In this paper, we discuss the sensitivity analysis of model response when the uncertain model inputs are not independent of one other. In this case, two different kinds of sensitivity indices can be evaluated: (i) the sensitivity indices that account for the dependence/correlation of an input or group of inputs with the remainder and (ii) the sensitivity indices that do not account for this dependence. We argue that this distinction applies to any global sensitivity measure. In the present work, we focus on the estimation of variance-based sensitivity indices which are based on the second-order moment of the model response of interest. In particular, we derive new strategies and new computationally efficient methods to assess them, which rely on the polynomial chaos expansion. Several numerical exercises are carried out to demonstrate the performance of the new methods, including a sensitivity analysis of a drainage model posterior to its statistical calibration.