  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.
