A two-dimensional hydrodynamic urban flood model based on equivalent drainage of manholes

Global climate change and rapid urbanization are increasing the prevalence of extreme climate phenomena, and frequent urban flood disasters cause huge economic losses and serious casualties (Ma et al. 2018; Xu et al. 2020a; Agonafir et al. 2023). At present, numerical simulations are vital in studying urban flooding, and accurate reproductions of the development of urban flooding based on numerical results are of great significance for urban construction and stormwater management (Wang et al. 2010; Moon et al. 2023). Insufficient drainage capacity is an important cause of urban flooding, and so drainage simulations are an indispensable part of the associated numerical simulations. Reasonable generalization and modeling of urban drainage is of great significance for improving the accuracy of urban flood simulations (Leitão et al. 2017).

Drainage simulations in hydrodynamic numerical models of urban flooding can be mainly divided into two categories. One approach is to directly simulate the runoff in the drainage pipeline network by solving the one-dimensional Saint–Venant equation, representing the surface inundation according to the overflow from the pipeline, or to accurately reflect the surface flood flow state by coupling with a two-dimensional algorithm (Cong et al. 2006; Fan et al. 2017). The second approach involves formulating an equivalent drainage scheme that does not require modeling of the pipe network system, with the drainage effect reflected through other generalized forms (Chen et al. 2015; Wang et al. 2018a; Luan et al. 2021). The Storm water management model (SWMM) and MIKE URBAN models include one-dimensional pipe network calculation modules that simplify or completely solve the Saint–Venant equations. This allows the runoff process of the drainage pipe network system to be calculated before simulating the surface water in combination with the surface hydrological module (runoff generation and confluence) (Li et al. 2018). Zhao et al. (2009) built a rainwater pipe network model based on real surveying and mapping data and used SWMM to simulate and analyze the service performance of regional rainwater drainage systems. One- and two-dimensional coupled hydrodynamic models involve complex hydrodynamic processes with strong physical support for both the surface confluence (two-dimensional) and underground pipe network (one-dimensional), resulting in highly accurate urban flood models (Xu et al. 2020b; Luo et al. 2022). Li et al. (2017) used the MIKE FLOOD software to couple one- and two-dimensional models in simulating the water depth of waterlogged and inundated areas under different rainfall recurrence periods based on pipe network information and topographic data. Leandro & Martins (2016) proposed a method of linking a two-dimensional surface flow model with the SWMM 5 stormwater channel model, enabling simulations of the two-way interaction between the two models, and verified the algorithm using a real case study.

Hydrodynamic models of pipe networks require significant amounts of pipe network data. Unfortunately, such data are often difficult to obtain. Moreover, the computational efficiency of one- and two-dimensional coupled models is low, and urban extreme rainstorms are generally short-duration events. Thus, researchers face the dual problems of data loss and low simulation efficiency of coupled models for large-scale urban area simulations. Therefore, when simulating urban flooding, the one-dimensional model is often replaced by an equivalent drainage mode in the two-dimensional hydrodynamic model to achieve a similar water loss effect to that of drainage via the pipe network (Bradbrook 2006). For example, in many studies, the discharge of the urban pipe network is distributed throughout the simulation area, and the water quantity is directly deducted from each calculation unit (Yu & Coulthard 2015; Yin et al. 2016). Alternatively, the method of deducting rainfall or continuous infiltration can be adopted to realize the water loss in the model, with the decrease in rainfall or increase in soil infiltration regarded as the pipe network displacement (Wang et al. 2018b). These two methods deviate from the actual operation mechanism of pipeline drainage; specifically, they only consider the displacement of the pipe network system and average deduction of surface water, while ignoring the location of drainage points and changes in displacement. In the actual situation, the pipeline and the ground are connected by manholes, and the accumulated water on the ground flows into the manholes and enters the pipeline to be discharged (Chang et al. 2018). In the one- and two-dimensional coupled model, the manholes are typically used as interaction points between the one- and two-dimensional models for water exchange. Xing et al. (2022) simulated drainage loss in a hydrodynamic model by deducting a constant inflow from inlets and manholes. However, in the actual situation, the displacement of inlets and manholes is not constant but is related to the confluence of the ground and the water accumulation in the underground pipeline. Therefore, the previous equivalent drainage model is either too simple, only considering the water loss; or the more advanced equivalent drainage model also only focuses on the location of drainage, but ignores that displacement is a dynamic process affected by flood conditions.

Aiming at the difficulty of obtaining underground pipe network data and the low computational efficiency of one- and two-dimensional coupled models in urban flood simulation, an equivalent drainage model based on a manhole is proposed in this paper. The model corrects the problem of insufficient consideration of drainage location and drainage process in traditional equivalent drainage simulation, so as to improve simulation accuracy and computational efficiency. The equivalent drainage model based on a manhole takes the manhole as the location of drainage and uses the displacement formula to calculate the displacement volume, to simulate the actual drainage mechanism, which has a certain physical basis. The calculation principle is simple and the calculation efficiency is improved. Compared with the simulation results of the classical one- and two-dimensional coupled models, the rationality of the equivalent drainage model proposed in this paper is verified.

In this study, the runoff curve method is used to calculate the flow-producing process of rainfall on the surface. The two-dimensional shallow water equation is used as the control equation for simulating the confluence process of surface runoff, and the displacement of a manhole is used as the source term of the two-dimensional hydrodynamic equation for drainage calculations. The displacement of the manhole is determined using the combined formula for weir flow and orifice flow and the drainage capacity of the pipe network system.

The CN value is determined according to the rainfall–runoff relationship diagram derived from the annual maximum rainfall and the resulting runoff. Thus, different land use types correspond to different CN values. According to the upper, lower, and median values of CN in Part IV Hydrology (NEH-4) of the ‘National Engineering Handbook’, the proposed model uses an empirical value of CN that regulates underground potential retention (Soil Conservation Service, U.S. Department of Agriculture (USDA-SCS) 1971).

The element-centered method is chosen for the spatial discretization of the control body, and the explicit finite volume method is used to solve the problem. Given the Riemannian discontinuity on both sides of the element interface, the Roe scheme is used to solve the element normal flux. The source terms can be divided into two categories: for topographic conditions, the source term has components relating to the bottom slope and friction; for the water balance in the flood process, the source terms include flow production and drainage. The bottom slope feature decomposition method is used to deal with the bottom slope source term, and the interface flux is balanced using an upwind treatment. The discrete friction source term has an explicit formulation. The runoff source term comes from the surface runoff calculated by the runoff generation module, and the drainage source term is the displacement calculated according to the equivalent drainage method.

Image data of the study area allow the locations of manholes to be identified. Thus, a grid containing the manholes can be determined on the corresponding two-dimensional grid. The grid is used to calculate the drainage source term, which is determined by the actual displacement of the manholes.

MIKE URBAN is a relatively mature urban flood model that has been widely used (Tong et al. 2019). This model is mainly applied to simulations of one-dimensional pipe network systems, providing accurate simulations of the flow inside the pipe network (Xu 2021). MIKE URBAN can invoke MIKE 21 to simulate the flow of the two-dimensional surface and is coupled with the one-dimensional MOUSE engine. This enables full one- and two-dimensional hydrodynamic simulations. Therefore, the results of MIKE URBAN for the two-dimensional surface are taken as the ground truth and used to verify the rationality of the equivalent drainage model based on manholes. MIKE URBAN uses pipe network data to construct a one-dimensional model in the drainage part, whereas the proposed equivalent drainage model uses an equivalent drainage algorithm. The two models are consistent in other parts of the flood simulation process, and so the rationality of replacing the one-dimensional model with the equivalent algorithm based on manholes can be verified by comparing their results.

The setup of the equivalent drainage model is consistent with that of the MIKE URBAN model. The boundary of the model is a fixed wall, and the rainfall is evenly distributed in space. The time step is 0.1 s, the simulation duration is 4 h (with rainfall only occurring in the first 2 h), and the roughness coefficient is 0.03. The drainage capacity in the equivalent drainage model is determined by the maximum displacement of the displacement port simulated by the MIKE URBAN model, which is 0.105 m³/s.

Taking the calculation results given by MIKE URBAN in the 5a scenario as the standard, the coefficients in the manhole displacement formula were adjusted to determine suitable parameters for the study area. The validity of the model was verified for the three rainfall conditions of 10a, 20a, and 50a.

Figure 3 shows that the two models are in good agreement regarding the submerged areas under different water depths for all four recurrence periods. In Table 1, for the different water depth thresholds is generally low, and the average relative error is 8% for 5a, 8% for 10a, 4% for 20a, and 6% for 50a. Analyzing the total relative errors in the submerged area under different recurrence periods, the simulation results for 20a are the closest to those of the MIKE URBAN model. This scenario gives the smallest overall average error and small relative errors for the different submersion depths. Additionally, the distribution of with different water depths exhibits the greatest level of consistency.

The submerged area above 0.05 m reaches the maximum at 1 h, when the absolute relative error is the largest, and the maximum relative error of −14% occurs in the 50a case. The areas submerged by water depths above 0.1, 0.2, 0.5, and 1 m reach the maximum when the rainfall ends at 2 h, and the maximum relative errors of each water depth at this time are 5% (10a), 4% (5a), 6% (5a), and 19% (10a). From the perspective of submerged water depth, the average values with water depths above 0.05, 0.1, 0.2, 0.5, and 1 m are 5, 2, 4, 6, and 15%, respectively. Considering the values across all recurrence periods, the smallest error occurs for a water depth of 0.1 m.

To further analyze the difference in submerged locations between the two models, the differences in water depth at various nodes are statistically analyzed. A total of 22,728 nodes are evenly distributed in the study area. The and of the water depth results simulated by the two models at different times under the four rainfall conditions are listed in Table 2. The proposed model performs well in terms of both and ⁠, with the average exceeding 85% for all four recurrence periods. The lowest average value of 88% occurs for scenario 5a, whereas the maximum of almost 99% is achieved in the case of 50a. The average is 0.055 m in scenario 5a and is less than 0.03 m in the other cases. The difference in the water depth distribution simulated by the two models is small, and the degree of fitting is high. increases with increasing recurrence period. is largest for the 5a scenario, and there is little difference with other recurrence periods. In conclusion, a longer recurrence period produces a better agreement in the water depth distributions of the equivalent drainage model and MIKE URBAN.

Table 2

Statistical analysis of water depth at nodes of MIKE URBAN model and equivalent drainage model

Time(h) .	5a .		10a .		20a .		50a .
	.	(m) .	.	(m) .	.	(m) .	.	(m) .
1:00	0.902	0.040	0.964	0.028	0.969	0.030	0.975	0.031
2:00	0.860	0.063	0.985	0.024	0.988	0.025	0.990	0.026
3:00	0.881	0.058	0.985	0.024	0.989	0.023	0.990	0.025
4:00	0.862	0.057	0.984	0.022	0.988	0.022	0.987	0.025
Average value	0.876	0.055	0.979	0.025	0.983	0.025	0.985	0.027

Time(h) .	5a .		10a .		20a .		50a .
	.	(m) .	.	(m) .	.	(m) .	.	(m) .
1:00	0.902	0.040	0.964	0.028	0.969	0.030	0.975	0.031
2:00	0.860	0.063	0.985	0.024	0.988	0.025	0.990	0.026
3:00	0.881	0.058	0.985	0.024	0.989	0.023	0.990	0.025
4:00	0.862	0.057	0.984	0.022	0.988	0.022	0.987	0.025
Average value	0.876	0.055	0.979	0.025	0.983	0.025	0.985	0.027

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The manhole information used in the proposed equivalent drainage model can be extracted from high-definition image data, which overcomes the difficulty of obtaining pipe network data. Even if sufficient pipe network data are available and a one-dimensional pipe network model can be established, the computational efficiency of coupling this to a two-dimensional model for large-scale urban flood simulations may be infeasible. At the same time, the construction of one- and two-dimensional coupled models covering large-scale areas is difficult, and the numerical simulations are prone to instability. Sometimes, one-dimensional models can be built to simulate urban flooding, but cannot simulate the movement of flood water on the surface, which is very important for the judgement of flood risk.

The calculation method of the equivalent drainage model in this paper is simple, and Compute Unified Device Architecture (CUDA) parallel technology is used to greatly improve the calculation rate. Taking the 20a scheme in this paper as an example, it takes only 18 s to simulate 4 h (the first 2 h of rain) using the equivalent drainage model, and 13 min for the one- and two-dimensional coupled simulation using the MIKE URBAN model. Both models were simulated on the same device, a win11 system with a 13th Gen Intel(R) Core(TM) i7-13700 processor on the CPU and NVIDA GeForce RTX 4080 on the GPU. Unlike previous oversimplified equivalent drainage models, the proposed model broadly reflects the actual drainage mechanism. The model combines the two-dimensional hydrodynamic model and the drainage formula of manholes, thus having hydraulic support, and the comparison results against MIKE URBAN verify that the model achieves good simulation accuracy.

This paper tested other equivalent drainage methods, which did not use the displacement formula to calculate the displacement of the manhole, but directly gave each manhole a fixed displacement (drainage capacity divided by the total number of manholes). It is found that the accuracy of the simulation results is lower than that of the equivalent drainage model in this paper. Taking a case in 20a as an example, the average and average of the node water depth between the other equivalent drainage algorithms and the MIKE URBAN model are 0.85 and 0.066, respectively, and the fitting degree is lower than the equivalent drainage model in this paper.

The equivalent drainage model based on manholes reflects that the surface flood discharge exceeds the bearing capacity of the pipeline system by reducing the displacement, and the actual displacement is less than the displacement calculated by the displacement formula. However, in actual life, when the drainage pressure of the pipeline system is too large, the phenomenon of water overflow from the pipeline to the ground may even occur. The one-dimensional pipeline model can simulate the overflow phenomenon, while the equivalent drainage model based on manholes cannot reflect the phenomenon due to the lack of hydraulic support of the underground pipeline.

The displacement of the manhole is determined jointly by the drainage formula and the drainage capacity of the pipe network system. When the sum of displacement of all manholes in the study area exceeds the drainage capacity, the formula displacement of each manhole is multiplied by a common reduction coefficient (drainage capacity divided by the sum of displacement of manhole) to limit the displacement, so as to ensure that the displacement does not exceed the drainage capacity. Because when the displacement exceeds the drainage capacity, the underground drainage pipe network is in the overloaded pressure state, which will produce resistance to the displacement, and the manhole cannot discharge according to the displacement calculated by the formula. This resistance is represented by the reduction coefficient in the equivalent drainage model based on the manhole. The internal pressure of the underground pipeline system is related to the external discharge and internal confluence, and the external discharge is determined by the surface confluence, which is closely related to geographical information such as elevation and slope. Similarly, the underground pipe network system receives different external inflow, and the internal convergence is also different. Therefore, manhole resistance is affected by geographic information, and manholes in different geographical locations shall have different reduction coefficients according to geographical locations, and incorporating geographical factors into reduction coefficients is a direction that can be discussed in the future.

To overcome the difficulty of obtaining drainage network data for urban flood simulations, this paper has proposed an equivalent drainage model based on manholes. The design of this model was intended to improve the part of the existing equivalent drainage method that is quite different from the actual drainage mechanism. The feasibility of the proposed equivalent drainage algorithm and the rationality of the model were evaluated through comparisons with the simulation results provided by MIKE URBAN.

• (1)

  According to the actual situation of urban drainage, manholes were considered as the key media, and an equivalent drainage model was constructed using easily available data such as manhole location information and the drainage capacity of the pipe network system. The equivalent drainage model is based on a two-dimensional hydrodynamic model and combines the weir flow and orifice flow formulas to fully reflect the two main flow modes of manhole discharge. The drainage capacity of the pipe network system was used to limit the displacement. Thus, the equivalent drainage model not only overcomes the difficulty of accessing pipe network data but also accelerates the calculation efficiency of the whole model for the simplicity of the algorithm. And the calculation mechanism accords with the actual situation of urban drainage, has a certain physical basis, and improves the accuracy of the simulation results. However, compared with the one-dimensional model, the equivalent drainage method in this paper cannot reflect the overflow situation of the pipeline. The study of incorporating geographical factors into the equivalent drainage model is a direction that can be continued in the future to further improve the accuracy of the model.

• (2)

  The proposed equivalent drainage model and MIKE URBAN were used to calculate the same four scenarios of rainfall events occurring once every 5 years, once every 10 years, once every 20 years, and once every 50 years. The differences in the simulation results given by the two models were then compared from the perspectives of the submerged area and submerged location. The average relative error between the equivalent drainage model and MIKE URBAN was found to be only 6%. The differences in water depths between the two models were statistically analyzed at multiple nodes, and the average and were found to be 96% and 0.03 m, respectively. The comparison results show that the equivalent drainage model and MIKE URBAN give highly similar outputs, which verifies the feasibility and rationality of the proposed model.

This study was supported by the National Natural Science Foundation of China (51979105, U2340204). We thank Stuart Jenkinson, PhD, from Liwen Bianji (Edanz) (www.liwenbianji.cn/) for editing the English text of a draft of this manuscript.

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

The authors declare there is no conflict.