Onset of transient natural convection in porous media due to porosity perturbations

Abstract

Onset of natural convection due to transient diffusion in porous media has attracted considerable attention for its applications to CO$_{2}$ sequestration. Stability analyses typically investigate the onset of such convection using an initial value problem approach in which a perturbation is introduced to the concentration field at an initial time $t=t_{p}$. This leads to debate concerning physically appropriate perturbations, the critical time $t_{c}$ for linear instability and the counter-intuitive notion of an optimal initial time $t_{p}$ that maximizes perturbation growth. We propose an alternate approach in which transient diffusion is continuously perturbed by small porosity variations. With this approach, instability occurs immediately ($t_{c}=0$) without violating any physical constraints, such that the concepts of initial time $t_{p}$ and critical time $t_{c}$ become irrelevant. We also argue that the onset time for nonlinear convection is a more physically relevant parameter, and show that it can be predicted using a simple asymptotic expansion. Using the expansion, we explore the onset of nonlinear convection due to porosity perturbations that vary sinusoidally in the horizontal and vertical directions, and show there are optimal wavelengths that minimize the onset time. Finally, we find simple relationships for these wavelengths as functions of perturbation magnitude. These show that even small porosity perturbations, typically considered negligible in previous literature, are sufficient to trigger nonlinear convection and thereby influence the rate of CO$_{2}$ dissolution within time scales comparable to previous analyses.