Quality modelling is a growing topic of research in water resource. Sensitivity analysis is an important topic which can help to investigate the stability of a process perturbed by parameter changes and uncertainty impacts. This work proposes to solve unsteady sensitivity equations for a complex looped pipe network. These equations are derived from and highly dependent on the unsteady advection-reaction equations for quality indicators. So, special attention is to be focused on the coupled version of these ones. In this purpose a Total Variation Diminishing (TVD) scheme based on a splitting and possessing very good quality of stability is set up and validated on a benchmark pipe networks. This scheme shows very good results.