Stormwater hydrographs from the highly urbanized catchment for synthetic rainfalls with linear patterns

The reliable design of stormwater drainage networks in urbanized areas presents significant challenges, especially in the background of global climate change (Fust & Schlecht 2022; IPCC 2023; Dong et al. 2024), population growth (Larson et al. 2024), increasing catchment imperviousness (Zhou et al. 2019; Oswald et al. 2023; Nodine et al. 2024), and other influencing factors. The primary objective of urban stormwater modelling is the reliable assessment of stormwater runoff to ensure the sustainable performance of drainage networks, thereby preventing flooding, soil erosion, and water pollution (Kordana & Słyś 2020; Zhuk et al. 2023a). The stormwater hydrograph is a complicated function of a range of factors, including parameters representing the rainfall events, topographic and landscape features of the catchment, structure, diameters, and longitudinal slopes of sections of the drainage network (Liu et al. 2015; Zhuk et al. 2021; Jeong & Kim 2023). Given the complex and unique rainfall–runoff relationships in different catchments, the use of specialized computer programs for modelling stormwater drainage is essential at all stages of urban planning and development (Marsalek et al. 1993; Haris et al. 2016; Ramovha et al. 2024). In recent decades, the stormwater management model (SWMM) developed by the United States Environmental Protection Agency (US EPA) is the most widely used for theoretical and practical drainage system modelling, due to its open-source code, usability, clear architecture, and worldwide validation for stormwater runoff simulations (Niazi et al. 2017; Marko et al. 2022; Giudicianni et al. 2024; Wang et al. 2024).

Rainfall intensity is a crucial input factor of rainfall − runoff modelling (Farina et al. 2023). An important feature of real rainfall events consists of permanent and unpredictable intensity variations over time (Guan et al. 2016; Maier et al. 2020; Czerpak & Widomski 2024). In most cases, to simplify the simulation, rains of constant intensity over time are used (Isidoro & de Lima 2015; Zhuk et al. 2023b, 2024). At the same time, it remains an open question how the irregularity of rainfall intensity affects the output parameters of runoff hydrographs for rains with the same layer depth (Szeląg et al. 2021; Ferrans et al. 2023). The most commonly studied are linearly changing synthetic rainfalls with maximum intensity at the beginning or at the end of a rainfall, as well as V-shaped and Λ-type hyetographs with minimal or maximal intensities in the middle part of a rainfall event (Alavinia et al. 2019; Ran et al. 2019; Gu et al. 2023).

An important derived parameter that describes the irregularity of hyetographs is the rate of intensity variation (RIV), which indicates the rate of change of rainfall intensity over time and is usually measured in mm·h⁻¹·min⁻¹ (Pritsis et al. 2024). Processing the data sources with short resolution times in the range of 1–15 s results in extremely high maximum RIV values above 500 mm·h⁻¹·min⁻¹ (Dunkerley 2011; Lyu et al. 2018). At the same time, for the time resolution of 1 min, which is most often used in the SWMM simulations, the maximum possible RIV for real rainfall events is orders of magnitude lower. In particular, for rising limbs of hyetographs, the reported maximum RIV values are in the range of 7.1–27 mm·h⁻¹·min⁻¹, while for descending limbs – from 4.7 to 36 mm·h⁻¹·min⁻¹ (Lanzinger et al. 2006; Yonese et al. 2018).

Rainfall intensity fluctuates continuously in natural events, but little information is available on these rates to guide rainfall simulation experiments. Recent simulations often use a ‘staircase’ pattern, applying constant intensities for discrete time periods and then varying them in fixed, step increments (Dunkerley 2021).

Accurately predicting stormwater runoff under varying rainfall patterns is essential for designing resilient urban drainage systems. A significant gap in current research lies in understanding how the rate of rainfall intensity variation influences critical peak flow rates, which are crucial for infrastructure design and flood risk assessment. Additionally, surface roughness and depression storage are key factors that significantly affect runoff generation, yet their combined effects on stormwater behaviour with varying rainfall patterns remain insufficiently explored. Despite advancements in modelling tools such as the SWMM, limited studies have numerically analysed the interplay of these parameters – particularly in highly urbanized, fully impervious small catchments. This gap underscores the need for detailed investigations to enhance the predictive accuracy of runoff models under dynamic rainfall conditions.

To address these gaps, this study was undertaken with the purpose of numerically analysing, using the SWMM tool, the influence of linear time-varying rainfall intensity parameters on stormwater runoff in a highly urbanized, fully impervious small catchment. The study places particular emphasis on assessing the combined effects of depression depth and Manning's roughness coefficient, along with the rate of rainfall intensity variation, on maximum and critical peak flow rates. The findings aim to improve stormwater runoff modelling and support the design of urban drainage systems capable of accommodating dynamic rainfall scenarios.

Limiting possible values of the dimensionless coefficient k′ by definition are −1 and +1 (Figure 1(b)). Stormwater hydrographs for the sets of rainfalls with intermediate values of k′ with a step of 0.25 were studied, including the base case of uniform rainfall with k′ = 0.

Surface runoff from each subcatchment was fully intercepted by the stormwater inlets 1–16, located in the lowest points of the subcatchments (Figure 2). The stormwater drainage network of the catchment is configured in a typical cross-rectangular scheme, consisting of a main sewer 8-12-16, as well as identical side sewers 5-8 and 13-16, and the same connections 1-5, 2-6, 3-7, 4-8, and 9-13, 10-14, 11-15. In order to run the SWMM hydraulic model, stormwater runoff from the entire catchment entering the drainage manhole No 16 was diverted to a sewer of significantly larger diameter through the outfall. Longitudinal slopes of all sections of the stormwater drainage pipeline were assumed to be equal to the corresponding surface slopes, i.e. S = 0.01. Diameters of the drainage pipeline correspond to the crucial rainfall events with a return period of about 20 years, ensuring that under the selected climate conditions, all sections are operating in a free surface mode, preventing the manholes from the surcharge and avoiding any flooding of the catchment area at basic tested return period P = 1 year.

This study was focused on single-event kinematic wave modelling of rainfall–runoff functions for a series of synthetic rainfall events corresponding to the climatic conditions of Lviv city (Ukraine), with a return period of P = 1 year.

The curve number (CN) method, developed by United States Department of Agriculture (USDA), assumes a single value of CN = 98 for all impervious surfaces, corresponding to an initial depression depth of 1.04 mm (TR-55 1986). However, current road surface materials exhibit a wide range of absolute surface roughnesses. The typical average absolute roughness k_av for the most commonly used asphalt and concrete coverings ranges from 1–2 mm to 8–10 mm (Mwendera & Feyen 1992; Krebs et al. 2013).

The value of Manning's roughness coefficient n = 0.01 was used in the case when h₀ = 0, based on the results reported by Skotnicki & Sowiński (2015) and Zakizadeh et al. (2022). The main parameters of the surface roughness are consolidated in Table 1.

A series of numerical simulations was performed for testing of the impervious catchment with parameters presented in the previous section (Section 2) and the corresponding hydrographs were obtained (Supplementary material, Figure S2). Six values of initial depression depth in the range of h₀ = 0 and h₀ = 5 mm were applied, with corresponding Manning's roughness coefficients ranging from 0.01 to 0.0180 (Table 1). Specifically, depression depths of 1 and 2 mm aligning the best with the typical asphalt and concrete coverings were used in contemporary road building (Szeląg et al. 2022; Matlai et al. 2024).

The influence of linearly variable rainfall patterns on the main parameters of runoff hydrographs for values of the dimensionless coefficient of linear intensity variability k′ in the range from −1 to +1 was studied. In the first approximation, the critical duration of rainfall events for all values of k′ was assumed to be equal to such duration for uniform rainfalls, when k′ = 0. Key parameters of the stormwater runoff for the variability coefficient k′ = 0 are presented in Table 2.

Additional parameters describing the influence of the intensity factor on rain hydrographs are peak time lag Δt_max = (t_r·max − t_cr), and dimensionless peak flow Q′‘_r·max = Q_r·max·k′/Q_r·max·0, where Q_r·max·0 corresponds to the maximum flow rate for rainfalls of constant intensity when k′ = 0 (Table 3). The difference between the maximum flow rates Q_r·max at limiting values k′ = 1 and k′ = −1 increase with increasing the depression depth equalling 20.7% at h₀ = 0, 25.9% at h₀ = 1 mm, 27.6% at h₀ = 2 mm, and as much as 36.9% at h₀ = 5 mm, where percentages correspond to the relevant maximum flow rates for the constant rainfalls. Detailed analysis of hydrographs revealed a slight decrease in peak flow rates when the intensity coefficient k′ changes from −1 to −0.67. The lowest peak discharges correspond to intensity coefficient k′ = −0.67 for all studied depression depth values in the range from 0 to 5 mm. The peak flow time is closest to rainfall duration when the intensity coefficient k′ ≈ −0.5. Peak time lags Δt_max in the range of k′ = −1… − 0.5 are negative, and vice versa are positive ones, for the rainfalls with intensity coefficients in the range of −0.5 and +1.

Dependencies of peak flow rates and peak flow times on the dimensionless intensity coefficient k′ exhibit similar patterns for various values of the depression depth. As noted by Giudicianni et al. (2024), the highest peak flow rates are observed at the smallest depression depth, whereas larger peak flow times occur when h₀ = 5. At the same time, the peak flow curves for all depression depths run almost parallel to each other; furthermore, for intensity coefficient values greater than –0.5, the dependencies become nearly linear (Figure 4(a)). The peak flow time graphs exhibit stretched S-shaped curves (Figure 4(b)), with the inflection points occurring within the range of k′ = −0.5 to −0.4, and with almost stable linear trends at k′ ≥ −0.4 (Figure 4(b)).

When designing drainage networks, the most crucial parameter is the maximum peak flow rate of stormwater runoff. The results of this study demonstrate that relying on standard engineering practices and designing stormwater systems with a single design storm does not consistently yield a robust system (Pritsis et al. 2024). It was shown previously that the maximum peak flow rate always corresponds to rains with the maximum increasing pattern, which corresponds to the dimensionless intensity coefficient k′ = +1. The same conclusion for linear rainfall patterns was obtained by Alavinia et al. (2019) and Ran et al. (2019). Similarly, for single-peak rainfall events, it was concluded that the later the peak intensity occurs, the higher the peak flow rate that was observed (Fatone et al. 2021; Chen et al. 2023).

Several significant limitations potentially influencing the quantitative outcomes of modelling design runoff hydrographs stem from specific SWMM software configurations. These include the usage of a nonlinear reservoir method for surface runoff from subcatchments, which precludes the consideration of surface flow concentration effects, alongside the temporal discretization of the design rainfall duration set at 1 min. Additionally, the study assumed a completely impervious catchment, an ideal scenario impossible in practice. Nonetheless, from a hydrological perspective, the impact of initial detention depth on surface runoff hydrographs is analogous to free infiltration. This study accounts for the initial rainfall detention depth on subcatchment surfaces, which primarily reflects the effect of surface retention in highly urbanized subcatchments with total imperviousness exceeding 0.8 (Zhuk et al. 2020). Hence, in terms of qualitative impact on runoff hydrographs, such representation should be considered equivalent, albeit with maximum emphasis.

The numerical study examining the impact of linear time-varying rainfall intensity parameters on stormwater hydrographs for a highly urbanized, fully impervious small catchment measuring 200 × 200 m, with depression depths ranging from 0 to 5 mm, has been conducted using the SWMM tool. In order to generalize the influence of linearly variable intensity on the key parameters of stormwater hydrographs, a dimensionless coefficient of linear intensity variability was introduced. The results indicate that, at constant depression depths, an increase in the intensity coefficient from −1 to +1 leads to both an enlargement of peak flow rates and an extension of peak flow time.

An increase in the intensity coefficient leads to larger increments in the rising limbs of the hydrographs. Although the falling limbs of all hydrographs exhibit minimal variation, there is a slight increase in increment for higher values of the intensity coefficient k′. The lowest peak discharges correspond to an intensity coefficient of k′ = −0.67. The maximum peak flow rate consistently corresponds to rainfall events with the maximum increasing pattern, indicated by a dimensionless intensity coefficient of k′ = +1. Overall, the maximum peak flow rate Q_max, associated with an intensity coefficient of k′ = +1, can be expressed as the product of the peak flow rate for design rainfall of constant intensity by the generalized adjustment factor K_tot, which has a clear linear correlation with depression depth using Equation (8).

The results obtained in this study enable numerical modelling of stormwater runoff from highly urbanized catchments, taking into account the potential worst-case rainfall scenarios. This approach enhances the reliability of such modelling, including the selection and design of stormwater management facilities. An important avenue for future research is the investigation of how rainfall patterns influence runoff hydrographs and pollutographs in urban catchments with variable imperviousness.

L.V. contributed to software, data curation, formal analysis, visualization, writing – review and editing. V.Z. contributed to conceptualization, methodology, supervision, validation, writing – original draft.

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

The authors declare there is no conflict.