In this work, we investigate numerically the perturbative effects of cell aperture in heat transport and thermal dissipation rate for a vertical Hele-Shaw geometry, which is used as an analogue representation of a planar vertical fracture at the laboratory scale. To model the problem, we derive a two-dimensional set of equations valid for this geometry. For Hele-Shaw cells heated from below and above, with periodic boundary conditions in the horizontal direction, the model gives new nonlinear scalings for both the time-averaged Nusselt number $\langle Nu\rangle _{\unicode[STIX]{x1D70F}}$ and dimensionless mean thermal dissipation rate $\langle \unicode[STIX]{x1D717}\rangle _{\unicode[STIX]{x1D70F}}$ in the high-Rayleigh regime. We demonstrate that $\langle Nu\rangle _{\unicode[STIX]{x1D70F}}$ and $\langle \unicode[STIX]{x1D717}\rangle _{\unicode[STIX]{x1D70F}}$ depend upon the cell anisotropy ratio $\unicode[STIX]{x1D716}$, which measures the ratio between the cell gap and height. We show that $\langle Nu\rangle _{\unicode[STIX]{x1D70F}}$ values in the high-Rayleigh regime decrease when $\unicode[STIX]{x1D716}$ grows, supporting the field observations at the fracture scale. When $\unicode[STIX]{x1D716}\ll 1$, our results are in agreement with the scalings found using the Darcy model. The numerical results satisfy the theoretical relation $\langle Nu\rangle _{\unicode[STIX]{x1D70F}}=Ra\langle \unicode[STIX]{x1D717}\rangle _{\unicode[STIX]{x1D70F}}$, which is obtained from the model. This latter relation is valid for all values of Rayleigh number considered. The perturbative effects of cell aperture are observed only in the exponents of the scalings $\langle Nu\rangle _{\unicode[STIX]{x1D70F}}\sim Ra^{\unicode[STIX]{x1D6FE}(\unicode[STIX]{x1D716})}$ and $\langle \unicode[STIX]{x1D717}\rangle _{\unicode[STIX]{x1D70F}}\sim Ra^{\unicode[STIX]{x1D6FE}(\unicode[STIX]{x1D716})-1}$.

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Thermally driven flows in fractures play a key role in enhancing the heat transfer and fluid mixing across the Earth's lithosphere. Yet the energy pathways in such confined environments have not been characterised. Building on Letelier et al. (J. Fluid Mech., vol. 864, 2019, pp. 746–767), we introduce novel expressions for energy transfer rates – energetics – of geometrically constrained Rayleigh–Bénard convection in Hele-Shaw cells (HS-RBC) based on two different conceptual frameworks. First, we derived the energetics following the well-established framework introduced by Winters et al. (J. Fluid Mech., vol. 289, 1995, pp. 115–128), in which the gravitational potential energy, $E_{\textit {p}}$ , is decomposed into its available, $E_{\textit {ap}}$ , and background, $E_{\textit {bp}}$ , components. Secondly, we derived the energetics considering a new decomposition for $E_{\textit {p}}$ , named dynamic, $E_{\textit {dp}}$ , and reference, $E_{\textit {rp}}$ , potential energies; $E_{\textit {dp}}$ is defined as the departure of the system's potential energy from the reference state $E_{\textit {rp}}$ , determined by the ‘energy’ of the scalar fluctuations. For HS-RBC, both frameworks lead to the same energy transfer rates at a steady state, satisfying the relationship $\langle E_{\textit {ap}} \rangle _{\tau } = \langle E_{\textit {dp}} \rangle _{\tau } + 1/6$ . Consistent with the work by Hughes et al. (J. Fluid Mech., vol. 729, 2013) on three-dimensional Rayleigh–Bénard convection, we report analytical expressions for the energetics and efficiencies of HS-RBC in terms of the Rayleigh number and the global Nusselt number. Additionally, we performed numerical experiments to illustrate the application of the energetics for the analysis of HS-RBC. Finally, we discuss the impact of the thermal forcing and the geometrical control exerted by Hele-Shaw cells on the development of boundary layers, protoplumes and the self-organisation of large-scale flows.

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Supercritical ${\rm CO}_2$ injection and dissolution into deep brine aquifers allow its sequestration within geological formations. After injection, ${\rm CO}_{2}$ gas phase is buoyancy-driven over the denser aqueous brine, reaching an apparent gravitational stable distribution. However, ${\rm CO}_2$ dissolution in brine propels convection since the mixture is even denser than the underlying brine. This process still needs to be characterised comprehensively. Here, we investigate the irreversible mixing of dissolved ${\rm CO}_2$ in brine through laboratory-scale numerical experiments utilising the Hele-Shaw model (Letelier et al., J. Fluid Mech., vol. 864, 2019, pp. 746–767) and a fully miscible two-fluid system. In this scenario, mixing the less dense fluid – mimicking ${\rm CO}_{2}$ gas phase – with the heavier fluid – representing aqueous brine – catalyses cabbeling-powered convection. Our numerical simulations recover the laboratory results in porous media by Neufeld et al. (Geophys. Res. Lett., vol. 37, issue 22, 2010, L22404) and may explain the scaling law obtained by Backhaus et al. (Phys. Rev. Lett., vol. 106, issue 10, 2011, 104501) in Hele-Shaw cells. More remarkably, we show that the mass flux between the two analogue fluids, characterised by the Sherwood number $ {{Sh}}$ , obeys the universal scaling law $ {{Sh}}\sim {{Ra}}\, \vartheta _{scalar}$ , with $ {{Ra}}$ the Rayleigh number and $\vartheta _{scalar}$ the mean scalar dissipation rate. This paper sheds light on the fluid dynamics and solubility trapping in geological carbon sequestration.