Directionality of gravitational and thermal diffusive transport in geologic fluid storage

Diffusive transport has implications for the long-term status of underground storage of hydrogen $\left(H_2\right)$ fuel and carbon dioxide $\left({\mathrm{CO}}_2\right)$, technologies which are being pursued to mitigate climate change and advance the energy transition. Once injected underground, ${\mathrm{CO}}_2$ and $H_2$ will exist in multiphase fluid-water-rock systems. The partially soluble injected fluids can flow through the porous rock in a connected plume, become disconnected and trapped as ganglia surrounded by groundwater within the storage rock pore space, and also dissolve and migrate through the aqueous phase once dissolved. Recent analyses have focused on the concentration gradients induced by differing capillary pressure between fluid ganglia which can drive diffusive transport ("Ostwald ripening"). However, studies have neglected or excessively simplified important factors, namely the nonideality of gases under geologic conditions, the opposing equilibrium state of dissolved ${\mathrm{CO}}_2$ and $H_2$ driven by the partial molar density of dissolved solutes, and entropic and thermodiffusive effects resulting from geothermal gradients. We conduct an analysis from thermodynamic first principles and use this to provide numerical estimates for ${\mathrm{CO}}_2$ and $H_2$ at conditions relevant to underground storage reservoirs. We show that while diffusive transport in isothermal systems is upwards for both gases, as indicated by previous analysis, entropic contributions to the free energy are so significant as to cause a reversal in the direction of diffusive transport in systems with geothermal gradients. For ${\mathrm{CO}}_2$, even geothermal gradients less than $10{}^{\circ }C/\mathrm{km}$ (far less than typical gradients of $25{}^{\circ }C/\mathrm{km}$) are sufficient to induce downwards diffusion at depths relevant to storage. Diffusive transport of $H_2$ is less affected but still reverses direction under typical gradients, e.g., $30{}^{\circ }C/\mathrm{km}$, at a depth of 1000 m. This reversal occurs independent of the solute's thermophobicity or thermophilicity in aqueous solutions. The entropic contribution also modifies the magnitude of flux where geothermal gradients are present, with the largest diffusive fluxes estimated for ${\mathrm{CO}}_2$ with a $30{}^{\circ }C/\mathrm{km}$ gradient, despite the higher diffusion coefficient of $H_2$. We find a maximum flux on the order of ${10}^{-13}$ $\mathrm{mol}/\left({\mathrm{cm}}^{2}s\right)$ for ${\mathrm{CO}}_2$ in the $30{}^{\circ }C/\mathrm{km}$ scenario; significantly lower than literature estimates for maximum convective fluxes in moderate to high permeability formations. Contrary to previous studies, we find that in diffusion and convection will likely work in concert—both driving ${\mathrm{CO}}_2$ downwards, and both driving $H_2$ upwards—for conditions representative of their respective storage reservoirs.