New Kriging-surrogate-model-based dynamic adaptive sampling methods are proposed for an accurate and efficient uncertainty quantification (UQ). The criteria for the proposed dynamic adaptive sampling are based on the combination of both the uncertainty and the gradient information of the Kriging predictors. The polynomial errors (related to Runge's phenomenon) appeared near the endpoints in the stochastic space are reduced by adding an extra error-estimate term (based on the difference of the Kriging predictors with different correlation functions) in the adaptive sampling criteria. The proposed Kriging-based dynamic adaptive sampling methods are tested on one-dimensional and two-dimensional analytic functions with smooth and non-smooth response surfaces. The method shows a superior performance to estimate the statistics of output solution in terms of efficiency, accuracy, and robustness regardless of the choice of initial samples and the smoothness and dimensionality of stochastic space compared to the existing criterion based on only the Kriging predictor uncertainty.