Data and Method

Three RDII sources were selected based on the type of flow paths: roof downspout, sump pump, and leaky lateral. Each flow path was characterized using physics-based models in a spatial domain of a simplified residential lot. The three RDII sources represent: flow through a direct connection from runoff catchments, flow through coarse porous media, and flow through compacted soil. These three flow paths can be simply referred to as fast, medium, and slow paths for convenience though it is ideal to differentiate them based on flow patterns and the medium that is involved in the processes.
The three IRFs are identified for the test sewershed that includes Hickory Hills, Palos Hills, and Bridgeview, Illinois (IL), where sewer system configurations and sewer flow monitoring data are available. Hickory Hills is a city in Cook County, IL, with a size of 7.33 km2 and a population of 14,049. The areal size of Palos Hills and Bridgeview, IL is 11.12 km2, and 10.75 km2, respectively, and the population of the cities is 17,484 and 16,446, respectively (U.S. Census Bureau, 2010).

2.1 Physics-based models

2.1.1 Roof connection model

The roof connection model consists of a sloped roof area, flat gutter, and vertical downspout. The roof area receives rainfall and conveys the flow to the rain gutter by gravity. The rain gutter is connected to a downspout(s) to transport flow to a drainage system. When the downspout is connected to a sewer system, it becomes RDII.
The flow from the roof is calculated using the one-dimensional kinematic wave model for rainfall-runoff. Two governing equations describe the rainfall-runoff process when using kinematic wave theory: one-dimensional continuity equation for unit width of sheet flow, and Manning’s equation as a momentum equation for one-dimensional steady uniform flow per unit width. The one-dimensional continuity equation is as follows:
\(\frac{\partial h}{\partial t}+\frac{\partial q}{\partial x}=I\)(1)
where h = water depth [L], t = time [T], q= flow rate per unit width [L2/T], x = distance in down slope (measured from upstream end of plane) [L],I = rainfall intensity [L/T].
Manning’s equation can be used as a momentum equation for one-dimensional steady uniform flow per unit width as following.
\(q=\frac{1.49}{n}S_{0}^{\frac{1}{2}}h^{\frac{5}{3}}\) (2)
where n = Manning’s roughness coefficient,S 0 = bottom slope.
The equation (1) and (2) can be expressed as one equation.
\(\frac{\partial q}{\partial x}+\alpha\beta q^{\beta-1}\frac{\partial q}{\partial t}=I\)(3)
where\(\alpha=\left(\frac{1.49}{n}S_{0}^{\frac{1}{2}}\right)^{-\beta}\)and β = 3/5, which is the governing equation of kinematic wave model with q as only dependent variable.
The gutter is treated as a simple bucket. The outlet of downspout is treated as a weir or orifice depending on the flow condition. The gutter is modeled using the standard level-pool routing method (Chow et al., 1988). Level-pool routing is a lumped flow routing method that is suitable for a case with a horizontal water surface in the storage unit. The storage is a function of its water surface elevation. By using the stage-storage relation of the rain gutter and the stage-discharge relation of the downspout, this equation can be solved. Stage-discharge relations of the rain gutter-outlet are derived using an orifice and a weir equation.

2.1.2 Sump pump connection model

To derive the IRF from a sump pump, the commercial software MIKE-SHE (DHI Software, 2007a;b) is used to model flow to the sump in the single residential lot. MIKE-SHE is a spatially distributed hydrologic model that simulates surface water flow and groundwater flow in the three-dimensional gridded form. The one-dimensional gravity flow equation in MIKE-SHE is selected as the unsaturated zone equation. The gravity flow equation is a simplified version of the Richards equation, which ignores the pressure head term. The vertical driving force is entirely due to gravity. By selecting the gravity flow module, the dynamics owing to capillarity in the unsaturated zone are ignored. This is typically a valid assumption for coarse soils, and drainage trench around a house is usually filled with coarse materials. This is suitable to calculate the recharge rate of groundwater and faster and more stable than the Richards equation (Graham & Butts, 2005). The governing equation for the Richards equation is presented as follows.
h = z + ψ (4)
Then the gravity equation drops the pressure term.
h = z (5)
where h is hydraulic head [L], z is gravitational head [L], and ψ is pressure head [L].
The vertical gradient of the hydraulic head is the driving force to transport water. Thus, for the Richards equation,
\(h=\frac{\partial h}{\partial z}\) (6)
and for the gravity equation,
\(h=\frac{\partial h}{\partial z}=1\) (7)
The volumetric flux that is obtained from Darcy’s law for the gravity equation is
\(q=-K\left(\theta\right)\frac{\partial h}{\partial z}=-K\left(\theta\right)\)(8)
where K (θ ) is unsaturated hydraulic conductivity [L3/T].
For incompressible soil matrix and soil water with constant density, the continuity equation is:
\(\frac{\partial\theta}{\partial t}=-\frac{\partial q}{\partial z}-S\left(z\right)\)(9)
where θ is volumetric soil moisture [L2] and S is root extraction sink term [L2/T]. The sum of root extraction over the entire root zone depth is equal to the total actual evapotranspiration. Direct soil evaporation is computed only in the first node below the surface.
Substituting equation (18) onto equation (19), the following expression is derived.
\(\frac{\partial\theta}{\partial t}=-\frac{\partial K\left(\theta\right)}{\partial z}-S\left(z\right)\)(10)
This can be also expressed using the soil water capacity,\(C=\frac{\partial\theta}{\partial\psi}\)
\(C\frac{\partial\psi}{\partial t}=\frac{\partial K\left(\theta\right)}{\partial z}-S\left(z\right)\)(11)
This is called the gravity equation. This equation is used to calculate the unsaturated zone flow into a sump pump, which is used to derive the sump pump IRF.
The drainage trench around the house enables surface water to percolate down to the bottom of the building then feeds into the sump pump. In MIKE-SHE, sink cells are placed under the building to mimic the sump pump behavior and extract the water from the foundation. Unsaturated zone flow at the foundation level of the drainage trench area is interpreted as the total sump pump flow from the house. When the outlet of this sump pump is connected to a sewer system, this becomes I&I.
The size of the computational domain of the sump pump model is 50 meter (m) lengthwise and 26 m widthwise. The cell size is 0.33 m x 0.33 m; thus, a total of 150 × 78 or 11,700 cells in total were created. The vertical cell height is 0.2 m. The vegetation was assumed as uniform grass with Leaf Area Index 5 and Root Depth 100 mm. The horizontal width of the drainage trench is assumed as 0.33 m, and the total number of cells in the horizontal domain is 149, which corresponds to a total 50 m length of the trench. The drainage trench goes down to the base level of the house, 4 m below the surface where the sump is located.
Three soil types are employed in the sump pump model: ambient soil, impermeable soil, and extremely permeable soil. The hydraulic conductivity of the ambient soil is calculated as the average hydraulic conductivity of soil in Hickory Hills, IL,K ambient = 2.19·10-7 meter per second (m/s; Natural Resources Conservation Service [NRCS], 2019). Hydraulic conductivity of impermeable soil is assumed as 1·10-12 m/s and that of extremely permeable soil is assumed as 1·100 m/s. The hydraulic conductivity value of the extremely permeable soil, which represents backfill in the drainage trench, is within the range of the hydraulic conductivity for gravels based on Chow et al. (1988). The Averjanov model (Vogel et al., 2000) is used to simulate a hydraulic conductivity curve that shows the relationship between soil moisture and hydraulic conductivity.
\(K\left(\theta\right)=K_{S}\left(\frac{\theta-\theta_{r}}{\theta_{S}-\theta_{r}}\right)^{m}\)(12)
where Ks is saturated hydraulic conductivity [L/T], θS is saturated water content [L3L-3],θr is residual water content [L3L-3], and m is an empirical constant. Following values are used for the sump pump connection model: saturated moisture content θS = 0.38, residual moisture content θr = 0.01, and empirical constant m = 13.
For the MIKE-SHE model setting, the Van Genuchten model (Van Genuchten, 1980) is used to estimate the retention curve, which is a relationship between moisture content and pressure.
\(\theta\left(\psi\right)=\theta_{r}+\frac{(\theta_{S}-\theta_{r})}{\left[1+{(\alpha\psi)}^{n}\right]^{1-1/n}}\)(13)
where θ (ψ ) is the water retention curve [L3L-3], ψ is suction pressure [L], α is an empirical constant as the inverse of the air entry suction (α > 0) [L-1], and n is a measure of the pore-size distribution (n > 1). Following values are used for the sump pump connection model: inverse of air entry suction α = 0.067, and pore-size distribution n = 1.446.
Bulk density of ambient soil and extremely permeable soil is assumed as 1,700 kilograms per cubic meter (kg/m3) and that of impermeable soil is assumed as 1,600 kg/m3. Manning’sn values for overland flow computation for each surface type are estimated as 0.013, 0.025, and 0.030 for concrete side walk, asphalt shingle rooftop, and grassed yard, respectively (Chow, 1959). Evapotranspiration rate is set as 2.76 millimeters per day (mm/d) which is a suggested value in the Chicago area according to Grimmond and Oke (1999).

2.1.3 Leaky sewer lateral model

Similar to the sump pump model, the leaky sewer lateral model is developed using MIKE-SHE. f

Input data

Rainfall data were obtained from the Illinois State Water Survey (ISWS) by averaging rainfall data from four nearby ISWS rain gages: G11, G12, G16, and G17 (Illinois State Water Survey, 2019). The sewer flow data were collected by the U.S. Geological Survey (USGS) at 17 monitoring locations in the spring and summer of 2009. Based on the data quality and the length, the site located on 104th Street and east of Terry Drive in a maintenance hole was selected for this study. This location receives sanitary sewer flow from Hickory Hills, Palos Hills, and Bridgeview, IL.
Both rainfall records and the sewer monitoring records are presented in Figure 2 in the period of April 17, 2009–August 3, 2009. The base flow shows the daily fluctuation of dry weather flow except when storm event occurs high flow peaks are observed, which tend to sync in time with the arrivals of rainfall peaks.
In order to only focus on the RDII portion of the sewer record, dry weather flow (DWF) needs to be estimated and separated from the sewer record. The average DWF was estimated using the DWF estimation component in Special Contributing Area Loading Program (SCALP), which is developed by Hydrocomp, Inc. (Hydrocomp 1979). SCALP is a flow routing model mainly developed for use in the Chicago area. DWF is determined on a per capita basis and distributed in time by coefficients: average DWF loading, monthly pattern, daily pattern, and hourly pattern using the following equation (Espey et al., 2009; Miller & Schmidt, 2010).
DWF = average DWF loading x monthly pattern x daily pattern x hourly pattern (14)
These DWF coefficients are estimated using data from a 14-day dry period from July 17, 2009, to July 31, 2009. The 14 days of DWF are averaged, and the set of best DWF coefficients is derived by adjusting each value until the best fit to the average DWF was achieved. Nash-Sutcliffe model efficiency coefficient is used to find the best fit (Nash & Sutcliffe, 1970).
The monthly pattern is the pattern describing the variability among months within a year. The monthly pattern values are all set to one throughout the year due to insufficient data to define them. The daily pattern describes the variability among days within a week, and the hourly pattern describes the variability among the hours of the day. The average DWF loading is calculated as 0.12 m3/s (4.40 ft3/s). The daily pattern shows that DWF is greater during weekends than on weekdays. The hourly pattern shows two peaks during a day: in mornings and evenings, and minimum DWF at 4 am.

2.3. Impulse Response Function derivation

A representative rainfall was introduced as model input, and three IRFs from the three physics-based modes were derived. Based on the rainfall record in Hickory Hills, IL, a total of 702 mm of rainfall was recorded from January 1 through July 31, 2009. Seventeen distinct storm events were identified manually during this period; hence the average rainfall volume for a single event was assumed as 41 mm (as 702 mm divided by 17). The maximum rainfall intensity during the same period is 14 mm/hr. Three hours of 14 mm/hr of rainfall produces a total of 42 mm of rainfall volume. Therefore, 3-hour 14-mm/hr uniform precipitation is selected as a representative rainfall. The representative uniform rainfall was used as an input of the three physics-based models to derive the IRF of each RDII process described in the models.
The representative rainfall can be used directly for the roof connection model because the antecedent moisture condition has a minimal effect on the flow response of the roof runoff. However, it cannot be used directly for the gravity flow models that are used to derive the sump pump IRF and leaky lateral IRF. Infiltration and runoff processes are affected by ground conditions, e.g., land cover, land use, soil type, vegetation, seasonality, antecedent moisture condition. In order to eliminate the variability of ground conditions, the representative rainfall was added to the actual rainfall hyetograph at random times, and the resulting RDII hydrograph was subtracted by the RDII hydrograph resulting from the unaltered rainfall record. The representative rainfall was added to the actual rainfall hyetograph at ten randomly selected times between June 1 and January 31, 2009, and the IRF was calculated by averaging the individual IRF, which is the difference between the hydrographs resulting from the altered and unaltered rainfall hyetographs.
Three IRFs derived from the roof downspout, sump pump, and leaky lateral models using the representative rainfall are presented in Figure 3. The flow discharge units are normalized using the contributing areas of each model so that effective flowrates can be compared among the models. The peak values of each IRF are 0.0942, 0.0427, and 0.00902 m3/day/m2 for the roof downspout, sump pump, and leaky lateral models, respectively. By integrating the flow over time, the resulting RDII volume per unit contributing area values are 0.0118, 0.0319, and 0.0842 m (m3/m2). The result indicates that the roof IRF sports the shortest response time, although the total RDII volume per unit area is the smallest. At the same time, the leaky lateral IRF shows the longest response time with the largest volume per unit area. The total volume of each IRF is 2.89, 1.54, and 1.63 m3, However, the values are not good indicators of showing the impact of each RDII source as the total volume is dependent on the size and the condition of each model domain. The order of total response time for each IRF was hours, days, and weeks for the roof downspout, sump pump, and leaky lateral, respectively.
To understand the long-term behavior of the three IRFs, each IRF is weighted based on the actual rainfall intensity record in the period of April 17–July 16, 2009. The total rainfall depth in this period was 372 mm, and the peak precipitation rate was 13 mm/hr. Based on the assumption such that resulting RDII hydrographs from each source are proportional to the rainfall, three independent hydrographs were created for the same time period. Then each hydrograph was divided by the effective contributing area to compare the net RDII volume. Figure 4 indicates the flow duration curves of the three RDII responses for the time period. The roof connection response presented in the solid black line shows a steep curve, which indicates a greater amount of RDII for a short period of time. This displays strong evidence that the flow is stormwater driven. The leaky lateral response, which is presented in a solid grey line, shows a flatter curve. This indicates that the leaky lateral IRF displays a longer flow duration than the roof IRF due to the delayed percolation through porous media. The sump pump IRF in the black dashed line falls between the roof IRF and the leaky lateral IRF. The sump pump flow path also involves flow through a porous media, but it is faster than the leaky lateral flow path as the travel distance of surface water in the sump pump model is shorter than that of the leaky lateral model, and the medium has a larger hydraulic conductivity. The shapes of the three IRFs are easily distinguishable from one another, which in turn makes them suitable as building blocks of an RDII hydrograph.