Direct comparison of density-driven convective mixing in a three-dimensional porous medium using experiments and simulation

We perform a direct comparison between experiment and simulation of density-driven convective mixing in three-dimensional (3D) porous media. We find excellent agreement between the experiment and the model in terms of both the convection fingering pattern and the average rate of fluid mixing. In particular, the experiment exhibits dynamic self-organization of columnar plumes into a reticular pattern, which, until now, had only been observed in 3D simulations. We also report good quantitative agreement between the experiment and simulation in the evolution of the state of mixing by comparing, over time, (i) the average concentration at depth, (ii) the variance of the concentration field, (iii) the scalar dissipation rate, and (iv) the dissolution flux. We derive a relation between the scalar dissipation rate and the dissolution flux in a closed system, and we show that the flux in a 3D system is approximately $\sim 30\%$ higher than in a 2D system, confirming previous numerical estimates.

Figure 1

A diagram that illustrates the initial setup and the dimensions of the convective dissolution problem in an enclosed cube. $H_T$ corresponds to the initial thickness of the MEG-brine layer and $H_B$ is the initial thickness of the MEG-free resident brine. In the experiment presented in this study, $H_T=2\mathrm{cm}, H_B=13\mathrm{cm}, W=L=15\mathrm{cm}$.

Figure 2

Right: 3D reconstruction of the simulated $c^{\prime }$ at $t=140\min$. Left: 2D reconstruction of fingering patterns at $z=4\mathrm{cm}$ and at 40, 80, and 140 min (top panel) and at $z=8\mathrm{cm}$ and at 100, 120, and 140 min (bottom panel). Within each panel, the top row is simulation and bottom row is experiment. Color map corresponds to the normalized MEG concentration ($c^{\prime }$).

Figure 3

The number of fingers (left) and the average area of a finger (right) for both the experiment (points) and simulation (solid lines). The results are shown at five depths where dark red is 3.8 cm, red is 4.4 cm, light red is 5.8 cm, orange is 7.4 cm, and yellow is 10.7 cm from the top.

Figure 4

Left: horizontally averaged normalized concentration $\langle c^{\prime }\rangle$ over depth ($z$) at various times from the experiment (cross) and the simulation (solid lines). Right: the same concentration profiles plotted as a function of the self-similarity variable $\zeta =(z-z_0)/(t-t_{\text{onset}}).$ The black dashed line marks $\zeta =0$.

Figure 5

(a) Mean normalized concentration over time at various depths where dark red is 3.8 cm, red is 4.4 cm, light red is 5.8 cm, orange is 7.4 cm, and yellow is 10.7 cm from the top; (b) time evolution of ${\sigma }^2$ and (c) time evolution of $\langle {\epsilon }^{\prime }\rangle$ calculated from experimental images (dots) and simulation (solid lines).

Figure 6

In 2D, at three different Ra (row), snapshot of $\epsilon ={\left|\mathbf{\nabla }c\right|}^2/\text{Ra}$ obtained from the numerical simulation at four different times (column). The color map is truncated at ${\epsilon }_{\text{threshold}}=0.1\text{max}\left(\epsilon \right)$ to accentuate the mixing volume (the black region).

Figure 7

In 3D, at $\mathrm{Ra}=6000$, snapshot of $\epsilon ={\left|\mathbf{\nabla }c\right|}^2/\text{Ra}$ obtained from the numerical simulation at four different times. The color map is truncated at ${\epsilon }_{\text{threshold}}=0.1\text{max}\left(\epsilon \right)$ to accentuate the mixing volume (the black region).

Figure 8

Time evolution of the dissolution flux using the interface-based method (red marker), and the $\langle {\epsilon }^{\prime }\rangle$-based method (black marker) applied to the experimental (diamonds) and simulation (lines) data. Inset: The time evolution of the interface position for the experiment (diamonds) and simulation (line).