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 refereed as fast, medium, and slow paths for convenience
though it is ideal to identify 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 the size of 7.33
km2 and the 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. 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,S0 = 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 and 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 horizontal water surface in 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
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 in the unsaturated zone caused by capillarity 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 Richards equation (Graham & Butts, 2005). The governing equation
for Richards equation is presented as following.
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 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 by 0.33 m thus
total 150 by 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 total 50 m length of
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, which is 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’s n 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. The equivalent medium approach from Carlier et
al. (2007) which was applied in agricultural drainage trenches is
adopted to simulate the actual flow into the sewer. According to the
equivalent medium approach, the leaky sewer pipe and surrounding
drainage trench area can be modeled as a single soil layer with an
equivalent uniform hydraulic conductivity. Hydraulic conductivity of
this layer is assumed as 1·100 m/s. Infiltrated flow
is estimated as the sum, over the length of the pipe line, of
unsaturated flow into the top of the equivalent porous medium
representing the leaky lateral. The same values of soil type, soil
property, Manning’s n values, and evapotranspiration rate were
used from the sump pump model.
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 U.S. Geological Survey (USGS) at 17 monitoring
locations in 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 manhole 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 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
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 is 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 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
To derive the three Impulse Response Functions (IRFs) from the three
models, a representative rainfall was introduced as a model input. A
common practice is to apply a unit rainfall input that occurs
instantaneously to the model, however, such input has a little impact on
the results of a groundwater model. Because of storage effect and
non-linearity of groundwater model domain such as a pervious soil layer,
direct application of unit hydrograph approach might not be suitable.
Since RDII is a problem that occurs during storm events, the IRFs in
this study were derived using a larger rainfall input that resembles a
typical storm event in the study area instead.
Based on the rainfall record in Hickory Hills, IL, a total of 702 mm of
rainfall was recorded in the period of January 1–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 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 the ground conditions
e.g. land cover, land use, soil type, vegetation, seasonality,
antecedent moisture condition, etc. To eliminate the variability of
ground conditions, the representative rainfall was added to the actual
rainfall hyetograph at random times and the resulted RDII hydrograph was
subtracted by the RDII hydrograph resulted from the unaltered rainfall
record. The representative rainfall was added to the actual rainfall
hyetograph at 10 randomly selected times during the period of June
1–January 31, 2009 and the IRF was calculated by averaging the
individual IRF which is the difference between the hydrographs resulted
from the altered and unaltered rainfall hyetographs. In case of the roof
connection model, antecedent moisture condition has a minimal effect on
the flow response of the roof runoff thus rainfall manipulation was not
necessary.
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, resulting RDII volume per unit contributing area are
0.0118, 0.0319, and 0.0842 m
(m3/m2). The result indicates that
the roof IRF sports the shortest response time while 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. 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 and superposed 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 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 black solid
line shows a steep curve, which indicates a greater amount of RDII being
contributed for a short period of time. This displays a strong evidence
that the flow is rain-caused. The leaky lateral response, which is
presented in a grey solid 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 a 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.