Gravity Driven Instabilities of Transient Diffusive Boundary Layers in Porous Media
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This study is motivated by the geological storage of carbon dioxide in subsurface saline aquifers. After injection in an aquifer, CO<sub>2</sub> dissolves in brine to form a diffusive solute boundary layer that is gravitationally unstable leading to natural convection within the aquifer. The exploration of the underlying hydrodynamic instability is of practical importance because of enhanced CO<sub>2</sub> dissolution and storage in aquifers. In comparison to the classical Rayleigh-Bénard convection in a heated fluid cell, the analysis is not straightforward because the CO<sub>2</sub> boundary layer is unsteady and nonlinear. The physics of the convective instability is described by a mathematical operator that is both non-autonomous and non-normal. Consequently, it is uncertain whether classical stability results for the onset of convection are valid. In addition, it is unclear how theoretical predictions compare with experiments. To explore these issues, the physical mechanisms and perturbation structures of transient, diffusive boundary layers are examined using multiple theoretical and computational tools. Traditional schemes based on linear stability theory, due to unique physical constraints, are unlikely to produce physically relevant perturbation structures. Therefore, a novel optimization procedure is formulated such that the optimization is restricted to physically admissible fields. The new method is compared with traditional stability approaches such as quasi-steady eigenvalue and classical optimization procedures along with fully resolved nonlinear direct numerical simulations. After establishing a suitable analytical framework, the role of viscosity and permeability variation is examined on the onset of natural convection. Onset of convection occurs sooner when viscosity decreases with aquifer depth. These effects of viscosity variation are in contrast to observations in classical viscous fingering problems. When the porous medium is horizontally layered, qualitatively different instability characteristics can occur depending on the relative length scales of the boundary layer and the permeability variation. For sufficiently high permeability contrast, small changes in the permeability field can lead to large variations in the onset times for convection. Resonance effects are observed only when the porous medium is vertically layered. The current study provides a framework to explore gravity driven instabilities arising both in pure fluid and porous media applications. The framework can be extended to study more complex systems such as those involving chemically reacting species and random anisotropic permeability fields.