Probabilistic modeling and global sensitivity analysis for $CO_2$   storage in geological formations: a spectral approach
  This work focuses on the simulation of $CO_2$ storage in deep underground formations under uncertainty and seeks to understand the impact of uncertainties in reservoir properties on $CO_2$ leakage. To simulate the process, a non-isothermal two-phase two-component flow system with equilibrium phase exchange is used. Since model evaluations are computationally intensive, instead of traditional Monte Carlo methods, we rely on polynomial chaos (PC) expansions for representation of the stochastic model response. A non-intrusive approach is used to determine the PC coefficients. We establish the accuracy of the PC representations within a reasonable error threshold through systematic convergence studies. In addition to characterizing the distributions of model observables, we compute probabilities of excess $CO_2$ leakage. Moreover, we consider the injection rate as a design parameter and compute an optimum injection rate that ensures that the risk of excess pressure buildup at the leaky well remains below acceptable levels. We also provide a comprehensive analysis of sensitivities of $CO_2$ leakage, where we compute the contributions of the random parameters, and their interactions, to the variance by computing first, second, and total order Sobol' indices. 
Bilal Saad and Alen Alexanderian and Serge Prudhomme and Omar M. Knio
stat.CO