Background leakages in water distribution networks (WDNs) may represent important economic and environmental losses, and significant ones should be integrated in hydraulic models to help utilities making good decisions for better rehabilitation and operational management of water-related infrastructures. Nowadays WDN models either do not take inertial effects into account or do not model background leakages explicitly. Moreover, not all of the current formulations have been tested on actual and large WDNs yet. Thus, there is a need to deeply analyse, test and compare these formulations, so as to clarify their range of validity and improve them taking the most benefit of their respective advantages. To achieve this goal we propose to analyse, compare and discuss the existing formulations which incorporate both inertia and background leakages in steady state and slow transient models. In particular, to reconcile computational efficiency and physical accuracy, we choose to consider background leakages as piecewise constant functions in time and streamline direction. We integrate the equations using a Rosenbrock method, we run the models on simplified and real WDNs, and we quantify the uncertainties of the models to assess their reliability and range of validity. Preliminary analysis of two existing formulations of background leakages shows strong similarities between inertial terms. Taking acceleration head and additional leakage convective inertia into account using slow transient models gives the most realistic results, and considering constant background leakages at the pipe scale provides a good representation while keeping the computation times acceptable. As a conclusion, this study is a first step toward the development and validation of slow transient models incorporating background leakages to simulate large WDNs. Future work will consist in the calibration and global sensitivity analysis of the models.