Urban waterlogging simulation of derived short-duration double-peak rainstorm pattern

Affected by climate changes and urbanization development, urban waterlogging induced by short-duration rainstorm events threatens people's life and property (Zhang et al. 2021; Yang et al. 2022; Wu et al. 2023; Xiao et al. 2023). On 20 July 2021, an extraordinary rainstorm extreme event happened in the city of Zhengzhou, which caused 292 deaths and 47 missing, and the direct economic losses reached 53.2 billion RMB (Xiong et al. 2022; Li et al. 2023a; Li et al. 2023b). During the extremely heavy rainstorm on 31 July 2023, in Beijing, the maximum rainfall reached 744.8 mm. It resulted in 33 deaths, 18 people missing, 59,000 houses collapsing, and 1.29 million people affected.

In the previous studies, short-duration single-peak rainstorm had been the focused issue for its larger proportion in urban rainstorm (Yang et al. 2022). However, further research works had found that two rain peaks existing in a short-duration double-peak rainstorm (SDDPR) may make a city suffer heavy rain twice and produce more waterlogging, thus much attention should be paid to the SDDPR pattern (Chen et al. 2023). However, few studies have been explored to date on the characteristics of SDDPR. Yuan-Yuan et al. (2020) converted double-peak rainstorm into single-peak in their research, but the method is simple and convenient with a certain precision. Then, Zhang et al. (2021) and (2022) proposed the concept of virtual rain peak to derive the SDDPR pattern by converting double peaks into a single peak (CDPISP) first and then into double peaks again, but they did not consider the probabilistic correlations of SDDPR characteristics (Zhang et al. 2022) (such as total rain volume (TRV), main peak rain, and secondary peak rain (SPR)). Actually, the probabilistic correlations of rainfall statistics influenced by the changing environment should be involved in the SDDPR pattern design to reveal the actual temporal distribution of rainfall (Chen et al. 2024). The copula function can link the correlated variables appropriately, thereby it can effectively present the probabilistic relations of rainfall statistics in SDDPR.

Numerous research works have demonstrated that the storm water management model (SWMM) is more suitable for urban rainfall–runoff (Sang et al. 2012; Zeng et al. 2021; Ferreira et al. 2023). Meanwhile, the non-source inundation method (NSIM) has been used to display all points lower than the specified elevation (Zhao et al. 2019; Xu et al. 2023), which are available more for the urban flood in the plain area (Xu et al. 2022; Liu et al. 2024). Usually, this method seldom considers the hydraulic connection of water flowing, which may bring about inaccurate inundation points, but with the geographic information system (GIS)-based SWMM, the NSIM is improved and can inundate the areas from lower to higher, so as to be consistent with the practical waterlogging situation (Huang & Jin 2019).

The objective of this paper is to propose a derivation methodology of SDDPR pattern with copula function by considering the rainfall statistic correlations of SDDPR and then to construct the GIS-based SWMM to simulate the hydrological and hydrodynamic characteristics with the input of derived SDDPR. Finally, heavy rainwater simulation is achieved to assess the waterlogging risk.

Four rain gauges are distributed in the Jinshui District involving Zhengzhou gauge, Huabeishuiyuan gauge, Shengwei gauge, and Laoyachen gauge. The rain data obtained by Zhengzhou meteorological Bureau are from 2009 to 2018; the land-use type is divided into six types: water body, forest land, grassland, farmland, resident land, unused land, and road; the Digital Elevation Model (DEM) elevation data adopt ASTER GDEM 30 m × 30 m resolution digital elevation data in geospatial data cloud, with which the elevation and slope data of junctions of drainage network nodes are obtained. The drainage network data and rainstorm ponding points in Jinshui District are derived from the regulatory drainage engineering planning of Zhengzhou City. There are 201 pipes with a total length of 142,265 m, of which the minimum diameter is 1.0 m and the maximum diameter is 3.6 m.

Three Archimedean copula families described in Table 1 are commonly used in hydrology and water resources fields.

Table 1

Three Archimedean copula families

Archimedean copula .	Expressions .	.
Gumbel Copula
Clayton Copula
Frank Copula

Archimedean copula .	Expressions .	.
Gumbel Copula
Clayton Copula
Frank Copula

View Large

The derivation steps of SDDPR can be described as follows:

• (1) Count the time periods with the highest occurrence frequency of the main peak and the secondary peak in the actual observed SDDPR as the time periods where the main peak and the secondary peak exist in the derived SDDPR pattern.

• (2) Calculate the virtual peak rain (VPR). Suppose to use one virtual rain peak to replace two rain peaks of SDDPR, its location should be between these two rain peaks. Assume that the main peak position is in the front, the virtual rain peak is calculated as follows:

  

  (6)

  where is VPR; n is number of time periods; a, b are coefficients of the main peak and the secondary peak, respectively; and are location coefficient of the main peak and the secondary peak, respectively; and are the actual main peak rain (MPR) and the actual secondary peak of SDDPR, respectively; is location coefficient of the virtual rain peak displayed as

  

  (7)

• (3) Select the suitable marginal distributions for the TRV, VPR, the MPR, and the SMPR, and then calculate the correlations between TRV and VPR, VPR and MPR, VPR, and SMPR.

• (4) Use the copula function to construct the joint probability distribution of TRV and VPR, VPR and MPR, VPR, and SMPR, thus for a given rainfall statistics probability combination (A, B), its VPR, MPR, and SMPR are achieved, and then put the achieved MPR and SMPR on their positions determined by Equation (4).

• (5) Compute and normalize the percentages of rain volume in other time periods except the location periods of MPR and SMPR. According to TRV, rain volumes in all other time periods are obtained and also placed in their corresponding positions to develop the final derived SDDPR.

SWMM can describe the hydrological and hydrodynamics characteristics of rainstorm waterlogging with three modules: surface runoff yield, surface confluence, and drainage network confluence. In the land surface runoff-yield module, the studied area can be classified into permeable area, impervious area with depressions, and impervious area without depressions. The runoff yield on permeable area equals the rainfall volume subtracting the depression detention and soil infiltration; the runoff yield on impervious area with depression is obtained by the rainfall volume minus the initial loss; and the runoff yield on impervious area without depressions refers to the amount of rainfall minus evaporation. In the surface confluence module, the outflow routing process in each sub-catchment is calculated by the continuous equation and Manning equation application with the supposed nonlinear reservoir model. In the drainage network confluence module, steady flow, kinematic wave, and dynamic wave methods are supplied to select, but the dynamic wave method is selected to use because it involves the pipe losses of inlet and outlet, and can be capable of simulating the pressure flow and complex and changeable flows in the closed pipe channels.

The parameters of SWMM are primarily determined by referring to the SWMM manual and related literatures combined with the practical situation. Deterministic parameters of the sub-catchment such as area, slope, and width are calculated by application of the GIS software; deterministic parameters of drainage network characteristics (i.e. length, slope, shape, and diameter) are obtained by the regulatory drainage engineering planning of Zhengzhou City, and their non-deterministic parameters such as roughness coefficient, export loss coefficient, and import loss coefficient are assigned by the related literatures as 0.011, 0.5, and 0.5, respectively; deterministic parameters (impervious surface area) and non-deterministic parameters of land-use type properties (i.e. Manning's n for impervious and pervious surfaces, depression storage for impervious and pervious surfaces, maximum soil infiltration rate, and minimum soil infiltration rate) are also presented by the related literatures.

Coupling SWMM with GIS to improve the NSIM can be used to explain that first some sub-catchment areas are classified along with the surrounding locations of waterlogging points and areas displayed by SWMM, and then using GIS technology these classified area ranges are submerged with the local equal-volume method to make their submerged volume equal to the overflow at junctions within these ranges. Thus, this improved NSIM with GIS-based SWMM is expected to substantially conform the simulated waterlogging points and areas to the actual ones.

To validate the effectiveness of the GIS-based SWMM model, four actual rainstorm events were input into the model for the 1-, 3-,5-, and 10-years RP. The simulated results show that the rational SWMM is presented with the consistency error of surface runoff ranging from 0 to 0.16% and flow calculus continuity error ranging from 0 to 0.03%. Further validation implemented by the integrated runoff coefficient method has also demonstrated the GIS-based SWMM performs reasonably well. Specifically, the integrated runoff coefficients of four storm events are 0.57, 0.63, 0.66, and 0.68, respectively, which is similar to the integrated runoff coefficient of dense urban areas in the regulatory drainage engineering planning of Zhengzhou City. Moreover, the simulated waterlogging points and areas are almost at the sites of the measured locations.

According to the field survey and observations, the rainstorm events resulting in urban flood fundamentally appeared in less than 2 h duration, but the total of 19 SDDPRs are mainly concentrated in less than 1.5 h duration with the front main rain peak.

It can be seen from Figure 3 that the VPRs by CDPISP occupied between 26.5 and 45.4% of the TRV, similar to the actual peak rain of the observed single rainstorm ranging within 25.7–42.6%. It was also found that the majority of VPRs fall in 6–8 mm except for four larger VPRS being beyond 14 mm.

From Figure 4, it can be seen that the empirical and theoretical distribution curves of TRV, VPR, MPR, and SMPR all fit well, preliminarily showing that the distribution can better reflect the actual changes in rainfall characteristics. The correlation coefficient above 0.8 of empirical and theoretical distributions of TRV, VPR, MPR, and SMPR presented reveals a preferred selected marginal distribution.

Three correlation indexes between TRV, VPR, MPR, and SMPR are calculated as presented in Table 2, with Pearson linear correlation coefficient R, Kendall rank correlation coefficient τ, and Spearman rank correlation coefficient ρ, respectively.

As can be seen in Table 3, larger positive values of these three correlation indexes mean that there exist favourable correlations between TRV and VPR, VPR and MPR, and also VPR and SMPR, so the copula function can be introduced to construct their joint probability distributions.

Considering three test indexes such as the Akaike Information Criterion (AIC), theRMSE, and the Bayesian Information Criterion (BIC), four commonly used copula functions (Frank, Clayton, Gumbel, and Gauss) in Archimedes copula family are chosen, as presented in Table 3.

The smallest test index implies the fittest copula, so Table 4 points out that the Frank copula with all three minimum test indexes is the best fit among these four copulas. Thereby, Frank's copula is chosen to derive the SDDPR by constructing the joint probability distributions of TRV and VPR, VPR and MPR, and also VPR and SMPR.

As can be seen in Figure 5, with the increase of TRV and VPR, VPR and MPR, or VPR and SMPR, their joint probability values all show an increasing trend. The correlation coefficients of empirical distribution versus theoretical distribution for these three rainfall statistics pairs are 0.92, 0.85, and 0.93, respectively. This indicates again the Frank copula is available to describe the rainfall statistics.

Based on the above-mentioned steps, the SDDPR event can be derived for different given rainfall statistic probability combinations. Moreover, the regulatory drainage engineering planning of Zhengzhou City stipulates the common area should resist flooding of the 1–2 years RP, important areas should resist 3–5 years RP, and especially important urban road overpasses, tunnels, and culverts should resist 10–20 years RP, so TRVs of 1-, 3-, 5-, and 10-years RP are considered in this study.

• (1) Rainfall statistics of derived SDDPR

It can be seen from Figure 6 that for a given TRV, the conditional probability of VPR exceeding a specific value (CPVPRESV) displays a decreasing trend along with the increase in VPR. At the same time, such trends decrease quickly in the early stage but become slow in the later stage; for a given VPR, larger RP indicates an increase of CPVPRESV; while for a given CPVPRESV, the same scenario of increasing VPR with increasing TRV occurs.

Focusing on the assumed scenarios as TRVs of 1-, 3-, 5-, and 10-years RP as well as CPVPRESV of 25%, 50%,70, and 90%, Figure 7 shows the changes of VPR, MPR, and SMPR in these 16 scenarios.

Figure 9 shows that similar MPR and SMPR exist in the rainfall intensity distributions of these four combination pairs versus the actual for the 1-, 3-,5, and 10-years RP. Meanwhile, the closest CPVPRESV and its corresponding rainfall intensity distribution to the actual can always be found under different RPs, rain volumes in other time periods allocated by proportion are favourable as well. This proves the accuracy of the derived SDDPR pattern.

Using the NSIM with GIS-based SWMM, urban waterlogging characteristics under the derived SDDPR for the total of 16 scenarios in Section 4.1.4 are simulated.

Under the same CPVPRESV as well as the increase in TRV and RP, the situation with increasing average ponding hour-duration and earlier ponding time represents more retention water can be produced quickly. Examples of 25% CPVPRESV condition reflect that the average ponding duration for 1-, 3-, 5-, and 10-years are different.

Therefore, it is observed that compared with rain pattern, TRV is more important to the ponding duration and ponding time. The urban waterlogging resist plan should give more priority to TRV than rain pattern to get more reasonable information.

It can be seen that for the same rainstorm event, the third outlet connecting the longest river to collect a great amount of rainwater owns the largest flow volume and peak flow; while the first outlet links the drainage pipe with the relatively smaller water storage capacity.

Under the same TRV and RP as well as increasing CPVPRESV condition, the total flow volume and peak flow decrease significantly, and an earlier peak flow time appears but its earlier time tends to become less. For the first outlet, the peak flow time occurs 11 min earlier in the Scenario (3a, 25%) compared with the Scenario (1a, 25%), 6 min earlier in Scenario (5a, 25%) compared with Scenario (3a, 25%), and 3 min earlier in Scenario (10a, 25%) compared with Scenario (5a, 25%). Otherwise, under the same CPVPRESV as well as with the increase in TRV and RP, the total flow volume and peak flow increase significantly, and the delayed peak flow time appears with the average delayed time less than 5 min. Generally, the lag time between the peak flow and peak rain is from 1 h,40 min to 2 h, 38min, and the greater flow volume and peak flow indicate earlier peak flow time.

According to the simulated results, for TRV with 1-year RP, the accumulated rainwater is mainly concentrated in the southeast with the smaller waterlogging area of 105,400 m² and lower ponding depth, so it dissipates quickly with less influence. The waterlogging characteristic changes induced by CPVPRESV can be ignored.

For TRV with 3-year RP, the original submerged water area expands besides the newly added waterlogging sections. The whole waterlogging area increased to 756,830 m² spreading from southeast to northeast with the deepest water approaching 30 cm, exerting some impacts on residents. However, the waterlogging areas tend to be less with little difference with the increase in CPVPRESV.

For TRV with 5-year RP, the waterlogging area expands continuously to 1,320,000 m². The 40-cm ponding depth appeared and more time is needed for the drainage of rainwater, which will get in the way of urban production. Similarly, with the 3-year RP, the waterlogging areas are gradually decreased.

For TRV with 10-year RP, a great amount of rainwater makes more waterlogging areas occur, amounting to 2,080,000 m², and especially the deepest ponding depth attains to 50 cm in the southeast, which results in a great number of residential areas and important roads being flooded.

It can be seen from Figure 14 that under the condition of total rainfall in the same return period, with the increase of conditional probability, although the number of waterlogged nodes will decrease, the number of waterlogged points in each conditional probability distribution has little difference. For different return periods, with the increase of the return period, the number of nodes with different risks will increase.

This study proposes a new method considering correlations of rainfall statistics to derive SDDPR. With the application of the copula function, the proposed derived SDDPR pattern with rainfall statistics probability combination is available. Especially in the main rain peak and secondary rain peak locations, the simulated rain volumes are similar to the actual, but some errors exist in some extreme rainstorm situations.

Using NSIM with GIS-based SWMM, the urban waterlogging simulation is performed to show the different characteristics including surface ponding at junctions, flow in conduits, and outflows in the outlets. It is found that compared with rain pattern, TRV is more important to the ponding duration and ponding time; RP has a greater influence on OCN, the rate of OCNFFTBB0.5H to total conduit numbers for 1-, 3-, 5-, and 10-years RP are 48%, 57, 64, and 78%, respectively; under the same CPVPRESV as well as with the increase in TRV and RP, the total flow volume and peak flow increases significantly, and the delayed peak flow time appears with the average delayed time less than 5 min.

Furthermore, urban waterlogging situations under the derived SDDPR with a total of 16 scenarios show that for the Jinshui District in Zhengzhou City, the larger RP along with more rainwater will result in larger waterlogging. For TRV with 1-year RP, the accumulated rainwater is 105,400 m² with lower ponding depth, and for TRV with 10-year RP, the depth and range of accumulated rainwater attains 50 cm and 2,080,000 m². Moreover, it is especially concentrated in the southeast. Thus, some measures to resist waterlogging should be undertaken in this area.

This research is supported by the National Natural Science Foundation of China (52379028) and the Natural Science Foundation of Henan Province (242300421007).

Data cannot be made publicly available; readers should contact the corresponding author for details.

The authors declare there is no conflict.