Estimating the probability of CO2 leakage using rare event simulation

Estimating the probability of rare events is an extremely challenging task. For example, estimating the probability of leakage of CO2 from a saline aquifer using a direct Monte-Carlo approach would in general require a number of simulations proportional to the inverse of the rare event probability. Since it is likely that any action will require a probability of failure of less than $10^{-6}$, requiring $10^7$ to $10^8$ simulations, it is understandable that few such studies have been published. In this paper, we propose a means of simulating such rare events using a multilevel splitting algorithm called subset simulation (SS)[Au and Beck, 2001]. SS is an iterative algorithm that introduces intermediate events which are easier to sample from, and then iteratively samples within each constrained region in the probability space until the rare event threshold is reached. We show results for a standard benchmark for CO2 leakage through a leaky well [Class et al., 2009]. In this test case, CO2 is injected into a deep aquifer, then spreads within the aquifer and, upon reaching an abandoned well, it rises to a shallower aquifer. We show that subset simulation is an effective algorithm for estimating the probability of rare events with significant computational advantages over a direct Monte-Carlo approach. The SS algorithm relies on two parameters that have to be adjusted at the start of the simulation; we show the effect of these parameters on the quality of estimated rare event probabilities and propose guidelines on how to select these parameters.