Bi-dimensional plume generated by the convective dissolution of an extended source of ${\mathrm{CO}}_2$

The dynamics of a bi-dimensional solutal plume generated by the convective dissolution of an extended source of carbon dioxide (${\mathrm{CO}}_2$) is studied from both experimental and numerical standpoints. In the experiments, the ${\mathrm{CO}}_2$ is injected into a 1-mm-thick water filled Hele-Shaw cell at pressures ranging between 1 and 5 bars. The plume is visualized using a $p\mathrm{H}$-sensitive dye, and the velocity field is measured by means of standard particle image velocimetry (PIV). Concentration and velocity fields are scrutinized over one order of magnitude in the Rayleigh number (Ra), and for two different values of the Darcy number (Da). In order to extend the explored range of these dimensionless numbers, experiments are complemented by a comprehenvive set of numerical simulations. At low Darcy numbers ($\mathrm{Da}<1$), the morphology of the numerical plume is fairly close to that of the analytical solution previously derived by Wooding [J. Fluid Mech. 15, 527 (1963)] for the case of a line source in a porous medium. At larger Darcy numbers ($\mathrm{Da}>1$), the numerical solution departs on several aspects from this solution and exhibits closer similarity to the analytical solution derived by Spalding and Cruddace [Int. J. Heat Mass Transfer. 3, 55 (1961)] for the case of a line source in a viscous fluid. As the introduction of an additional length scale (the lateral size of the source) in the problem allows for the definition of a length-based Rayleigh number Ra, the respective amplitudes of the plume velocity $w$, plume width $\sigma$, and dimensionless total injection rate (i.e., the Nusselt number Nu) can be explicitly obtained as functions of Ra and Da. The scaling laws are reasonably close to those obtained from simple mass conservation considerations: (i) $w\sim \mathrm{Ra},\sigma \sim {\mathrm{Ra}}^{-1/2}$, and $\mathrm{Nu}\sim {\mathrm{Ra}}^{1/2}$ for $\mathrm{Ra}\mathrm{Da}<1$, and (ii) $w\sim {\left(\mathrm{Ra}/\mathrm{Da}\right)}^{1/2},\sigma \sim {\left(\mathrm{Ra}/\mathrm{Da}\right)}^{-1/4}$, and $\mathrm{Nu}\sim {\left(\mathrm{Ra}/\mathrm{Da}\right)}^{1/4}$ for $\mathrm{Ra}\mathrm{Da}>1$. Although the concentration field measurements are marred by a large uncertainty due to the logarithmic nature of the $p\mathrm{H}$-concentration relationship, the experimental data obtained from the PIV measurements show good agreement with the numerical results.

Figure 1

Experimental setup. (a) Hele-Shaw cell and masking glass slides. (b) Side view of the setup. HS: Hele-Shaw cell; RES: reservoir; CH: pressure chamber; V: ball valve; PCH: pressure sensor; PDIF: differential pressure sensor; TH: thermocouple. (c) Top view of the PIV and $p\mathrm{H}$-sensitive dye visualization systems (see the text for further explanations).

Figure 2

(a) Raw image of the plume obtained for $L_s=4\mathrm{mm}$ and $P_{{\mathrm{CO}}_2}=1\mathrm{bar}$. (b) Color-to-$p\mathrm{H}$ calibration curves used to translate the raw image into a $p\mathrm{H}$ map. (c) Corresponding map $p\mathrm{H}(x,y)$ [rectangled region in (a)]. (d) Concentration map $C(x,z)$ extracted from (c) via the simplified relationship $p\mathrm{H}=(pC+p{K}_a)/2$, where $p{K}_a\simeq 6.4$.

Figure 3

(a) Expanded view of the bi-dimensional mesh in the vicinity of the source (b) convergence of the simulation with respect to the mesh size $\mathrm{\Delta }x$ for $\mathrm{Ra}={10}^6$ and $\mathrm{Da}=1$. (c) Time profile of the velocity for $\mathrm{Ra}={10}^4$ (thin line) and $\mathrm{Ra}={10}^5$ (thick line) at $\mathrm{Da}=0.01$.

Figure 4

(a) Velocity field $\langle w\rangle$ averaged over the period of time (0.05–0.025) for the dimensionless parameters $\mathrm{Da}=5.2\times {10}^{-3}$ and $\mathrm{Ra}=2973(E=1,L_s=4\mathrm{mm},P_{{\mathrm{CO}}_2}=4\mathrm{bars}$). The space scale is provided by the width of the image which is equal to 1 (width of the source in dimensionless units). (b) Time profile of the maximum velocity measured on transverse profiles at $z=1$. (c) Average transverse profile $\langle w\rangle$ measured at $z=1$ (thick solid line). The fitting function of the form $\langle w_0\rangle {[cosh(1.085x/\langle {\sigma }_w\rangle )]}^{-2}$ is plotted with a thin solid line, whereas the Gaussian function of same curvature on the axis is plotted with a dashed line. (d) Axial $z$-profile $\langle w_0\rangle$ of the maximum velocity obtained from the averaged field $\langle w\rangle$ shown in (a). (e) Axial $z$ profile of the width $\langle {\sigma }_w\rangle$.

Figure 5

(a) Numerical velocity and concentration fields obtained for $\mathrm{Ra}={10}^3$ and $\mathrm{Da}={10}^{-4}$. (b) Normalized transverse profile of the concentration as a function of the normalized $x$ coordinate, based on solution (10). (c) Normalized transverse profile of the vertical component of the velocity as a function of the normalized $x$ coordinate, based on solution (10). In both (b) and (c) the numerical data (open circles) correspond to the fields shown in (a), and the theoretical profiles corresponding to Eq. (10) are plotted as a solid line with $\mathrm{Nu}=16.8$ obtained from (11). The Gaussian function of same curvature on the axis is plotted as a dashed line.

Figure 6

(a) Axial $z$ profile of the maximum concentration $c_0$. (b) Width of the numerical concentration profile $\sigma$ as a function of $z$. (c) Axial $z$ profile of the maximum velocity $w_0$. (d) Width of the numerical transverse profile of the vertical velocity component ${\sigma }_w$ as a function of $z$. The numerical data are plotted for $\mathrm{Ra}={10}^3$ and four different values of the Darcy number (open colored symbols). The theoretical scalings (solid black line) correspond to the low Darcy solution (10) derived by Wooding [23] in the case of a line source.

Figure 7

(a) Maximum of the vertical velocity component (on the plume axis), (b) width of the transverse velocity profile, (c) width of the concentration profile (the maximum value of the transverse concentration profile is shown in the inset), and (d) Nusselt number as functions of the Rayleigh number. In (a)–(d) each quantity (extracted from the numerical calculation at $z=1$) is plotted for all the values of the Darcy number listed in Table 2.

Figure 8

(a) Numerical velocity and concentration fields obtained for $\mathrm{Ra}={10}^6$ and $\mathrm{Da}=1$. (b) Normalized transverse profile of the concentration as a function of the normalized $x$ coordinate, based on solution (13). (c) Normalized transverse profile of the vertical component of the velocity as a function of the normalized $x$ coordinate, based on solution (13). In both (b) and (c) the numerical data (open circles) correspond to the fields shown in (a) and the theoretical profiles [Eqs. (13)] are plotted with $\mathrm{Nu}=19$ as given by (11) in the numerics.

Figure 9

(a) Axial $z$ profile of the maximum concentration $c_0$. (b) Width of the numerical transverse concentration profile $\sigma$ as a function of $z$. (c) Axial $z$ profile of the maximum velocity $w_0$. (d) Width of the numerical transverse velocity profile ${\sigma }_w$ as a function of $z$. The numerical data are plotted for three different combinations of the Rayleigh and Darcy numbers (open colored symbols). The theoretical scalings (solid black line) correspond to the large Schmidt solution (13) derived by Spalding and Cruddace [24] for a line source.

Figure 10

Simplified picture of the plume suitable for the derivations of scaling laws of the vertical velocity and concentration width and Nusselt number with respect to the Rayleigh and Darcy numbers.

Figure 11

Rescaled (a) maximum of the vertical velocity, (b) width of the velocity profile, (c) width of the concentration profile, and (d) Nusselt number, plotted with respect to the key parameter of the problem Ra Da . Experimental data are plotted using dark gray ($\mathrm{Da}=5.2\times {10}^{-4}$) and light gray ($\mathrm{Da}=1.3\times {10}^{-3}$) triangles. All the quantities are extracted/measured at $z=1$. The rescaled maximum concentration is plotted in the inset of (c) for information.

Figure 12

(a) Concentration map of the plume for $E=1\mathrm{mm},P_{{\mathrm{CO}}_2}=1\mathrm{bar}$. The length of the white scale bar is 4 mm. (b) Corresponding transverse profile at $z=L_s$. The dashed line corresponds to a Gaussian fit $\propto exp(-x^2/{\sigma }^2)$ whereas the solid line corresponds to a fit $\propto {[cosh(1.085x/\sigma )]}^{-2}$. (c) Width $\sigma$ of the concentration profile as a function of the altitude.