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Aldrees, A.; Dan’azumi, S. Analytical Probabilistic Models in Urban Runoff Control Systems. Encyclopedia. Available online: https://encyclopedia.pub/entry/43482 (accessed on 01 July 2024).
Aldrees A, Dan’azumi S. Analytical Probabilistic Models in Urban Runoff Control Systems. Encyclopedia. Available at: https://encyclopedia.pub/entry/43482. Accessed July 01, 2024.
Aldrees, Ali, Salisu Dan’azumi. "Analytical Probabilistic Models in Urban Runoff Control Systems" Encyclopedia, https://encyclopedia.pub/entry/43482 (accessed July 01, 2024).
Aldrees, A., & Dan’azumi, S. (2023, April 25). Analytical Probabilistic Models in Urban Runoff Control Systems. In Encyclopedia. https://encyclopedia.pub/entry/43482
Aldrees, Ali and Salisu Dan’azumi. "Analytical Probabilistic Models in Urban Runoff Control Systems." Encyclopedia. Web. 25 April, 2023.
Analytical Probabilistic Models in Urban Runoff Control Systems
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This entry is adapted from the peer-reviewed paper 10.3390/w15091640

Urban stormwater is known to cause a myriad of problems, ranging from flooding to water quality degradations. analytical probabilistic model (APMs) are closed-form mathematical expressions representing a long-term system’s output performance derived from the probability distribution of the system’s input variables. Once derived, the APMs are easy to handle, allow for sensitive analysis, and can be co-opted into optimization frameworks. The implementation of APM in the planning and design of runoff control systems will not only help address the runoff quantity and quality problems of urban stormwater, but will also go a long way in optimizing the benefits derived from the systems. 

best management practices low-impact development water-sensitive urban design

1. Detention Ponds/Stormwater Tanks

Detention ponds involve the temporary storage of runoff in ponds, basins or even underground containers, and are meant to control the quantity as well as quality of urban runoff downstream of a catchment [1][2]. The purpose of stormwater detention is to reduce the flood damages caused by increased runoff due to imperviousness by limiting post-development peak discharges to be less than or equal to pre-development runoff [3], or to a rate based on other criteria specified by the stormwater authorities in charge [4]. Furthermore, stormwater detention improves the quality of stormwater runoff in addition to reducing the peak discharge [5]. The residence time resulting from stormwater detention allows for the suspended particulate matter and adsorbed contaminants to settle [6][7]. As a BMP, detention ponds can help limit the pollutants loaded into receiving water bodies.
Many researchers have dedicated much attention to the application of APMs in detention basins. Papa et al. (1997) [8] derived APM expressions for the pollution control performance of detention ponds for different combinations of active to permanent pool volumes. The results of the study have been compared to those simulated using SWMM software. It was found that the degree of suspended solid removal in both cases was comparable, with a difference of only 5 to 10% in extended dry ponds and 10 to 30% in wet ponds. Guo and Adams (1999a) [9] derived analytical expressions for the probability distribution of peak outflow rate from flood control detention ponds. The derived analytical expressions were used to determine the storage–discharge relationship required to achieve the specified level of flood control at the facility. Using the runoff volume and peak outflow rate presented in [10][11], the runoff rate exceedance probability per rainfall event was derived based on different combinations of storage and outflow. Comparisons were made between the results obtained from the analytical probabilistic model and similar results obtained from SWMM software, and the results were found to be in good agreement. Guo and Adams (1999b) [12] also used the expressions previously developed in [10][11] to derive APM expressions for the long-term performance of a stormwater quality control pond. The expression of flow capture efficiency was derived from the total spill volume, while the volume-weighted average detention time of the basin was derived by taking into account the variable inflow and outflow rates and the inter-runoff event time. The APM expressions describing the detention time and the statistical solution of flow capture efficiency were compared with similar values obtained from SWMM, and the results were found to be in close agreement, thus confirming the validity of the assumptions made in deriving the models.
Li and Adams (2000) [13] used an analytical probabilistic approach to derive runoff quantity and quality control performances for urban runoff control systems. Rainfall was first transformed to runoff, and the runoff transformed to overflow using the derived analytical expressions. The runoff volume was also transformed to runoff pollution mass load using the EMC concept, and was later transformed to total pollution mass discharge load. The APM expressions for fraction of runoff overflow and total pollution mass discharge load were used to derive closed-form APM expressions for the long-term runoff control and long-term pollution control performances of the stormwater storage and treatment systems. Comparisons between the runoff control performances predicted with the analytical model (coded in computer programs called SUDS and EXSUDS) and those obtained using a continuous simulation model STORM were conducted, and the results were in reasonably good agreement.
Analytical expressions for runoff control performances using different forms of rainfall–runoff transformations were developed [14][15][16][17][18][19]. Chen and Adams (2005a) [16] modified the rainfall–runoff transformation to consider infiltration rather than a common runoff coefficient, and developed closed-form analytical expressions for runoff control performances, including exceedance probability of a spill volume, expected value of a spill volume, average annual volume and number of spills, and runoff capture efficiency. The performance of the modified analytical model developed was tested against values obtained from continuous simulation using SWMM and the analytical models developed earlier by [20] for rainfall–runoff transformation (called ASTORM models), and good similarities between the three results were obtained. Chen and Adams (2005b) [17] also used the extended version of rainfall–runoff transformation, which divides the catchment into pervious and impervious areas with different depression storages and runoff coefficients, to develop APM expressions for the average annual number and volume of spills and the runoff control efficiencies. The results of the extended analytical model were compared with those from ASTORM and SWMM, and the results were in good agreement, with the extended model outperforming the ASTORM rainfall–runoff conversion model.
Chen and Adams (2006a) [18] used two types of rainfall–runoff transformations, ASTORM and the extended ASTORM, to derive analytical expressions for stormwater quality control based on build-up and wash-off functions. The appropriate models for pollutant build-up and wash-off (designated as Type 1 and Type 2) were chosen, and were combined to formulate the pollutant load model. Finally, the system quality control measures were derived, which are closed-form expressions that can be used to evaluate the long-term system behavior. Comparisons were made between the quality control models developed with observed values, and the values predicted using SWMM gave good estimates of system performance. Chen and Adams (2006b) [19] also used the derived analytical expressions based on three different rainfall–runoff transformations (i.e., ASTORM, Type 1 and Type 2) to derive APM expressions for stormwater quality control measures.
Chen and Adams (2007a) [15] used the ASTORM rainfall–runoff transformation, extended ASTORM and the modified rainfall–runoff transformation to develop analytical expressions for average annual runoff volume from an urban catchment. In the second case, the Horton’s infiltration equation was slightly modified, in that the rainfall duration was assumed to be a temporarily averaged constant. Model verification showed that both of the two analytical models compared favorably with results obtained from SWMM. Chen and Adams (2007b) [14] also used rainfall–runoff transformations, and pollutant build-up and wash-off functions, to derive analytical expressions of the cumulative density function (CDF) of pollutants load, as well as the expected value of pollutant EMC and average annual pollutant EMC. In the rainfall–runoff transformation, two types of models were proposed: the lumped parameter rainfall–runoff, and its extended form [13][17]. Two forms of pollutant load model (Type 1 and Type II) were obtained, and the expected pollutants’ EMC and average annual pollutants’ EMC values were derived. The pollutant load models were compared with observed values, and a good agreement was obtained. However, the Type II load model was found to outperform Type I in the estimation of average annual pollutants’ EMC.
Apart from the tremendous contributions made to the development of APM in relation to detention pond’s analysis and design, coming from Canada and USA, some important contributions coming from Italy are noticeable. Becciu and Raimondi (2014) [21] derived APM expressions for the overflow spill of stormwater detention ponds. Two management rules regarding the emptying of the pond were considered. Likewise, the probabilities for spilled volumes varied from zero to one, corresponding to no spill and a spill volume equal to the storage capacity of the pond, respectively. Data of rainfall series from Milano-Monviso, Italy, were used. The resulting analytical expressions can act as very valuable tools that can be used to estimate the overflow probability and the probability of a specific spilled volume. Raimondi and Becciu (2015) [22] used rainfall statistics, detention pond outlet operation rules, storage volume and maximum outflow to derive APM expressions for the pre-filling probability of detention ponds. As in their previous paper, the same management rules regarding the pond’s emptying were considered. The results can be used to estimate the pond’s volume and out flow rate as a function of pre-filling probability. A comparison of the analytical results with continuous simulation, using case study rainfall data from Monviso, Milano (Italy), showed a very good agreement, thus confirming the applicability of the method in the design and performance assessment of stormwater detention basins. Becciu and Raimondi (2015) [23] derived a similar expression for the PDF of a detention pond’s spilled volume in order to evaluate its efficiency. Becciu et al. (2015) [24] also derived APMs of retention time in stormwater detention ponds. The analytical formulae developed can be used for the design of pond storage corresponding to a specified retention time that ensures some pollutants are removed from the pond. The APM expressions were validated against results from a continuous simulation using the case study in Monviso, Milano (Italy), and were found to fit very well. Raimondi et al. (2022) [25] derived APM expressions for the probability of runoff volume and residual storage in sustainable urban drainage systems. The models were applied to two catchments in Genova and Milano (Italy) using rainfall data from Monviso station. In both cases, the results were compared with those from continuous simulation, and were found to be accurate.
Due to the shorter rainfall durations compared to the corresponding dry spell between the rainfall events, some researchers have considered rainfall arrival as a marked Poisson’s process, and modeled rainfall characteristics stochastically [26][27][28][29]. Wang and Guo (2019) [30] used analytical stochastic models (ASM) to describe the runoff capture efficiency of detention ponds as a power function, rather than linear. The ASM results were compared with the results of an SWMM continuous simulation using a case study catchment area located in Jackson, Mississippi. The values of the root mean square error (RMSE), Nash–Sutcliff efficiency (NSE) and correlation coefficient (R) for runoff capture efficiency were 0.021, 0.994 and 0.9983, while these values for average pond fullness level were 0.012, 0.998 and 0.9997, respectively. This indicates the applicability of ASMs.
Stormwater retention basins can also be analyzed by using stochastic water balance to develop analytical models. Parolari et al. (2018) [31] developed a stochastic water balance model of stormwater retention ponds under passive and active outlet conditions. Analytical expressions of the steady-state and joint PDF of water level and valve closure time, which can be used to define the water level and flow duration curves of the basin, were derived. The model’s performance was tested by taking observations of water levels from a retention pond located in Ann Arbor, MI, USA. He results show that the model accurately predicts the water level PDF, which can be used to form a basis for evaluating the changes in rainfall–runoff due to climate change and land-use.
Stormwater detention tanks are used to mitigate the impact of sewer overflow. Balistrocchi et al. (2009) [32] applied APMs to develop a CDF of the overflow volume and pollutant load distribution of a sewer tank. Weibull distribution was used to model rainfall characteristics. Analytical expressions of performance indices such as the decrease in the annual runoff volume and ratio of pollutant load captured by the tank were derived. The model was verified with SWMM continuous simulation, using the urban catchment of Brescia, Italy, and the results were found to be satisfactory. Andres-Domenech et al. (2010) [33] derived analytical PDFs of the number of overflows, volume of overflows and overflow reduction efficiency of a stormwater tank. Rainfall data from Valencia and Santander, Spain using different probability distributions were tested. Pareto and Gamma-2 PDFs were found to fit well. The analytical results regarding long-term volumetric flow and overflow reduction efficiencies were compared with those of IW continuous simulation, and were found to be similar. Becciu and Raimondi (2012) [34] developed APM expressions for the pre-filling probability of stormwater tanks. The effects of minimum inter-event time definition on outflow rate and storage volume were investigated using rainfall data from Monviso, Milano, Italy. The results of the APM were compared to the results of continuous simulation, and it was shown that the APM underestimated the pre-filling probability due to some assumptions made in the development of the model. Thus, the model needs to be refined further. Stormwater tanks, designed using APM, have also been found to be capable of improving the quality of sewer discharges from catchments along the Tyrrhenian coast of Italy [35].
Detaining runoff in stormwater detention ponds for a longer period improves the quality of the treated runoff, but this poses the risk of overflow from subsequent rainfall, which may generate runoff. There is an optimal detention time in the facilities such that the trade-off between runoff and pollution control is addressed [36]. There is also a need to minimize the cost of building the facility, while at the same time achieving the objectives. Papa and Adams (1997) [37] used APM expressions to develop a dynamic programming model for the optimization of the cost of building detention ponds in multiple parallel catchments, subject to meeting runoff quality control constraints.

2. Rainwater Harvesting System/Rainwater Tanks

Rainwater tanks, consisting of rain-barrels and cisterns, are rainwater harvesting systems (RHS) that store rainwater for household use and reduce the volume of runoff generated from urban surfaces. The use of rainwater tanks reduces water consumption from municipal supply, and thus reduces the water bill. The water stored in the tank can be used for gardening and toilet flushing, thus reducing municipal water consumption. Some rainwater tanks have two compartments: the rainwater tank itself and an infiltration facility, which aids in groundwater recharge [38][39].
Raimondi and Becciu (2014a) [38] developed APM expressions to estimate the probability of meeting the water demand using rainwater tanks as a function of household population and number of storm events occurring, using long-term rainfall data from 35 years at the Milano-Monviso station. The results of the study can be used to determine whether it is efficient to use rainwater harvesting alone, or in combination with municipal water supply. Raimondi and Becciu (2014b) [39] developed APMs for the design of multi-use rainwater tanks. These rainwater tanks were designed to have two basins: a rainwater basin and an infiltration basin. A trade-off between the risk of water shortage in the basin and the risk of overflow was studied. The results of a case study in a catchment in Milan, Italy, show that the probability of complete rainwater use in a household depends on the period of regulation, with weekly regulation yielding a higher probability compared to daily regulation. Additionally, the probability of overflow was high for a small storage volume and low infiltration rate. Becciu et al. (2016) [40] improved on their previous models by considering the effect of re-filling during the regulation period, and developed an analytical expression to estimate the CDF of active storage in the rainwater tank. The results were compared with those of a continuous simulation model using data from Milano, Italy, and there was a good agreement.
Guo and Baetz (2007) [27] derived an analytical expression that could be used to design rainwater storage units in green buildings, focusing on the rate of water use in the building, the climate characteristics of the area and the reliability of the system. The APM was applied to a hypothetical catchment in Chicago and Montana, USA, and it was shown that the APM provided an efficient approach to designing the system. De Paola and De Martino (2013) [41] studied the efficiency of four stormwater tank configurations using SWMM, and applied the semi-probabilistic approach to determine the qualitative and quantitative stormwater capture efficiencies of the most efficient tank configuration. It was concluded that the analytical approach provided similar results to continuous simulation. Kim et al. (2012) [42] used mass balance equations for each component of a rainwater tank to develop APM expressions for the rainfall–runoff reduction in an RHS. The PDF and CDF of runoff from the catchment and the RHS were derived, and the expected value of runoff volume was determined. The model was applied to a dormitory building in Seoul (Korea) to design an RHS and to estimate the runoff reduction achieved as a result of it. Di Chiano et al. (2023) [43] used APM expressions to derive the CDF of active storage in RHS. Active storage was considered as a function of rainfall moments, water demand and mean number of chained events under deficit conditions. The results of the model were compared with those of continuous simulation, using rainfall data from Monviso, Milano (Italy), focusing on a case study of RHS in Milan. An average normalized RMSE of 0.033, under three demand conditions, was obtained between the APM and the continuous simulation, suggesting a very good prediction.
Stochastic mass balance equations of RHS have been used to develop analytical models for RHS systems. Guo and Guo (2018a) [44] derived an ASM that could be used to determine the size of an RHS using a differential mass balance equation. Analytical expressions of a rainwater tank’s efficiency in terms of water supply reliability, required storage volume and its runoff reduction benefits were derived. The stochastic models, developed using rainfall data from five different climates (Atlanta, Concord, Detroit, Flagstaff and Billings) in the USA, were validated against the results obtained from SWMM continuous simulation and also those of Guo and Baetz (2007) [27]. The values of mean Nash–Sutcliffe efficiency (NSE), root mean square error (RMSE) and correlation coefficient of 0.98, 0.035 and 0.99, respectively, were obtained, indicating a good result. Pelak and Porporato (2016) [45] modeled rainfall as a marked Poisson’s process, and developed an analytical expression that optimizes the volume of a rainwater harvesting system at minimum cost. The volume was expressed as a function of rainfall parameters, roof area, water use rate, and the cost of the cistern and that of the external water source. The cost consists of fixed and distributed costs. The result of the study can be used to size an RHS in any climate. This will help reduce urban stormwater runoff and water consumption from public mains. Sim and Kim (2020) [46] used stochastic mass balance to develop an analytical model for the quantification of the water supply and stormwater interception efficiency of an RHS. In the study, the sensitivity of the RHS to climate change was evaluated, and the model was assessed using rainfall data from Busan (Korea). The results of the analytical model were compared with those derived using multiple regression. The R2 and RMSE values for water supply and stormwater interception efficiency ranged from 0.91 to 0.96 and 0.026 to 0.033, respectively. Cheng et al. (2021) [47] also used water balance to develop a stochastic model of an RHS. Due to the random occurrence of rainfall, the reliability of the model was expressed in terms of the fraction of time for which the RHS satisfies water demand. The model was applied to three RHSs in Toronto, Canada, and was found to have high accuracy.

3. Green Roofs

A green roof is a rooftop garden. These are used to provide shade, reduce the temperatures of the roof surface and surrounding air, and to moderate the heat island effect [48]. Green roofs comprise four layers: a vegetation layer, a substrate layer, a drainage layer and a waterproof layer. Some green roofs have a water storage layer combined with the drainage layer for holding more rainwater. Vegetation is planted on top of the substrate layer, where rainwater is retained. Excess rainwater from the roof is drained through the drainage layer [49].
Researchers also explored the application of APMs to green roof design and analysis. Zhang and Guo (2013a) [28] derived analytical expressions for runoff generation from green roofs. The results obtained from the analytical models were compared with those of continuous simulation using the LID module of SWMM, and also from the field results derived from a real case study in Portland, USA. The results of the APM were found to be in good agreement with both. Additionally, Guo et al. (2014) [50] derived analytical expressions for long-term average runoff reduction rates (defined as the ratio of total runoff captured to that of total runoff generated) and the irrigation water requirement of green roofs. The performance values of the APM in terms of runoff reduction rates and irrigation time fraction at different growing medium depths under semi-arid climate (Atlanta, USA) and humid climate (Billings, USA) conditions were compared with those from continuous simulation using SWMM, and it was concluded that the APM can be used as an alternative to SWMM in the planning, design and management of green roofs.
Guo (2016) [51] further refined the work of [50] by considering rainfall occurrence as a stochastic process to derive a stochastic differential equation of green roofs. The stochastic water balance equation was formulated to determine the mean and PDF of the moisture contents of green roofs. The accuracy of the model, in terms of runoff reduction rates and irrigation time fraction, was evaluated by testing the results against those of SWMM continuous simulation using four sets of rainfall data from Billings, Phoenix, Atlanta and Boston (USA), and using sandy and loamy soils as growing media. The comparison of results between SWMM and the stochastic model implied the good correlation coefficients of 0.993 and 0.995, respectively, for runoff reduction rates and irrigation time fraction. Most studies assume that the RHS is dry at the beginning of the rainfall event. However, some moisture retention is possible in the roof at the beginning of the next rainfall period. Raimondi and Becciu (2020) [49] considered the possibility of pre-filling from previous rainfall events to develop an APM for the design of green roofs. The APM was tested against the results of continuous simulation, using rainfall data from the Milano Monviso (Italy) station. The results show that the model compared well with continuous simulation. Thus, the APM can be used for the optimization of the design of green roofs. Raimondi et al. (2022a) [52] extended their work from 2020 to develop an APM that could be used to determine the thickness of the substrate layer of green roofs as a function of runoff reduction. The results of the study compare well with those obtained from continuous simulation.

4. Filtration Practices

4.1. Bioretention Cells/Biofilters/Rain Gardens/Impervious Area Disconnection

Bioretention systems are shallow landscaped depressions, commonly located in parking lots or within residential land-use areas, that are designed to incorporate many of the pollutant removal mechanisms that are operated in forested ecosystems. They are also known as biofilters or rain gardens [53]. Stormwater treatment in a bioretention cell is achieved through sedimentation, filtration, soil adsorption, micro-biological decay processes and the uptake of pollutants by plants [54]. The components of a bioretention area include a grass buffer strip, planting soil, plant material, a ponding area with surface mulch, an underground sand bed, an organic layer and infiltration chambers [55].
The resilience and reliability of using bioretention cells as runoff control systems was studied by [56]. APM expressions were used to evaluate the resilience indices related to the system’s robustness, rapidity and serviceability in the context of extreme runoff events. The results of the APM were compared with those generated using the continuous simulation SWMM. Resilience indices of 0.66 to 1.0 and 0.73 to 1.0, respectively, were observed for the APM and SWMM. The reliability index found ranged from 56% to 100% and 60% to 100% for the APM and SWMM, respectively.
Impervious area disconnection is a system that works in a similar way to bioretention cells. Runoff from urban surfaces (roof tops, pavements) is made to pass through pervious surfaces (grassed area), where processes such as infiltration and pollutant removal occur, thus reducing the volume of surface runoff. The time of concentration in the catchment is also reduced, thereby reducing the peak discharge from the catchment. Wang et al. (2019) [30] determined the effect of impervious area disconnection on runoff reduction from two urban catchments in the USA. Two different catchments’ soils (sandy and loamy) were used. The runoff reduction due to impervious area disconnection was examined using different imperviousness ratios. The results of the APM and SWMM were compared, and in all cases, impervious area disconnection was found to significantly reduce the volume of runoff to the sewer system, and the APM results compared very well with those of the SWMM. Zhang and Guo (2013b) [29] studied the hydrologic operations of a rain garden to derive analytical expressions for its long-term runoff capture efficiency. The results from the APM model were compared with those of SWMM simulations, and a very good agreement between the APM and continuous simulation results was observed. The APM was applied to rain gardens in Atlanta and Flagstaff in USA to demonstrate the sensitivity of runoff capture efficiency to specific model parameters.

4.2. Infiltration Trenches/Basins

Infiltration trenches are rectangular excavations with void-forming materials, such as gravel aggregates, which are designed to receive, filter, store and infiltrate urban stormwater. They aid in reducing urban runoff and improving groundwater recharge. They also assist in sediment and heavy metal removal from stormwater [57]. Guo and Gao (2016) [58] derived analytical expression for the total annual overflow volume and total runoff reduction rate of infiltration basins. The results of the APM were compared with those of SWMM continuous simulation using rainfall data from Atlanta and Billings (USA), and the results were found to be consistent, with a relative difference of less than 10%. Guo and Guo (2018b) [59] derived APM expressions for the overflow frequency and stormwater capture efficiency of non-vegetated infiltration facilities, such as infiltration trenches, infiltration chambers, dry wells, etc. In deriving the expressions, infiltration was assumed to occur at the bottom only. The results from the analytical expressions were compared with those of SWMM simulations in relation to a case study of a catchment in Concord, New Hampshire (USA), using sandy and loamy soils, and the two sets were found to be in good agreement. The average absolute difference and average relative difference between the APM and SWMM were found to be 0.04% and 5%, respectively. Wang and Guo (2020a) [60] analyzed the water balance of infiltration-based BMPs by considering infiltration through their sides and bottom, in an attempt to overcome the shortcomings of [61]. The mean degree of saturation and mean runoff capture efficiency were derived, and the results of the analytical model were compared with those of SWMM. Two soils, sandy and loamy, were used, and the rainfall data from two climate conditions (Billings and Jackson) were used to develop the APM model. The results were found to be reasonably comparable, with the largest absolute relative difference being less than 10%.
Following the design guidelines released by the Atlanta and New Durham authorities, Wang and Guo (2020b) [62] applied the analytical models they had developed earlier in [60] to a practical design analysis of infiltration trenches. Runoff values, generated using rainfall data from hypothetical catchments, in the two locations were assumed, and the performance of the trench was assessed as a function of its soil type, footprint dimensions, drain time and infiltration conditions. The results of the runoff reduction ratio indicate that the conditions of infiltration through the sides, the bottom, and both combined have profound effects on the runoff reduction ratio, by up to 15%. The runoff reduction ratio was found to be most highly affected by changes in soil type and trench dimensions.

4.3. Permeable Pavements

Pervious pavements consist of pavements made with porous blocks or porous asphalt that permits water to infiltrate. Pervious pavements may also be made from impervious blocks that are fitted in such a way that water can pass between them. They can be used in road surfaces with light traffic or in car parks. The infiltration rate through the pavement may be as high as 1000 mm/h in new developments, although this value may reduce to 10% of the original value over the lifetime of the pavement [63]. Zhang and Guo (2015) [64] derived analytical expressions for the runoff capture efficiency of a permeable pavement as an LID system to mitigate the impact of urban stormwater. SWMM simulations were run on the modeled pavement in order to validate the APM expressions, and the results showed little discrepancy. It is recommended that the APM results be compared with those of a case study on real-life pavements.
Stochastic differential equations of permeable pavements were used by Guo et al. (2018) [65] to model the dynamic water balance of their system. Rainfall and the corresponding net inflow were represented as a marked Poisson process to develop the PDF of inflow volume, and to derive analytical expressions for the long-term stormwater capture efficiency and moisture content of permeable pavements. The results of the APM were compared with those simulated using SWMM using data from the four climates of Atlanta, New Durham, Charlotte and Flagstaff (USA), and were found to be very similar.
Three runoff control systems—bioretention cell, permeable pavement and green roofs—were compared to determine the most cost-effective. The runoff reduction efficiencies and licecycle costs of implementing each of them were considered. APM expressions were combined with a genetic algorithm for the optimization. The objective function was to maximize runoff reduction capacity and minimize the lifecycle cost. The results show that the bioretention cell had the greatest runoff reduction capability, but given the high land cost in urbanized areas, permeable pavements are the most reasonable option [66].

5. Vegetated Open Channel Practices

These are systems designed to treat stormwater runoff in a swale or channel formed by check dams or other processes. Usually, they do not allow for quantity control, and are therefore combined with other stormwater BMPs to meet regulations. These systems directly receive runoff from an impervious surface; they have a temporary ponding time of less than 48 h and feature a 6-inch drop onto a protected shelf to minimize the clogging potential of the inlet [67].
According to [68], grassed swales are broad, shallow earthen channels used to treat stormwater runoff using flood-tolerant and erosion-resistant grasses. Filtering via these practices occurs through the vegetation, a subsoil matrix, and infiltration into the underlying soils. Grassed swales feature gentle longitudinal slopes, with check dams perpendicular to the flow so as to slow down the flows and allow the particulates to settle. There are two types of grass swales—dry swales, which have a filter bed of prepared soil laid over an under-drain system, and wet swales, which are designed to sustain moisture conditions that support wetland vegetation [69].
Grassed channels are used in pretreatment practices that provide nominal treatment, because they lack the filter media present in grassed swales. They act by allowing the infiltration of some runoff from small storms into areas with pervious soils, and are therefore most highly applicable to other structural stormwater BMPs [68]. They help in reducing the effect of imperviousness, and provide aesthetic benefits. Grassed channels are designed for use on <4% flat slopes with infiltration rates greater than 0.27 inches per hour. The stormwater runoff takes an average of 5 min to flow from the top to the bottom of the channel. For efficient usage, the channel should be used to treat small drainage areas of less than 5 acres. For the effective removal of particles, the grass of the channel should be maintained at a height of 3 to 4 inches [67].

6. Other Stormwater BMPs

Other types of stormwater BMPs that are used to control urban stormwater runoff include: constructed wetlands, dry wells, artificial marshes, oil/greet separators, catch basins, etc. [63][70]. Their effectiveness can be represented via a decrease in the SCS curve number of the basins. Perez-Pedini et al. (2005) [71] determined the optimal number and location of infiltration facilities in a watershed for the purpose of peak flow reduction at the watershed outlet. The watershed was discretized into 4553 hydrologic response units, whereby each unit represents a 120 × 120 m plot of the watershed. Different types of infiltration-based BMPs were conceptualized as binary integers that decrease the curve number of hydrologic response unit by five. The results of the optimization show that the optimal number and location of infiltration-based BMPs depends on various factors, such as flow travel time, catchment network connectivity, land-use, contributing area, and distance to the channel. APMs of stormwater management can be applied to constructed wetlands, dry wells, artificial marshes, catch basins, etc., in future studies.

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