The 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 (H$_2$) fuel and carbon dioxide (CO$_2$), technologies which are being pursued to mitigate climate change and advance the energy transition. Once injected underground, CO$_2$ and H$_2$ will exist in multiphase fluid-water-rock systems: being partially-soluble, injected fluids can flow through the porous rack 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. 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 non-ideality of gases under geologic conditions, the opposing equilibrium state of dissolved 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 at conditions relevant to underground storage reservoirs. We show that 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 CO$_2$, even geothermal gradients less than 10 C/km induce downwards diffusion at depths relevant to storage. Diffusive transport of H$_2$ is less affected, but still reverses direction under typical gradients. Contrary to previous studies, we find that in diffusion and convection will likely work in concert - both driving CO$_2$ downwards, and both driving H$_2$ upwards - for conditions representative of their respective storage reservoirs.