3 Results and discussion

In this section we will assess potential consequences of the assumption of Darcy flow by examining the role of the cell width when it is used as the parameter to increase the Rayleigh–Darcy number. However, it is instrumental in our analysis to examine the behaviour of dissolution flux in time for cases in which the Darcy flow assumptions can be safely applied (we will demonstrate this later). For experimental convenience, we will examine the solute flux behaviour for different Rayleigh–Darcy numbers: for this assessment, we increase $Ra$ by increasing the domain height  $H^{\ast }$ , as from (1.1), but fixing the gap thickness to $b^{\ast }=0.30~\text{mm}$ . The evolution of the dimensionless time-dependent dissolution rate, $F(t)$ , computed as in (2.3), is shown in figure 2. The line at $Ra=6.3\times 10^{5}$ is black as a reference for later discussion. It is apparent from the plateaux exhibited by the different experiments that a constant flux regime is found and it is later followed by a shutdown regime. The value of the plateau is similar for all the different cases and appears independent of $Ra$ . This behaviour was found also, with quantitatively similar results, by a number of numerical studies performed under Darcy flow assumptions (Pau et al. Reference Pau, Bell, Pruess, Almgren, Lijewski and Zhang2010; Hidalgo et al. Reference Hidalgo, Fe, Cueto-Felgueroso and Juanes2012; Slim Reference Slim2014; De Paoli, Zonta & Soldati Reference De Paoli, Zonta and Soldati2017; Amooie, Soltanian & Moortgat Reference Amooie, Soltanian and Moortgat2018; Wen et al. Reference Wen, Akhbari, Zhang and Hesse2018a ).

Figure 2. Time-dependent dimensionless dissolution rate $F(t)$ for different Rayleigh–Darcy numbers, as indicated in the figure. All experiments refer to the same gap thickness ( $b^{\ast }=0.30~\text{mm}$ ), whereas $H^{\ast }$ varies. It is apparent here that a constant flux regime is found and it is later followed by a shutdown regime. The black line at $Ra=6.3\times 10^{5}$ corresponds to the experiment shown in figure 3(a). The time-averaged value of the dissolution rate during the constant flux regime, averaged over three experiments, is reported in figure 4(a) (red squares).

Figure 3. Effect of the gap thickness on the flow. Concentration fields are reported for (a)  $b^{\ast }=0.30~\text{mm}$ ( $Ra=6.3\times 10^{5}$ ), (b)  $b^{\ast }=0.50~\text{mm}$ ( $Ra=1.8\times 10^{6}$ ), (c)  $b^{\ast }=0.80~\text{mm}$ ( $Ra=4.5\times 10^{6}$ ) and (d)  $b^{\ast }=1.0~\text{mm}$ ( $Ra=7.0\times 10^{6}$ ). Larger gap thicknesses correspond to stronger effects of dispersion. When gap thickness is increased, the shape of the fingers is less defined and the concentration gradients across the interface of the fingers reduce.

To focus now on the role of the domain gap, $b^{\ast }$ , we consider experiments performed in a cell with constant domain height $H^{\ast }=343~\text{mm}$ , and variable gap thickness  $b^{\ast }$ . Four snapshots of the solute concentration are shown in figure 3, and correspond to four different cases of increasing gap width ( $b^{\ast }=0.30$ , 0.50, 0.80 and $1.00~\text{mm}$ ), i.e. of increasing $Ra$ ; the experiment shown in figure 3(a) corresponds to the black line in figure 2. The four snapshots of figure 3 are taken at the time at which fingers have impinged on the lower boundary. This time precedes the end of the different plateaux in figure 2, just before the beginning of the shutdown phase. While in figure 3(a) sharp concentration gradients (highlighted by the pronounced colour gradients) are noticeable, we observe from figure 3(b–d) that increasing $Ra$ via increasing the gap width corresponds to a smoothing of the concentration gradients, and fingers appear progressively less and less coherent. Responsible for this smoothing of the concentration gradients is the shear (or hydrodynamic) dispersion (Taylor Reference Taylor1953), which occurs when a velocity profile advects scalar species at different streamwise velocity. Increasing the gap width corresponds to a decrease of the resistance experienced by the fluid flowing down the cell and, for the same driving force (i.e. density difference) we should expect a larger solute flux, since the convective velocity ${\mathcal{W}}^{\ast }$ scales with  $(b^{\ast })^{2}$ , see (2.2). However, since the hydrodynamic dispersion varies with the velocity gradient of the Poiseuille flow profile, which scales as ${\mathcal{W}}^{\ast }/b^{\ast }\sim b^{\ast }$ , increasing the gap width produces an interplay between convective velocity and hydrodynamic dispersion; in this way, the flow in the cell can be influenced by the three-dimensional spurious effects, which can prevent the application of the Darcy flow hypothesis to analyse the flow features. This will be further discussed later in this section.

Figure 4. (a) Time-averaged dissolution rate, $\langle F\rangle$ , as a function of the Rayleigh–Darcy number, $Ra$ . Here, $Ra$ is changed varying gap thickness $b^{\ast }$ and domain height $H^{\ast }$ . A difference in the behaviour of $\langle F\rangle$ is observed between low and high values of the gap thickness. (b) Low-thickness experiments, corresponding to three values of thickness  $b^{\ast }$ , and six values of the domain height $H^{\ast }=104~\text{mm}$ , 132 mm, 164 mm, 201 mm, 236 mm and 343 mm. Sherwood number is shown as a function of the Rayleigh–Darcy number. Results are fitted within the same value of the gap thickness. Best fitting functions $Sh=\unicode[STIX]{x1D6FD}Ra^{\unicode[STIX]{x1D6FC}}$ computed for constant $b^{\ast }$ are shown as dashed lines. Scaling exponents found via data fitting and limited to each subdataset are also reported. The curve corresponding to the exponent found in numerical simulations, $\unicode[STIX]{x1D6FC}=1$ , is also shown (solid line).

To fully appreciate the important role of hydrodynamic dispersion on this type of experiment, we present in figure 4 a global view of all our measurements of $\langle F\rangle$ , which is the time average of the solute flux $F(t)$ during the constant flux regime. We underline here that Letelier et al. (Reference Letelier, Mujica and Ortega2019) used the scalar dissipation rate rather than $\langle F\rangle$ , but as discussed by De Paoli et al. (Reference De Paoli, Giurgiu, Zonta and Soldati2019), for the present configuration, $\langle F\rangle$ is equivalent to the mean scalar dissipation rate. In figure 4(a), we present $\langle F\rangle$ as a function of $Ra$ , which is varied both by changing $H^{\ast }$ and  $b^{\ast }$ , see (1.1). The figure shows five datasets, corresponding to five values of  $b^{\ast }$ . We can observe that data corresponding to small values of $b^{\ast }$ ( ${\leqslant}0.50~\text{mm}$ ) have a distinctly different behaviour from those obtained for the two larger values of  $b^{\ast }$ . In particular, for $b^{\ast }=0.80~\text{mm}$ and 1.00 mm, corresponding to a decrease of resistance to the flow, there is an important decrease of $\langle F\rangle$ , while hypothetical two-dimensional simulations would give a constant $\langle F\rangle$ . The case of the reported strong reduction of $\langle F\rangle$ in our experiments may be interpreted as the combined action of spurious transversal solute fluxes in the Hele-Shaw cell, which produce the dispersion effects as discussed by Letelier et al. (Reference Letelier, Mujica and Ortega2019), and the transition towards to the three-dimensional flow regime. Numerical simulations can replicate a perfect two-dimensional situation and therefore will be used here to benchmark and analyse our data in the limit of small thicknesses. Numerical simulations are commonly analysed in terms of the Sherwood number, $Sh=\langle F\rangle Ra$ which, in the instance of heat transfer problems, is called the Nusselt number. In the papers by Slim (Reference Slim2014) and Wen et al. (Reference Wen, Akhbari, Zhang and Hesse2018a ), a linear scaling was found for the Sherwood number as $Sh=\unicode[STIX]{x1D6FD}Ra^{\unicode[STIX]{x1D6FC}}$ with $\unicode[STIX]{x1D6FC}=1$ and $\unicode[STIX]{x1D6FD}=0.017$ . Similarly, a linear scaling is obtained by Hidalgo et al. (Reference Hidalgo, Fe, Cueto-Felgueroso and Juanes2012) and Amooie et al. (Reference Amooie, Soltanian and Moortgat2018). With the aim of emphasising the effect of the experimental parameters on $Sh$ , in figure 4(b) we show the behaviour of the Sherwood number as a function of $Ra$ in the following way: we grouped the data into subdatasets of six experiments obtained for the same gap, and for each subdataset, $Ra$ was varied by changing only the domain height  $H^{\ast }$ . In figure 4(b), the scaling exponents found via data fitting and limited to each subdataset are also reported: the value of the exponents is in very good agreement with those found in the literature, which are summarised in table 2. To find best fit power laws, a regression model is applied on the data in the form $\log Sh=\unicode[STIX]{x1D6FC}\log Ra+\log \unicode[STIX]{x1D6FD}$ .

Figure 5. Time-averaged dissolution rate, $\langle F\rangle$ , as a function of the Rayleigh–Darcy number, $Ra$ . Experiments for all the values of height considered, $H^{\ast }$ , and small thicknesses ( $b^{\ast }=0.15$ , 0.30 and 0.50 mm) are shown. Best fitting functions $\langle F\rangle \sim Ra^{\unicode[STIX]{x1D6FC}-1}$ computed for constant $H^{\ast }$ are shown as dashed lines. Scaling exponents found via data fitting and limited to each value of domain height are also reported. The curve corresponding to the mean value of best fit exponents, $\unicode[STIX]{x1D6FC}=0.85$ , is also shown (solid line).

Table 2. Summary of scaling exponents, $\unicode[STIX]{x1D6FC}$ of equation $Sh=\unicode[STIX]{x1D6FD}Ra^{\unicode[STIX]{x1D6FC}}$ , as found in previous numerical and experimental works. We consider here only experiments performed in Hele-Shaw cells and two-dimensional Darcy simulations in homogeneous and isotropic porous media. In the experimental works, an aqueous solution of PPG has been used. In the numerical works, different discretisation techniques have been adopted. The range of Rayleigh–Darcy numbers considered is also reported.

If we compare now our situation to previous experimental campaigns, we observe that Backhaus et al. (Reference Backhaus, Turitsyn and Ecke2011) and Tsai et al. (Reference Tsai, Riesing and Stone2013) report $\unicode[STIX]{x1D6FC}=0.80$ and 0.76, respectively. In both papers $\unicode[STIX]{x1D6FC}$ was identified as a best fit exponent on the entire dataset, but while Tsai et al. (Reference Tsai, Riesing and Stone2013) varied $b^{\ast }$ only, Backhaus et al. (Reference Backhaus, Turitsyn and Ecke2011) varied both $b^{\ast }$ and  $H^{\ast }$ . To emphasise clearly the effect of changing $b^{\ast }$ and $H^{\ast }$ on our data, in figure 5 we show the behaviour $\langle F\rangle$ as a function of $Ra$ and we find the best fit exponents $\unicode[STIX]{x1D6FC}$ on different subdatasets which, this time, group data points with the same  $H^{\ast }$ . We plot $\langle F\rangle$ rather than $Sh$ because effects emerge in a more evident manner, and therefore we now look for the scaling

(3.1) $$\begin{eqnarray}\langle F\rangle =\frac{Sh}{Ra}\sim Ra^{\unicode[STIX]{x1D6FC}-1}.\end{eqnarray}$$

As shown clearly by the value of the slopes reported in figure 5, in this case we find values for $\unicode[STIX]{x1D6FC}\approx 0.85$ , much closer to those obtained in previous experiments by Backhaus et al. (Reference Backhaus, Turitsyn and Ecke2011) and Tsai et al. (Reference Tsai, Riesing and Stone2013). It seems therefore clear that the scaling of $\unicode[STIX]{x1D6FC}$ found in previous experimental works includes also some effect of the spurious transversal currents.

Figure 6. Time-averaged dissolution rate, $\langle F\rangle$ is here reported. Experiments are grouped by gap thickness  $b^{\ast }$ . (a) Here, $\langle F\rangle$ is shown as a function of the anisotropy ratio, $\unicode[STIX]{x1D716}$ . Results are grouped by regime as Hele-Shaw and three-dimensional regime. Here, (b)  $\langle F\rangle$ is shown as a function of $\unicode[STIX]{x1D716}^{2}Ra$ . In this case, the regime classification is more evident. A drop of the dissolution rate is observed in correspondence of $\unicode[STIX]{x1D716}^{2}Ra=1$ , where the transition from Hele-Shaw flow to three-dimensional flow occurs. The Darcy regime, which represents a theoretical limit for the experiments and corresponds to $\unicode[STIX]{x1D716}^{2}Ra\rightarrow 0$ , is also indicated.

To analyse in detail, and also in an effort to quantify the influence of the geometry of the Hele-Shaw cell, which is crucial for the evolution of the flow, we use the cell anisotropy ratio, $\unicode[STIX]{x1D716}=b^{\ast }/\sqrt{12}H^{\ast }$ , which scales as the ratio of the characteristic time of transverse diffusion, $(b^{\ast })^{2}/D$ , to the longitudinal advection time, $H^{\ast }/{\hat{w}}^{\ast }$ , being ${\hat{w}}^{\ast }$ the vertical velocity averaged across the gap $b^{\ast }$ (see Bae, Beta & Bodenschatz Reference Bae, Beta and Bodenschatz2009; Kirby Reference Kirby2010). The behaviour of $\langle F\rangle$ as a function of the anisotropy ratio is shown in figure 6(a). As discussed by Letelier et al. (Reference Letelier, Mujica and Ortega2019), a Hele-Shaw cell behaves as a two-dimensional domain only in the limit of $\unicode[STIX]{x1D716}\rightarrow 0$ , which is also the limit for Darcy flow. Indeed, an increase of the gap width produces an increase of vertical velocity, while the characteristic diffusion time scale remains unchanged. Since diffusion is responsible for the homogenisation of the solute distribution in the wall-normal direction, we cannot assume that the solute concentration is uniform across the gap. An increase of $\unicode[STIX]{x1D716}$ implies the occurrence of transverse effects which will lead the behaviour of the cell to Hele-Shaw regime and further to three-dimensional regime as proposed by Letelier et al. (Reference Letelier, Mujica and Ortega2019). The application of their classification to our data emerges very clearly from figure 6(b), where we plot $\langle F\rangle$ as a function of the dimensionless number $\unicode[STIX]{x1D716}^{2}Ra$ : if $\unicode[STIX]{x1D716}^{2}Ra$ is small, the cell behaviour is closer to the Darcy flow behaviour, whereas if $\unicode[STIX]{x1D716}^{2}Ra$ is large the flow exhibits three-dimensional effects which cannot be described with the Darcy equations, and can cause discrepancies of the scaling exponents. This is perhaps rather obvious, but it is here quantified precisely via experiments for the first time. We observe a drop of $\langle F\rangle$ after the threshold $\unicode[STIX]{x1D716}^{2}Ra=1$ , which represents the threshold value identified theoretically for the transition to the three-dimensional flow (Letelier et al. Reference Letelier, Mujica and Ortega2019). Our current dataset does not allow us to investigate whether there is a sharp or a smooth transition in correspondence of $\unicode[STIX]{x1D716}^{2}Ra=1$ . We can, however, confirm that the transition to the three-dimensional regime occurs for the subdatasets $b^{\ast }=0.80$ and 1.00 mm, corresponding to $\unicode[STIX]{x1D716}^{2}Ra\gg 1$ . These results provide a further assessment of the gap-induced dispersion effects, which represent one of the possible differences in the flow behaviour observed numerically and experimentally. Clearly, a transition of the cell behaviour from Hele-Shaw to three-dimensional regime, will require using the full Navier–Stokes equation set rather than a Darcy model (plus corrective terms) to analyse the flow. From the experimental viewpoint, a satisfactory analysis of full three-dimensional regime would require further observations in the out-of-plane direction. Phenomena like multiple finger in the thickness of the cell could not, otherwise, be accurately quantified.

Figure 7. (a) Concentration field over a small portion of the domain. The solid, black line identifies the position of the interface between the solute-rich mixture (aqueous solution of KMnO₄) and the pure fluid (water). The dashed line shows the position of the averaged fingertip, identified as the value of $z^{\ast }$ at which $C^{\ast }/C_{s}^{\ast }=1/50$ , and with indication of the fingertip velocity, $w_{f}^{\ast }$ (vectors). (b) Horizontally averaged concentration profile (black, solid line) used to identify the fingertip position. (c) Gap-based Reynolds number, $Re$ , estimated for all the experiments considered; when $Re\leqslant 1$ the system is in the Hele-Shaw regime, the transition to the three-dimensional flow occurs for $Re\gg 1$ .

The effect of inertia in the Hele-Shaw regime was examined theoretically by Letelier et al. (Reference Letelier, Mujica and Ortega2019), who proposed to add extra terms to the Darcy model. In our experiments, these extra terms scale as $\unicode[STIX]{x1D716}^{2}Ra/Sc$ , where $Sc$ is the Schmidt number, i.e. the ratio of momentum to mass diffusivity, and is defined here as $Sc=\unicode[STIX]{x1D707}/\unicode[STIX]{x1D70C}D$ . We have that in our experiments $Sc\sim O(10^{3})$ . Since inertial terms are multiplied by the coefficient $\unicode[STIX]{x1D716}^{2}Ra/Sc$ (Letelier et al. Reference Letelier, Mujica and Ortega2019), we can neglect these terms when $\unicode[STIX]{x1D716}^{2}Ra\leqslant 1$ , which in our experimental dataset corresponds to $b^{\ast }\leqslant 0.50~\text{mm}$ . Our aim here is to evaluate the Reynolds number $Re$ to confirm the predictions of Letelier et al. (Reference Letelier, Mujica and Ortega2019). In this context, we assume as velocity scale $\langle w_{f}^{\ast }\rangle$ , defined as the mean value during the constant flux regime of the average fingertip velocity $w_{f}^{\ast }$ , as sketched in figure 7(a), and we compute the Reynolds number $Re$ as follows:

(3.2) $$\begin{eqnarray}Re=\frac{\unicode[STIX]{x1D70C}_{s}^{\ast }\langle w_{f}^{\ast }\rangle b^{\ast }}{\unicode[STIX]{x1D707}}.\end{eqnarray}$$

To find $w_{f}^{\ast }$ we use the horizontally averaged concentration profile

(3.3) $$\begin{eqnarray}\overline{C^{\ast }}(z^{\ast },t^{\ast })=\frac{1}{L^{\ast }}\int _{0}^{L^{\ast }}C^{\ast }(x^{\ast },z^{\ast },t^{\ast })\,\text{d}x^{\ast }.\end{eqnarray}$$

We customarily set the location of the average fingertip as the vertical coordinate $z^{\ast }$ of the cell where

(3.4) $$\begin{eqnarray}\overline{C^{\ast }}(z^{\ast },t^{\ast })/C_{s}^{\ast }=1/50,\end{eqnarray}$$

as illustrated in figure 7(b). In figure 7(c) we report the values of $Re$ for our dataset. We can observe that the Reynolds number $Re$ increases for increasing the cell gap width, $b^{\ast }$ . The smallest gap-width experiments ( $b^{\ast }=0.15~\text{mm}$ ) give a value of $Re$ lower than  $10^{-1}$ , which is a good experimental approximation of Darcy flow conditions. For $Re\leqslant 1$ (i.e.  $b^{\ast }\leqslant 0.50~\text{mm}$ ) inertial effects are of the same order of viscous effects. However, we have shown via detailed analysis of the dissolution rate that these experiments are in the Hele-Shaw regime. This indicates that the Reynolds number is not sufficient for the evaluation of inertia, and a detailed analysis of the local flow velocities may be required. Finally, for the larger gap-width experiments $Re\gg 1$ (i.e.  $b^{\ast }\geqslant 0.80~\text{mm}$ ), where three-dimensional effects are important, we can observe that also inertial effects are becoming important, with $Re$ value of the order of 10.