First-order approximations have been used with some success for criticality analysis, sensitivity analysis of physical networks, such as water distribution systems, and uncertainty propagation of model parameters. Certain limitations have been reported regarding the accuracy of results, particularly when non-linearity is dominant. In this paper, we show how to efficiently derive the first and second order sensitivities with respect to the variation of their parameters. This makes it possible to improve the first order estimate when necessary. The method is illustrated on a small example system.