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AbstractBesides collecting legitimate wastewater, municipal sanitary sewer systems, even when separated from storm sewer systems, also receive unwanted inflow and infiltration (I/I) that adversely impact their sizing, economics, and operation. Although several statistical models have been proposed to quantify I/I, their advanced methods and intricate data requirements may be unrealistic for practitioners who need to investigate I/I for individual sewer systems. Here, a practical regression model of daily sewer flow is developed with discrete terms for sanitary flow (based on winter water use), groundwater infiltration (based on sinusoidal seasonality), direct inflow (based on same-day precipitation), and delayed inflow (based on multiday moving-average precipitation). The terms are intuitive and the model performs well with flow observations from a case study. The model can help practitioners separate I/I from other wastewater flows, customize measures to control I/I, and improve sewer system performance.IntroductionThe 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 (I/I) from stormwater and groundwater, respectively. Even with separate storm and sanitary sewer systems, it is impossible to completely eliminate I/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, I/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 I/I will directly affect its sustainability across economic, social, and environmental dimensions.Past research has produced several worthwhile methods to analyze I/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 I/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.Table 1. Some regression models for estimating inflow and infiltrationTable 1. Some regression models for estimating inflow and infiltrationReferenceData requirementsCrawford et al. (1999) and USEPA (2021)Synthetic unit hydrographs; graphical fitting; sewershed delineation; precipitation depths from individual storm events (method used in SSOAP Toolbox)Karpf et al. (2007)Soil hydraulic conductivity; groundwater hydraulic gradient; wetted area; height of flood water; creek runoff at each time stepZhang (2005, 2007)Autoregressive error model; site-specific sewer flow time series before and after pipe rehabilitationBuilding 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 I/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 I/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 I/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 I/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 I/I. Beheshti et al. (2015) evaluated available methods of localizing I/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 I/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 I/I, the industry could benefit from a simplified, data-light approach that still offers some statistical rigor and useful fractionation of I/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.Methods and DataDefinitionsTotal flow in a sewer system is the sum of sanitary flow, groundwater infiltration, direct inflow, and delayed inflow as follows: (1) Wastewater flow=Sanitary flow+Groundwater infiltration+Direct inflow+Delayed inflowEach term is described subsequently using definitions adapted from USEPA (2014a, b). Their mathematical representations are also given for the purposes of a regression model, where the coefficients are to be determined by the method of ordinary least squares (OLS).Sanitary FlowSanitary flow is the portion that includes domestic, commercial, institutional, and industrial wastewater. This is legitimate wastewater from sinks, showers, toilets, bathtubs, and others. Sanitary flow is relatively constant, but the flow on weekdays is typically more than on weekends and holidays (Butler 1993), as is readily observed in dry conditions.In this study, sanitary flow is taken as follows: (2) where C1 = average weekend/holiday sanitary flow (m3/day); β1 = regression coefficient (m3/day) adjusting for the weekday sanitary flow; and W = weekday indicator (W=1 for weekday and W=0 for weekend or holiday). The analysis requires that each day of the year be categorized as weekday or weekend/holiday and assigned a value of W accordingly.A good surrogate for the sanitary flow is the winter water use. One may reasonably assume that in the winter, water is used only indoors and all indoor water goes into the sewer system. Although the winter water use (and therefore sanitary flow) for a mix of weekdays, weekends, and holidays may be known, the constant C1 is not yet known but will be calculated subsequently.Groundwater InfiltrationGroundwater infiltration is water that enters a sewer system from the ground through defective pipes, pipe joints, connections, or manholes, but not because of immediate wet periods.According to Darcy’s law, groundwater infiltration into the pipe network is proportional to the head of the water table above it as well as the hydraulic conductivity of the soil. Local groundwater level data are preferred for this part of the analysis, but they are often unavailable in the proper spatial or temporal resolution. Instead, due to their recurring seasonal patterns, long-term levels of shallow groundwater are often represented as constant plus a sine or cosine wave (Tison 1965; Brombach et al. 2002; Weiss et al. 2002; Ashland et al. 2005; Cuthbert 2010; Mackay et al. 2014). The same may be applied for groundwater infiltration in the present situation. For a 365-day year with a single seasonal groundwater peak, the corresponding form is (3) Groundwater infiltration=C2+β2cos(2π364(D−Dpeak))where C2 = constant for the average groundwater infiltration (m3/day); β2 = regression coefficient (m3/day) for the amplitude of the cosine; 2π/364 = factor (day−1) to adjust the cosine period to 1 year; and D = day of the year (0–364). The variable Dpeak shifts the assumed groundwater peak by Dpeak days from the beginning of the year (e.g., Dpeak=0 days for January 1 peak and Dpeak=181 days for July 1 peak), which will depend on each sewer system’s local groundwater pattern. The constant C2 is unknown at this point but will be calculated subsequently.Because the second term of Eq. (3) is cumbersome to insert directly into a regression model, it helps to define a dimensionless groundwater seasonality factor G which, after Dpeak is chosen, depends only on the day of the year (4) Then, Eq. (3) simplifies to (5) Groundwater infiltration=C2+β2GThis particular G-factor function may not be suitable for all areas, for example, areas with year-round rain (rainforest), nearly no rain at all (Atacama Desert), or more than one rainy season (monsoonal influence or irrigation-induced groundwater rises). For these cases, the approach described here may necessitate an informed modification to suit the regional hydrology.Direct InflowDirect inflow is stormwater runoff that directly enters the sewer system from downspouts, manhole covers, cross connections from a storm drain system, and others. Direct inflow is the immediate response to precipitation and can rapidly and dramatically increase sewer flows for short periods.Direct inflow is proportional to precipitation occurring at approximately the same time. Brombach et al. (2002) suggested that evaporation rates and air temperature could play an important role in addition to the precipitation. Here, direct inflow is taken as a portion of the same-day precipitation when the minimum ambient temperature is above 0°C (32°F) and liquid runoff can occur (6) where β3 = regression coefficient (m3/mm) converting precipitation depth into daily flow volume; and P0 = same-day precipitation (mm/day) when the temperature is above freezing.Depending on the sewer system and storm characteristics, direct inflow causes peak flows lasting minutes to hours, but here only the daily volume is of interest. Unlike sanitary flow and groundwater infiltration, there is no constant for direct inflow because it occurs only in conjunction with a precipitation event.Delayed InflowDelayed inflow is stormwater that enters the sewer system after a significant time delay from the beginning of a storm. It is sometimes called stormwater infiltration to distinguish it from groundwater. Delayed inflow includes runoff transmitted through the soil into the sewer system as well as contributions from foundation drains and sump pumps connected to the sewer system. Delayed inflow comes in between the fast response of direct inflow and slow response of groundwater infiltration.Delayed inflow is perhaps the most difficult component of the I/I profile to distinguish because it is obscured by the other components and responds differently in each system. To bridge this gap, a mathematical representation of delayed inflow must consider both the magnitude and duration of a storm. The variable proposed here is the moving-average precipitation. Other research has linked moving-average precipitation to shallow groundwater levels and infiltration potential because it captures both the amount and the extent of recent precipitation over several days or weeks (Changnon et al. 1988; Chen et al. 2002; Gardner and Heilweil 2008; Karpf and Krebs 2011; Smail et al. 2019). Accordingly, one may define delayed inflow as follows: (7) where β4 = regression coefficient (m3/mm); and PA = moving-average precipitation (mm/day) over the A days leading up to the day of interest. Like Dpeak for groundwater infiltration, the time period A for the moving-average precipitation may be selected by the modeler with some trial and error to achieve the best fit. Unlike sanitary flow and groundwater infiltration, there is no constant for delayed inflow because it occurs only after a precipitation event.Other mathematical expressions including linear and exponential decay functions were evaluated to test whether a better fit exists for delayed inflow. It was found that although better fits exist, the increase in correlation was minimal (R2 from 0.91 to 0.92). To maintain ease of application for the design users, and because more complicated expressions would only slightly improve the correlation, delayed inflow is recommended as found in Eq. (7).Regression ModelThe foregoing expressions for individual components of the wastewater balance (including I/I) may be combined into a complete set for regression to determine the coefficients. Substituting Eqs. (2) and (5)–(7) into Eq. (1) defines the daily wastewater flow as a linear combination of four known daily variables (W, G, P0, and PA) and a few unknown coefficients and constants (8) Wastewater flow(m3/day)=(C1+β1W)+(C2+β2G)+(β3P0)+(β4PA)In this expression, the four terms in parentheses on the right-hand side separate the contributions of sanitary flow, groundwater infiltration, direct inflow, and delayed inflow as described previously.Whereas Eq. (8) has two constants, C1 and C2, the regression gives only a single intercept, β0, as follows: (9) Wastewater flow (m3/day)=β0+β1W+β2G+β3P0+β4PAFirst, one must determine C1, the sanitary flow associated with weekends and holidays. The difference in flow between weekdays and weekends/holidays is not known prior to regression; the coefficient β1 is the additional sanitary flow associated with weekdays. In a 365-day year, the constant C1 is (10) where U = average winter (indoor) water use (m3/day); and n = number of weekdays in the year (about 250 in the US) after removing weekends and holidays (about 115 in the US).Second, in Eq. (9), β0 is the intercept, not a coefficient scaling other variables, so it is simply the sum of C1 and C2 in Eq. (8). Then (11) After regression, Eq. (8) can be constructed from Eqs. (9)–(11) to indicate discrete contributions of sanitary flow, groundwater infiltration, direct inflow, and delayed inflow, as demonstrated in the following case study.Study AreaThe foregoing statistical approach was tested using case study data. The study area (Fig. 1) is in the northern portion of the City of South Salt Lake, Utah. The City operates its own sewer system for this area, covering about 8.4  km2 (3.2 sq mi) and serving about 15,000 residents plus several large industrial and commercial zones, whereas another sewer district serves the rest of the city. The City’s sewer system contains some 680 manholes and 60 km (37 mi) of pipe, most of which is 200-mm (8-in.) clay pipe, although sizes up to 750 mm (30 in.) are present. Most of the collection occurs by gravity, but some areas require lift stations and force mains. The system collects to a single outfall at which total flows are measured by the receiving treatment facility.Data SourcesDaily Sewer FlowsThe authors obtained average daily sewer flow rates for 2019 from the wastewater treatment plant that measures and receives the City’s wastewater. Flows were verified by a third party prior to this study. The data set constitutes the daily observations against which to test the regression model. Ideally, multiple years of data should be used, but due to data quality problems with the master sewer meter prior to 2019, only 1 complete year was available for this study.Winter Water UseWinter water use (U) of 4,280  m3/day [1.13 million gallons per day (MGD)] was determined from water sales in the study area in February 2019 and was taken as the year-round sanitary flow. This average includes weekdays and weekends and is all the indoor water use that could be accounted for.Weekday EffectsIn 2019, there were 115  weekends/holidays and 250 weekdays in the study system.Groundwater PeakThe authors reviewed well logs, water rights, and groundwater monitoring sites in and near the study area but found no significant data for the surficial aquifer that would be pertinent to the I/I analysis. In the study area, it is assumed that surficial groundwater levels peak at the same time as mountain snowpack, being around April 1 each year (Barandiaran et al. 2017). Prior sewer flow data in the study area consistently indicated a spring high and a fall low. Accordingly, an April 1 maximum (Dpeak=90  days) was defined in the groundwater function, G. This will vary in other sewer systems.PrecipitationA rooftop rain gage, part of the Salt Lake County watershed monitoring program, was located within 1.0 km (0.62 mi) of the study area and contained a daily history for 2018 and 2019 (Salt Lake County 2020). The 2019 data appear in Fig. 2 and correspond to observed flow peaks in the sewer system. The year 2019 was ideal for analysis of the system’s I/I because there were several extended wet periods and several extended dry periods which could be compared. Further, 2019 was the wettest year since 1998 (judging by other local rain gages), so the analysis happened to capture what is likely an extreme case for I/I. Daily minimum temperature records for 2019 were acquired from a weather station at the Salt Lake City International Airport (NOAA 2021) and paired with the daily precipitation. Precipitation depths on above-freezing days were assigned as P0 for the direct inflow.The period A of the average moving precipitation, PA, used in the delayed inflow was tested from 1 to 30 days, and the best fit was chosen as 14 days (i.e., P14). One clue for selecting this value was the observed recession of flows after an extended wet period in May and June 2019, suggesting an effect that outlasted the storms by several days but not permanently (Fig. 2). The period will vary in other sewer systems.ResultsFig. 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: (12) Wastewater flow (m3/day)=7,760  m3/day+(688  m3/day)W+(2,360  m3/day)G+(86.2  m3/mm)P0+(1,020  m3/mm)P14The regression model had an adjusted R2 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, C1=3,810  m3/day and C2=3,950  m3/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: (13) Wastewater flow (m3/day)=[3,810  m3/day+(688  m3/day)W]+[3,950  m3/day+(2,360  m3/day)G]+[(86.2  m3/mm)P0]+[(1,020  m3/mm) P14]With this fractionation of individual flow components, one can now visualize their relative contributions on daily and annual scales, as in Fig. 3.DiscussionThe 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  m3/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 I/I interventions. If the goal is to reduce overall I/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 I/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 I/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 R2=0.85 compared with South Salt Lake’s 0.91). As expected, different timing of peak groundwater (Dpeak=135  days, compared with South Salt Lake’s 90 days) and different period of moving-average precipitation (A=30  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 I/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 I/I and enhance sewer system performance.ConclusionEngineers 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.Data Availability StatementAll data, models, or code that support the findings of this study are available from the corresponding author upon reasonable request.References Ashland, F. X., R. E. Giraud, and G. N. McDonald. 2005. Ground-water-level fluctuations in Wasatch Front landslides and adjacent slopes, Northern Utah. Open-File Rep. No. 448. Salt Lake City: Utah Geological Survey. 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