An improvement in drainage flow equation of EPA SWMM green roof module

Rapid urbanization is a major threat to the environment (Hibbs & Sharp 2012). Among the most severe adverse effects of urbanization are the high ratio of impervious surfaces, and wash-off of accumulated pollutants on buildings, roadways, and parking lots into the streams (Grimm et al. 2008). The said conditions result in an increase in stormwater runoff which generally leads to an increase in severe flood events. High flood peaks and low flood peak lag times are observed in urbanized regions (Hibbs & Sharp 2012). Thus, the development of innovative stormwater management techniques is necessary to mitigate these problems. Low impact development (LID) best management practice (BMP) is an emerging stormwater management method which relies on various planning tools and control practices to preserve the natural hydrologic functions of the site (Prince George County 1999; Eckart et al. 2017). Examples of LID types are green roofs, rain gardens, bioretention cells, swales, permeable pavements and infiltration trenches (Eckart et al. 2017).

Green roofs are one of the most common types of LID which consist of three main layers: drainage layer, soil layer for growing plants (substrate) and surface layer for vegetation. There are two major types of green roof drainage layers: (1) drainage mats, which are made of high-strength plastic materials and contain cells for water storage, and (2) drainage granular materials, which have large pores for water storage (Vijayaraghavan 2016). The soil layer directly influences the plant growth and performance of green roofs. The soil used as a growth media for green roofs is different from natural soils and there is a gap in the literature on the standard agronomic properties of such mixtures (Rossman & Huber 2016a). For the vegetation layer, the plant type should be selected based on the local climatic conditions, nutrient content, and the impact of the plant on the ecosystems (Vijayaraghavan 2016). The type and thickness of the soil layer, the type and storage capacity of the drainage layer, vegetation type and coverage, rain event and time of previous dry period, and slope of green roof are the effective parameters on green roof performance (Vijayaraghavan 2016).

Green roofs sustain stormwater management by retention of rainfall, which allows for evapotranspiration (ET) of rainwater in the long run and no contribution to runoff or drainage flow, and detention of runoff, which allows for the temporary storage of rainfall as it passes through the roof layers (De-Ville et al. 2018). Green roof performance has been evaluated in various studies (Stovin et al. 2012; Carson et al. 2013; Johannessen et al. 2018; Palermo et al. 2019; Sims et al. 2019). Some of these studies focus on the behavior of green roof layers and identify features that can increase the performance of the green roof (Vesuviano & Stovin 2013; Hill et al. 2017; Bollman et al. 2019; Peng et al. 2019; Cristiano et al. 2020; Ekşi et al. 2020).

The U.S. Environmental Protection Agency Stormwater Management Model (EPA SWMM) is one of the most widely used software for green roof hydrology simulations. SWMM simulates the flow over and through the green roof layers and calculates the surface runoff, the flow between the layers, i.e., infiltration and percolation, and drainage flow. In recent years, different studies have evaluated the hydrological performance of green roofs using SWMM's green roof module (Abualfaraj et al. 2018; Palla et al. 2018; Hamouz & Muthanna 2019). Krebs et al. (2016) evaluated the runoff simulation ability of SWMM from monitored small-scale green roof test beds under Nordic climate conditions. Peng et al. (2017) modeled an extensive green roof test bed using SWMM under both annual time series for predicting long-term volumetric retention and significant rainfall events for predicting detention. Liu & Chui (2019) evaluated the effects of the hydraulic conductivity and thickness of the growing medium on the hydrological performance of the green roof under various rainfall conditions using SWMM. Johannessen et al. (2019) investigated the performance of SWMM for several roofs with varying roof buildups, sizes, slopes, and climates.

Most studies evaluated the overall green roof modeling performance of SWMM. However, few studies focused on the improvement of equations used in green roof modeling. Some studies suggested improvements in the calculation of ET in SWMM (Krebs et al. 2016; Peng et al. 2017; Palla et al. 2018). Peng et al. (2017) highlighted the need for robust models for soil layer and drainage layer modeling of green roofs. In this study, the green roof modeling capability of SWMM is investigated by exploring the details of each flow modeling component and improvements are suggested. The SWMM version 5.1 is employed in the analyses of this study. The answers to two questions are sought for: (i) what are the appropriate values of the decay constant HCO in the percolation equation for different green roof substrates?; (ii) how can Manning's equation be adopted by SWMM for the calculation of drainage flow through green roof be improved? For this purpose, Manning's equation used for drainage flow calculation in the green roof module is improved by introducing the drainage mat void thickness and adapting a new value for the exponential coefficient of depth. Finally, the SWMM model results and results of the improved equation are compared with the existing experimental data.

In these equations, i is the rainfall intensity falling directly on the surface layer; e₁ is the surface ET rate; e₂ is the soil layer ET rate; e₃ is the drainage layer ET rate; ϕ₁ is the void fraction of any surface volume (i.e., the fraction of freeboard above the surface not filled with vegetation); d₁ is the depth of water stored on the surface (L); D₂ is the thickness of the soil layer (L); θ₂ is the soil layer moisture content; d₃ is the depth of water in the drainage mat (L) and ϕ₃ is the void fraction of the storage layer (void volume/total volume). K_2s is the soil's saturated hydraulic conductivity (L/T); ϕ₂ is the porosity (void volume/total volume) of the soil layer; ψ₂ is the suction head at the infiltration wetting front formed in the soil (L); and F is the cumulative infiltration volume per unit area over a storm event (L). HCO is a decay constant derived from moisture retention curve data; θ_FC is the soil's field capacity (FC) moisture content, n₁ is the surface roughness coefficient; S₁ is the surface slope; W₁ is the total length along edge of the roof where runoff is collected (L); D₁ is the surface depression storage depth (L); A₁ is the roof surface area (L²); c is a conversion factor between SI and English units. The unit of the length dimension is meter for SI system and feet for English system and the unit of the time dimension is second.

If the moisture content θ is less than or equal to FC θ_FC, then the percolation rate becomes zero. The coefficient HCO characterizes the exponential decrease in hydraulic conductivity with decreasing moisture content. In order to estimate HCO for a given soil accurately, measured data of hydraulic conductivity K as a function of soil moisture content θ should be used. HCO values for some soil classes such as sand, sandy loam, and silty loam are available in the literature, but these values may not work well for green roof substrates since soil used as growth media for green roofs is different from natural soils and there is a limited amount of information on the standard agronomic properties of such mixtures (Rossman & Huber 2016a).

SWMM models the drainage flow using Manning's equation (Equation (7)) and this equation does not take the storage capacity of the drainage mat into account. However, the said storage capacity may have an important effect on the drainage flow and thus should be considered. Therefore, as the first step of the improvement, the storage capacity of the drainage mat is represented by introducing the drainage mat void thickness D_s3 into Manning's equation. As the second step of the improvement, a new value for the exponential coefficient of depth is sought. This coefficient takes the value of 5/3 in the original Manning's equation. However, the drainage flow is not well represented for green roofs with this value (Peng et al. 2017). In order to carry out these improvements, the experimental data by Vesuviano (2014) is employed. The new value of the exponential coefficient was determined by using the nonlinear curve fitting algorithm embedded in Matlab and named as lsqcurvefit, which solves data-fitting problems in the least-squares sense. The best fit of the calculated drainage flow to the measured drainage flow is satisfied with the value of 2 for the exponential coefficient. Moreover, there is no need for a conversion factor between SI and English units in the improved equation anymore as the fundamental quantity of length to the power one is obtained on the right-hand side of the equation.

The improved equation (Equation (10)) is obtained by employing the experimental data of the experiments conducted using independent drainage mats without part of a green roof. In the next section, the improved equation is incorporated into the entire green roof system.

Two different green roofs used in the existing experimental studies are modeled with SWMM and with the improved method. Then, the drainage flow calculated with SWMM, the drainage flow calculated with the improved equation and the measured drainage flow are compared.

Three sets of data available in the literature were used in this study. The first set of data belongs to a study by Peng et al. (2020) and consists of measurements of unsaturated hydraulic conductivities with respect to soil moisture content for four representative green roof substrates. The measurements were taken using steady state and transient techniques by using the test column with a height of 540 mm and a diameter of 300 mm. In addition, they determined the substrate characteristics including porosity, maximum water holding capacity (MWHC) and saturated hydraulic conductivity (Ks). The tested green roof substrates are Heather with Lavender Substrate (HLS), Sedum Carpet Substrate (SCS), Marie Curie Substrate (MCS) and New Substrate Mix (NSM) substrate. In this study, the steady-state data are employed to obtain HCO values in the percolation equation for different substrates.

The second set of data belongs to a study by Vesuviano (2014) and consists of inflow and drainage flow collected using different types of drainage mats. The experiments were conducted using a rainfall simulator chamber which consists of a channel base with a length of slightly over 5 m and a width of slightly over 1 m. The tested drainage mat types were ZinCo Floradrain FD 25, FD 40 and ZinCo Floraset FS 50. The drainage mat lengths are 2 and 5 m and drainage mat surface slopes are 2 and 18%. ZinCo Floradrain FD 25 and FD 40 drainage layers consist of a thin layer of hard plastic (high density polyethylene) molded into a regular three-dimensional pattern containing cup-like storage receptacles and channels. FS 50 type of drainage mat has deep polystyrene modules with water storage receptacles formed in the upper surface profile (Vesuviano 2014). A collection barrel with a pressure transducer was used for measurements. The momentum of flow entering the barrel resulted in instabilities in the recorded data. In addition, the pressure transducer generates its own noise (Vesuviano 2014). Therefore, some oscillations are observed in the measurements. The roughness coefficient n of high-density polyethylene and expanded polystyrene are given as 0.012 and 0.014, respectively. For all types of drainage mats, void fraction ϕ₃ was 0.5. In this study, the experimental data by Vesuviano (2014) is used for improving the green roof drainage flow equation.

The third set of data belongs to green roof experimental studies done by Sims et al. (2019) and Peng et al. (2019). Peng et al. (2019) conducted experiments using a green roof test bed (Green Roof A) located on a terrace of a building at University of Sheffield, UK, and has dimensions of 3 m × 1 m with 1.5° slope. Sims et al. (2019) conducted experiments using a single module drainage unit (Green Roof B) with dimensions of 0.3 m × 0.3 m located on a building at Western University in London, Ontario. The green roofs used in the experimental studies are modeled with SWMM and with the improved model. The data sets consist of rainfall and drainage flow. The effective design parameters, which need to be defined in SWMM, are provided by Peng et al. (2019), Peng et al. (2020), and De-Ville et al. (2018) for Green Roof A and by Sims et al. (2019) for Green Roof B. The values of the rest of the parameters, which need to be defined in SWMM, are selected according to the possible range of values provided by SWMM (Table 1).

Linear regression was performed between experimental and calculated data for nine experiments. As shown in Table 2, the mean correlation coefficient for Manning's equation changes from 0.33 to 0.891 whereas it takes values of 0.865–0.991 for the improved equation. Thus, there is a significant improvement in the results when calculated with the improved drainage flow equation.

The drainage flow data of the experiments conducted by Peng et al. (2019) and Sims et al. (2019) are compared with the results of green roof model of SWMM and with the improved drainage flow equation. For this purpose, the Green Roof A used in the experimental studies by Peng et al. (2019) and Green Roof B used in the experimental studies by Sims et al. (2019) are modeled with SWMM by using the parameters provided in Table 1. Then, the drainage flow calculated with SWMM, the drainage flow calculated with the improved equation and the measured drainage flow are compared.

Linear regression was performed between experimental and calculated data for seven experiments. As shown in Table 3, the mean correlation coefficient for SWMM changes from 0.29 to 0.874 whereas it takes values of 0.549–0.97 for the improved equation.

The rate of percolation f₂ through the soil layer is modeled using Darcy's Law in SWMM (Rossman & Huber 2016a). In the case of an unsaturated flow, the hydraulic conductivity K is a function of the moisture content θ, and the percolation rate is equal to the unsaturated hydraulic conductivity.

The method that considers K to be an exponential function of θ, gives relatively accurate and reproducible hydraulic conductivity results for steady-state conditions during which the soil surface is ponded with water (Libardi et al. 1980). However, this assumption may not be valid for green roof substrate which generally has different agronomic properties from natural soils. Figure 3 presents the results for four different substrates, among which two of them match well. However, the behavior of the rest of the two substrates did not fit to an exponential decrease in hydraulic conductivity with decreasing moisture content. Therefore, there is a need for more experiments using a wider range of green roof substrates to evaluate and improve the used percolation equation.

Manning's equation is originally derived to present flow in prismatic channels which is then adapted to present the drainage flow in green roofs and gives the storage depth-discharge rate relationship as a function of drainage length, roof slope and the roughness coefficient n. Since the shape of drainage mats is like an egg box, the Manning equation cannot represent the flow through drainage mats well in its original form. The value of the exponential coefficient of depth, which is 5/3 in the original Manning's equation and valid for prismatic channels, becomes 2 in the improved equation which is determined by the best fit of the experimental data to the calculated data. In addition, the storage capacity of the drainage mat is represented by introducing the drainage mat void thickness D_s into the improved equation.

Better results are obtained with the improved equation without losing the simplicity of the original equation. In particular, the start time of the rising limb, the peak flow arrival time, and the end time of the recession limb of the hydrographs obtained with the improved equation show a better match with the observations than the hydrographs obtained with the original equation (Figure 4). The start and end time of drainage and the peak drainage rate arrival time play an important role in the detention performance of green roofs and therefore should be determined accurately.

SWMM green roof model gives acceptable results for the most part of this study. In some cases, the SWMM results cannot catch the rising limb of the hydrograph which may be attributed to inaccurate measurement of initial moisture content. In addition, there are a few outliers in the peak drainage flow values which may be attributed to the inaccurate rainfall measurement in a few time steps. Comparison of the measured hydrographs with the simulated hydrographs shows little difference in first flow and peak flow arrival and recession time and the peaks. These issues have been fixed to some extent with the improved drainage flow equation.

While doing the assessment of SWMM green roof modeling capacity, additional details are noticed: SWMM models green roofs in three layers. However, in some cases, green roofs contain a retention layer as a fourth layer for more water retainment. Such a layer cannot be taken into account in SWMM modeling. Thus, future studies may focus on including a fourth layer in the SWMM green roof model. Moreover, some suggestions were made by Peng et al. (2017) aimed at improving SWMM green roof modeling. The ET rate should be calculated by taking the water content in the substrate into account. And, modeling efforts should focus on improving soil layer and drainage layer models (Peng et al. 2017).

In this study, the green roof modeling capacity of EPA SWMM is evaluated. In particular, an improved equation representing the drainage flow is developed. In addition, the green roof percolation model is examined and HCO values for four types of substrates were determined. Experimental data sets obtained in several former studies are employed to determine the HCO values and improve the drainage flow equation. A comparison of the results show that the drainage flow values obtained with the improved drainage flow equation are in better agreement with the experimental findings than the drainage flow obtained with Manning's equation. Furthermore, a comparison of the measured drainage flow with the calculated drainage flow using SWMM and the improved equation for different rainfall events also support this outcome. Additional experiments with a wider range of conditions are needed for further comparison to have a better support for this outcome.

The authors would like to thank Virginia Stovin and Denis O'Carroll for sharing their experimental data. The authors also would like to express their gratitude to the anonymous reviewers and the editor for their excellent suggestions, which strengthened the paper.

All relevant data are included in the paper or its Supplementary Information.

The authors declare there is no conflict.