  Deep saline aquifers used for CO$_2$ sequestration are commonly made of
sedimentary formations consisting of several layers of distinguishable
permeability. In this work, the effect of a non-monotonic, vertically varying
permeability profile on the onset of convective instability is studied
theoretically using linear stability analyses. The onset time depends on the
interaction between the permeability profile and the location of the
concentration perturbation peak beyond which the concentration of CO$_2$
decays. A thin low-permeability layer can either accelerate or delay the onset
time of the convective instability depending on the nature of the permeability
variation - whether the permeability transition is smooth or layered, the
Rayleigh number (Ra), and the location of the permeability change ($\hat{a}$)
relative to the perturbation peak ($\hat{a}_{c^{*}}$), which scales as
$\hat{a}_{c^{*}}\approx 14Ra^{-1}$ for homogeneous systems. However, the low
permeable layer has no effect on the onset time when it is near the lower
boundary of a medium with sufficiently large Ra ($\hat{a}_{c^{*}} \ll
\hat{a}$). This nontrivial dependence highlights the implication of ignoring
geological features of a small spatial extent, indicating the importance of a
detailed characterization of CO$_2$ storage sites.
