A heavy tailed distribution for Combined Sewer Overflow spill durations driven by infilitration

Combined sewer overflows (CSOs) are a major source of untreated wastewater discharge into urban water bodies during periods of excess flow, posing significant environmental and public health risks. Using approximately four million CSO spill events recorded by Event Duration Monitors across England (2020-2024), we analyze the statistical distribution of spill durations to better understand the stresses applied to urban watersheds from CSOs. Our results reveal a strongly heavy-tailed distribution: while spills exceeding two hours represent only ~10% of events, they account for ~85% of total spill time, indicating potentially disproportionate ecological impact from long-duration spills. Likelihood ratio tests confirm that this distribution is best described by a stretched exponential (Weibull) model with a shape parameter of 0.15, a finding consistent across multiple subsettings of our dataset. Periodic deviations from this trend correspond to diurnal water-use cycles, with elevated probabilities of a spill lasting near integer multiples of 24 hours. Hydraulic modeling reproduces the observed heavy tail only when groundwater infiltration is included, suggesting that prolonged spills are primarily driven by infiltration into sewer networks rather than extreme precipitation. Furthermore, we show that the observed scaling can be approximated statistically by first passage times of a sewer head modeled as fractional Brownian motion. Given the outsized environmental impact of long-duration spills, we recommend incorporating tail behavior explicitly into hydraulic model calibration and propose using stretched exponential parameters as robust metrics for CSO performance assessment.