A number of groups have reported that the stress-induced leakage current (SILC) follows a power law dependence on the stress time. In this study, we observed that the power-law behaviour is only an approximation of the fast rising part of a more complex behaviour. SILC rises during the initial stress stage and saturates after a long stress time. Based on the trap-assisted tunneling (TAT) model, we show that the stress time dependence of SILC is better described as the depletion of multi-precursors of traps. Although the new model involves many fitting parameters, we show that the fitting results are consistent with the physical interpretation of these parameters. To further support the physical interpretation, we examined these parameters with annealing experiments.