The objective of reliability sensitivity analysis is to determine input variables that mostly contribute to the variability of the failure probability. In this paper, we study a recently introduced method for the reliability sensitivity analysis based on a perturbation of the original probability distribution of the input variables. The objective is to determine the most influential input variables and to analyze their impact on the failure probability. We propose a moment independent sensitivity measure that is based on a perturbation of the original probability density independently for each input variable. The variables providing the highest variation of the original failure probability are settled to be more influential. These variables will need a proper characterization in terms of uncertainty. The method is intended to work in applications involving a computationally expensive simulation code for evaluating the failure probability such as the CO2 storage risk analysis. An application of the method to a synthetic CO2 storage case study is provided together with some analytical examples