Modeling the pipe failure phenomenon is essential for any water utility. The linear extension of the Yule process (LEYP), pro-posed in 2009, was intended to be a synthesis of the pipe failure models available at the time. Since then, two major improvements have beenproposed. First, in 2012 and 2014, several papers introduced time-dependent factors into the LEYP model. Second, in 2016, the LEYP withSelective Survival (LEYP2s) was proposed to deal with the selective survival bias. The objectives of this paper were (1) from a theoreticalpoint of view, to show how temporal covariates also can be introduced into the LEYP2s model; and (2) from a practical point of view, to provethe feasibility and interest of a LEYP2s model with time-dependent variables, using a case study. The mathematical consequences of takinginto account temporal factors within the LEYP2s model are presented, together with the solutions found that allow the calculation of themodel likelihood and thus the estimation of the model parameters. These solutions were applied to the black polyethylene service linenetwork of the city of Bordeaux, France. Three time-dependent covariates were considered: air temperature, pressure regime, and typeof disinfectant. This case study proved the feasibility of a LEYP2s model with temporal variables. The results showed that the time-dependentfactors considered did not improve the ranking of the service lines, but did improve the annual prediction of the network failure rate