Unstable density-driven groundwater flow and solute transport (i.e., free convection) leads to spatiotemporal variations in pressure. Specific storage (So) indicates the capability of a confined aquifer to release or store groundwater associated with a pressure change. Although So is known to dampen pressure propagation, So has been implicitly assumed to have a negligible impact on the unstable free convective process in prior studies. This work explores the effect of So on both the classic onset criterion and the fingering process using numerical models. Results show that the classic onset criterion is applicable when So is smaller than 10-1 m-1. Results also demonstrate that So does not play a significant role in the free convective fingering process unless it is greater than 10-3 m-1. For most practical purposes in hydrogeology (large Rayleigh number and small So), the implicit assumption of small or zero So is appropriate.