A Generalization of the Drainage Capacity in Data-Scarce Urban Areas: An Improved Equivalent Infiltration Method

Abstract

This article addresses the challenge of simulating rainstorm waterlogging in urban-scale areas where reliable drainage pipe network data are often lacking. Although methods have been developed to tackle this issue, there remains a gap in their effectiveness. We present a novel approach, the modified equivalent infiltration (MEI) method, by building upon the foundation of the Equivalent Infiltration (EI) method. This study focuses on the outer ring area of Shanghai, utilizing data from the “In-Fa” typhoon period for simulation and comparison. Our findings reveal that the MEI method, requiring the same data inputs as the EI method, surpasses its predecessor in both principle and simulation results. Additionally, the MEI method demonstrates robustness in handling rainstorm waterlogging scenarios.

1. Introduction

In recent years, global warming-induced temperature rise has increased the occurrence of extreme precipitation events, coupled with urban development, leading to heightened runoff volumes and resulting in more frequent and severe urban flooding globally [1]. With ongoing urbanization and economic growth, the damage caused by stormwater waterlogging is expected to rise [2,3], prompting a need for effective preventive strategies [4,5]. Addressing waterlogging involves a comprehensive approach, including pre-flooding forecasts, emergency measures, and post-flooding resilience solutions. Central to this is the development of urban rainstorm waterlogging models and numerical simulations to identify high-risk areas and implement timely measures [6,7]. These models provide vital theoretical support for decision making in urban stormwater flood prevention and mitigation. Generally, coupled models for simulating rainfall waterlogging can be categorized into those simulating flow production processes [8] and those focusing on water distribution based on flow production [9], with the latter requiring indispensable pipe network data.

When conducting research on community-scale areas, it is easy to obtain pipe network data and input them into the model, and a number of researchers have carried this out. For instance, Li et al. [10] investigated the Yangmei River basin in Tianhe District, Guangzhou City. Similarly, Niu’s study [11] encompassed the flight area of an airport, and Qiu et al. [12] concentrated on an old district in Shanghai. Szeląg et al. [13], on the other hand, examined the vicinity of the Si9 canal in Kielce, Poland. However, when expanding the research area into a city-scale area, some encounter severe problems [14]; for example, credible pipeline network data are neither readily available nor easy to input into the model in urban-scale studies.

When dealing with pipe network data in a city-scale area, the problem is actually twofold. On one hand, network data are hard to obtain. In a large number of urban areas, there are no complete data on the pipeline network [15,16]. There are a number of reasons for this, including the fact that relevant information has been missing from the outset or has been lost over a long period of time for one reason or another, especially since, in some places, the pipeline network was constructed decades ago [17]. Another reason is that the actual layout of the pipe network does not match the design drawings for various reasons [18] or that there are unrecorded blockages and damage in the network [19,20]. Furthermore, it is quite expensive and difficult to measure the drainage network system onsite [18]. On the other hand, in a city-scale area, the complexity of the network data increases, significantly reducing the computational speed of the model. Therefore, in attempts to simulate urban-scale study sites, consideration needs to be given to trade-offs in the pipe network, such as generalizing the main pipe network [21,22]; by using this approach, the computational costs can be significantly reduced [23]. For example, Wang et al. [24] selected main drainage pipes and simplified branch ones in their simulation of Zhengzhou City, China. Yang et al. [25] employed a similar approach in their simulation of rainwater waterlogging in Fengxi New City near Xi’an City in Shaanxi Province, China, simplifying minor pipes while retaining the major drainage pipes. Shrestha et al. [26] followed a similar approach in their study of Phoenix City. This approach is justified by the fact that the data requirements of the coupled model for the simulation of the pipe network are limited to the overflow process of the pipe network and not to the entire convergence process [27].

In response to the many problems faced when using pipe network data in the field of urban-scale research, some researchers have attempted to use alternative methods rather than recovering pipe network data, such as developing synthetic sewerage networks to complement the representation of sewerage systems in urban flood models [28], using the empirical calibration method of the rainfall comprehensive runoff coefficient [29], or using community mapping [30]; some other researchers have used crowd-sourced data to build models [31]. However, these methods are lacking in hydrology and hydrodynamics principles; therefore, many researchers have proposed corresponding methods based on coupled hydrology–hydrodynamic models, and coincidentally, many researchers have replaced the specific drainage network with a parameter called drainage capacity. However, researchers have proposed different approaches on how to substitute this parameter into the model. Some researchers opt to subtract the drainage capacity from rainfall [32], while others treat it as an infiltration process; thus, three methods are widely used: the rainfall reduction method, constant infiltration method, and equivalent infiltration method. In practice, Chen et al. [33] utilized rainfall discounting in their simulations. Chang et al. [34] and Leandro et al. [35] used the method of adding the drainage capacity into the infiltration process. Wang et al. [36], on the other hand, compared the rainfall discounting method with the constant infiltration method and concluded that the latter produces superior results.

As a matter of fact, all three methods mentioned above are problematic to a certain extent as far as the principle is concerned. The rainfall reduction method, for instance, produces a rainfall curve that cannot reach a minimum value below zero [36]. This oversimplification disregards the gradual recession of water on the site after the rainfall peak, leading to significant simulation errors. On the other hand, the constant infiltration method converts the drainage capacity into a fixed infiltration capacity. While this method can simulate the water recession process following the rainfall peak, it overlooks the change in the site infiltration capacity with time. Therefore, in the case of land with high permeability (e.g., green space) or a site with a low drainage capacity, this approach may produce inaccurate results. As for the equivalent infiltration (EI) method, this approach allows the drainage capacity of the pipeline to be influenced by the infiltration curve of the site, which leads to errors that will be explained later.

After analyzing the principles, it can be found that, among the three methods mentioned above, the equivalent infiltration method is superior to the remaining two. Therefore, this paper chooses to improve the equivalent infiltration method on the basis of the existing research [37] in order to solve the interference brought to the model by the influence of the site infiltration process on the drainage capacity of the pipe network and proposes a modified equivalent infiltration method (MEI). By using virtual drainage pipes and virtual drainage wells, the MEI method has the same data requirements as the equivalent infiltration method, but with better performance. The method proposed in this study helps to simulate storm waterlogging at urban scale sites where pipe network data are difficult to obtain, and it provides a theoretical basis for storm waterlogging prevention and mitigation.

The structure of this article is as follows: Section 2 presents the principle of the MEI method and the evaluation indicators, including the difference between the EI and MEI methods; Section 3 discusses the research site and data preprocessing content. In Section 4, the error of the model, the comparison between the EI and MEI methods, and the influence of well spacing on the simulation results are discussed. And Section 5 presents the conclusion.

2. Materials and Methods

2.1. Principle of the MEI

The MEI method used in this paper also generalizes the suppression of site waterlogging by the drainage network into a drainage capacity, like the existing methods. But unlike those methods, the modified equivalent infiltration method does not superimpose the drainage capacity on the rainfall or infiltration curves; instead, it assumes that there is a virtual drainage pipe in each sub-catchment with an upper flow limit. The upper flow limit is equal to the capacity of the drainage system in the sub-catchments. The drainage pipes in the different sub-catchments are independent, and this generalization allows the principles of the waterlogging simulation model to be fitted as closely as possible in the absence of network data. The difference between the EI method and MEI method is shown in Figure 1.

Considering that infiltration in the sub-bedding surface is not uniform, a parabola is used for the area distribution of different infiltration intensities [38], and considering Horton’s law, the joint infiltration rate ${\mathrm{I}}_{\mathrm{r}}$ is shown as follows:

$$I_r={\left.\frac{{\left(1-\frac{f_c}{f_m}\right)}^{n+1}·f_m}{n+1}\right|}_{f_c=0}^{f_c=i}=\frac{f_m}{n+1}·\left[1-{\left(1-\frac{i}{f_m}\right)}^{n+1}\right]$$

(1)

where ${\mathrm{f}}_{\mathrm{m}}$ is the maximum infiltration rate, ${\mathrm{f}}_{\mathrm{c}}$ is the point infiltration rate, i is the precipitation rate, and n equals 2.5.

In Equation (1), ${\mathrm{I}}_{\mathrm{r}}$ can be considered as a function of the precipitation rate, I, and the infiltration rate, ${\mathrm{f}}_{\mathrm{m}}$, i.e., ${\mathrm{I}}_{\mathrm{r}}=\mathrm{g}(\mathrm{i},{\mathrm{f}}_{\mathrm{m}})$. The joint infiltration rate, ${\mathrm{I}}_{\mathrm{r}}$, under the combined effect of site infiltration and pipe network drainage can be expressed as $\mathrm{g}(\mathrm{i},\mathrm{f}+\mathrm{p})$ in the equivalent infiltration method, while in the modified method, the combined effect of the two is expressed as $\mathrm{g}(\mathrm{i},\mathrm{f})+\mathrm{p}$, where p is the drainage capacity, as shown in Equations (2) and (3):

$$T_1=g(i,f+p)=\frac{f+p}{n+1}\left[1-{\left(1-\frac{i}{f+p}\right)}^{n+1}\right]$$

(2)

$$T_2=g(i,f)+p=\frac{f}{n+1}\left[1-{\left(1-\frac{i}{f}\right)}^{n+1}\right]+p$$

(3)

In the equations above, ${\mathrm{T}}_1$ is the combined effect of EI, while ${\mathrm{T}}_2$ is that of MEI. While assuming the constant intensity of infiltration, the relative error between ${\mathrm{T}}_1$ and ${\mathrm{T}}_2$ was calculated, and the results are shown in Figure 2.

In Figure 2, it can be seen that as the drainage capacity and rainfall intensity vary, the gap between the joint value of the EI and MEI methods changes as well. As can be seen in line A-B in Figure 2, the relative error remains at zero when p is equal to zero, which is well understood because the EI method is indistinguishable from the MEI method when the site drainage intensity is zero. After executing typical scenarios, depending on the drainage capacity and rainfall intensity, this gap varies from over 10% to negative values; thus, the error due to combining the drainage strength with infiltration strength is not negligible, as seen in point C, where the relative error is about −20%.

Figure 2 further demonstrates that the MEI method shows a reduced overflow rate and an elevated joint value during periods of low rainfall intensity, given that the value of the difference function is negative. However, as the intensity of the rainfall increases, there is a reversal in the relationship between the magnitude of the joint value, and the joint value of the MEI method becomes lower at this point. Considering the lag in the response of overflow to site waterlogging, this analysis provides insight into the higher total waterlogging area observed for the MEI method after the rainfall peak. Additionally, it explains the observation that the equivalent infiltration method results in more water waterlogging compared to the MEI method prior to the occurrence of the first major rainfall peak. The two phenomena mentioned above will be dealt with later on.

2.2. Evaluation Indicators

In this paper, the simulation results are validated using three indicators: the Nash–Sutcliffe efficiency coefficient (NSE), the coefficient of determination (R2), and the Root Mean Squared Error (RMSE). These three indicators are calculated as follows:

$$NSE=1-\frac{{\sum }_{i=1}^n{(d_{ob,i}-d_{sim,i})}^2}{{\sum }_{i=1}^n{(d_{ob,i}-\overline{d_{ob}})}^2}$$

(4)

$$R^2=\frac{{\left({\sum }_{i=1}^n(d_{sim,i}-\overline{d_{sim}})(d_{ob,i}-\overline{d_{ob}})\right)}^2}{{\sum }_{i=1}^n{(d_{ob,i}-\overline{d_{sim}})}^2{(d_{ob,i}-\overline{d_{ob}})}^2}$$

(5)

$$RMSE=\sqrt{\frac{1}{n}{\sum }_{i=1}^n{(d_{ob,i}-d_{sim,i})}^2}$$

(6)

where ${\mathrm{d}}_{\mathrm{o}\mathrm{b},\mathrm{i}}$ and ${\mathrm{d}}_{\mathrm{s}\mathrm{i}\mathrm{m},\mathrm{i}}$ are the observed and simulation values of waterlogging depth for site i, and $\overline{{\mathrm{d}}_{\mathrm{o}\mathrm{b}}}$ and $\overline{{\mathrm{d}}_{\mathrm{s}\mathrm{i}\mathrm{m}}}$ are the averages of the observed value and simulation value.

Of these three parameters, the NSE as well as the R2 are used to assess the simulation accuracy of the method, and the RMSE is used to quantify the assessment error.

2.3. Preprocessing of Data

Pipe network data include parameters such as the size and length of underground pipes and parameters related to the location of surface overflow wells. Thus, the lack of credible pipe network data also leads to a lack of well data. For the urban-scale study area where the flood model was applied, the researcher was also no longer able to use specific well locations for the sub-basin delineation of the study site using the Thiessen polygon method. In an urban-scale study area, the subsurface is predominantly composed of human-made elements; however, it is important to recognize that the presence of other land use types (e.g., forests and grasslands) cannot be ignored. Therefore, considering that the basic condition for the use of the Thiessen polygon method is the isotropic nature of the subsurface of the study site, the division was firstly based on the type of underlayment, and then on a secondary level based on selected key points. Given that the distribution of major drainage networks often corresponds to major urban roads [39], the intersections of major roads in the study area were selected as key points for secondary classification.

It is important to note that while the intersections of major roads serve as pivotal points in sub-catchment delineation, they cannot be considered as all well points. This limitation arises due to the increase in flow generated within the sub-catchment as its area expands. If only pivotal points were designated as wells, excessive overflow would occur from a single point, resulting in water depths around the overflow location being higher than they actually are and the simulated water area being smaller than the real water area. This issue becomes more pronounced with larger sub-catchment areas and higher precipitation intensities.

To address this issue, virtual wells were strategically placed at fixed distances along roads within the study area. This arrangement is based on the assumption that wells are generally located along major roadways and are evenly distributed throughout the site within the city limits. By dispersing the overflow through these virtual wells, the flood distribution across the site is very close to that of the actual situation, thus reducing the discrepancies mentioned above. The differences in simulation results due to different distribution distances will be discussed later. Therefore, the flow chart of this study is shown in Figure 3.

3. Case Study

Research Site

Located in the Yangtze River Delta, Shanghai is subject to a humid subtropical monsoon climate, with abundant precipitation throughout the year and a low-lying, gentle terrain. Heavy rainfall and typhoons have caused varying degrees of flooding and damage to properties in Shanghai. In 2005, the “Metsa” typhoon caused heavy rainfall of over 200 mm in Xuhui, Changning, and Hongkou, resulting in 238 roads in Shanghai being waterlogged to a depth of 20–30 cm; from the 6–8 October 2013, the “Fate” typhoon caused severe flooding in Shanghai, paralyzing traffic and affecting 97,000 people. Furthermore, the “In-Fa” typhoon led to the emergency relocation of 362,000 people from Shanghai and the cancellation of all flights at the Pudong and Hongqiao airports.

The research site is the region within the outer ring of Shanghai (Shown in Figure 4), with an area of approximately 620 square kilometers and a high degree of urbanization, which is Shanghai’s central part. The sub-surface is mostly concrete in built-up areas and asphalt on roads, and the overall permeability of the site is low.

The rainfall data used in the model were measured at various rainfall stations in Shanghai during the landfall of “In-Fa” typhoon at the end of July 2021. A total of 19 rainfall stations within and outside of the Outer Ring Road were involved in the study site. The slope of the sub-catchment was derived from the Shanghai DEM map; the Manning’s coefficient of the site was derived from the vector map of Shanghai land use types. In the calculation of the percentage of imperviousness in the sub-catchment area, it was assumed that the road and building coverage area is impervious and the rest of the area is permeable. The road coverage area was calculated by generalizing the road width according to the national standard for classification of Classes 1–4, and then superimposed with the building coverage area to calculate the impervious area.

Other parameters, such as permeability and the amount of water stored in impermeable depressions, as shown in

Table 1. Partial parameter rate values.

Name	Description	Values
D-Imperv	Impermeable depression storage	1.905 mm
D-Perv	Water storage in permeable depressions	3.81 mm
MaxRate	Maximum infiltration rate	76.2 mm/h
MinRate	Minimum infiltration rate	2.54 mm/h
Decay	Attenuation factor	6

, were calibrated from the SWMM user manual and other research results available in Shanghai [40].

In addition to the above parameters, one of the most important parameters of this paper is the drainage capacity of each part of the site. Although Shanghai has already planned the drainage capacity of the urban pipe network, as said before, due to the force majeure effects such as the presence of debris in the pipe network, the clogging of the drainage wells, and the aging of the pipe network, it is inevitable that the actual drainage capacity of the site will be lower than the theoretical value in the case of heavy rainfall, and therefore, it is necessary to calibrate the drainage capacity of the site by using the following equations:

$$L={\int }_0^{t_0}ldt$$

(7)

$$l=i-T_2$$

(8)

where L is the total overflow volume (${\mathrm{m}}^3$), l is the overflow volume at a given moment in the sub-catchment (${\mathrm{m}}^3/\mathrm{s}$), i is the precipitation volume at a given moment in the sub-catchment, and ${\mathrm{T}}_2$ is the joint value at a given moment in the sub-catchment (${\mathrm{m}}^3/\mathrm{s}$), calculated using Equation (3). From this, the inundation process during the first flood of the precipitation process can be used to calibrate the drainage capacity, p, of different areas of the site in combination with the reported depth of water in each waterlogged cell and the time of occurrence of the waterlogging.

4. Results and Discussion

4.1. Model Validation

The “In-Fa” typhoon impacted Shanghai during 25–28 July 2021. The simulation process commenced at 2:30 a.m. on 25 July and concluded at 18:30 p.m. on 26 July, encompassing a 40 h period of continuous rainfall. By comparing the occurrence of waterlogging reported by the plots within the study site during the precipitation period with the corresponding locations in the simulation results at the corresponding time, a reasonable agreement was observed between the simulation results and the reported waterlogging depths on 25 July. Furthermore, a total of 28 reports of waterlogging on the research site were recorded from 25 July to 26 July. Among these reports, 12 were documented on 25 July, and 16 were documented on 26 July. Specifically, 11 reports corresponded to 25 July, and 14 reports corresponded to 26 July. Considering that the reported time may not precisely align with the exact time of the waterlogging occurrence, the simulated results for the half-hour period before and after the reported time for certain identified waterlogging areas between 25 July and 26 July were aggregated, leading to the results shown in

Table 2. Validation of errors.

No.	Reporting Depth (cm)	Average Depth (cm)	Relative Error (%)
(I)	5	2.8	44
(II)	5	3	40
(III)	20	15	25
(IV)	5	5.8	16
(V)	10	7.1	29
(VI)	2	1.1	45
(VII)	3	3.7	23
(VIII)	1	1.6	60
(IX)	3	3.8	27
(X)	3	1.8	40
(XI)	20	13.4	33
(XII)	25	23.1	10
(XIII)	5	4.2	22
		Average Relative error	31.2

.

The error validation results show that the simulation results have some deviations compared with the actual simulation data, and the average relative error of the waterlogging depth is about 31.2%. The NSE, R2, and RMSE values during the “In-Fa” typhoon were calculated to be 0.818, 0.945, and 2.741, respectively. It can be concluded that the model is well coupled with the measured data and can effectively simulate the distribution of water accumulation during the rainfall period.

4.2. Comparison of EI and MEI

Figure 5 shows the change in the waterlogging area over time; it can be seen that the model successfully simulates the 40 h rainfall process, capturing four distinct peaks, which are named rain peaks A/B/C/D in Figure 5, respectively. Peaks A and C exhibit lower intensities, while peaks B and D demonstrate higher intensities. Figure 6 shows the results obtained after counting waterlogging on the site using the equivalent infiltration method with the same parameter sets as those used in the MEI method and differentiated by the same water depth intervals.

In Figure 6, it can be seen that the trend of change in the waterlogged area of the site obtained using the EI method is similar to that of the MEI method (in Figure 5), but at the peaks of the rainfall, it is different from that of the MEI method, especially at peaks B and C.

To further compare the difference between the EI and MEI methods in terms of waterlogging area, the total waterlogging area maps obtained from both methods were extracted for comparison, and the obtained results are shown in Figure 7.

There is no significant difference in the total flooded area between the two methods at the beginning of the rainfall period (beginning of peak A). However, the difference in the total flooded area between the EI method and the MEI method becomes progressively more pronounced after the onset of rainfall peak A. The EI method produces a smaller flooded area, while the MEI method produces a larger flooded area. This difference persists for the next three rainfall peaks, and this phenomenon arises because the synergistic value calculated using the EI method is larger than that calculated using the MEI method during higher rainfall, and therefore, the total overflow volume during the rainfall peaks is smaller for the EI method, which, in turn, leads to a smaller total waterlogging area, as can be seen in Figure 2 in Section 2.1. It is worth noting that the flooded area from the EI method shows a platform during peak C, the second small rainfall peak, during which no significant peaks in the flooded area can be observed. From this, it can be tentatively concluded that the difference in the results obtained using the two methods is heavily influenced by the rainfall intensity.

Based on the three evaluation indicators mentioned in Section 2.2, the simulation results of the EI method and MEI method are compared, as shown in Figure 8.

As can be seen from the above figure, of the results obtained using the two methods, the water depth results obtained using the MEI method are closer to the reported water depth than those obtained using the EI method. After conducting a calculation by using the data shown in Figure 8, the NSE and R2 values of the results obtained using the MEI method (0.818, 0.945) are better than those obtained using the EI method (0.699, 0.918); the RMSE for the EI method is 3.436, and that for the MEI method is 2.741. The findings suggest that the results derived through the MEI method more closely align with the empirically measured values, indicating enhanced reliability, particularly when applied to identical drainage capacity distributions.

4.3. Effect of Spatial Distribution of Virtual Well

The MEI method was used in this study to address the lack of pipe network data. This method involves the construction of a virtual pipe within sub-basins to simulate the role of the actual underground drainage network in receiving surface runoff from the site. As stated previously, this method requires the placement of virtual overflow well points within the site at regular intervals to refine the hydraulic interaction between the surface and the pipe network. As a parameter at the discretion of the researcher, this section discusses the effect of the distance of virtual well placement on the simulation results.

To assess the influence of well point placement, virtual wells were positioned at distances of 200 m, 300 m, and 400 m for the simulation and subsequent comparison. Figure 9 below shows the final results of these simulations, categorized in terms of overflow point distances (m) and waterlogging depth intervals (cm).

The comparison of the different scenarios indicates that the waterlogging area exhibits little variation across depth intervals when the spacing of virtual overflow points is set at 200 m, 300 m, and 400 m. Moreover, since the total hydraulic interaction within the sub-catchment area remains unaffected by the change in the spacing of points, it can be inferred that the distribution of waterlogging areas demonstrates strong similarity among these scenarios. Consequently, when considering the need to enhance the calculation efficiency, a spacing of 300 m is selected for the placement of virtual overflow points in the modified equivalent infiltration method.

In simulations of rainstorm waterlogging at sites with incomplete pipe network data, it is inevitable that the placement distance of virtual wells will be determined by the researcher’s own experience. In the case of other sites, the researcher can either use existing pipeline network data or make their own walk-through observations of the site to determine the placement distance of the virtual wells. The comparison of the aforementioned scenarios demonstrates that the MEI method consistently produces similar and stable results, even when researchers employ different spacing for virtual wells, so long as the proper distance is used. This highlights the method’s robustness.

5. Conclusions

In this paper, the MEI method, as an improvement of the equivalent infiltration method, is proposed to address the issue of missing data on urban drainage networks. This method involves extrapolating the drainage ability of a specific pipe network to the drainage capacity of the entire area and replacing specific wells with virtual wells distributed along the roads. The paper includes both qualitative and quantitative tests based on the measured data of the “In-Fa” typhoon. Compared with the equivalent infiltration method, the improved method can better simulate water-prone areas and the severity of water accumulation under rainstorm conditions with the same data conditions, both qualitatively and quantitatively. The MEI’s average relative error of water depth is approximately 31%, with an NSE (Nash–Sutcliffe efficiency) of 0.818, an R2 (coefficient of determination) of 0.945, and an RMSE (Root Mean Squared Error) of 2.741.

Additionally, by comparing different virtual well distances, it was found that the simulation results of the waterlogging simulation model constructed using the MEI method remained consistent when varying the parameter of the virtual wells’ deployment distance in a suitable range, demonstrating the model’s stability.

By employing the model using the MEI method, it becomes possible to simulate the spatial distribution and temporal evolution of flooding more accurately. This capability provides theoretical support for various subsequent studies, including accessibility analysis and other related research areas. Addressing these issues and further refining the model will contribute to its overall effectiveness and utility in practical flood control and urban planning applications.