2.1. Governing equations

The dynamics of the velocity field $\boldsymbol {u}$ is governed by Darcy's law and the continuity equation. Strictly speaking, the fluid viscosity is not constant due to the dissolution of CO$_2$. For instance, at the temperature of 323 K and pressure of 30 MPa the viscosity of an aqueous solution of CO$_2$ increases from $5.64\times 10^{-4}$ to $5.73\times 10^{-4}\,{\rm Pa}\,{\rm s}$ when the CO$_2$ mole fraction increases from 0.0086 to 0.0168 (McBride-Wright, Maitland & Trusler Reference McBride-Wright, Maitland and Trusler2015). Nevertheless, this change in viscosity is relatively small and in the current study we employ a constant viscosity for the whole reservoir. The temperature $T$ and concentration $S$ obey the advection–diffusion equations. We denote the vertical coordinate by $z$ and the two horizontal coordinates by $x$ and $y$, respectively. Then the full governing equations read

(2.1a)\begin{gather} \boldsymbol{\nabla} \boldsymbol{\cdot} \boldsymbol{u} = 0, \end{gather}

(2.1b)\begin{gather}\boldsymbol{u} ={-}\frac{K}{\mu}[\boldsymbol{\nabla} p+(\beta_S S - \beta_T T) \rho_0 g \boldsymbol{e}_z], \end{gather}

(2.1c)\begin{gather}\frac{(\rho c)_m}{(\rho c_p)_f}\frac{\partial T}{\partial t} ={-}\boldsymbol{u} \boldsymbol{\cdot} \boldsymbol{\nabla} T+\kappa_m \nabla^{2} T, \end{gather}

(2.1d)\begin{gather}\phi \frac{\partial S}{\partial t} ={-}\boldsymbol{u} \boldsymbol{\cdot} \boldsymbol{\nabla} S+\phi \kappa_S \nabla^{2} S. \end{gather}

Here, $p$ is the pressure, $g$ is the gravitational acceleration, $\boldsymbol {e}_z$ is the vertical unit vector, $\mu$ is fluid viscosity, $c_p$ is the specific heat of the fluid at constant pressure, $c$ is the specific heat of the solid, $\kappa _m$ is the overall thermal diffusivity and $\kappa _S$ is the molecular diffusivity of the concentration field, respectively. The subscript ‘$m$’ stands for the effective properties of the whole porous medium, including both solid and fluid. The boundary conditions at the top and bottom boundaries are

(2.2a)\begin{gather} u_z = 0,\quad T = T_{bot},\quad \partial_z S = 0,\quad\mbox{at}\ z=0, \end{gather}

(2.2b)\begin{gather}u_z = 0,\quad T = T_{top},\quad S = S_{top},\quad\mbox{at}\ z=H. \end{gather}

Note that we do not introduce boundary conditions for the tangential components of the velocity at the boundaries since, at the top and bottom boundary, the tangential velocity is balanced by the tangential gradient of pressure. Periodic boundary conditions are applied in the horizontal directions for all quantities.

The above governing equations are non-dimensionalized by using the reservoir height $H$, the temperature difference ${\rm \Delta} T$, the concentration at the top boundary $S_{top}$, the characteristic velocity $U_c=Kg\rho _0\beta _S S_{top}/\mu$ and the characteristic time scale $t_c = \phi H / U_c$, respectively. Then the non-dimensionalized equations are

(2.3a)\begin{gather} \boldsymbol{\nabla} \boldsymbol{\cdot} \boldsymbol{u} = 0, \end{gather}

(2.3b)\begin{gather}\boldsymbol{u} ={-}\boldsymbol{\nabla} p+ (\varLambda T -S)\boldsymbol{e}_z, \end{gather}

(2.3c)\begin{gather}\sigma \frac{\partial T}{\partial t} ={-}\boldsymbol{u} \boldsymbol{\cdot} \boldsymbol{\nabla} T+ \frac{{\textit{Le}}}{{\textit{Ra}}_S}\nabla^{2} T , \end{gather}

(2.3d)\begin{gather}\frac{\partial S}{\partial t} ={-}\boldsymbol{u} \boldsymbol{\cdot} \boldsymbol{\nabla} S+ \frac{1}{{\textit{Ra}}_S}\nabla^{2} S . \end{gather}

From now on, all the flow quantities are referred to their non-dimensionalized values unless otherwise mentioned. The dimensionless control parameters include the heat capacity ratio $\sigma$, the Lewis number ${\textit {Le}}$, the Rayleigh number of concentration field ${\textit {Ra}}_S$ and the thermal Rayleigh number ${\textit {Ra}}_T$, which are respectively defined as

(2.4a–d)\begin{equation} \sigma = \frac{(\rho c)_m}{\phi(\rho c_p)_f},\quad {\textit{Le}} = \frac{\kappa_m}{\phi\kappa_S}, \quad {\textit{Ra}}_S = \frac{KHg\rho_0\beta_S S_{top}}{\kappa_S \mu \phi}, \quad {\textit{Ra}}_T = \frac{KHg\rho_0\beta_T{\rm \Delta} T}{\kappa_m\mu}. \end{equation}

The density ratio $\varLambda =(\beta _T{\rm \Delta} T)/(\beta _S S_{top}) = {\textit {Ra}}_T {\textit {Le}} / {\textit {Ra}}_S$ will also be used below to measure the strength of the temperature difference relative to the initial concentration difference. The non-dimensionalized boundary conditions are the top and bottom boundaries are

(2.5a)\begin{gather} u_z = 0,\quad T = 1,\quad \partial_z S = 0,\quad \mbox{at}\ z=0, \end{gather}

(2.5b)\begin{gather}u_z = 0,\quad T = 0,\quad S = 1,\quad \mbox{at}\ z=1. \end{gather}

2.2. Physical properties of the fluid and reservoirs

In order to determine the parameter range of the current study, we review the typical reservoir conditions reported in the existing literature. Considering the reservoirs as a porous medium, the permeability is usually in the range of $K=10^{-15}\unicode{x2013}10^{-12}\,\mathrm {m^2}$ (Kopp, Class & Helmig Reference Kopp, Class and Helmig2009; Huppert & Neufeld Reference Huppert and Neufeld2014; Emami-Meybodi Reference Emami-Meybodi2017) and the porosity in the range of $\phi =0.05\unicode{x2013}0.4$ (Bachu & Adams Reference Bachu and Adams2003; Van Der Meer Reference Van Der Meer2005; Hassanzadeh, Pooladi-Darvish & Keith Reference Hassanzadeh, Pooladi-Darvish and Keith2006; Kopp et al. Reference Kopp, Class and Helmig2009; Emami-Meybodi Reference Emami-Meybodi2017). The reservoir thickness can vary from $H=10$ to $300\,\mathrm {m}$ (Bachu & Stewart Reference Bachu and Stewart2002; Bachu & Adams Reference Bachu and Adams2003; Ennis-King & Paterson Reference Ennis-King and Paterson2003; De Silva, Ranjith & Perera Reference De Silva, Ranjith and Perera2015; Emami-Meybodi Reference Emami-Meybodi2017). Typical values of the viscosity and density of the pore fluid are $\mu =10^{-4}\unicode{x2013}10^{-3}\,\mathrm {Pa}\,{\rm s}$ and $\rho _0=945\unicode{x2013}1273\,\mathrm {kg}\,{\rm m}^{-3}$, respectively (Bachu & Carroll Reference Bachu and Carroll2005; Huppert & Neufeld Reference Huppert and Neufeld2014). The solubility of CO$_2$ in the pore fluid depends on the pressure, temperature and salinity of the brine (Bentham & Kirby Reference Bentham and Kirby2005; Bachu Reference Bachu2015; De Silva et al. Reference De Silva, Ranjith and Perera2015; Luo et al. Reference Luo, Li, Chen, Zhu and Peng2022), and the density increment due to the CO$_2$ dissolution can be from $0.1\,\%$ up to approximately $3\,\%$ (Bachu & Adams Reference Bachu and Adams2003; Pau et al. Reference Pau, Bell, Pruess, Almgren, Lijewski and Zhang2010; Luo et al. Reference Luo, Li, Chen, Zhu and Peng2022), which gives a density difference of approximately ${\rm \Delta} \rho _S=1\unicode{x2013}30\,\mathrm {kg}\,{\rm m}^{-3}$. In situ measurements indicate the typical geothermal gradient in the range of $G=20\unicode{x2013}65\,^\circ \mathrm {C}\,{\rm km}^{-1}$ (Rao et al. Reference Rao, Subrahmanyam, Sharma, Rastogi and Deka2001; Bachu & Stewart Reference Bachu and Stewart2002; Nordbotten et al. Reference Nordbotten, Celia and Bachu2005; Kopp et al. Reference Kopp, Class and Helmig2009; Yang et al. Reference Yang, Bai, Tang, Shari and David2010; De Silva et al. Reference De Silva, Ranjith and Perera2015). Then, taking a thermal expansion coefficient $\beta _T$ in the range $3\times 10^{-4}\unicode{x2013}8\times 10^{-4}\,{\rm K}^{-1}$ (Javaheri et al. Reference Javaheri, Abedi and Hassanzadeh2010), the density difference induced by the temperature difference can be estimated as ${\rm \Delta} \rho _T=0.1\unicode{x2013}10\,\mathrm {kg}\,{\rm m}^{-3}$. The molecular diffusivity and overall thermal diffusivity are typically of the order of $10^{-9}$ and $10^{-7}\,{\rm m}^2\,{\rm s}^{-1}$, respectively (Hassanzadeh et al. Reference Hassanzadeh, Pooladi-Darvish and Keith2006; Javaheri et al. Reference Javaheri, Abedi and Hassanzadeh2010; Emami-Meybodi Reference Emami-Meybodi2017).

Given the typical values of physical and reservoir properties discussed above, the non-dimensional parameters defined in (2.4a–d) can be readily estimated. The concentration Rayleigh number ${\textit {Ra}}_S$ can reach over $10^5$, while for a basin with a large thermal gradient and thickness, the thermal Rayleigh number ${\textit {Ra}}_T$ can be as high as $10^3$. It should also be pointed out that, for the current flow configuration, the density difference induced by the concentration field is determined by the constant concentration $S_{top}$ which is related to the dissolution process of supercritical CO$_2$, and that induced by the temperature difference is determined by both the thermal gradient and the reservoir thickness. Therefore, even for a fixed thermal gradient, the density ratio $\varLambda$ increases as the thickness becomes larger since $S_{top}$ should not depend on thickness. Taking all these and the computing resources into consideration, our simulations explore the parameter range with $10^3\le {\textit {Ra}}_S\le 10^4$ and $0.1\le \varLambda \le 5$, with the highest thermal Rayleigh number ${\textit {Ra}}_T=300$. We also fix $\sigma =1$ and ${\textit {Le}}=100$ for all simulations. The parameter space is shown in figure 1. Note that in the figure we also show the critical value ${\textit {Ra}}^{{cr}}_T=4{\rm \pi} ^2$ predicted by the linear instability analysis for a convection flow solely driven by a constant temperature difference between the top and bottom boundaries (Horton & Rogers Reference Horton and Rogers1945). Therefore, for some cases ${\textit {Ra}}_T$ is below ${\textit {Ra}}^{{cr}}_T$ while for the others it is above ${\textit {Ra}}^{{cr}}_T$.

Figure 1.

The parameter space explored in the current simulations. For all cases $\sigma =1$ and ${\textit {Le}}=100$, and for each ${\textit {Ra}}_S$ the corresponding case with ${\textit {Ra}}_T=0$ is also simulated. The red vertical dashed line marks the critical value ${\textit {Ra}}^{{cr}}_T=4{\rm \pi} ^2$ predicted by linear instability analysis for convection driven by a constant temperature difference.

2.3. Numerical methods

The governing equations (2.3) are numerically solved using our in-house code which was originally designed for wall-bounded convection turbulence (Ostilla-Mónico et al. Reference Ostilla-Mónico, Yang, van der Poel, Lohse and Verzicco2015). The code employs a second-order finite difference scheme for spatial discretization with staggered grids. For the time advance of the advection–diffusion equations for $T$ and $S$, a second-order Runge–Kutta method is used with the nonlinear advection terms treated by a scheme similar to the explicit Adams–Bashforth method and the diffusion terms by a scheme similar to the semi-implicit Crank–Nicolson method. To solve the velocity and pressure fields at each time step, we take the divergence of (2.3b) and use the continuity equation (2.3a), which gives the following Poisson equation for pressure:

(2.6)\begin{equation} \boldsymbol{\nabla}^2 p = \varLambda \partial_z T - \partial_zS, \end{equation}

with the boundary conditions

(2.7)\begin{equation} \partial_z p = \varLambda T - S,\quad \mbox{at}\ z=0,1. \end{equation}

Once the pressure field is obtained, the velocity field can be readily computed from (2.3b). Moreover, to efficiently solve the concentration field with relatively small diffusivity, the multiple grid strategy is used as in Ostilla-Mónico et al. (Reference Ostilla-Mónico, Yang, van der Poel, Lohse and Verzicco2015). Specifically, the concentration field is solved on a refined mesh, and the other quantities on a base mesh.

Initially, the velocity is set to zero and the temperature has a vertically linear distribution. The initial concentration field is uniform and equal to a small positive value within the domain to avoid any unphysical negative value during the simulation. A hyperbolic tangent function is applied to introduce a smooth transition between the concentration at the top boundary and the initial value at the interior of the domain. Random perturbations with a magnitude of $10^{-3}$ are introduced to trigger the convective motions. With these initial conditions, the 3-D simulations are conducted systematically for the parameters stated in figure 1. The details of the numerical settings are summarized in the Appendix. As a validation of our numerical method, we conducted a simulation of RDC at ${\textit {Ra}}=10^4$ with the same flow configuration as Pirozzoli et al. (Reference Pirozzoli, De Paoli, Zonta and Soldati2021). By using the same definition of the Nusselt number, our numerical measurements show ${\textit {Nu}}=98.44$, which is close to $99.84$ given by Pirozzoli et al. (Reference Pirozzoli, De Paoli, Zonta and Soldati2021).

4.1. Evolution of Nusselt numbers

One of the key questions for the current flow is the rate at which the dissolved CO$_2$ is transferred downwards. Since the bottom boundary is impermeable for the CO$_2$ concentration, the concentration flux is measured at the top boundary as, by the non-dimensional quantities,

(4.1)\begin{equation} {\textit{Nu}}_S(t) = \langle \partial_z S|_{z=1} \rangle_h =\int_0^1\int_0^1 \partial_z S|_{z=1} \mathrm{d}x \,\mathrm{d}y. \end{equation}

Meanwhile, the heat flux is calculated as the mean of the fluxes through the top and bottom boundaries as

(4.2)\begin{equation} {\textit{Nu}}_T(t) = \tfrac{1}{2} (\langle \partial_z T |_{z=1} \rangle_h + \langle\partial_z T|_{z=0} \rangle_h). \end{equation}

In figure 7 we plot the complete time evolution of ${\textit {Nu}}_S(t)$ and ${\textit {Nu}}_T(t)$ for fixed ${\textit {Ra}}_S=10^4$ and increasing ${\textit {Ra}}_T$. For the case with ${\textit {Ra}}_T=0$, i.e. without a temperature difference, the time history of ${\textit {Nu}}_S(t)$ is very similar to those reported in Hewitt et al. (Reference Hewitt, Neufeld and Lister2013). Initially, the concentration flux is dominated by the diffusion process and remains at a small value. When the convective motions start to develop at the top boundary at $t\approx 1$, ${\textit {Nu}}_S$ increases exponentially and reaches a maximum at approximately $t=2.5$. After this peak ${\textit {Nu}}_S$ decreases and then maintains a nearly constant value until $t=15$, and the flow is in an intermediate quasi-steady state. Then, ${\textit {Nu}}_S$ continues to slowly decrease until the end of the simulation at $t=30$. This last stage with decreasing ${\textit {Nu}}_S$ is identified as the shutdown process by Hewitt et al. (Reference Hewitt, Neufeld and Lister2013).

Figure 7.

Temporal evolution of (a) concentration Nusselt number ${\textit {Nu}}_S(t)$, (b) temperature Nusselt number ${\textit {Nu}}_T(t)$ and (c) normalized concentration Nusselt number for ${\textit {Ra}}_S=10^4$. Colours from cold to warm correspond to increasing ${\textit {Ra}}_T$.

When the temperature difference is introduced across a layer of porous medium, different evolution behaviours are observed for different ${\textit {Ra}}_T$. For small ${\textit {Ra}}_T$, the time history of ${\textit {Nu}}_S$ does not change much compared with the case without a temperature difference, which is expected. The value of ${\textit {Nu}}_T$ increases slightly from unity when the convective motions induced by the concentration field become apparent. For large ${\textit {Ra}}_T$, the temperature difference alone is strong enough to drive convective motions, and ${\textit {Nu}}_T$ can be higher than unity after a certain time. The value of ${\textit {Nu}}_T$ first quickly increases and then oscillates around a final value. The initial increase of ${\textit {Nu}}_T$ happens earlier and the final value becomes larger for higher ${\textit {Ra}}_T$. This second scenario occurs when ${\textit {Ra}}_T$ is considerably larger than the critical value ${\textit {Ra}}^{cr}_T$, which we refer to as the high ${\textit {Ra}}_T$ cases.

The comparison between the time history of ${\textit {Nu}}_S$ and that of ${\textit {Nu}}_T$ reveals that, for the high ${\textit {Ra}}_T$ cases, ${\textit {Nu}}_S$ starts to increase at an earlier time than ${\textit {Nu}}_T$ does. Together with the flow evolution shown in the previous section, one discovers that the initial increase of ${\textit {Nu}}_S$ corresponds to the finger structures driven by the concentration gradient near the top boundary, while the initial increase of ${\textit {Nu}}_T$ happens roughly at the same time as when the large-scale rolls driven by the temperature difference emerge. The appearance of these large-scale rolls also destroys the quasi-steady state which exists in the low ${\textit {Ra}}_T$ cases and the flow directly enters the final shutdown process, which will be further discussed in § 4.2.

The above discussions suggest that the large-scale rolls driven by the temperature difference have two opposite effects on the concentration transport. At the early stage, the appearance of large-scale rolls greatly enhances the downward transport rate of the concentration field. At the later stage, however, the non-dimensional concentration flux is suppressed for the cases with high ${\textit {Ra}}_T$. This latter effect is due to the fact that the upwelling currents of the large-scale rolls can prevent the descent of high concentration fingers near the top boundary, which can be observed in figure 3(c,d). Moreover, the larger concentration flux carried by the downwelling currents of large-scale rolls accelerates the accumulation of high concentration at the bottom region and the transition towards the shutdown process.

During the final shutdown stage, the time history of ${\textit {Nu}}_S$ exhibits self-similarity for different ${\textit {Ra}}_T$. To demonstrate this, we normalize the time $t$ by $t_m$ when ${\textit {Nu}}_T$ reaches the first maximum, and ${\textit {Nu}}_S$ by the value ${\textit {Nu}}_S(t_m)$. The rescaled plot is shown in figure 7(c). One observes that all the curves for different ${\textit {Ra}}_T$ collapse to certain extent for the range $t/t_m>1$. This enables us to develop a theoretical model for the shutdown stage as described in the following subsection.

As a final remark on the evolution of the fluxes, we plot the time history of ${\textit {Nu}}_S$ and ${\textit {Nu}}_T$ for three cases with fixed $\varLambda =2$ and increasing ${\textit {Ra}}_S$ in figure 8. The thermal Rayleigh number ${\textit {Ra}}_T$ then increases accordingly. For the smallest ${\textit {Ra}}_S=10^3$, ${\textit {Ra}}_T={\textit {Ra}}_S\varLambda /{\textit {Le}}=20$ which is below ${\textit {Ra}}_T^{cr}$ and the temperature gradient shows a very minor effect on the fluxes. As ${\textit {Ra}}_T$ becomes higher than ${\textit {Ra}}_T^{cr}$ for the two cases with ${\textit {Ra}}_S=5\times 10^3$ and $10^4$, similar behaviours are observed as those shown in figure 7 with large ${\textit {Ra}}_T$. For larger ${\textit {Ra}}_T$, the large-scale convection rolls develop within a shorter time period and ${\textit {Nu}}_T$ starts to rise earlier, which causes the faster transition of the flow into the final shutdown stage.

Figure 8.

The time evolution of fluxes for three cases with fixed $\varLambda =2$ and increasing ${\textit {Ra}}_S$.

4.2. A theoretical model for the shutdown stage

Following the procedure in Hewitt et al. (Reference Hewitt, Neufeld and Lister2013), we present a theoretical model to describe the temporal variations of concentration flux and volume averaged concentration during the final shutdown stage. We define the instantaneous volume averaged concentration as

(4.3)\begin{equation} \mathcal{S}(t)=\int_0^1 \int_0^1 \int_0^1 S(x,y,z,t)\,{{\rm d}x}\,{{\rm d}y}\,{\rm d} z. \end{equation}

At any given time $t$, if we integrate the concentration equation (2.3d) over the entire domain, then by using the boundary conditions and the definition of ${\textit {Nu}}_S(t)$ it is easy to obtain

(4.4)\begin{equation} \frac{{\rm d} \mathcal{S}}{{\rm d} t} =\frac{1}{{\textit{Ra}}_{S}} {\textit{Nu}}_S(t). \end{equation}

The above ordinary differential equation can be closed by an appropriate relation between ${\textit {Nu}}_S(t)$ and $\mathcal {S}(t)$. Similar to Hewitt et al. (Reference Hewitt, Neufeld and Lister2013), we use the relation between the Nusselt number and the Rayleigh number for the convection in a porous medium driven by a constant concentration difference across the layer. Since the current flow is constantly evolving, the instantaneous Nusselt and Rayleigh numbers must be defined properly. At any given time, the actual concentration difference, which provides part of the driving force, can be approximated as $1-\mathcal {S}(t)$, while the constant temperature difference also contributes to the driving force of the system. Then an effective total Rayleigh number, which measures the strength of the actual driving force, can be defined as

(4.5)\begin{equation} {\textit{Ra}}_e(t) = {\textit{Ra}}_S[1-\mathcal{S}(t)]+{\textit{Ra}}_T. \end{equation}

The effective Nusselt number at any time, which should be rescaled by the actual concentration difference, can be calculated as

(4.6)\begin{equation} {\textit{Nu}}_e(t)=\frac{{\textit{Nu}}_S(t)}{1-\mathcal{S}(t)}. \end{equation}

Then, as suggested by Hewitt et al. (Reference Hewitt, Neufeld and Lister2013) and Hewitt et al. (Reference Hewitt, Neufeld and Lister2014), a linear scaling can be applied to the relation between the effective Nusselt number and the effective Rayleigh number as

(4.7)\begin{equation} {\textit{Nu}}_e(t)=4\alpha {\textit{Ra}}_e(t), \end{equation}

where $\alpha$ is a scaling coefficient and will be discussed later.

With the help of (4.6), (4.5) and (4.7), equation (4.4) can be closed and gives

(4.8)\begin{equation} \frac{{\rm d}\mathcal{S}}{{\rm d} t} =\frac{1-\mathcal{S}}{Ra_{S}} 4\alpha\left[ Ra_S (1-\mathcal{S})+Ra_T\right]. \end{equation}

The solution of the above equations can be readily obtained. For the case without a thermal gradient, for ${\textit {Ra}}_T=0$, the solution reads

(4.9)\begin{equation} \mathcal{S}(t)= 1-\left(C_0+4\alpha t\right)^{{-}1}, \end{equation}

which is the same as that given in Hewitt et al. (Reference Hewitt, Neufeld and Lister2013). Here, $C_0$ is some integral constant and will be determined from the simulation results. The corresponding Nusselt number is

(4.10)\begin{equation} {\textit{Nu}}_S(t)=4\alpha {\textit{Ra}}_S\left(C_0+4\alpha t\right)^{{-}2}. \end{equation}

When ${\textit {Ra}}_T>0$, the solution takes a more complex form as

(4.11)\begin{equation} \mathcal{S}(t)=1-\frac{{\textit{Ra}}_T}{{\textit{Ra}}_S}\left[C_0 \exp\left(\frac{4\alpha {\textit{Ra}}_T t}{{\textit{Ra}}_S}\right)-1\right]^{{-}1}. \end{equation}

And the Nusselt number is

(4.12)\begin{align} {\textit{Nu}}_S(t)= 4\alpha \frac{{\textit{Ra}}_T^2}{{\textit{Ra}}_S}\left[C_0 \exp\left(\frac{4\alpha {\textit{Ra}}_T t}{{\textit{Ra}}_S} \right)-1\right]^{{-}1} \left[ \left(C_0 \exp\left(\frac{4\alpha {\textit{Ra}}_T t}{{\textit{Ra}}_S}\right)-1\right)^{{-}1}+1\right]. \end{align}

The above solutions contain two parameters, namely $\alpha$ and $C_0$, which need to be determined from the numerical results.

We first look at the coefficient $\alpha$. Note that the linear scaling (4.7) is adopted from the RDC driven by a constant temperature difference. Here, the value of $\alpha$ will also be affected by the strength of thermal difference. In order to determine the value of $\alpha$, one notices that, by using the expressions for $\mathcal {S}(t)$ and ${\textit {Nu}}_S(t)$,

(4.13)\begin{equation} \alpha=\frac{{\textit{Nu}}_S}{4(1-\mathcal{S})\left[{\textit{Ra}}_S (1-\mathcal{S})+{\textit{Ra}}_T\right]}. \end{equation}

The time variations of $\alpha$ for all the cases with ${\textit {Ra}}_S=10^4$ are shown in figure 9. For all cases $\alpha$ is almost constant during the last ten time units, especially for the cases with small ${\textit {Ra}}_T$. We therefore calculate the mean $\bar {\alpha }$ over the time period $20 \le t \le 30$. Then the value of $C_0$ is then determined by fitting the curve of $\mathcal {S}(t)$ over the same time period. The values of $\bar {\alpha }$ and $C_0$ are summarized in table 1. In figure 10 we compare the theoretical model with the numerical results for all the cases with ${\textit {Ra}}_S=10^4$. The agreement between the model and the numerical curves is very good, especially during the final time period. Numerical results also indicate that a similar agreement is achieved for the cases with ${\textit {Ra}}_S=10^3$ and $5\times 10^3$.

Figure 9.

Time evolution of $\alpha$ for all the cases with ${\textit {Ra}}_S=10^4$.

Figure 10.

Solid lines show the variations of concentration Nusselt number ${\textit {Nu}}_S(t)$ and the volume averaged concentration $\mathcal {S}(t)$ vs time, while the dashed lines are the predictions of model (4.9) and (4.11). Panels (a–i) show ${\textit {Ra}}_T=0$, $10$, $50$, $80$, $100$, $120$, $150$, $200$ and $300$, respectively; ${\textit {Ra}}_S=10^4$ for all cases.

Table 1.

Summary of simulations. Columns from left to right are the concentration Rayleigh number, thermal Rayleigh number, density ratio, resolutions in the $x$-axis, $y$-axis, $z$-axis (with refinement factors for multiple resolutions), total CO$_2$ transferred to the domain at the end time of the simulations, averaged values of $\alpha$ from $t=20$ to $30$, fitted integral coefficient, predicted time to reach $\mathcal {S}=0.9$.

The variation of $\bar {\alpha }$ vs ${\textit {Ra}}_T$ is plotted in figure 11 for all cases. At small ${\textit {Ra}}_T$, $\bar {\alpha }$ is nearly independent of ${\textit {Ra}}_T$ but decreases as ${\textit {Ra}}_S$ increases. For the two higher ${\textit {Ra}}_S$ considered here, $\bar {\alpha }$ is very close to those reported in Hewitt et al. (Reference Hewitt, Neufeld and Lister2014). When ${\textit {Ra}}_T$ is large enough, $\bar {\alpha }$ exhibits a consistent dependence on ${\textit {Ra}}_T$ for all the cases with different ${\textit {Ra}}_S$, gradually decreasing with ${\textit {Ra}}_T$. For the highest ${\textit {Ra}}_T=300$, $\bar {\alpha }$ is below $0.005$. One possible explanation for the decrease of $\bar {\alpha }$ can be attributed to the difference between the flow morphology at small ${\textit {Ra}}_T$ and that at large ${\textit {Ra}}_T$. As shown in figures 3 and 4, at small ${\textit {Ra}}_T$ the concentration fingers develop over the whole domain and the flow is fully three-dimensional. While at large ${\textit {Ra}}_T$, fingers mainly develop in the region the downwelling currents induced by the large-scale convection rolls driven by the thermal gradient. That is, the flow morphology is more similar to the quasi-2-D flow over the narrow sheet-like region of downwelling currents. Previous studies suggest that the scaling coefficient $\alpha$ takes different values for 2-D and 3-D flows. For 3-D flows, Hewitt et al. (Reference Hewitt, Neufeld and Lister2014) suggests $\alpha \approx 0.0096$, while for 2-D flows $\alpha$ is smaller and around $0.0069$ (Hewitt, Neufeld & Lister Reference Hewitt, Neufeld and Lister2012). These values are very similar to those at small ${\textit {Ra}}_T$ and large ${\textit {Ra}}_T$ in the current flow.

Figure 11.

The coefficient $\bar {\alpha }$ vs the thermal Rayleigh number ${\textit {Ra}}_T$ for all cases.

4.3. Dissolved carbon dioxide

In the previous section we discussed the evolution of flow structures and fluxes. In this subsection we focus on the total carbon dioxide which is transported during the entire simulation time. The total CO$_2$ transferred through the top boundary into the domain at given time $t$ can be calculated by the following integration:

(4.14)\begin{equation} M_S(t) = \int_0^t {\textit{Nu}}_S(\tau)\,{\rm d}\tau. \end{equation}

Figure 12(a) shows $M_S(t)$ for all cases with ${\textit {Ra}}_S=10^4$; $M_S(t)$ exhibits non-monotonic variations as ${\textit {Ra}}_T$ increases. Let $M^0_S(t)$ denote the case with ${\textit {Ra}}_T=0$, then for small ${\textit {Ra}}_T$, namely ${\textit {Ra}}_T \le 100$ in the current study, $M_S(t)$ is larger than $M^0_S(t)$ during the entire simulation time $0< t<30$. When ${\textit {Ra}}_T$ is high enough, first $M_S(t)$ is larger than $M^0_S(t)$ and then smaller than $M^0_S(t)$ at the final stage. These variations with increasing ${\textit {Ra}}_T$ can be seen more clearly in figure 12(b) where the same functions are normalized by $M^0_S(t)$. The transition from $M_S>M^0_S$ to $M_S< M^0_S$ occurs at later time as ${\textit {Ra}}_T$ becomes smaller. Due to the limitation of the current simulation time, we do not observe this transition for ${\textit {Ra}}_T\le 100$. For larger ${\textit {Ra}}_T$, there exist two peaks in the time history of $M_S/M_S^0$. Comparisons with the time history of the two Nusselt numbers shown in figure 7 reveal that the first peak of $M_S/M^0_S$ for different ${\textit {Ra}}_T$ occurs at almost the same time of around $t=2.5$, which corresponds to the end of the exponential growth of ${\textit {Nu}}_S$. During this initial period, the temperature field does not have significant effects on the fluxes, and the flow is mainly driven by the concentration field. The amplitudes of first peaks vary for different ${\textit {Ra}}_T$ because of different initial perturbations before the rapid increase of ${\textit {Nu}}_S$. The second peak is noticeable for cases with ${\textit {Ra}}_T \ge 100$ in figures 7(a) and 12(b). For each ${\textit {Ra}}_T$ the second peak of $M_S/M^0_S$ is related to the second rise of ${\textit {Nu}}_S$. At this stage, ${\textit {Nu}}_T$ is evidently larger than unity at higher ${\textit {Ra}}_T$. The presence of large-scale rolls changes the instantaneous concentration flux at the top boundary, and therefore the total transferred CO$_2$. It takes less time for the flow to form these rolls at larger ${\textit {Ra}}_T$, which corresponds to an earlier occurrence of the second peak of $M_S/M^0_S$.

Figure 12.

(a) The accumulative transfer of CO$_2$ as the function of time for the cases with ${\textit {Ra}}_S=10^4$. (b) The same functions normalized by that of the case with ${\textit {Ra}}_T=0$. Here, ${\textit {Ra}}_T$ increases as the line colour changes from blue to red.

We then look at the total CO$_2$ transported into the domain at the end of the simulations. Figure 13 displays the ratio $M_S/M^0_S$ at $t=30$ vs ${\textit {Ra}}_T$ and $\varLambda$ for all simulated cases. For all three concentration Rayleigh numbers, the ratio first increases then decreases with $\varLambda$, see figure 13(b). For the smallest ${\textit {Ra}}_S=10^3$, the largest thermal Rayleigh number ${\textit {Ra}}_T=50$ is only slightly higher than the critical value ${\textit {Ra}}_T^{cr}$ and the convective motions driven by the thermal gradient are not strong. Still, the additional temperature difference across the domain has noticeable effects on the accumulative transfer of CO$_2$. As ${\textit {Ra}}_S$ increases, the ratio $M_S/M^0_S$ at $t=30$ reaches the peak value at larger ${\textit {Ra}}_T$. Moreover, when ${\textit {Ra}}_T$ is large enough, the ratio drops below unity, indicating that the total CO$_2$ transported downwards at $t=30$ is less than that without the presence of a temperature difference. For the case with ${\textit {Ra}}_S=5\times 10^3$ and ${\textit {Ra}}_T=200$, the total amount of CO$_2$ transferred into the domain at $t=30$ can be reduced by over $17\,\%$ compared with the case with ${\textit {Ra}}_T=0$. Interestingly, if the ratio $M_S/M_S^0$ at $t=30$ is plotted against the density ratio $\varLambda$, see figure 13(b), one observes that the ratio reaches the maximum at around $\varLambda =1$ for all three ${\textit {Ra}}_S$. The maximal increment in $M_S$ induced by the temperature difference is over $3\,\%$ for the two larger ${\textit {Ra}}_S$. This non-monotonic trend can be understood as follows. Two opposite effects are introduced by the unstable temperature gradient. On one hand, the vertical flow motions are stronger due to the extra buoyancy force of the thermal field. On the other hand, the concentration fingers are clustered by the large-scale rolls driven by the thermal field. The former will enhance the downward transport while the latter will suppress the transport because of the reduction of the horizontal area with larger concentration.

Figure 13.

The ratio $M_S/M^0_S$ at $t=30$, i.e. the end of the simulation, vs (a) the thermal Rayleigh number ${\textit {Ra}}_T$ and (b) the density ratio $\varLambda$.

To quantitatively demonstrate these two effects, we employ the r.m.s. value $u^{rms}_z$ of the vertical velocity and the total area $\varOmega$ of the regions where $S'>S_{std}$ and $u_z'<-u^{std}_z$. Here, $S'$ and $u_z'$ are the deviations from the horizontally averaged concentration and vertical velocity, while $S_{std}$ and $u^{std}_z$ are the corresponding standard deviations. The two values $u^{rms}_z$ and $\varOmega$ are calculated over the horizontal plane at the height with maximal $u^{rms}_z$ and then averaged over the time period $20\le t \le 30$. In figure 14 we plot the dependence of $\overline {u^{rms}_z}$ and $\bar {\varOmega }$ on the density ratio $\varLambda$ for fixed ${\textit {Ra}}_S=10^4$. Clearly, $\overline {u^{rms}_z}$ increases with $\varLambda$, while $\bar {\varOmega }$ decreases. The two curves intersect with each other at some moderate density ratio slightly above unity. Therefore, the maximal $M_S/M_S^0$ at moderate density ratio is very likely caused by these two competing effects of the thermal field.

Figure 14.

The averaged $\overline {u^{rms}_z}$ and $\bar {\varOmega }$ over the time period $20\le t \le 30$. For all cases ${\textit {Ra}}_S=10^4$.

For the current flow configuration, the whole domain will eventually have $\mathcal {S}=1$ when the time approaches infinity. The theoretical model constructed in the previous section can provide some indications about the long-term behaviours of the total dissolved carbon dioxide. By using (4.9) and (4.11) and the coefficients determined from the numerical results, we estimate the time $t_{90}$ when $\mathcal {S}=0.9$, i.e. where the volume averaged concentration equals to $90\,\%$ of the concentration at the top boundary. The values of $t_{90}$ are listed in table 1, and their dependence on $\varLambda$ is plotted in figure 15. Note that $t_{90}$ is in the non-dimensional form. Figure 15 suggests that overall $t_{90}$ increases with ${\textit {Ra}}_S$. For fixed ${\textit {Ra}}_S$, however, $t_{90}$ generally first decreases and then increases with $\varLambda$ or equivalently ${\textit {Ra}}_T$. It reaches the minimum around $\varLambda =1$ for all three ${\textit {Ra}}_S$ considered here.

Figure 15.

The time $t_{90}$ when the volume averaged concentration $\mathcal {S}$ reaches $90\,\%$ of the top value, as predicted by the theoretical models (4.9) and (4.11).

4.4. Implications for CO$_2$ sequestration

Above findings reveal that, for high ${\textit {Ra}}_S$, which is very likely the case in a real geological reservoir, a weak to moderate geothermal gradient may increase the total carbon dioxide transferred into the saline aquifer due to convective dissolution during a certain time period, while a strong geothermal gradient may have the opposite effect. To demonstrate this in the context of a realistic situation, we take a typical geological reservoir with porosity $\phi =0.2$, permeability $K=1.25\times 10^{-13}\,\mathrm {m}^2$, viscosity $\mu =2\times 10^{-4}\,{\rm Pa}\,{\rm s}$, overall thermal diffusivity $\kappa _m=10^{-7}\,\mathrm {m}^2\,{\rm s}^{-1}$ and concentration diffusivity ${\kappa _S=5\times 10^{-9}\,\mathrm {m}^2\,{\rm s}^{-1}}$. For brine saturated with CO$_2$, the density increment is approximately $8\,\mathrm {kg}\,{\rm m}^{-3}$. Then a $200\,\mathrm {m}$ aquifer will give a concentration Rayleigh number ${\textit {Ra}}_S=10^4$ if roughly taking $g=10\,\mathrm {m}\,{\rm s}^{-2}$, and the non-dimensional simulation time $t=30$ corresponds to approximately $760$ years. Assuming a geothermal gradient of $50\,^{\circ }\mathrm {C}\,{\rm km}^{-1}$, the temperature difference and thermal Rayleigh number across the aquifer are ${\rm \Delta} T=10\,^{\circ }\mathrm {C}$ and ${\textit {Ra}}_T=100$ by using $\beta _T=8\times 10^{-4}\,\mathrm {K}^{-1}$, respectively. Therefore, under this circumstance, the density ratio is $1.0$. For an aquifer with an area of $100\,\mathrm {km}^2$, approximately $2.4\times 10^6$ tons of extra CO$_2$ will be transferred into the aquifer from the perspective of convective dissolution with a geothermal gradient during the time period of $760$ years.

Furthermore, our results suggest that the large-scale convection rolls driven by the thermal gradient can strongly alter the horizontal pattern of the finger structures. In actual environments, finger structures mainly happen at the interface region between the supercritical CO$_2$ layer and the brine layer below. However, the large-scale convection rolls may already exist over a layer with much larger thickness and therefore much higher thermal Rayleigh number. Then, the upwelling and downwelling currents of the large-scale rolls can still affect the horizontal distribution of the finger structures, even though the local thermal Rayleigh number across the interface region is much lower. Also, the geothermal gradient may play a non-negligible role in many CO$_2$-sequestration sites.