A Practical Statistical Method to Differentiate Inflow and Infiltration in Sanitary Sewer Systems

The collection, conveyance, and treatment of municipal wastewater constitute a vital chain of services that protects public health and the environment and enables economic activity. Designing and operating a municipal sewer system comes with many challenges, including how to manage unwanted inflow and infiltration ($\mathrm{I}/\mathrm{I}$) from stormwater and groundwater, respectively. Even with separate storm and sanitary sewer systems, it is impossible to completely eliminate $\mathrm{I}/\mathrm{I}$. These flows occur on top of legitimate sanitary flows and decrease the system’s capacity and performance, sometimes leading to expensive capital interventions or sanitary sewer overflows (SSOs). For these reasons, $\mathrm{I}/\mathrm{I}$ studies of individual systems, besides being required by some regulators, are often undertaken to inform design standards, construction methods, mitigation measures, and capital spending. Accordingly, the degree to which a sewer utility controls $\mathrm{I}/\mathrm{I}$ will directly affect its sustainability across economic, social, and environmental dimensions.

Past research has produced several worthwhile methods to analyze $\mathrm{I}/\mathrm{I}$. Crawford et al. (1999) highlighted unit rates, runoff coefficients, streamflow coefficients, regression analysis, and unit hydrographs. Regression analysis has been popular among those seeking to separate $\mathrm{I}/\mathrm{I}$ components from other flows and relate them to weather (Crawford et al. 1999; Karpf et al. 2007; Zhang 2005, 2007; Zhang et al. 2011) (Table 1). Of particular interest is the model used in the USEPA’s Sanitary Sewer Overflow Analysis and Planning (SSOAP) Toolbox (USEPA 2021; Lai 2008), which can partition flows using unit hydrographs and individual storms. Although helpful, its data requirements are very specific and it gives only total rainfall-derived inflow and infiltration without differentiating the respective contributions.

Building on these regression models, Karpf et al. (2011) prepared a hydrodynamic model that considers surface water and groundwater boundary conditions in addition to precipitation. Karpf and Krebs (2011, 2013) combined multiple models to predict $\mathrm{I}/\mathrm{I}$, integrating sewer operation models with surface water models and MODFLOW groundwater models. More recently, some teams took a physical rather than statistical or computational approach, successfully distinguishing sanitary flow from $\mathrm{I}/\mathrm{I}$ based on pollutant fluxes (Bareš et al. 2009), biological markers (Shelton et al. 2011), conductivity measurements (Zhang et al. 2018a, b), and thermal differences (Beheshti and Sægrov 2018),

More practical research has focused on not just quantifying but also mitigating $\mathrm{I}/\mathrm{I}$. De Mosanbert and Thornton (1997) proposed a least-cost solution to optimize repair alternatives. Karpf and Krebs (2011) and Wirahadikusumah et al. (2001) related $\mathrm{I}/\mathrm{I}$ to pipe condition, and a case study by Staufer et al. (2012) concluded that pipe rehabilitation reduced groundwater infiltration by 24%. Pawlowski et al. (2014), studying an Ohio city, found that residential downspouts and laterals contributed a significant portion of $\mathrm{I}/\mathrm{I}$. Beheshti et al. (2015) evaluated available methods of localizing $\mathrm{I}/\mathrm{I}$ in sewer systems to prioritize rehabilitation.

Although this research is welcome, the proposed methods and data requirements may be infeasible for the public works employees or consulting engineers who typically carry out $\mathrm{I}/\mathrm{I}$ studies of individual systems. Accordingly, the methods and data must be accessible to these practitioners (Sowby and Walski 2021). Given the importance of quantifying and controlling $\mathrm{I}/\mathrm{I}$, the industry could benefit from a simplified, data-light approach that still offers some statistical rigor and useful fractionation of $\mathrm{I}/\mathrm{I}$ components. The proposed model can join the array lineup of other existing models to be chosen according to a given project’s goals, data resources, and staff preferences.

This work presents a novel and practical linear model based on daily sewer flows that discretely accounts for sanitary flow, groundwater infiltration, direct inflow, and delayed inflow. The model is developed and demonstrated in the context of a case study. By quantifying the individual contributors to total sewer flow in this manner, practitioners can customize practices to manage them and extract other insights into more sustainable sewer system operation.

Fig. 2 shows daily values of observed and predicted flows, along with precipitation increments, for the study area in 2019. The regression model provided the following equation:

The regression model had an adjusted $R^2$ of 0.91, meaning that it explained 91% of the variation in the daily sewer flows. All variables were individually significant at the 99% confidence level ($p<0.01$), meaning there was less than a 1% chance that the relationship was random.

The intercept was divided into two pieces as per Eqs. (10) and (11); therefore, $C_1=\mathrm{3,810}{\mathrm{m}}^3/\mathrm{day}$ and $C_2=\mathrm{3,950}{\mathrm{m}}^3/\mathrm{day}$. Separating individual components as in Eq. (8) was now possible, yielding a breakdown of the total flow profile into discrete terms for daily sanitary flow, groundwater infiltration, direct inflow, and delayed inflow as follows:

With this fractionation of individual flow components, one can now visualize their relative contributions on daily and annual scales, as in Fig. 3.

The regression model, built on separate terms for groundwater infiltration, sanitary flow, direct inflow, and delayed inflow, performed well with the case study data. Although not as sophisticated as other statistical models, it is intuitive, simple, and accurate, lending itself well to use by practitioners as well as researchers (Sowby and Walski 2021). Aside from a few justifiable assumptions about groundwater, nothing in the model itself is specific to the study area, but it relies on inputs of local sewer system data and hydrologic data to function.

According to the flow profile of Fig. 3, in the study area in 2019, only 42% of the annual volume was legitimate sanitary flow. Although this would be apparent just by analyzing the winter water use, further breakdown of the remaining flow is the most illuminating. Groundwater infiltration was the next largest contributor, accounting for 39% of the annual volume. It peaked in the spring but was always present. In the sanitary flow, the sawtooth pattern indicated weekdays and weekends/holidays; on average, flow was ${688\mathrm{m}}^3/\mathrm{day}$ more on weekdays. Delayed inflow made up 18% of the annual volume and corresponded to the 14-day moving-average precipitation that was elevated in the spring, contributing additional loads beyond sanitary flow and groundwater infiltration. Brombach et al. (2002) observed similar seasonal highs.

Direct inflow was a function of the same-day precipitation above freezing and contributed only 1% of the annual volume. This result may seem curious when one imagines the sudden flow of runoff into a sewer system, but the impact is short, especially compared with the daily resolution of this model, and the volume is minimal. Direct inflow may be responsible for peak flows on the scales of minutes or hours and may very well govern pipe sizing, but those effects do not appear in this daily model. For these same reasons, the model seems to underpredict daily flows during such events.

The breakdown just described is pertinent to prioritizing $\mathrm{I}/\mathrm{I}$ interventions. If the goal is to reduce overall $\mathrm{I}/\mathrm{I}$ volume in the study system, the focus should be on preventing groundwater infiltration and delayed inflow instead of direct inflow. The first two enter the pipe system by the same mechanism (through pipe cracks and joints via the subsurface), whereas the latter comes from runoff and other immediate discharges. Accordingly, pipe and manhole inspection, rehabilitation, and/or replacement would be prudent interventions, whereas disconnecting sump pumps and downspouts from the system would not (although it may help reduce peak flows). A different distribution of individual $\mathrm{I}/\mathrm{I}$ contributors may suggest different actions.

To be clear, the flow profile of Fig. 3 was estimated, not observed. Total flow and a few other variables were known, and the regression analysis gave this breakdown as the best fit for the given data and problem definition. It offers helpful insight into the relative magnitudes and timing of $\mathrm{I}/\mathrm{I}$ in the overall wastewater balance.

Although not detailed here, the authors applied the same approach on a neighboring sewer system for the same year and same precipitation and achieved comparable accuracy (adjusted $R^2=0.85$ compared with South Salt Lake’s 0.91). As expected, different timing of peak groundwater ($D_{\mathrm{peak}}=135\mathrm{days}$, compared with South Salt Lake’s 90 days) and different period of moving-average precipitation ($A=30\mathrm{days}$, compared with South Salt Lake’s 14 days) produced the best fit. The differences in parameter values, compared with each other and the case-study system, are attributed to unique system conditions such as soil type, hydraulic conductivity, water table levels, pipe condition, pipe depth, and impervious cover that cannot be directly modeled. Still, the model performs well in both systems.

A key advancement of this study was including moving-average precipitation as a predictor of delayed inflow. The term greatly improved the model’s ability to predict sewer flows during and after storms, separate from the baseline effects of sanitary flow and groundwater infiltration and the short response of direct inflow. Without it, the model performed poorly, especially after storms. Moving-average precipitation should be considered in future $\mathrm{I}/\mathrm{I}$ studies to fill this gap.

The data required for the model described here are generally available in the US. Water/wastewater utilities can provide the daily wastewater flows and winter water use, and daily precipitation and air temperature data are available from agencies like the National Weather Service. As such, the method should be replicable in many locations. Ideally, multiple years of data should be used.

The primary limitation of this research is when physical processes are poorly represented by mathematical expressions like those described previously. For example, this research may not apply well when a region has groundwater variations that are unlike a sinusoidal function. However, the strength of this research is its practicality; it can be applied quickly to individual systems to validate its applicability. Other limitations may include data availability and capacity to model exfiltration and intraday peak flows.

Further work should test the general approach used here on study areas with different climates, pipe network conditions, flow rates, soils, and other system properties from those found in Utah. Additionally, a variety of analysis period durations should be tested to validate impact on goodness of fit. It would be worthwhile to see how the effect of moving-average precipitation varies across such settings and what assumptions about groundwater infiltration are useful for model fitting in the absence of actual groundwater observations. The results should then inform actions to mitigate $\mathrm{I}/\mathrm{I}$ and enhance sewer system performance.

Engineers and sewer system employees need practical ways to differentiate sanitary flow, groundwater infiltration, direct inflow, and delayed inflow in a wastewater balance. This work proposed a regression model containing discrete terms for each of these components. Sanitary flow was estimated from winter water use, groundwater infiltration was approximated with a sinusoidal pattern and an assumed seasonal maximum, direct inflow was modeled as a function of same-day precipitation above freezing, and delayed inflow was modeled as a function of the moving-average precipitation to capture both the magnitude and the duration of recent storms. The input data (winter water use, daily wastewater flow, daily precipitation, and daily air temperature) are generally available for sewer systems in the US, so the analysis may be applied across various locations. The model performed well with a case study of 2019 daily sewer flows in South Salt Lake, Utah, and a neighboring sewer system. Further work should extend the analysis to other study areas (perhaps dividing larger systems into smaller, metered sewersheds), test the limits of the approach, and integrate the results with other sewer system planning and rehabilitation efforts.

All data, models, or code that support the findings of this study are available from the corresponding author upon reasonable request.