Onset of convective instability in an inclined porous medium

TL;DR

This study analyzes how the inclination angle of a porous medium affects the onset of convective instability during solute diffusion, revealing that stability depends on tilt angle and Rayleigh number, with implications for geological CO2 sequestration.

Contribution

It provides a linear stability analysis of convective onset in inclined porous media, highlighting the influence of inclination angle and flow shear on stability thresholds.

Findings

Onset time increases with inclination angle until a cut-off angle.

The cut-off angle depends on the Rayleigh number.

Instabilities can occur in gravitationally stable configurations due to shear.

Abstract

The diffusion of a solute from a concentrated source into a horizontal, stationary, fluid-saturated porous medium can lead to a convective motion when a gravitationally unstable density stratification evolves. In an inclined porous medium, the convective flow becomes intricate as it originates from a combination of diffusion and lateral flow, which is dominant near the source of the solute. Here, we investigate the role of inclination on the onset of convective instability by linear stability analyses of Darcy's law and mass conservation for the flow and the concentration field. We find that the onset time increases with the angle of inclination ($\theta$) until it reaches a cut-off angle beyond which the system remains stable. The cut-off angle increases with the Rayleigh number, $Ra$. The evolving wavenumber at the onset exhibits a lateral velocity that depends non-monotonically on $\theta$ and linearly on $Ra$. Instabilities are observed in gravitationally stable configurations ($\theta \geq 90^{\circ}$) solely due to the non-uniform base flow generating a velocity shear commonly associated with Kelvin-Helmholtz instability. These results quantify the role of medium tilt on convective instabilities, which is of great importance to geological CO$_2$ sequestration.