A comparative study of the time of concentration methods for designing urban drainage infrastructure

Urban drainage systems are a vital part of the water infrastructure of a city (Wang et al. 2021). Most Indian cities are facing the issue of urban flooding due to the paucity of adequate drainage facilities (Bisht et al. 2016). Although it is a crucial flood mitigation strategy (Day & Sharma 2020), it often receives less priority than mega infrastructures, namely, reservoirs, dams, and seawalls, resulting in inadequate and craptacular drainage systems in cities (CPHEEO 2019). Thus, these systems often fail to discharge the runoff volume generated during heavy storms, even of shorter duration, and lead to urban floods (See et al. 2020; Guo & Wang 2022; Tansar et al. 2022).

Time parameters are the significant components of flood-flow hydrologic design and modeling (McCuen et al. 1984). Commonly used time parameters are the lag time, the time of concentration (ToC), the time−area curve, the time to equilibrium, and the time to peak. One or more time-scale parameters are required as an input in almost all hydrologic analyses (Wu & Wang 2022; Yan et al. 2022). The ToC is the most significant one among all the time parameters (McCuen et al. 1984; Wong 2009) as it reveals the speed at which the catchment reacts to rainfall events (Pavlovic & Moglen 2008; Sharifi & Hosseini 2011). The ToC is a key parameter in synthetic unit hydrographs such as Soil Conservation Services Dimensionless Unit Hydrograph and Clark's Unit Hydrograph (Liang & Melching 2012). Information on the ToC is essential for various water resource infrastructures such as water supply systems, stormwater management systems, irrigation systems, spillways, and bridge/culvert openings (Sharifi & Hosseini 2011; Kaufmann de Almeida et al. 2017).

Notably, the ToC is an important parameter in designing an urban drainage system (UDS) as it is required to estimate the design storm (Wadhwa & Kummamuru 2021) and set up the simulation duration. While using the rational method in designing an UDS, the ToC is required to select the design rainfall intensity from the intensity–duration–frequency (IDF) curves (Fang et al. 2008). Knowledge of the behavior of a drainage system vis-a-vis ToC is essential for determining peak discharge of urban flood, the arrival of the peak, flood forecasting, and flood warning systems (Kang et al. 2008; Piadeh et al. 2022). Fang et al. (2008) emphasized the significance of precision in estimations of ToC, as the underestimation of ToC leads to the overestimation of peak discharge and vice versa.

Acknowledging the significance of time parameters in hydrologic design and analysis, researchers have developed various physical, analytical, semi-empirical, and empirical methods for determining the ToC in a natural catchment. In physical methods, chemical (Calkins & Dunne 1970) and radioactive (Pilgrim 1976) tracers are used to observe the ToC directly. Liang & Melching (2012) used the experimentally obtained hydrograph to determine the ToC for a V-shaped watershed experiment system. The ToC can be determined using a hyetograph and hydrograph in the graphical technique as the time interval measured from the end of rainfall to the end of surface runoff (Kaufmann de Almeida et al. 2017). Empirical and semi-empirical formulas derived based on the results of the physical and analytical methods are commonly used to estimate the ToC in watersheds. The most commonly used empirical methods for an urban watershed by numerous researchers for ToC estimation are the Kirpich method, Hathway formula, Giandotti formula, and Schaake equation (Fang et al. 2008; Preti et al. 2011; Sharifi & Hosseini 2011; Radice et al. 2012; Leitão et al. 2013; Kusumastuti et al. 2015; Manawi et al. 2020; Wadhwa & Kummamuru 2021). A limited number of input parameters required for the applicability of these empirical methods make these methods extremely popular.

Wong (2009) categorized the ToC estimation methods into two categories: (i) lumped approach and (ii) distributed approach. The ToC of a watershed in the lumped approach is determined using a single formula that effectively incorporates flow times through the drainage channel and on the overland surface. The lumped approach that mainly includes empirical methods is often established using the regression approach, for instance, Kirpich (1940), Carter (1961), and McCuen et al. (1984) methods. Contrary to lumped methods, in the distributed approach, the flow time of each flow regime is estimated independently and then aggregated to obtain the ToC (Kibler & Aron 1983). For the estimation of ToC in an UDS, researchers have employed lumped methods such as the Kirpich method, U.S. Federal Aviation Administration (FAA) equation, Kerby equation, Hathway formula, Schaake equation, and Desbordes method (Lhomme et al. 2004; Karamouz et al. 2011; Hsu et al. 2013; Leitão et al. 2013; Kusumastuti et al. 2015) as well as a distributed method such as Natural Resource Conservation Service (NRCS) method and Storm Water Drainage Manual (SWDM) method (Mugume 2015; Zhou et al. 2018; Osheen et al. 2022).

In modern hydrology, the ToC remains one of the challenging parameters to estimate (Grimaldi et al. 2012), as numerous estimation procedures and definitions are present in the literature (McCuen 2009; Michailidi et al. 2018). An exact global method for determining the ToC does not exist, and different methods lead to significantly different design values (Kang et al. 2008). However, owing to its importance in hydrological design pertaining to urban drainage, it is imperative to understand different ToC estimation approaches and their effect on the final outcomes to make an informed decision. There have been earlier attempts to evaluate and compare the accuracy of a method with other methods to minimize the uncertainty involved in the computation of ToC. McCuen et al. (1984) evaluated and compared the ToC obtained by 11 empirical equations with a velocity-based method for an urban watershed. Wong (2005) compared nine methods of overland flow estimation subjected to uniform rain with the experimental values attained for grass and concrete surfaces under similar circumstances. Liang & Melching (2012) compared the experimentally determined ToC for a V-shaped laboratory-based watershed with the Kinematic wave-based method.

Although ToC is not a direct input parameter in Storm Water Management Model (SWMM), the underestimation or overestimation of ToC affects the quantification of runoff volume (González-Álvarez et al. 2020). Bondelid et al. (1982) found that errors in the ToC can lead to nearly 75% of the total error in estimating the peak discharge. Salimi et al. (2017) stated that the accurate estimation of ToC is necessary to obtain precise peak discharge. Hence, the inaccurate estimation of ToC will lead to an imprecise drainage system.

Therefore, considering the importance of ToC in UDS design and the light of the discussion made previously, the present study was conceptualized with the following two key objectives: (i) to understand and compare the different approaches for ToC estimation in an urban catchment and (ii) to study the effect of variation in the ToC on flood hydrograph parameters and outflow at the outlet during the evaluation of an UDS analysis.

In the subsequent sections, the descriptions of the study area, data used, methodology, and results are presented in a detailed manner.

The city has an area of 475 km² and a population density of about 1,241/km². The climate of Gurugram is generally dry, with May and June as the hottest months when the mean daily maximum temperature touches about 43 °C. The topography of the study area is flat. The average annual rainfall of the area is nearly 773 mm, and most of the rainfall occurs in the monsoon season. The land use/land cover (LULC) distribution of the study area is as follows: built-up area (26%), wasteland (26%), agricultural land (23%), bare land (22%), scrub forest (1.5%), natural vegetation (1.4%), and shallow water body (0.1%) (Figure 1).

The city has two main natural drains: the Najafgarh drain and the Badshahpur drain. Rainwater flows off the Aravallis in the east, enters the main city through the Badshahpur drain, and moves toward the Najafgarh drain. The capacity of the Badshahpur drain is reduced to one-third of its required capacity due to factors such as encroachments, solid waste disposal, and siltation (Rawat et al. 2021). The obstructions in the flow transfer between the Badshahpur drain and the Najafgarh drain lead to overflowing drains. The Badshahpur drain, Ghata Jheel, and other natural water bodies are struggling for their existence under the pressure of frantic construction and encroachment. Due to the abovementioned factors, flooding has become a significant problem for city residents in recent years. A few recent acute flooding events were reported on July 28, 2016, August 27, 2018, and August 19, 2020 when the rainfall depth was recorded to be 58, 130, and 95 mm, respectively (Kansal & Osheen 2019; Rawat et al. 2021). These acute flooding events interfere with commercial activities, affect day-to-day public life, and make the city gridlocked and waterlogged.

In the lumped approach, the ToC of a catchment is determined using a single formula that effectively incorporates flow times through the drainage channel and on the overland surface. A total of eight methods based on the lumped approach of ToC estimation briefly presented in Table 1 are the Eagleson Lag method, Espey-Winslow, Kirpich method, Carter method, Desbordes method, Schaake method, McCuen method, and Soil Conservation Service (SCS) lag method (Silveira 2005; Kaufmann de Almeida et al. 2014; Salimi et al. 2017; Azizian 2019). The inputs required for estimating the ToC are catchment size, slope, water input, and flow resistance (McCuen et al. 1984). Here, ‘catchment size’ reflects the area of catchment and the length of conduit (i.e., open conduit/channel or closed conduit/pipe); ‘slope’ represents the slope of overland flow or channel. The water input can be represented by the hydraulic radius, rainfall intensity, or the volume of surface runoff, whereas the flow resistance is represented by Manning's roughness coefficient, the percentage of imperviousness, the SCS runoff curve number (CN), runoff coefficient of the rational method, or the land cover type.

Lumped methods for estimating the ToC can also be classified based on input data requirement and dominant-flow regime (McCuen et al. 1984). There are three possible flow regimes in a catchment, i.e., overland flow, channel flow, and pipe flow. Most of the methods are designed based on one of the three flow regimes; however, there are also the ‘mixed methods,’ which include different flow regimes. Kirpich method, Carter method, and SCS lag method are the methods in which one flow regime is dominant, whereas Eagleson Lag, Espey-Winslow, and McCuen methods are examples of mixed methods.

In the present study, all eight methods were used to estimate the ToC for the case area's existing LULC conditions. The average acquired values of various parameters and land cover coefficients for various equations are provided in Table 2.

In the distributed approach of ToC estimation, the flow time for the overland flow regime and the channel flow regime are determined separately and then aggregated to compute the ToC for the drainage system. Two methods based on the distributed approach of ToC estimation briefly presented in Table 1 are the NRCS method and the SWDM method. Sharifi & Hosseini (2011) asserted that of the two approaches, i.e., lumped and distributed, the distributed approach is more likely to produce better ToC estimates as it requires a higher number of input parameters. Furthermore, in the distributed approach, as the time of flow for two flow regimes is calculated separately, it is observed to provide better estimates of ToC (Yen 1982; Sharifi & Hosseini 2011). As the distributed approach is asserted to yield better results, two methods of the distributed approach, i.e., the NRCS method and the SWDM method, are discussed in detail in this section.

Soil Conservation Services introduced the NRCS method in 1972, in which the flow time is calculated for three flow regimes, i.e., sheet flow, shallow concentrated flow, and channel flow. Another distributed approach, namely the SWDM method, was introduced by the Ministry of Housing and Urban Affairs and CPHEEO of the Government of India in 2019. In the SWDM method, the flow time is calculated for two regimes, i.e., overland flow and channel flow.

The NRCS method is the most reliable and commonly used method for estimating the ToC in natural and urban catchments as it relies on a robust hydraulic basis for determining flow velocity (Fang et al. 2007; Sharifi & Hosseini 2011). As per the NRCS method, travel time is a function of flow velocity and flow length of runoff (Fang et al. 2007). McCuen et al. (1984) observed a close concurrence between the ToC estimated from rainfall and corresponding runoff data and the NRCS method.

In the NRCS method, the ToC is calculated by splitting the longest flow path into small parts and adding the flow time of each part. Flow regimes are classified into three types: sheet flow, shallow concentrated flow, and channel flow. Further equations are described for determining the velocity or travel time.

Shallow concentrated flow length (L_sc) to connect the overland flow with the main channel flow is inferred to have a channel length of about 304.8 m, i.e., 1,000 feet (Fang et al. 2007). V can be estimated using the chart in the TR-55 manual to estimate average velocities for travel time for shallow concentrated flow (NRCS 1986).

The SWDM method for ToC estimation is described in the manual on stormwater drainage systems by CPHEEO (2019). This method is frequently used to determine the ToC in urban catchments. In this method, ToC consists of two components:

• i.

  t_o is the time for the surface flow to reach the first inlet.

• ii.

  t_f is the flow time through the storm drainage system to the outlet or point of consideration.

For the urban areas, t_o lies in the range of 5–30 min (CPHEEO 2019).

Here, n is Manning's roughness coefficient of the channel; R represents the hydraulic radius of drains, and S_c is the average slope of the main channel.

Modelers, in general, struggle with various available choices of ToC determination formulas and methods and eventually adopt a method without comparing and analyzing its accuracy to that of other methods. The first move in determining the most suitable ToC estimation method would be to choose a method that can give accurate estimates of this time parameter and is reliable enough to be used as the true value. Sharifi & Hosseini (2011) considered the ToC determined using the NRCS method to select and evaluate the best-performing method among nine empirical methods. González-Álvarez et al. (2020) compared the statistical performance of 10 ToC estimation methods for 15 watersheds by considering the ToC by the NRCS method as a ‘true’ value.

The NRCS method is the most accurate, reliable, and commonly used method for determining the ToC in natural and urban watersheds (Fang et al. 2007; Sharifi & Hosseini 2011; González-Álvarez et al. 2020). Therefore, in the present paper, following Sharifi & Hosseini (2011), ToC obtained using the NRCS method is considered a ‘reference’ value for evaluating the methods among different lumped and distributed methods discussed herein. The differences between the ToC values obtained using the NRCS method and those obtained using the discussed equations (Table 1) were evaluated with percentage deviation measures.

ToC is an important parameter for the analysis of an UDS. The under- or overestimation of ToC affects the computation of design runoff volume. The effect of the variation of ToC on the flood volume, flood peak, time to peak, flooding duration, and the number of flooded nodes must be analyzed to understand the need for accuracy in the estimation of ToC. Fang et al. (2008) observed that the underestimation of ToC leads to the overestimation of peak discharge and vice versa. To demonstrate the effect of ToC on outflow and flood hydrograph, the drainage model of the study area was set up in SWMM (detailed in Section 3.4.1). The model was run for each ToC value obtained after employing the afore-discussed 10 methods (Table 1). The effect of the variation of ToC on outflow at the outlet and the flood volume, flood peak, time to peak, flooding duration, and the number of flooded nodes was analyzed and are discussed in Section 4.5.

SWMM is a frequently used tool for designing, planning, and evaluating an UDS (Rossman 2010). The model simulates the rainfall−runoff response, runoff generation, and runoff transport across the drainage network, including pipes, channels, pumps, storage tanks, regulators, and equipment. The details of the SWMM model setup for the study area can be referred from Osheen et al. (2022). The whole area is divided into 90 sub-catchments. The area of sub-catchments ranges from∼4 to ∼178 ha. The drainage network of the study area had 187 nodes, and the invert levels of nodes were extracted from the drainage map. The length of the drainage channel is ∼115 km. The longest channel flow path starts from Node W5, as this is the most distant point in the outlet of the area (Figure 2). The length of this path is 12,245 m and consists of 13 conduits, out of which 1 is a circular conduit and 12 are rectangular conduits.

For flow routing, the dynamic wave routing method was applied (Bisht et al. 2016). The dynamic wave routing method provides the most theoretically reliable results as it completely solves the one-dimensional Saint-Venant flow equations (Rossman 2010). The one-factor-at-a-time (OAT) technique of local sensitivity analysis, a model evaluation approach to determine the parameters to which the model findings are most sensitive, has been used to overcome the limitation of calibration (Mugume 2015; Osheen et al. 2022).

This section summarizes the values of various parameters used to calculate ToC in the lumped and distributed methods. The method to estimate flow length, land slope, runoff coefficient, hydraulic radius, flow velocity, conduit slope, and channelization factor is explained in detail, and the opted values of the study area's drainage system parameters for the calculation of ToC are summarized in Table 2.

As per the guidelines of UDFCD (2016), the maximum overland flow length for urban areas is recommended as 300 feet, and in rural areas, it is 500 feet. Therefore, L_o is considered 300 feet.

The GIS-based terrain pre-processing is carried out with the ‘Slope’ tool in ‘Spatial Analyst Tools’ in the ArcGIS Desktop. The average slope for the case area is ∼2.08%, and the mean slope of sub-catchments determined with the ‘Zonal Statistics as Table’ tool varies from 0.64 to 2.15%.

The runoff coefficient (C) reflects the relationship between rainfall and runoff (Machado et al. 2021). It depicts the consolidated effect of catchment losses and thus depends on the slope as well as the nature of the surface. C is extracted for each sub-catchment using the results of sub-catchment runoff details from SWMM simulations and ranges from 0.1 to 0.7.

The hydraulic radius, R, of the conduits in the longest flow path (i.e., W5 to outlet) is estimated at individual sections to determine the flow velocity at the respective sections. R is estimated at different depth levels (0.1 to 0.9d). The size of conduits and depth intervals taken are summarized in Table 3. For the circular conduit, R is calculated as 0.115 m at 0.1d and 0.542 m at 0.9d, whereas it ranges from 0.158 to 1.603 m for rectangular channels.

Table 3

Average channel flow velocity

Conduit .	Dimensions (Dia.a/w×db) (m) .	.	.	.	.	.	.	.	.	.	Average (V m/s) .	/ (h) .
Circular	1.82	0.79	1.21	1.53	1.78	1.97	2.11	2.21	2.25	2.22	1.78	0.26
Rectangular	2.43×1.82	0.59	0.86	1.04	1.18	1.29	1.38	1.46	1.52	1.57	1.21	0.23
Rectangular	2.43×1.82	0.59	0.86	1.04	1.18	1.29	1.38	1.46	1.52	1.57	1.21	0.12
Rectangular	3.04×1.82	0.40	0.59	0.73	0.83	0.92	0.99	1.04	1.09	1.14	0.86	0.38
Rectangular	3.04×1.82	0.40	0.59	0.73	0.83	0.92	0.99	1.04	1.09	1.14	0.86	0.23
Rectangular	3.04×1.82	0.40	0.59	0.73	0.83	0.92	0.99	1.04	1.09	1.14	0.86	0.14
Rectangular	3.04×1.82	0.60	0.88	1.09	1.25	1.37	1.48	1.57	1.64	1.71	1.29	0.28
Rectangular	4.0×2.43	0.46	0.68	0.83	0.95	1.05	1.13	1.20	1.25	1.30	0.98	0.21
Rectangular	4.0×2.43	0.48	0.71	0.88	1.00	1.11	1.19	1.26	1.32	1.37	1.04	0.11
Rectangular	4.50×2.43	0.48	0.72	0.90	1.03	1.14	1.23	1.31	1.37	1.43	1.07	0.13
Rectangular	4.50×2.43	0.46	0.69	0.85	0.98	1.08	1.17	1.24	1.30	1.36	1.01	0.23
Rectangular	4.50×2.43	0.48	0.72	0.90	1.03	1.14	1.23	1.31	1.37	1.43	1.07	0.04
Rectangular	12.0×2.43	0.40	0.63	0.80	0.95	1.07	1.19	1.29	1.38	1.46	1.02	0.75

Conduit .	Dimensions (Dia.a/w×db) (m) .	.	.	.	.	.	.	.	.	.	Average (V m/s) .	/ (h) .
Circular	1.82	0.79	1.21	1.53	1.78	1.97	2.11	2.21	2.25	2.22	1.78	0.26
Rectangular	2.43×1.82	0.59	0.86	1.04	1.18	1.29	1.38	1.46	1.52	1.57	1.21	0.23
Rectangular	2.43×1.82	0.59	0.86	1.04	1.18	1.29	1.38	1.46	1.52	1.57	1.21	0.12
Rectangular	3.04×1.82	0.40	0.59	0.73	0.83	0.92	0.99	1.04	1.09	1.14	0.86	0.38
Rectangular	3.04×1.82	0.40	0.59	0.73	0.83	0.92	0.99	1.04	1.09	1.14	0.86	0.23
Rectangular	3.04×1.82	0.40	0.59	0.73	0.83	0.92	0.99	1.04	1.09	1.14	0.86	0.14
Rectangular	3.04×1.82	0.60	0.88	1.09	1.25	1.37	1.48	1.57	1.64	1.71	1.29	0.28
Rectangular	4.0×2.43	0.46	0.68	0.83	0.95	1.05	1.13	1.20	1.25	1.30	0.98	0.21
Rectangular	4.0×2.43	0.48	0.71	0.88	1.00	1.11	1.19	1.26	1.32	1.37	1.04	0.11
Rectangular	4.50×2.43	0.48	0.72	0.90	1.03	1.14	1.23	1.31	1.37	1.43	1.07	0.13
Rectangular	4.50×2.43	0.46	0.69	0.85	0.98	1.08	1.17	1.24	1.30	1.36	1.01	0.23
Rectangular	4.50×2.43	0.48	0.72	0.90	1.03	1.14	1.23	1.31	1.37	1.43	1.07	0.04
Rectangular	12.0×2.43	0.40	0.63	0.80	0.95	1.07	1.19	1.29	1.38	1.46	1.02	0.75

Note:^aDiameter of circular drain, and ^bw×d represents width×depth of rectangular drain.

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For estimating the channel flow in the NRCS method and the time of flow in the SWDM method, average velocity along the longest channel flow path is required. Therefore, the flow velocity (discussed in Section 3.2.2) is determined at different depth levels (d), 0.1−0.9d, to determine the mean velocity of flow in the system. Flow velocity (in m/s) is estimated using Manning's equation (Equation (7)), and the results are tabulated in Table 3.

The longitudinal slope, S_c, is calculated for each conduit as where is the invert level at the higher and is the invert level at the lower end of the conduit, respectively. It varies in the range of 0.0003−0.0027 m/m.

Some of these methods are designed for catchments with the overland flow as a dominant regime, whereas others have mixed regimes. Furthermore, area-specific methods are designed according to particular area characteristics. Hence, arbitrary selection of specific methods over any catchment may lead to errors. Therefore, the similarity of area characteristics and the assessment of an area's flow regime are two criteria that must be assessed before applying a lumped method to a catchment.

It can be noted that most empirical methods are derived by evaluating the data assembled from a specific geographic region with specific rainfall patterns and physical attributes. Additionally, the application of each method is always constrained by certain limitations, e.g., the dominant-flow regime (i.e., sheet, shallow concentrated, channel, or mixed flow), range of basin area, main channel slope, basin slope, and the regional location of the basin. The results of the present study, as well as previous studies (Sharifi & Hosseini 2011; Kaufmann de Almeida et al. 2017; González-Álvarez et al. 2020), indicate that if an empirical method's limitations are ignored and applied to a basin under conditions different than the developed ones, the ToC estimated will be either under- or overestimated.

The distributed methods are commonly considered the most accurate for ToC estimation for urban and natural catchments (Fang et al. 2007; Sharifi & Hosseini 2011). The time estimated for each flow regime is detailed in the subsequent section.

As discussed in the NRCS method, the sheet flow (t_s) is estimated using Equation (1), and the time of shallow concentrated (t_sc) flow is estimated using Equation (2). For estimating the time of channel flow, Equation (3) (t_cl), velocity is calculated at different depth levels, i.e., 0.1−0.9d, and inscribed in Table 3. The time of surface flow (t_s), time of shallow concentrated flow (t_sc), and time of channel flow (t_cl) are determined to be 0.22, 0.09, and 3.12 h, respectively. The ToC is, therefore, found to be 3.43 h using Equation (4). Usually, the results estimated by velocity methods are referred to as the ‘true’ value of ToC (McCuen et al. 1984; Sharifi & Hosseini 2011).

As discussed in the SWDM method, the surface flow (t_o) is determined using Equation (5). The time of flow (t_f) is determined using Equation (6), for which the flow velocity (Equation (7)) is calculated at different depth levels (0.1–0.9d) and inscribed in Table 3. The time of surface flow (t_o) is computed as 0.09 h, and the time of flow (t_f) is computed as 3.12 h. Therefore, the ToC is found to be 3.21 h (Equation (8)) for the study area.

Previous studies have highlighted the need for a prior assessment of the accuracy of empirical methods by exhibiting the variation in ToC values obtained by the equations. According to Fang et al. (2008), the mean relative differences ranged from −38 to 207% for ToC estimates made using various empirical methods under similar basin characteristics. Kaufmann de Almeida et al. (2017) observed a percentage deviation in the range of −91 to 107% from the mean ToC value obtained from the graphical method. Similar inferences have been made by González-Álvarez et al. (2020).

ToC estimated employing the Schaake method, which calculates the lag time (i.e., the time between the centroid of the rainfall hyetograph and the centroid of the runoff hydrograph), is not found to be in good agreement with other approaches, particularly with the NRCS method. The reason can be attributed to the fact that Schaake et al. (1967) aim to achieve the rainfall averaging time in the rational formula; however, rainfall averaging time is the average of lag time and is not similar to ToC. Furthermore, the Schaake method has the following three conditions: (1) the drainage area imperviousness must be greater than 8%; (2) the average slope of the main drainage channel must lie in the range of 0.5−6%, and (3) the length of the main drainage channel must lie in the range of 45−1,829 m. Here, the third condition could not be satisfied for the study area as the length of the main channel is 12.245 km, which is considerably larger than the Schaake method. The Eagleson Lag method is primarily developed for an urban area with a sewer system. The only restriction for applying this method is that the sewer must be assumed to flow full over the length. This method was originally derived to estimate the lag time, and then a conversion coefficient is used to convert lag time to ToC. It is to be noted that the coefficient is taken as 1.67 (following Fang et al. (2008)), which may result in a significant deviation from the actual value as it was originally derived based on theoretical calculations for the specific region (Eagleson 1962; McCuen et al. 1984). The Espey-Winslow equation is primarily derived from data from 20 urban watersheds and incorporates the urban factor to account for the effect of a sewer system. To apply the Espey-Winslow method, following three criteria must be met, i.e., (1) the length of the main channel must be ranged from 61 to 16,703 m, (2) the slope of the channel must be in the range of 0.0064−0.0104 m/m, and (3) the imperviousness fraction must be in the range of 0.027−1 (Espey et al. 1966). Notably, the slope of the channel may not lie in the prescribed range and may impact final outcomes. Furthermore, this formula was originally derived to estimate the time to peak and a conversion factor of 1.67 (for multiplication) is used to obtain ToC. Since the conversion factor is empirical in nature, it can result in imprecision in the calculated ToC value when applied to different areas. Contrary to the size of the study area, i.e., ∼30 km², the Kirpich equation was derived for very small watersheds with an area of 0.004−0.45 km² (McCuen et al. 1984; Sharifi & Hosseini 2011); therefore, owing to the inherent limitation in the derivation of the method itself, the universal application of the Kirpich equation may introduce error if applied for a large watershed. Interestingly, the Carter method, which provided ToC estimate in close agreement with the ‘true method’, was derived from similar watershed conditions as the case area for the present study, i.e., an urban watershed of ∼30 km² and ∼11.26 km main channel length. The SCS Lag method was primarily developed for lag time estimation of a watershed with dominant overland flow conditions. However, later, the method was modified by incorporating the CN to account for the effect of urbanization. Owing to the fact that in an urban setup, pipe and channel flows are the dominant-flow regimes in contrast to the overland flow regime in a natural catchment, the SCS lag method is prone to error (McCuen et al. 1984). ToC estimated employing the McCuen method is not in good agreement with the NRCS method. The reason can be attributed to the fact that the slope of the basin may not lie in the prescribed range, and the nature of urban catchment changes from one place to another (McCuen et al. 1984). Desbordes formula that estimated lag time was primarily derived using 21 urban catchments of France, Europe, and the United States. As reported by Lhomme et al. (2004), the empirical nature of the Desbordes formula may have regional effects, which could potentially contribute to the overall error of ToC calculation. Interestingly, the SWDM method, which gave ToC values in good agreement with the true method, was primarily designed for Indian cities and specifically for the UDS of Indian cities (CPHEEO 2019).

The possible errors while computing ToC should be minimized. The main source of error is the estimation of the variables associated with a particular equation (slope, land cover, area, rainfall, flow path length, etc.). More input variables increase the likelihood of errors being introduced into the model, especially in hydrology, where the estimation of the basin's morphometric parameters relies on map scales and the interpretation of satellite imagery.

An important parameter for designing an UDS is the estimated design flow from a rainfall−runoff model. The estimated design flow depends on ToC since the duration of the design storm depends on ToC. As a result, any tendency to under- or overestimate the ToC will affect the design runoff. The inherent implications of this can be obtained on the design of hydraulic structures, which might lead to reduced infrastructure life and economic losses.

The ToC is one of the fundamental parameters of hydrology. Realizing the significance of ToC in hydrological designs, particularly in the urban setup, various methods have been devised for ToC estimation. In the present study, a total of eight methods, namely, Eagleson Lag, Espey-Winslow, Kirpich method, Carter method, Desbordes method, Schaake method, McCuen method, and SCS lag method based on the lumped approach and two methods, namely the NRCS method and the SWDM method based on the distributed approach, were used to estimate ToC for the drainage system of the study area. For evaluating the effect of variation in the ToC on the performance of an UDS, the model has been set up in SWMM.

Due to the lack of observed flow records, calibration and validation of model setup in SWMM could not be carried out; instead, sensitivity analysis, a model evaluation technique, is performed to overcome the limitation of calibration. It is also worth mentioning that ToC estimation methods in the lumped approach are region-specific as different methods were calibrated for specific regions with a limited range of input parameters, potentially introducing errors in estimated ToC values. Furthermore, input parameters like Manning's roughness coefficient, overland flow length in Sheet, and shallow flow conditions in the NRCS method are significantly dependent on the modeler's assumptions and expertise; therefore, informed decision on the modeler's part is also a determining factor for the overall accuracies of the entire exercise.

The comparative analysis of selected 10 different methods of estimating ToC indicates wide variation (0.19−6.14 h) in the computed ToC values for the study area, which infers that the suitable method of ToC estimation is essential to predict the design storm. Among the analyzed methods, taking NRCS as the reference method, the SWDM and Carter methods among distributed and lumped methods, respectively, yielded results closer to the true value computed using the reference method. To study the effect of variation in the ToC on flood hydrograph parameters, the model runs in SWMM were carried out for different ToC values. It was observed that flood volume increases from 89.85 to 380.71 m³, flood duration increases from 0.48 to 3.52 h, and time to peak increases from 0.5 to 3.5 h if ToC increases from 0.19 to 6.21 h. Therefore, to avoid major alteration in design values obtained through model simulation, it is imperative that ToC estimations are carried out carefully to select the design storm duration and simulation run period.

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

The authors declare there is no conflict.