# Multidimensional Observations of Dissolution-Driven Convection in Simple Porous Media Using X-ray CT Scanning

## Abstract

We present an experimental study of dissolution-driven convection in a three-dimensional porous medium formed from a dense random packing of glass beads. Measurements are conducted using the model fluid system MEG/water in the regime of Rayleigh numbers, \(Ra=2000{-}5000\). X-ray computed tomography is applied to image the spatial and temporal evolution of the solute plume non-invasively. The tomograms are used to compute macroscopic quantities including the rate of dissolution and horizontally averaged concentration profiles, and enable the visualisation of the flow patterns that arise upon mixing at a spatial resolution of about (\(2\times 2\times 2)\,\hbox {mm}^3\). The latter highlights that under this *Ra* regime convection becomes truly three-dimensional with the emergence of characteristic patterns that closely resemble the dynamical flow structures produced by high-resolution numerical simulations reported in the literature. We observe that the mixing process evolves systematically through three stages, starting from pure diffusion, followed by convection-dominated and shutdown. A modified diffusion equation is applied to model the convective process with an onset time of convection that compares favourably with the literature data and an effective diffusion coefficient that is almost two orders of magnitude larger than the molecular diffusivity of the solute. The comparison of the experimental observations of convective mixing against their numerical counterparts of the purely diffusive scenario enables the estimation of a non-dimensional convective mass flux in terms of the Sherwood number, \(Sh=0.025Ra\). We observe that the latter scales linearly with *Ra*, in agreement with both experimental and numerical studies on thermal convection over the same *Ra* regime.

## Keywords

Solute mixing 3D imaging Porous media## 1 Introduction

The study of convective mixing in porous media continues to find applications in both traditional and emerging engineering problems, many of which occur in natural environments (Gebhart and Pera 1971; Diersch and Kolditz 2002). We focus here on density-driven *free* convection to highlight that the mixing process is induced and sustained by a buoyancy effect, in the absence of advective flows that are introduced by, e.g. an external pressure gradient. One particular application that has increased the interest in this phenomenon is geologic carbon sequestration (GCS) (Huppert and Neufeld 2014), because of its potential impact on the dissolution rate of \(\hbox {CO}_2\) into formation fluids. In this scenario, the buoyancy effect may be due to the natural occurring geothermal gradient in the reservoir, but also -and primarily- to varying composition of the aqueous phases (Lindeberg and Wessel-Berg 1997). In fact, \(\hbox {CO}_2\), dissolution into brine leads to a local density increases on the order of 0.1–1% [depending on pressure and temperature (Efika et al. 2016)], which is sufficient to create a buoyant instability that in turn induces a convective overturn in the brine; the denser \(\hbox {CO}_2\)-rich aqueous mixture flows downwards and pushes fresh brine up towards the \(\hbox {CO}_2\)-brine interface. The ability of \(\hbox {CO}_2\)-saturated brine to sink deeper into the aquifer reduces the likelihood of \(\hbox {CO}_2\) leakage, thereby increasing long-term storage security. Dissolution of \(\hbox {CO}_2\) is considered a key trapping mechanism in GCS (Benson and Cole 2008) and convective mixing is expected to contribute largely to this process (Ennis-King and Paterson 2005), partly because mass transfer by diffusion, despite being ubiquitous, is very slow. Recent surveys of potential storage sites around the world suggest that the conditions are often met for convective mixing to occur [e.g. Sathaye et al. (2014) using data compiled in Szulczewski et al. (2012)]; however, estimates on its actual contribution towards storage, its spatial footprint and its timescale are still largely uncertain, because of the lack of direct observations at representative subsurface conditions and the intrinsic difficulty in estimating dimensions and properties in heterogeneous rock formations (Sathaye et al. 2014).

There have been numerous experimental studies where density-driven convection has been investigated in the context of GCS. These efforts may be broadly divided in two categories, namely (i) studies using high-pressure blind PVT cells and (ii) those using 2D transparent Hele-Shaw cells. The former can be operated with representative fluids (e.g. supercritical \(\hbox {CO}_2\) and brine) and the rate of dissolution is typically inferred from pressure decay (Yang and Gu 2006; Farajzadeh et al. 2009; Khosrokhavar et al. 2014) and/or changes in weight (Arendt et al. 2004) inside the closed reactor, or is measured directly by recording the make up liquid volume needed to maintain a constant pressure in the system (Newell et al. 2018). For data reconciliation, some authors have applied the diffusion equation with an effective diffusion coefficient (Yang and Gu 2006; Moghaddam et al. 2012), while others have used more rigorous mathematical models that account for both mass and momentum conservation in the liquid phase (and that use the bulk molecular diffusivity) (Khosrokhavar et al. 2014; Farajzadeh et al. 2009). Results from these studies consistently show that under the convective regime the mass-transfer rate across the \(\hbox {CO}_2\)/brine interface is indeed much faster than that predicted by Fickian diffusion (with an effective diffusion coefficient that is one to two orders of magnitude larger than the (bulk) molecular diffusivity, depending on the initial gas pressure and salt concentration in the brine). Unfortunately, the majority of these observations still refer to the dissolution of \(\hbox {CO}_2\) into bulk brine and experiments using porous media have just begun (Newell et al. 2018; Nazari Moghaddam et al. 2015). Also, only in rare cases did the experiments enable direct visualisation of convective patterns (through an embedded optical side cell) (Khosrokhavar et al. 2014; Arendt et al. 2004).

With the intention of visualising the convective process, several authors have made use of Hele-Shaw cells, albeit with analogue fluid-pairs [e.g. MEG-water (Neufeld et al. 2010), water-PG (Backhaus et al. 2011; Tsai et al. 2013; Macminn and Juanes 2013; Agartan et al. 2015), gaseous \(\hbox {CO}_2\)-water (Kneafsey and Pruess 2010) and \(\hbox {KMnO}_4\) in water (Slim et al. 2013; Ching et al. 2017). By enabling direct access to local measures of convection (e.g. wavelength of the instability, vertical plume velocity, plume width and their statistics) (Slim et al. 2013; Ecke and Backhaus 2016), these experiments have been pivotal in supporting the significant effort that has been dedicated to the study of density-driven convection in porous media by means of numerical simulations [see the recent review Emami-Meybodi et al. (2015) and references therein]. Evidence (Hidalgo et al. 2012; Raad and Hassanzadeh 2015; Jafari Raad et al. 2016) now exists that results using analogue fluid pairs may not be directly applicable to the subsurface \(\hbox {CO}_2\)/brine system (see also Sect. 5); these studies have also demonstrated that despite its inherent chaotic nature, the process of convective mixing can be parameterised in terms of useful macroscopic variables, such as the Rayleigh, *Ra*, and Sherwood number, *Sh* (or its counterpart for heat transfer studies, the Nusselt number, *Nu*). Nevertheless, because the convection process in a porous medium is three-dimensional, concerns have been raised with regard to the inherent limitations of two-dimensional experiments (or of their numerical counterparts) (Lister 1990) and to the applicability of the obtained scaling laws. On the one hand, some authors have proposed that for \(Ra>2000\), a scaling relationship exists for the convective mixing process that is universal, both in two and three dimensions (Fu et al. 2013). On the other hand, results from numerical simulations suggest that in three dimensions (i) the dissolution flux is 25–40% larger than in two dimensions (Pau et al. 2010; Hewitt et al. 2014), (ii) stronger dispersion occurs (thus leading to weaker flow) and (iii) fingers grow bigger (thus leading to faster penetration) (Knorr et al. 2016). Experimental validation of these findings is still lacking.

Because of the inherent difficulty of imaging the convective process within an opaque medium non-invasively, very limited experimental observations exist of density-driven convection in three-dimensional porous media. In their early seminal work, Bories and Thirriot (1969) used photographs of the top free surface of a liquid-saturated (\(70\times 50\times 8)\,\hbox {cm}^3\) rectangular beadpack to infer fluid movements within the medium itself; most significantly, they demonstrated that cellular structures appear with length scale \(\mathcal {O}(l)\sim 10\,\hbox {cm}\), which are not possible in two-dimensional settings, as the latter limit the growth of the plume to two orthogonal directions. These findings were later confirmed by Lister (1990), who used a similar experimental approach and extended these observations to the regime \(\mathcal {O}(Ra)\sim 1000\). The first images of the convection pattern *within* a porous medium were reported only a few years later by Howle et al. (1993, 1997) using a shadowgraphic techniques and by Shattuck et al. (1995) using magnetic resonance imaging (MRI) for both regular and disordered packings. In these experiments, observations were limited to \(\mathcal {O}(Ra)\sim 100\) and two two dimensions (horizontal flow patterns), and the mixing process was driven by temperature gradients, rather than dissolution. Nevertheless, by demonstrating a novel ability to image the convective process within opaque media non-invasively, these studies have provided direct evidence that the structure of the medium plays a fundamentally important role in the determination of the flow pattern.

In this study, we build on the findings above by presenting multidimensional observations of convective dissolution in simple porous media using X-ray computed tomography (X-ray CT) for the MEG-brine fluid pair. Together with the recent work by Nakanishi et al. (2016) and Wang et al. (2016), we provide what are, to our knowledge, the first non-invasive determinations of three-dimensional patterns in opaque, random porous media. Experiments are carried out in the regime \(\mathcal {O}(Ra)\sim 1000\) and the mixing process is quantified using various metrics, including the rate of dissolution and effective diffusion coefficients. Observations are compared to the limiting case of a purely diffusive scenario, which further enables the investigation of a \(Sh{-}Ra\) scaling law and its comparison with results reported in the literature using a similar fluid pair.

## 2 Experimental

### 2.1 Porous Medium and Fluids

^{®}, 99%, Sigma Aldrich) to achieve high X-ray imaging contrast for the experiments. Only one brine solution is used that contains 6 wt% sodium chloride (NaCl, \(>99\)%, Sigma Aldrich) in distilled water. The density of the pure solutions and of their mixtures have been measured using an oscillating U-tube density meter (DM5000 by Anton Paar) at 20\(^\circ \)C and 1 atm. For each measurement, approximately 3 mL of solution was used and the density was taken to be the average of three repeated measurements. The obtained density curves are shown in Fig. 2a as a function of wt% MEG,

*w*, where the experimental values (symbols) are plotted alongside fitted polynomial curves (parameters provided in “Appendix A1”). Error bars are not shown in the figure, because they are smaller than the symbols. These curves present a characteristic non-monotonic profile with a maximum at intermediate MEG concentrations (\(w=0.4-0.5\)) and a density larger than that of pure brine (\(\rho (w)>\rho _2\)), whereas at larger concentrations (\(w>0.7\)) the solution becomes buoyant (\(\rho _1<\rho (w)<\rho _2\)). The key characteristic properties of the solutions are summarised in Table 1, together with estimates of the Rayleigh number,

*Ra*. The latter is calculated as:

Characteristic metrics of the density curves that represent the three solution pairs used in this study, namely maximum density difference between the two solutions (\(\Delta \rho _\mathrm{max}/\rho _2\), where \(\rho _2=1.040\) g/mL is the initial density of the brine), weight fraction at maximum density (\(w^\mathrm{max}\)) and at neutral buoyancy \(w^0\). The Rayleigh number, *Ra*, is calculated from Eq. 1

Solution | \(\rho _1\) (g/mL) | \(\Delta \rho _\mathrm{max}/\rho _2\) (%) | \(w^\mathrm{max}\) | \(w^0\) | |
---|---|---|---|---|---|

MEG55/brine | 1.018 | 0.4 | 0.41 | 0.68 | 2150 |

MEG57/brine | 1.025 | 0.6 | 0.47 | 0.78 | 3230 |

MEG59/brine | 1.032 | 0.9 | 0.50 | 0.88 | 4610 |

### 2.2 Experimental Procedure and Imaging

Summary of experiments conducted in this study. The parameters listed in the table have been estimated upon following the procedure described in Sect. 2.3. \(M_1\) and \(M_2\) are the mass of solution 1 (MEG) and 2 (brine) with estimated uncertainty, \(\sigma _\mathrm{M}\); \(V_\mathrm{T}\) and \(V_\mathrm{B}\) are the volumes of the top and bottom sections of the bowl and \(\widehat{w}_\mathrm{B}(t_\mathrm{f})\) is the mass fraction of solute in the bottom section of the bowl at the end of the experiment. For each experiment, \(H_\mathrm{T}=2\,\hbox {cm}\) and \(H_\mathrm{B}=13\,\hbox {cm}\)

Solution | \(M_1\) (g) | \(M_2\) (g) | \(\sigma _\mathrm{M}\) (g) | \(V_\mathrm{T}\) (mL) | \(V_\mathrm{B}\) (mL) | \(\widehat{w}_\mathrm{B}(t_\mathrm{f})\) |
---|---|---|---|---|---|---|

MEG55/brine | 88.7 | 792.4 | 4.9 | 242.2 | 2116 | 0.112 |

97.8 | 788.7 | 5.4 | 267.0 | 2107 | 0.124 | |

MEG57/brine | 79.4 | 808.2 | 5.3 | 215.2 | 2159 | 0.098 |

72.4 | 815.3 | 6.8 | 196.1 | 2178 | 0.089 | |

MEG59/brine | 83.3 | 799.1 | 7.9 | 224.1 | 2134 | 0.104 |

103.9 | 784.0 | 10.2 | 279.6 | 2094 | 0.133 |

### 2.3 Image Processing

*i*at given time

*t*, \(CT_i(t)\), is expressed as the linear combination of the CT numbers associated with the volume and mass fractions of each of its components:

*N*in each section, i.e. \(V_j=N_jV_\mathrm{vox}\), where \(j=T,B\). The corresponding total mass of solution 1 and 2 can be readily computed as \(M_1= \phi V_\mathrm{T}\rho _1\) and \(M_2= \phi V_\mathrm{B}\rho _2\). At the end of the experiment (\(t=t_\mathrm{f}\)), solution 1 (MEG) has completely dissolved and the top section of the bowl (\(V_\mathrm{T}\)) contains only solution 2 (brine, \(\widehat{w}_\mathrm{T}(t_\mathrm{f})=0\)); the corresponding value in \(V_\mathrm{B}\) is obtained from the following material balance,

*i*in the top, \(w_{\mathrm {T},i}(t)\), and bottom sections of the bowl, \(w_{\mathrm {B},i}(t)\), can therefore be computed as follows:

*i*, while \(\widehat{CT}_\mathrm{B}\) and \(\widehat{CT}_\mathrm{T}\) represent the average of all voxel CT values in the bottom and top sections of the bowl at the initial and final time (\(t_{0}\) and \(t_\mathrm{f}\), respectively). The latter are associated with the CT numbers of the pure liquid solutions and are obtained for each experiment independently, e.g. for the top section in Eq. 2: \(\widehat{CT}_\mathrm{T}(t_\mathrm{f})-\widehat{CT}_\mathrm{T}(t_\mathrm{0}) = \phi (CT_2-CT_1)\). Equations 4a and 4b are applied on a voxel scale (as shown in Fig. 3c for the central slice of the bowl) and the operation is repeated for each slice in the bowl to enable the three-dimensional reconstruction of the temporal and spatial evolution of the process of convective mixing (Fig. 3d). As an important component in the analysis that follows, the temporal evolution of the total mass of solute in each section (top,

*T*, and bottom,

*B*) is estimated as:

where the density of the mixture is computed from the parameterisation of the curves shown in Fig. 2 as a function of the average mass fraction of the solute in the given section of the bowl, \(w_\mathrm{B}(t)\) or \(w_\mathrm{T}(t)\), which are estimated by using section-averaged *CT*(*t*) numbers in Eqs. 4a and 4b.

## 3 Modelling

*c*is the concentration of MEG in the brine solution, \(\phi \) is the porosity, \(\mathscr {D}\) is the molecular diffusion coefficient, and

*z*and

*t*are the spatial (vertical) and temporal coordinates. The cross-sectional area can be conveniently described as a function of

*z*, i.e. \(A(z) = \pi (z+h)[d-(z+h)]\) for \(-h\le z\le d-h\), where

*d*is the diameter of the sphere,

*h*is defined so that \(A(z=0)=\pi {d_\mathrm{t}}^2/4\) and

*z*increases downwards (see Fig. 1). Equation 6 can be simplified further to give,

## 4 Results

### 4.1 Extent of Dissolution and Mixing Regimes

Macroscopic measures of convective mixing extracted from the experiments carried out in this study. Rayleigh number (*Ra*), effective diffusion coefficient achieved in the convective regime (\(\mathscr {D_\mathrm{eff}}\)), onset time of convection (\(t_\mathrm{c}\)) and time of convective shutdown (\(t_\mathrm{s}\)). The molecular (bulk) diffusion coefficient takes the value \(\mathscr {D}=1\times 10^{-5}\,\hbox {cm}^2\)/s. The parameters and their uncertainties have been obtained using standard relationships for weighted linear regression (Taylor 1997)

Solution | | \(\mathscr {D}_\mathrm{eff}/\mathscr {D}\) | \(t_\mathrm{c}\) (min) | \(t_\mathrm{s}\) (min) |
---|---|---|---|---|

MEG55/brine | 2150 | \(79\pm 7\) | \(50\pm 9\) | \(363\pm 38\) |

\(67\pm 7\) | \(57\pm 13\) | \(423\pm 54\) | ||

MEG57/brine | 3230 | \(78\pm 9\) | \(32\pm 9\) | \(314\pm 46\) |

\(69\pm 11\) | \(21\pm 9\) | \(300\pm 58\) | ||

MEG59/brine | 4610 | \(115\pm 22\) | \(15\pm 8\) | \(190\pm 44\) |

\(105\pm 20\) | \(19\pm 9\) | \(215\pm 49\) |

The experimental data confirm the expected positive trend in the onset and subsequent rate of dissolution with increasing Rayleigh number. The time for the onset of the convective regime, \(t_\mathrm{c}\), has been estimated by identifying the point at which the experimental measurements depart from the model-predicted diffusive line. For each scenario, this point is denoted in Fig. 4 by the black circle, which has been obtained upon extrapolation of the trend predicted by the pseudo-diffusive regime back to \(m_j/M_1=1\); the obtained values are \(54\pm 16\) min (MEG55), \(26\pm 13\) min (MEG57) and \(17\pm 12\) min (MEG59), and are additionally plotted in Fig. 5 as a function of the Rayleigh number, *Ra*. It can be seen that our experimental observations compare favourably with results from numerical studies reported in the literature and summarised in Emami-Meybodi et al. (2015), where it is shown that \(t_\mathrm{c}\sim \mathrm {Ra}^{-2}\). As it can be inferred from the figure, the determination of the onset of convection is affected by a significant degree of uncertainty (deviations of up to one order of magnitude are seen among the trends predicted by the numerical simulations). The latter results from the presence of perturbations at the interface, which need to be imposed artificially (in numerical simulations) or are naturally introduced by packing heterogeneities (as it is the case of our experiments). As shown in Fig. 4, the convective regime is followed by a gradual slow down of the dissolution rate that eventually approaches a value near zero. In our system, this shutdown appears because of the depletion of the MEG plume; accordingly, because of the trend in the rate of dissolution described above, the time to attain convective shutdown (\(t_\mathrm{s}\) in Table 3) decreases with increasing *Ra* number, i.e. \(t_\mathrm{s}=393\pm 66\) min (MEG55), \(307\pm 74\) min (MEG57) and \(202\pm 66\) min (MEG59).

### 4.2 Horizontally Averaged Concentration Profiles

In Fig. 6, vertical profiles are presented of the mass fraction of solute, *w*, at various times and for the three systems investigated, namely MEG55, MEG57 and MEG59. The profiles have been computed upon using *CT* numbers in Eq. 4 that represent the average of all voxels in each 2 mm-thick horizontal section of the bowl. To facilitate comparison among observations with different MEG solutions (and, accordingly, *Ra* numbers), profiles are shown in the figure for CT scans that have been acquired at similar values of the dimensionless time, \(\tau =\mathscr {D_\mathrm{eff}}t/H^2_\mathrm{B}\approx 0.01-0.6\). Results are also shown in the rightmost panel of the figure for the purely diffusive case and for which \(\tau =\mathscr {D}t/H^2_\mathrm{B}\). In each plot, the black solid line represents the position of the interface at the start of the experiment. The experimental results obtained for different Rayleigh numbers show a significant degree of similarity in terms of both the temporal and spatial evolution of the dissolved plume: for \(\tau <0.08\) (red profiles), the MEG/brine interface recedes gradually, while the solute plume moves downwards in the bowl, because of its larger density as compared to fresh brine; at \(\tau \approx 0.1\), the pure MEG solution has almost completely dissolved and for \(\tau >0.1\) the solute plume begins accumulating at the bottom of the bowl (blue profiles). Notably, this results in the reversal of the concentration gradient along the bowl, with the mass fraction of MEG in brine now increasing with the distance from the top. At the end of the experiment (\(\tau \approx 0.7\)), the mass fraction of MEG increases from \(w\approx 0\) at \(z=0\,\hbox {cm}\) to \(w\approx 0.25\) at \(z=13\,\hbox {cm}\). This late-time distribution of the solute differs from the corresponding profile predicted by the model that describes a purely diffusive scenario (rightmost panel), where -as expected- the solute reaches a uniform distribution along the entire length of the bowl. In other words, convection precludes a perfect dilution of the plume as it moves downwards and the resulting (stable) density gradient once convection ceases is such that diffusion remains the only mechanism to achieve complete mixing. Two additional observations arise from the comparison between the experiment and the diffusion model. First, because density is constant in the model and diffusion is ubiquitous, the MEG/brine interface does not show the characteristic receding behaviour observed in the experiments, where \(\rho _1<\rho (w)\). In this context, despite being fully miscible with brine, the lower density of MEG acts towards stabilising the interface, while maintaining a much steeper concentration gradient across it. Second, prior to the cessation of convection the behaviour of the solute plume underneath the interface is similar to the one observed in the diffusion model, thus supporting the findings discussed above on the establishment of a pseudo-diffusive regime in the experiments.

### 4.3 Three-Dimensional Imaging and Convective Patterns

*w*, has been calculated using Eq. 4 and the dimensionless time, \(\tau \), is again chosen to facilitate the analysis and comparison of experiments conducted at different Rayleigh numbers (\(\tau \approx 0.008-0.17\)). In particular, the following regimes are identified from the 3D images: at early times (\(\tau <0.01\), first column), a large number of small-scale finger projections (\(\mathcal {O}(l)\sim 1\,\hbox {cm}\)) are seen just underneath the MEG/brine interface; upon further dissolution (\(0.01<\tau <0.1\), columns 2-4), the MEG layer continues to retract and the fingers continue to grow until they reach the bottom of the bowl (\(\mathcal {O}(l)\sim 10\,\hbox {cm}\)). Closer inspection of the images indicates that the mass fraction of MEG vary considerably among the different finger projections reaching values as high as \(w=0.6-0.7\) in the centre of some of the fingers. By the time the MEG layer has completely dissolved (\(\tau >0.15\), last column), the plume has reached the bottom of the bowl, where the solute accumulates. At this stage of the dissolution process, although a concentration gradient is still present, the associated density gradient is such that the system is stable and further mixing can be achieved only by diffusion (see also one-dimensional profiles shown in Fig. 6). We note that the regimes just described are observed in each experiment conducted in this study and their dynamics are very similar when the dimensionless time \(\tau \) is considered. Accordingly, the 3D maps shown in Fig. 7 are strikingly similar in terms of the development and propagation of the fingers. These observations provide further support to the existence of a pseudo-diffusive regime throughout a large portion of the dissolution process with a characteristic time-scale \(\tau =\mathscr {D}t/H^2_\mathrm{B}\). In agreement with previous studies on density-driven convection [e.g. Riaz et al. (2006)], we also observe that the number of fingers increases with

*Ra*.

## 5 Discussion

### 5.1 Rate of Convective Dissolution and Mass Flux

*Ra*, because of the effects introduced by the characteristic shape of the \(\rho (w)\) curve of the three fluid pairs (Jafari Raad et al. 2016). Nevertheless, the observed trend closely reflects the attainment of the three regimes discussed in Sect. 4, namely diffusive, convection-dominated (or pseudo-diffusive, with onset-time \(t_\mathrm{c}\) shown by the crosses) and shutdown. As expected, with increasing Rayleigh number the experimental curves depart sooner from the diffusive regime and they also reach a larger (and earlier) maximum dissolution rate, \(r_\mathrm{max}\). For the three MEG systems, the obtained estimates are \(r_\mathrm{max}=0.37\pm 0.06\) g/min (MEG55, at 186 min), \(r_\mathrm{max}=0.43\pm 0.06\) g/min (MEG57 at 124 min) and \(r_\mathrm{max}=0.61\pm 0.11\) g/min (MEG59 at 74 min). Interestingly, in all scenarios the time to reach maximum dissolution rates is about four times larger than the time required for the onset of convection, i.e. \(t(r_\mathrm{max})\approx 4t_\mathrm{c}\). The black circles in Fig. 9 represent the rates of dissolution achieved by diffusion at equivalent (absolute) time and take values \(r_\mathscr {D} = 0.033\) g/min (MEG55), \(r_\mathscr {D} = 0.040\) g/min (MEG57) and \(r_\mathscr {D} = 0.052\) g/min (MEG59), respectively. These dissolution rates are approximately one order of magnitude smaller than the corresponding values achieved in the presence of convection. We also note that by using a fixed boundary (i.e. the original interface) in our calculations the amount of MEG dissolved over time is underestimated. Accordingly, one should refer to a rate of MEG removal, rather than dissolution. This rate of removal combines two contributions: the rate of change in mass of buoyant solution (which is, effectively, the rate of dissolution) and the rate of change in the mass of non-buoyant solution (\(w<w_0\)). In our experiments, the latter is expected to be significantly smaller than the former, because, while some dissolved MEG does accumulate (temporarily) above the initial interface, a given amount also leaves the volume by convection. This last process is quite fast and effectively minimises the accumulation of solute above the interface. Accordingly, in our experiments the rate of MEG removal approaches the rate of dissolution. The latter is estimated with an uncertainty in the order of 15–20%, which we consider to be larger than any error introduced by using a fixed boundary in the calculations.

*Sh*, represents a non-dimensional measure of the convective mass flux and can be estimated from the ratio of the maximum convective dissolution rate computed above to the corresponding value in the presence of diffusion alone, while accounting for the appropriate length scales, i.e.

### 5.2 Sherwood–Rayleigh scaling and geological \(\hbox {CO}_2\) storage

*Sh*vs.

*Ra*plots aimed at identifying scaling laws that can be used to relate laboratory observations to field settings. The use of these dimensionless numbers is also needed as a means to compare observations from laboratory studies using different fluid pairs and geometries (e.g. 2D vs. 3D). In Fig. 10, we attempt this comparison by presenting the results from this study (circles) together with a selection of data and correlations found in the literature (details given in the figure caption). In the figure, the Rayleigh number has been normalised by its critical value, \(Ra_\mathrm{c}\), defined as the

*Ra*value for which

*Sh*(or

*Nu*in heat transfer studies) departs from the value 1 (Nield and Bejan 2006). It has been shown by numerous experimental studies that for convective flow to occur in a porous medium, \(Ra>Ra_\mathrm{c}=4\pi ^2\approx 40\) (Katto and Masuoka 1967). We also purposely focus here on the range \(\widetilde{Ra}=Ra/Ra_\mathrm{c}=1{-}300\) (\(Ra=40{-}12000\)), as this is the regime that is more likely to be expected at depth in potential geologic carbon sequestration sites (Sathaye et al. 2014), should the process of convective dissolution occur. We provide further support to this last observation with the bar chart also shown in Fig. 10 where data from 38 aquifers around the world are sorted according to the expected Rayleigh number. These include 11 major saline aquifers in the USA [\(\widetilde{Ra}\sim 1{-}100\), 21 reservoirs in total compiled in Szulczewski et al. (2012)], 13 injection sites in the Alberta Basin (\(\widetilde{Ra}\sim 1{-}10\)) (Hassanzadeh et al. 2007) and the Sleipner site in the North Sea [\(\widetilde{Ra}\sim 100{-}1000\), 4 cases depending on the assumed pressure/temperature conditions (Lindeberg and Wessel-Berg 1997)]. While these estimates must be used with some precaution due to the intrinsic difficulty in estimating suitable mean permeabilities and dimensions in heterogeneous reservoirs, the perception is that the condition \(\widetilde{Ra}<100\) (\({Ra}<4000\)) may be typical in geologic reservoirs.

The data plotted in Fig. 10 evidence two aspects. First, the available data set is still quite scarce, particularly for \(\mathcal {O}(Ra)\sim 1000\). A significant body of the literature exist on observations at low Ra values (\({Ra}<1000\), corresponding to \(\widetilde{Ra}<25\) in Fig. 10), including early studies on thermal convection in porous media [see a collection of more than 100 data points in Xie et al. (2012)] and more recent ones on dissolution-driven convection (Slim et al. 2013; Agartan et al. 2015). Others have focused on the high Rayleigh number regime (\(\mathcal {O}(Ra)\sim 10^4{-}10^6\)) (Neufeld et al. 2010; Kneafsey and Pruess 2010; Backhaus et al. 2011; Tsai et al. 2013; Ching et al. 2017; Nakanishi et al. 2016) and their observations fall outside the bounds of Fig. 10. Second, there is a significant degree of scatter among the reported results, which may be due to the use of 2D vs. 3D geometries, as well as of different model fluids. As discussed in the following, both aspects contribute to additional uncertainty on the fundamental behaviour of the dissolution flux and its dependence on the system parameters, such as the Rayleigh number.

The experiments carried out in the present study (circles) are well within the range expected in potential \(\hbox {CO}_2\) storage sites lie near the identity line, suggesting that in this regime the dissolution flux increases linearly with Ra as \(Sh=\alpha Ra\) with \(\alpha \approx 1\). However, they disagree considerably with results reported on a supposedly similar experimental system, i.e. MEG/brine in a packed bed imaged by X-ray CT, for which a significantly larger dissolution has been reported (\(\alpha \approx 4\), blue crosses in the figure) (Wang et al. 2016). We attribute this differences to the distinct shape of the density-concentration curve, in particular with the position of the maximum and cross-over points (\(w^\mathrm{max}\) and \(w^0\) in Fig. 2), which in Wang et al. (2016) are shifted towards larger MEG concentration values (\(w^\mathrm{max}\approx 0.6\) and \(w^0>0.9\)). This further implies that the range of concentration values over which the MEG solution is no longer buoyant is wider and mixing rate is thus enhanced. This pattern has been quantitatively demonstrated by means of numerical simulations (Jafari Raad et al. 2016). It may not be surprising therefore that experimental data acquired on a different model fluid pair (Backhaus et al. 2011; Tsai et al. 2013), namely propylene glycol (PG) and water (red crosses in the figure), lie on the opposite corner of the diagram and suggest that the dissolution flux is significantly smaller (3–4 times when compared to our data at \(\widetilde{Ra}=115\)). In fact, for PG-water mixture the maximum and cross-over points of the density curve are shifted towards much *lower* values (\(w^\mathrm{max}\approx 0.25\) and \(w^0\approx 0.5\)) when compared to the systems above (Dow Chemical 2017) and the mixing rate is thus expected to be significantly smaller (Hidalgo et al. 2012). For the PG-water system, the Sh-Ra correlation was also found to be nonlinear (\(\mathrm {Sh}\sim Ra^{0.76}\), red dashed-line) with parameters affected by a relatively large uncertainty (as represented by the red shaded region in the figure). Interestingly, our results seem to follow more closely the correlation found in another study that used the MEG/brine system with similar density curves (Neufeld et al. 2010), although also in this case the scaling of the flux was found to be nonlinear (\({Sh}\sim Ra^{0.84}\), blue dashed-line) and the uncertainty on the obtained parameters is admittedly large (represented by the blue shaded region in the figure).

As anticipated above, one of the key observations from the results obtained in study is the attainment of a linear \(Sh\sim Ra\) scaling. Interestingly, this behaviour has been observed in studies on thermal convection in porous media, including observations from experiments (Xie et al. 2012) and numerical simulations in both two- (Hewitt et al. 2013) and three dimensions (Hewitt et al. 2014). The latter are shown in the plot with the dash-dotted line and predict a flux that is approximately three times smaller than the values observed in this study (\(\alpha =0.379\)). We note that the linear scaling is specific to Rayleigh numbers that are relatively small (\(Ra_\mathrm{c}<Ra<\mathcal {O}(Ra)\sim 10^3\)), while *Sh* is expected to become independent of *Ra* for \(Ra>\mathcal {O}(Ra)\sim 10^4\) (Hidalgo et al. 2012; Slim 2014; Ching et al. 2017). Most significantly, our data seem to extend the results from one of the (very) few experimental studies reported in the literature where density-driven convection was investigated in a three-dimensional porous medium (grey-shaded square symbols) (Lister 1990). More observations within this important regime of Rayleigh numbers are needed to corroborate these findings, because at this stage we cannot exclude a priori that our data are affected by the characteristic density behaviour of aqueous MEG solutions (Jafari Raad et al. 2016). Nevertheless, the conclusion can be drawn that in the regime \(1<\widetilde{Ra}<100\) (\(40<Ra<4000\)) and irrespectively of the chosen model fluid (pair), the dissolution flux increases linearly with *Ra* reaching values that are \(40-100\) times larger than predictions based on diffusion alone. In the context of geological \(\hbox {CO}_2\) storage, this could result in a reduction in the time scale for dissolution from \(\sim 80{,}000\) years down to \(\sim 1500\) years in a 50 m-thick permeable aquifer.

## 6 Concluding Remarks

We have presented an experimental study on dissolution-driven convection imaged by X-ray CT in a uniform porous medium with MEG-water as model fluid pair. We obtain very good experimental reproducibility in terms of macroscopic measures of mixing, such as onset time of convection, maximum dissolution rate and averaged concentration profiles. Together with the recent work by Nakanishi et al. (2016) and Wang et al. (2016), we provide what are, to our knowledge, the first non-invasive determinations of three-dimensional patterns in opaque, random porous media in the regime \(\mathcal {O}(Ra)\sim 1000\). The tomograms reveal the emergence and evolution of characteristic concentration structures, which are imaged at a resolution of 10 \(\hbox {mm}^3\) from the onset of convection until its shutdown. The experimental observations are compared to the limiting numerical case of a purely diffusive scenario and are well described by a relationship of the form \(Sh=0.025Ra\) for \(Ra<5000\).

In agreement with previous findings, the comparison with results from other experimental studies suggests that the extrapolation of observations on analogue model fluids to the \(\hbox {CO}_2\)/brine system should be done with caution, due to effects introduced by the characteristic shape of the density-concentration curve. We contend that similar risks are posed by the use of simplified two-dimensional systems to mimic a porous medium and to model a process that is inherently three-dimensional. We also observe that there is a lack of direct experimental observations in the regime \(\mathcal {O}(Ra)\sim 100{-}1000\), where subsurface processes are very likely to operate. We demonstrate that X-ray CT allows for precise imaging of solute concentrations at a resolution of about (\(2\times 2\times 2)\,\hbox {mm}^3\), thus providing highly resolved spatial and temporal information on the fundamental behaviour of the convective process. This novel ability is key towards providing more realistic estimates on the extent of dissolution-driven convection in natural environments, because their inherent heterogeneity is likely to play a fundamentally important role in the determination of the convective flow pattern.

## Notes

### Acknowledgements

This work was performed as part of the PhD thesis of Rebecca Liyanage, funded by a departmental scholarship from the Department of Chemical Engineering, Imperial College London, provided by EPSRC (Award Ref. 1508319). Jiajun Cen thanks the Natural Environment Research Council (NERC) for funding a PhD scholarship as part of the Science and Solutions for a Changing Planet (SSCP) Doctoral Training Partnership. Experiments were performed in the Qatar Carbonates and Carbon Storage Research Centre at Imperial College London, funded jointly by Shell, Qatar Petroleum, and the Qatar Science and Technology Park. The tomograms associated with this work may be obtained from the UKCCSRC data repository (Data set ID 13607381): http://www.bgs.ac.uk/services/ngdc/accessions/index.html#item118273.

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