Δρgc is the one that drives the migration of the buoyant gravity current.. The dimensionless brine density is always ρ(c=)= and the dimensionless density maximum is always ρ(c=cm)=. We represent the density law with a polynomial of degree three, ρ(c) = 6.9 c c c, which has neutral concentration cn=.56, a density maximum at cm=.6, and a dimensionless CO density of ρ(c=)=-3.6. This density law is qualitatively and quantitatively similar to the true density law for mixtures of propylene glycol (c=, brine analogue) and water (c=, CO analogue) []. We choose an exponential constitutive law for the dimensionless viscosity, μ(c)=e [R(cm-c)], where we have scaled μ (c) by characteristic viscosity μm so that μ(c=cm=.6)=. The parameter R=ln(M), where M= μbrine/μco= μ(c=)/μ(c=) is the mobility ratio. This viscosity law is qualitatively and quantitatively similar to the true viscosity law for mixtures of propylene glycol and water for R~3.7 []..3 Governing equations We consider a two-dimensional aquifer in the x-z plane, with dimensional length Lx and uniform dimensional thickness Lz. The aquifer is tilted by an angle θ relative to horizontal. This can be viewed as a cross-section of a sedimentary basin taken perpendicular to a line-drive array of injection wells [, 3]. We assume that the aquifer is homogeneous and with isotropic permeability. We use the classical model for incompressible fluid flow and advective-dispersive mass transport under the Boussinesq approximation, modeling hydrodynamic dispersion as a Fickian process with a velocityindependent diffusion--dispersion coefficient. The governing equations for this model in dimensionless form are [4] uu = () uu = μμ ( ρρ(cc)ee gg ) () PANACEA - 89 Deliverable D3.4 - Version. Page 5 of 35

6 = uu + RRRR cc (3) where p is the scaled pressure deviation from a hydrostatic datum, u is the scaled Darcy velocity, and is the unit vector in the direction of gravity. ρ(c) and μ(c) are the dimensionless density and viscosity as functions of the scaled concentration c. The Rayleigh number Ra is given by RRRR = ΔΔρρ mmggggll zz φφdd mm μμ mm (4) where g is the body force per unit mass due to gravity, φ is porosity, k is the aquifer permeability, Dm is the diffusion-dispersion coefficient, Δρm is the characteristic density difference driving convective dissolution, and μm is the characteristic viscosity. To study convective dissolution from a gravity current, we solve the governing equations numerically in a rectangular domain of dimensionless height and length A=Lx/Lz=. We discretize the equations for flow and transport in space using nd-order finite volumes and 6th-order compact finite differences (4th order for boundary conditions), respectively, in a domain of x5 grid blocks. We evolve this system in time using an explicit 3rd-order Runge-Kutta scheme. Perturbations are triggered by small numerical errors [5]. We prescribe the pressure along the right boundary and take the other boundaries to be impervious. Initially, the region x 4 is filled with CO. A sequence of snapshots from a typical simulation is shown in Figure.. These results are qualitatively similar to the fingering patterns observed in experiments using water and propylene glycol, although those fluids have a much higher value of R~3.7 [, 9]. Figure.: Sequence of snapshots from a high-resolution simulation of convective dissolution from a buoyant current in a sloping aquifer for Ra=5, R=, and θ=.5 o (not shown) at dimensionless times, 3, 9, and 7. The domain extends to x=, but only x 5 is shown here. The red line marks the contour of neutrally buoyant concentration c=cn, which separates the buoyant current from the sinking fluid (Fig..)..4 Effect of dissolution on CO migration We quantify the evolution of the buoyant current with four macroscopic quantities: its mass, its length, the total dissolution rate of CO into the brine, and the average dissolution flux per unit length of the PANACEA - 89 Deliverable D3.4 - Version. Page 6 of 35

7 current. These quantities characterize the spreading and migration of the current and the effectiveness of dissolution trapping, which have implications for planning and risk assessment[6,7]. We define the dissolution flux via the non-monotonic behavior of fluid density with concentration. Since mixtures with concentration c=cn are neutrally buoyant relative to the ambient fluid, this concentration can be used to define a neutral contour separating the buoyant, mobile CO (c cn) from the dense brine with dissolved CO (c<cn; Fig..). This is an unstable equilibrium point and any perturbation of concentration causes significant buoyancy forces that trigger convection. To define the dissolution flux, we first compute the mass of buoyant fluid as MM bb (tt) = c dddd, ΩΩ ΩΩ bb(tt) = {(xx, zz): cc(xx, zz, tt) > cc nn }(Fig..3a). We bb(tt) then define the total dissolution rate as ddmm bb (tt) dddd(fig..3b). By dividing this quantity by the length of CO-brine interface, which we measure as the length of the neutral contour (Fig..3c), we obtain the average dissolution flux (Fig..3d). Both the total dissolution rate and the average dissolution flux evolve as the buoyant current migrates (Fig..3b,d). Much like for a stationary layer of CO dissolving into brine [4, 5,,, 8, 9], we distinguish three distinct regimes in convective dissolution from the migrating current: a diffusive regime at early times, a constant-flux regime during intermediate times, and a decay at late times. Both the dissolution rate and dissolution flux are small at early times as the CO-brine interface tilts from its initial, vertical orientation and diffusion-dispersion dominates. After the onset of convection (t~), the dissolution flux becomes roughly constant (t~-4), as expected for a stationary layer, and the growth of the interface slows down. Before the fingers interact significantly with the bottom boundary, our computed dissolution flux exhibits the same qualitative behavior as has been observed previously for dissolution of a stationary layer [5, 9, ]. The total dissolution rate grows strongly during this period since the interface length grows rapidly (Fig..3c) while the flux remains roughly constant. At later times (t>5), the accumulation of dissolved CO under the leftmost part of the current begins to suppress further convective dissolution there and the average dissolution flux begins to decay (Fig..3d) [8,9]. The total dissolution rate also decays (Fig..3b) even though the length of the interface continues to increase (Fig..3c), reflecting the fact that the accumulation of dissolved CO is suppressing convective dissolution along a progressively larger fraction of the interface (Fig..). PANACEA - 89 Deliverable D3.4 - Version. Page 7 of 35

8 Figure.3: We characterize the dynamics of convective dissolution from a migrating gravity current with the time evolution of four macroscopic quantities: (a) the remaining buoyant mass,, (b) the total dissolution rate, ddmm bb (tt) dddd, (c) the length of the CO-brine interface, L(t), measured as the length of the neutral contour, and (d) the average dissolution flux per unit interface length, ( LL)ddMM bb (tt) dddd. Results shown here are for R=, θ=.5 o, and several values of Ra, as indicated. As Ra increases, we find that the dynamics of this process converge to a common high-ra limit, indicating that relevant macroscopic quantities are independent of Ra for Ra~5 and higher [8]..5 Conclusions Using high-resolution numerical simulations, we have studied the detailed dynamics of convective dissolution from a buoyant current of CO in a sloping aquifer. We have found that, much like for a stationary layer of CO dissolving into brine, the dissolution flux from a buoyant current is characterized by three regimes: an early-time diffusive regime before the onset of convection, an intermediate constant-flux regime, and a late-time decay as convection is suppressed by the accumulation of dissolved CO in the brine. We have found, further, that these dynamics are independent of Ra for Ra~5 and higher (Fig..3)..6 References CSIC [] G. J. Weir, S. P. White, W. M. Kissling, Reservoir storage and containment of greenhouse gases, Transport in Porous Media 3 (996) [] E. Lindeberg, D. Wessel-Berg, Vertical convection in an aquifer column under a gas cap of CO, Energy Conversion and Management 38 (997) S9 S34. [3] J. Ennis-King, I. Preston, L. Paterson, Onset of convection in anisotropic porous media subject to a rapid change in boundary conditions, Physics of Fluids 7 (5) 847. [4] A. Riaz, M. Hesse, H. A. Tchelepi, F. M. Orr Jr., Onset of convection in a gravitationally unstable diffusive boundary layer in porous media, Journal of Fluid Mechanics 548 (6) 87. PANACEA - 89 Deliverable D3.4 - Version. Page 8 of 35

9 [5] K. Pruess, J. Nordbotten, Numerical simulation studies of the long-term evolution of a CO plume in a saline aquifer with a sloping caprock, Transport in Porous Media 9 () [6] S. E. Gasda, J. M. Nordbotten, M. A. Celia, Vertically-averaged ap- proaches for CO migration with solubility trapping, Water Resources Research 47 () W558. [7] C. W. MacMinn, M. L. Szulczewski, R. Juanes, CO migration in saline aquifers. Part. Capillary and solubility trapping, Journal of Fluid Me- chanics 688 () [8] C. W. MacMinn, J. A. Neufeld, M. A. Hesse, H. E. Huppert, Spread- ing and convective dissolution of carbon dioxide in vertically confined, horizontal aquifers, Water Resources Research 48 () W56. [9] C. W. MacMinn, R. Juanes, Buoyant currents arrested by convective dis- solution, Geophysical Research LettersDoi:./grl In press. [] J. A. Neufeld, M. A. Hesse, A. Riaz, M. A. Hallworth, H. A. Tchelepi, H. E. Huppert, Convective dissolution of carbon dioxide in saline aquifers, Geophysical Research Letters 37 () L44. [] S. Backhaus, K. Turitsyn, R. E. Ecke, Convective instability and mass transport of diffusion layers in a Hele-Shaw geometry, Physical Review Letters 6 () 45. [] J.-P. Nicot, Evaluation of large-scale CO storage on fresh-water sections of aquifers: An example from the Texas Gulf Coast Basin, International Journal of Greenhouse Gas Control (8) [3] M. L. Szulczewski, C. W. MacMinn, H. J. Herzog, R. Juanes, Lifetime of carbon capture and storage as a climate-change mitigation technology, Proceedings of the National Academy of Sciences 9 () [4] M. Ruith, E. Meiburg, Miscible rectilinear displacements with gravity override. Part. Homogeneous porous medium, Journal of Fluid Me- chanics 4 () [5] J. J. Hidalgo, J. Carrera, Effect of dispersion on the onset of convection during CO sequestration, Journal of Fluid Mechanics 64 (9) [6] E. J. Wilson, S. J. Friedmann, M. F. Pollak, Research for deployment: Incorporating risk, regulation, and liability for carbon capture and se- questration, Environmental Science & Technology 4 (7) [7] C. J. Seto, G. J. McRae, Reducing risk in basin scale CO sequestration: A framework for integrated monitoring design, Environmental Science & Technology 45 () [8] J. J. Hidalgo, J. Fe, L. Cueto-Felgueroso, R. Juanes, Scaling of convec- tive mixing in porous media, Physical Review Letters 9 () [9] A. C. Slim, M. M. Bandi, J. C. Miller, L. Mahadevan, Dissolution-driven convection in a Hele-Shaw cell, Physics of Fluids (To appear). [] G. S. H. Pau, J. B. Bell, K. Pruess, A. S. Almgren, M. J. Lijewski, K. Zhang, High-resolution simulation and characterization of density- driven flow in CO storage in saline aquifers, Advances in Water Resources 33 () PANACEA - 89 Deliverable D3.4 - Version. Page 9 of 35

10 . Dissolution-driven porous-medium convection in the presence of chemical reaction (UNOTT). Introduction Motivated by processes during CO sequestration in an underground saline aquifer (for more details see [] and []), we examine two-dimensional convection in a finite-depth porous medium induced by a solute introduced at the upper boundary. The solute concentration is assumed to decay via a firstorder chemical reaction, restricting the depth over which solute can penetrate the domain. Using spectral and asymptotic methods, we explore the resulting convective mixing using linear stability analysis, computation of nonlinear steady solution branches and time-dependent simulations, as a function of Rayleigh number, Damköhler number and domain size. Long-wave eigenmodes show how deep recirculation can be driven by a shallow solute field while explicit approximations are derived for the growth of shirt-wave eigenmodes. Steady solution branches undergo numerous secondary bifurcations, forming to an intricate network of mixed states. Although many of these states are unstable, some play an important role in organising the phase space of time-dependent states, providing approximate bounds for time-averaged mixing rates. When supercritical CO is injected into an underground aquifer, dissolution in brine causes an increase in the density of the mixture. The resulting convection enhances mixing [8, 3, 3], reducing reservoir mixing times from thousands to hundreds of years [4]. To investigate this process, researchers have drawn on, and significantly extended, the substantial literature on Rayleigh Bénard convection in porous media. Recent attention has focussed on the onset-time of convection following the introduction of solute from an upper surface [7, 3, 8] and the asymptotic transport properties of high-rayleighnumber convection [4, 3, 7, 5, 6]. Associated experiments have exploited the analogy between porous-medium flow (described most simply by the Darcy equation coupled to a solute transport equation) and flow in a Hele Shaw cell [, 4, 9, 3, 3]. The chemical interaction between CO and brine is complex, with a sequence of reactions taking place leading to acidification and ultimately to a reaction with the host rock. The reaction timescales span many orders of magnitude [], which makes great demands of computational studies that attempt to integrate flow and chemistry. This has motivated fundamental studies where chemistry is represented in a highly simplified manner, either as a first-order reaction (where the dissolved CO reacts with an abundant substrate; []) or a second-order reaction (where the substrate may be depleted at the leading edge of the advancing solute field; [9, ]). Related studies, addressing the effect of chemical reaction on Rayleigh Taylor instability, have employed nonlinear reaction terms that allow for chemical front propagation [6, 7]. It is well established that consumption of dissolved CO prevents the solute from penetrating deep into the underlying fluid and delays the onset of convection [9]. However the resulting convective flows remain poorly characterised. In the present study, we consider two-dimensional porous-medium convection in the presence of a first-order reaction. This is typically described by three parameters (a Rayleigh number, a Damköhler number and the domain aspect ratio), although for sufficiently deep layers two parameters suffice []. We keep the description of the chemistry deliberately simple, in order to understand how a linear reaction term influences the structure of the resulting dynamical system. We focus first on computing steady solutions and their stability, comparing the structure of primary and secondary solution branches to the reaction-free case in a box of finite depth [8, 9]. We then show how the pattern of steady solutions provides a framework for understanding the long-term mixing processes at high Rayleigh number when the system is inherently unsteady. Numerous physical features of real porous media are neglected, including heterogeneity and disorder [5, 6], anisotropy of the medium [], hydrodynamic dispersion of the solute, capillary trapping [4] and three dimensionality [5]. We consider a planar domain with the solute concentration held constant at the upper surface. Previous studies have focused on the stability of the thickening boundary layer []; here we examine the PANACEA - 89 Deliverable D3.4 - Version. Page of 35

11 stability of the horizontally uniform steady state. We first conduct a linear stability analysis ( 3) and derive asymptotic predictions for 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 accuracy in space; arc-length continuation and numerical bifurcation tracking demonstrate how mode 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 solvent into the system fluctuates between values associated with a subset of the steady states.. Models and Methodologies.. Conceptual model We consider the dissolution of a solute in a two dimensional homogeneous porous medium. The solute undergoes a first-order A B reaction, where species A (but not B) increases solution density. The equations governing this system are Darcy s law (), the convection-diffusion-reaction equation () and the continuity equation (3): * K * * u = ( p ρgc zˆ ), () µ * * * * * ϕc t * + u C = ϕd C αc, () * u =. (3) where C *, p * * * * and u ( u, w ) are the concentration of the species A dissolved in the solution, pressure and velocity respectively. The parameters K, µ, D, ϕ, α, g and ρ are respectively the medium permeability, solvent viscosity, solute diffusion coefficient, medium porosity, reaction rate of the solute, gravity and the density increase per unit concentration of solute. We assume the solution density is linear in C * and that ρ is sufficiently small to apply the Boussinesq approximation. Equations ()-(3) are defined on a domain with horizontal length L and vertical depth H, i.e. x [, L ] and z [ H, H ], where gravity acts in the z direction. The initial solute distribution is assumed known, with C * = Ĉ * at time t * =. Equations ()-(3) are subject to the boundary and initial conditions * * * * * * * C ( x, H, t ) = C, C ( x, H, t ) =, (4) z * * * * * * * w ( x, H, t ) =, w ( x, H, t ) =, (5) * * * * * * * * * * * C * (, z, t ) = C * ( L, z, t ) =, u (, z, t ) = u ( L, z, t x x * ) =, (6) * * * ˆ * * * * C ( x, z,) = C, u ( x, z,) =. (7) According to the above boundary conditions, the solute concentration is held constant at the upper boundary and there is no flux of solute across the lower and lateral boundaries. Equations ()-(3) are non-dimensionalized using ( x, z) = (x * *, z )/ H = Hg ρc * p * and u=u * µ / Kg C * ρ, yielding, t = t * * ρkgc / Hϕµ, C = C u = p + Czˆ, (8) C t + u C = Ra C DaC, (9) u =, () PANACEA - 89 Deliverable D3.4 - Version. Page of 35 * /C *, p

12 subject to the boundary conditions and initial condition C( x,, t) =, C ( x,, t) =, () z w( x,, t) =, w( x,, t) =, () C (, z, t) = C ( L, z, t) =, u(, z, t) = u( L, z, t) =, (3) x x Cˆ ( x, z,) = C, u =, (4) where L =L/H is the aspect ratio of the box and C = Ĉ * * / C. The Rayleigh and Damköhler numbers are defined here as * HKgC ρ αhµ Ra =, Da =. (5) * ϕµ D KgC ρ As a result, (8)-() with boundary conditions ()-(3) and initial condition (4) produce a system dependent on Ra, Da, and L. When RaDa = α H / Dϕ = O(), the time taken for the solute to diffuse across the box is comparable to the reaction time. For sufficiently large RaDa, therefore, we may expect the domain depth to have negligible influence on the convection. To describe this deep-domain case, we introduce the H- independent parameters Ra KgC ρ β = =, Λ = RaDaL, (6) Da αdϕ µ with which we define the new variables (x,z ) = Da ( x, ( z + ) ) z ), t =tra/ β, (u,w ) = β (u,w), p =pra. Equations ()-(3) become The boundary conditions become Ra, C = β C, C = β C, =( / x, / u ' = p + C zˆ, (7) C t + u' C = C C, (8) u' =. (9) C ( x,, t ) =, C as z, β () z w ( x,, t ) =, w as z, () C x (, z, t ) = C x ( Λ, z, t ) =, u (, z, t ) = u ( Λ, z, t ) = () with initial condition C ˆ'( x, z,) = C'. (3) The boundary conditions (-a) and () depend only upon the parameters β and Λ ; a similar reduction was employed by []. We define the Sherwood number, Sh, a dimensionless measure of the solute flux across the upper boundary, by Sh = L C z z = dx. (4) * The corresponding dimensional flux per unit length is D C Sh. ; when Sh is proportional to Ra, the net flux is independent of molecular diffusivity D, and when proportional to Ra/Da, it is independent of both D and H. The total mass of dissolved solute, M ( t) = L Cdxdz, satisfies PANACEA - 89 Deliverable D3.4 - Version. Page of 35

13 Sh M t = DaM. (5) Ra The mean square concentration, obtained by multiplying (8) by C and integrating over the domain, satisfies Sh M t = M N, (6) Ra L L where M = C dxdz and N = C dxdz. Multiplying (9) by u and integrating over the domain implies L L dxdz = wcdxdz, u u (7) Identities (5 7) provide useful computational validation. In the deep-domain case we define Λ Sh' = C z z dx = (8) where Sh = Sh' / β... Linear stability of the equilibrium state Equations (8)-(3) admit the no-flow steady solution cosh RaDa ( z) C z) =, cosh( RaDa ) for which Sh = L RaDa tanh( RaDa ). ( ) =, ( u (9) We investigate small disturbances to this state of the form C( x, z, t) = C( z) + Re[ C( z)cos( kx) exp( σt) ], (3) u( x, z, t) = u( z) + Re[ u( z)sin( kx) exp( σt) ], (3) where σ and k are the growth rate and wavenumber of perturbations respectively; we assume k = nπ/l for n =,,3, to ensure disturbances fit within the domain, and allow σ to be complex. Writing (8) in terms of the stream function, such that z = u and x = w, eliminating the pressure and linearising about the base state, gives k σ C k C z = Czz C DaC, Ra Ra (3) zz k kc =, (33) subject to the boundary conditions C ) = C () =, ( ) = () =. (34) ( z This is an eigenvalue problem for = σ ( k, Ra, Da) (k, Ra, Da) = σ for any k. σ. We define Ra c (Da) as the smallest Ra satisfying In the deep-domain case ( RaDa >> ), for which we might expect the solution to be confined to a boundary layer at the top of the domain, the base state plus perturbations are with σ =σ Ra/ [ ], [ z x σ t ], = β and k= Ra Da C = exp( z ) + Re βc( z )cos( k x ) exp( σ t ) ( u w ) = Re (, ') sin( k x ) exp( ) β (35), (36) β, k. Linearising in C and, we obtain from (7)-(8) PANACEA - 89 Deliverable D3.4 - Version. Page 3 of 35

14 ( + k ), σ ' C + k exp( z ) = C z z C (37) k k =. (38) z z ' βc In this approximation, the boundary conditions are given by C = () = ; C and as z. (39) () z This gives an eigenvalue problem for σ = σ (k', β which σ ( k, β ) =. ). We define c β as the smallest β for any k for.3 Numerical Methods The linear stability problem (3)-(34) was solved numerically using a spectral method. C and were approximated in z as a truncated sum of N Chebyshev polynomials and were evaluated at the ( j ) π Chebyshev collocation points k j = cos, for j=...n, yielding an eigenvalue problem of the N form C C σb = A, (4) for some N N matrices A and B. The spectral accuracy of the approximation was validated by comparison wtih the exact solution for k =, namely σ = π /(6Ra) Da. The non-linear equations (8) and () subject to relevant boundary conditions were also solved using a spectral method, using N Chebyshev polynomials (satisfying the vertical boundary conditions) in z and a truncated cosine series (using M modes) in x. The problem was formulated as follows. Equations (8) and () imply that p = C, with boundary conditions () becoming p z ( x, ±, t) = C( x, ±, t). To z pdxdz = specify a unique solution, we imposed the additional constraint, allowing the pressure to be considered as a function of concentration, p = p( C). Thus (8) yields u = u( C), making (9) a non linear evolution equation for C. Steady solutions ( Cs, s ) were obtained using Newton s method, yielding the concentration at each of the N M collocation points with spectral accuracy; the Crank Nicholson method was employed to evolve the solution in time. The linear stability of a steady solution ( Cs, s ) was calculated by forming an eigenvalue problem using Chebyshev differentiation operators in the z-direction and cosine series differentiation operators in the ( j) x-direction. In particular, the steady state (9) in a box of width L yields eigenmodes C which can be labelled by j =,,3, according to the number of half-wavelengths of disturbance in the box. As we illustrate below, it is helpful to project nonlinear solutions C onto these eigenmodes (evaluated at convenient nearby parameters) using L ( j) ( j) C C C z jπ x L f g (4) =, ( )cos( / ), where, L fgdxdz. Such projections help reveal the symmetries of nonlinear solution branches. We used arc-length continuation to determine how nonlinear steady solutions change with a parameter, allowing for solution branches with turning points. In addition to solving (8) and () at each parameter value using Newton s method, we introduced an arc-length parameter, λ, satisfying ( C C ) C i i i ( λi λi ) λi = δs, (4) PANACEA - 89 Deliverable D3.4 - Version. Page 4 of 35

15 where ( C i, previous step ( C i, λ i ), and δ s is the step size. This approach accounts for the curvature of the solution branch and allows continuation around turning points. λi ) is the tangent to the solution branch with respect to the stepping parameter at the Computing the number of unstable eigenvalues along each step of the solution branch also allowed us to locate secondary bifurcations and turning points with respect to a bifurcation parameter. The principle of exchange of stabilities then suggested whether the bifurcation diagram is consistent, and helped determine whether all secondary bifurcations had been found for a given parameter range. At bifurcation points, linear eigenvectors were used to perturb the solution, providing a means of locating the bifurcating solution branch. We were able to validate these numerical techniques by reproducing the bifurcation diagram of the Rayleigh- Bénard problem in a square cavity presented by [8]..4 Results.4. Numerical solution of the linear stability of the equilibrium state The eigenvalue problem (3)-(34) was solved numerically to obtain the growth rates of perturbations. Figure. shows how, for fixed Da, increasing Ra leads to a band of wavenumbers for which perturbations grow. The most unstable wavenumber increases with Ra. For Da =., for example, the critical Ra for the onset of instability is Ra 4.6 c. Figure.: Growth rates of the perturbations about the uniform steady state for Da=. and indicated k and Ra. Neutral stability curves in Figure.( a ) illustrate the values of Ra and Da for fixed k at which the onset of linear instability occurs. For parameters above a neutral stability curve, perturbations of that wavenumber are linearly unstable. The neutral stability envelope bounding all such curves (Figure /.a), Ra = Ra c (Da), increases like Da for Da << and like Da for large Da. The corresponding wavenumber k (Da c ) of neutral modes, satisfying σ ( k c,ra c,da) =, tends to a constant for Da << and rises like Da for large Da (Figure.b). For given Da and Ra > Ra c (Da), there are two wavenumbers for which σ = bounding the range of unstable eigenmodes, corresponding to a shortwave and a long-wave neutral mode; these cut-off wavenumbers are represented by the intersection of two lines k = constant in the (Da,Ra)-plane. For given k, we find that there is a maximum Da above PANACEA - 89 Deliverable D3.4 - Version. Page 5 of 35

16 which longer-wavelength modes are stable; thus larger Da favours modes of shorter wavelength, corresponding to shrinkage of the underlying concentration boundary layer. (a) (b) Figure.: (a) Neutral stability envelope (bold line) in ( Da,Ra) -space, plus asymptotic limits (dashed, showing (5) for Da << and (43) for RaDa >> ), and neutral stability curves for given wavenumbers. Points below the neutral stability envelope are linearly stable. Dots show the turning point of the neutral curve for given k with the solid straight line giving the asymptotic approximation (45). (b) Critical wavenumber at which linear instability first occurs (solid), and showing the asymptotic limits for Da << (5) and for RaDa >> (43). Figure.(a) gives stability boundaries for the problem with no-flux boundary conditions in a box of width L for k = nπ/ L, n =,,3,. Figure.3 shows the corresponding effect of varying L for fixed Da =.. For small L, any mode becomes unstable for sufficiently large Ra, whereas for sufficiently large L lower-order modes remain linearly stable. For L = π, for example, mode solutions are always stable (Figure.a), but higher modes bifurcate from the no-flow state in the sequence shown in Figure.3; low-order modes restabilise when Ra is sufficiently large. Modes j and j + bifurcate simultaneously when L.595π ( j = ), L.65π ( j = ); below, we will explore how finiteamplitude solution branches interact in the neighbourhood of these crossing points. As we saw for Da =., i.e. Figure., the lowest value at which any mode becomes unstable is Ra 4.6 PANACEA - 89 Deliverable D3.4 - Version. Page 6 of 35 c

17 (corresponding to the neutral stability envelope); this threshold is indicated by the circular symbols in Figure.3. Figure.3: Neutral stability curves showing the critical Ra for the onset of linear instability as L varies for Da=. and indicated mode number. Dots denote the modal onset points for increasing Ra. The horizontal line shows the approximate stability envelope (43) with approximate modal minima (44) shown with squares. The turning points for given k (asterisks) are approximated by (46, open circles). The asymptotic approximation (48) for the onset of convection for small L and large Ra is illustrated by the dotted line..4. Asymptotic limits: deep domain To understand the structure of the solutions arising at high Ra, we turn to the stability problem (37)-. Figure.4(a) shows how (39), governed by the single parameter β, applying in the limit RaDa >> the growth rate σ depends on wavenumber k in this case: an unstable band of wavenumbers exists for β > 9.9, and the neutral wavenumber at onset (satisfying k, β ) = ) is k.. Thus β c σ ( c c the envelope of neutral curves and critical wavenumbers in Figure.(a,b) asymptote respectively to Ra= β Da, k = k Da (RaDa >> ). (43) c β c c This turns out to provide a good estimate for the minimum Ra for stability even when Da =., as shown in Figure.3. Likewise, we can estimate the domain lengths at which mode n becomes unstable at this minimum Ra, namely L n = ( n =,,3, ), (44) π β c k c Da points that are indicated with squares in Figure.3. c PANACEA - 89 Deliverable D3.4 - Version. Page 7 of 35

18 k ' resulting in Figure.4: (a) Growth rates obtained from (37)-(39), showing the values of instabilities for values of β as indicated. (b) Neutral stability envelope obtained from (37)-(39) showing the critical value of β resulting in the onset of linear instability, along with the asymptotic limits for k << and k >> from (59) and (8) respectively. The dot illustrates -. β and k at which the gradient is Dots on Figure.( a ) show turning points on the neutral stability curves, occurring when σ / Ra = for constant k. This translates to the point where d log /d log k = which β = 34.4 and k = k.38. Thus the turning points lie on the asymptote β β (marked on Figure.4(b )) at Ra = β Da, (45) as illustrated in Figure.( a ), sitting at Da = k/( β k ). Correspondingly, the the turning points of the neutral curves in Figure.3 are well approximated by nπ L =, Ra = β Da ( n =,,3, ). (46) Daβ k This defines the maximum domain length in which mode n is linearly unstable. For large β, Figure.4(b) shows the existence of long- and short-wave neutral modes for which = ( β / / 3 /4 3 /4 k O ) and k = O( β ), corresponding to k = O((RaDa ) ) and k = O((Ra Da) ) respectively. Thus sufficiently far above Ra c in the ( log Da,log Ra) -plane (Figure.a), with RaDa >>, lines of constant k intersect with slope -3 and -/3 respectively, representing the long- and short-wave cut-offs 4 respectively. The corresponding eigenmodes for β = are illustrated in Figure.5; recall from (35)- (36) that the base state is proportional to exp( z ). The long-wave neutral mode (Figure.5 a ) has its concentration perturbation confined to a depth comparable to the base state, but it drives a deeper recirculating flow which penetrates a distance comparable to the wavelength of the perturbation. In contrast, the short-wave mode (Figure.5b ) has concentration and streamfunction perturbations that are almost identical, but which are confined to a thin boundary layer that is much shorter than the penetration depth of the base state. PANACEA - 89 Deliverable D3.4 - Version. Page 8 of 35

19 4 Figure.5: (a) Concentration and streamfunction profiles of neutrally stable eigenmodes for β = and (a) k =.3 and (b) = 93.. Solutions to (37)-(39) are shown with lines and asymptotic c k c approximations (6), (83) with dots. The maximum concetration is set to unity in each case. Further analysis (for details see Appendix 8) reveals that σ ( k, β ) has four distinguished asymptotic limits when β >>, identified as regions Ia, Ib, and I in Table, two of which capture the neutral modes. In each distinguished limit, the -parameter eigenvalue problem (3)-(34) can be reduced to a distinct -parameter problem. In the overlaps between these regions, σ can be captured by simpler zero-parameter approximations (after appropriate rescaling). The predictions of σ in each distinguished limit are compared to a numerical prediction of the growth rate in Figure.6, showing good agreement for large β. Table : Summary of the asymptotic approximation of (37)-(38) for β >> for varying orders of k. Four regions (Ia, Ib,, I) are identified along with intermediate limits, where ˆ σ Ib.697, and η.338. σ Ib and σ are solutions to the eigenvalue problems (58) and (65) respectively. The table shows the approximation of σ, the length scales of the concentration and stream function profiles, and the magnitude of, given that maxc =. k σ z Ia O ( β ) 3 4 4βk + 4k β, β β / k / 4 β <<< k << β + 4k β, / β k, / k /( β k 3 ) Ib / / / / O ( β ) σ Ib ( k β ), β β / β << k << ˆ σ Ib βk, / k β k O () βσ (k ) β / /3 << k << β β ( ηk ) k /3 β / k I / O ( β ) /3 β / β /3 /3 k + β β k ηβ k PANACEA - 89 Deliverable D3.4 - Version. Page 9 of 35

20 Figure.6: Growth rates obtained from (37)-(39) (solid) compared with the asymptotic 4 approximations for β >> shown in Table for β =, the constant c =. is used to offset the growth-rate before plotting its log. In regions Ia and Ib, where perturbations are long compared to the penetration depth of the base state, the eigenmodes have a boundary-layer structure, for which the concentration field decays to zero much more rapidly than the streamfunction as z increases. The eigenmode illustrated in Figure.5( a ) is well captured by the region-ib approximation (6). The corresponding criticality condition (59) provides the long-wave asymptote shown in Figure.4(b ). The condition for the recirculation depth of the neutrally stable long-wave mode to be less than the depth of the domain is Ra >> Da, consistent with the underlying assumption of large β. In Region, with k = O(), which represents rapidly growing modes, the stream function and concentration vary over the same vertical lengthscale. However diffusion and reaction of the solute perturbation do not arise at leading order, so that the eigenvalue problem reduces to second order (65) and the predicted asymptotic growth rate βσ (k ) increases monotonically with k. However, once / k = O( β ) (Region I), solute diffusion becomes important and we can recover the short-wave neutral mode asymptotically using a WKB method, as illustrated in Figure.5(b ). The eigenmode forms /6 shallow and slender recirculation cells with depth-to-width aspect ratio scaling like β. The prediction (8) for the wavenumber of the short-wave neutral mode, which can be written /6 / ηβ k c = β + ( β >> ), (47) /3 4 where η.338 is the largest zero of Ai( η ), is shown in Figure.4(b ). Equation (47) also captures the onset condition for convection in a narrow domain at high Ra, given by nπ / = (RaDa) k c, (48) L which is illustrated in Figure.3 for n =. Region I also contains the most rapidly growing modes, for 3/8 which (from (8)) k = ( η β/3), with eigenmodes resembling those in Figure.5(b ). From this we can estimate that, for RaDa?, the number of recirculation cells n f initially arising in a large domain is approximately 3/8 L η /6 5/6 n f Ra Da. (49) π 3 PANACEA - 89 Deliverable D3.4 - Version. Page of 35

21 .4.3 Asymptotic limits: shallow domain When RaDa <<, vertical diffusion dominates reaction and the solute can penetrate to the base of the domain. The steady state is almost uniform with a small parabolic deviation due to reaction. We set Da = γ Ra and σ = ~ σ / Ra, where γ and ~ σ are order unity as Da. Then (9) may be approximated by C z ( γ /Ra)( z), so that (3), (33) becomes ~ σ C + γ k( z) = C k, (5) zz C zz k =, kc (5) with boundary conditions (34). The reaction term affects the base state but not the perturbations in (5)-(5). Solving the eigenvalue problem for ~ σ ( k, γ ), we see how increasing γ destabilizes the linear system, increasing the range of unstable k and maximum growth rate of perturbation (Figure.4a below). Instability arises for γ > γ c where γ c 7.733, with onset wavenumber k c.. Thus the envelope of neutral curves and critical wavenumbers asymptote respectively to as Da, as illustrated in Figure.(a,b). γ c Ra =, k = Da For γ >>, we see from Figure.4(b ) below how the short- and long-wave neutral modes have / / / k γ = (RaDa ) and k γ = (RaDa / ) respectively, implying that lines σ = for constant k all have slope in the ( log Da,log Ra) -plane (Figure. a ). The long-wave neutral mode occupies the full depth of the domain, whereas the short-wave mode is confined to a boundary layer near the upper surface (Figure.7). An analysis of (5)-(5) for γ >>, outlined in Appendix 9, reveals that ~ σ ( k, γ ) has three distinct asymptotic limits, labelled I I in table, with regions I and I capturing the neutral modes. The mathematical structure of regions and I corresponds closely to that identified in the deep-layer case; however the boundary-layer behaviour seen at smaller wavenumbers (regions Ia and Ib in table ) does not arise. The asymptotic predictions of the neutral eigenmodes and growth rates 4 are accurate for γ = (Figure.7 ( a, b) ). The lower boundary has an influence only in regions I and ; the modes in region I are identical up to rescaling with those in the deep-layer case. k c (5) Figure.7: (a) Concentration and stream function profiles of neutrally stable eigenmodes, showing 4 solutions to (5)-(5) (lines) and asymptotic approximations (84)-(9) (dots) for = and k = 44.4 γ and k =.. The long-wave mode occupies the whole domain while the short-wave mode is confined to a boundary layer near the upper surface. (b) Growth rates obtained from (3.) with the 4 asymptotic approximations for γ >> (dashed, see table ) for γ = and c =.. PANACEA - 89 Deliverable D3.4 - Version. Page of 35

22 Table : Summary of the asymptotic approximation of (5)-(5) for neutral modes with γ >> where ˆ σ I.4 and.338 η. σ I and σ are solutions to the eigenvalue problems defined in (84) and (88) respectively. The table shows the approximation of σ ~, the length scales of the concentration and stream function profiles, and the magnitude of relative to C. k ~ σ z I / / O ( γ ) σ I ( kγ ) / γ / γ << k << ˆ σ I k / k O () γσ ( k) / << k << γ /3 /3 γ ( η / k ) /3 k k I O ( γ / ) /3 /3 k + γ γ k η γ k /3 γ / γ.4.4 Numerical solution of the nonlinear steady states Figure.8 shows the ratio of Sherwood number (4) and Ra for steady solutions of (8)-(4) arising in a box of width L = π for Da =.. At the anticipated bifurcation points (see Figure.3), mode-,3,4, solutions bifurcate supercritically from the base state. 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 flow is upward (downward) adjacent to the right-hand wall of the box. Figure.8 reveals additional mixed-mode steady solutions arising via secondary bifurcations, and turning points in the primary solution branches. For example, the mode- branch does not extend beyond Ra 67. Secondary bifurcations of the mode-5,6,7, branches are not shown in Figure.8. Equation (6) shows that, for steady states, Sh/Ra is a positive-definite functional of C s and Cs ; it is straightforward to show that solutions bifurcating from the uniform state must increase Sh. Recirculation enhances mixing, although we found no clear relationship between Sh and mode number or Ra, at least over the range of parameters investigated. The figure highlights the complexity of the system even at moderately low Ra. We now examine in more detail the stability and structure of the primary and mixed states. PANACEA - 89 Deliverable D3.4 - Version. Page of 35

23 Figure.8: Change in the Sherwood number for steady states with different modes as Ra varies. The steady states are calculated for Da=. and L = π using the no-flux boundary conditions. This shows steady-states with indicated modes bifurcating from the uniform state along with secondary bifurcation branches. To explore the bifurcation structure of the system, we project the solution branches onto the underlying eigenmodes using (4). Figure.9 shows the projections of the steady solutions illustrated in Figure.8, with thick lines denoting stable states. Of the two primary mode- solutions, that for which fingers descend adjacent to the walls of the domain is stable as far as the turning point at Ra 67, whereas the solution with a finger descending in the middle of the domain loses stability to a mixed mode at Ra 5.5. This is because the no-flux boundary conditions suppress some classes of disturbances to the wall-bound fingers. A mixed mode (illustrated by example A-) arises subcritically and coexists with an unstable solution of opposite parity (A+); the latter has a turning point, regaining stability at Ra 46.6 before turning into a pure mode-3 state. This mode-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 strong differences between states with fingers attached to the walls of the () (4) domain ( C >, C > ; example C), those with fingers not adjacent to boundaries (example B) and () (4) those with mixed properties ( C <, C >, example D). A pure mode-4 state is stable for 6.4 < Ra < 9, which in turn loses stability to a new mixed state as Ra increases beyond 9 (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.9 show a notable difference from the bifurcation structure of the Rayleigh Bénard problem [8], illustrated for comparison in Figure. (this problem has Da= in () and the no-flux condition in () replaced by C ( x,, t) =, making the problem symmetric about z = ). The two primary solution branches of each mode in the Rayleigh Bénard problem have the same linear stability, whereas solutions of (8)-(4) lack this vertical symmetry of problem. The inclusion of the chemical reaction does not alter the (3) (3) symmetry of odd modes (solutions with C <, for example, resemble those with C > after reflection about x = L/ ), whereas it breaks the symmetry of even modes (so that solutions with () (4) () C > or C > have descending fingers attached to no-flux boundaries, unlike the cases C < or (4) C < ). Furthermore, inclusion of the reaction term truncates bifurcations from the base state at lower Ra, while for L = π we also find the reaction term prevents the existence of the mode- state present in the Rayleigh Bénard problem. PANACEA - 89 Deliverable D3.4 - Version. Page 3 of 35