Development and application of a mathematical model for calculating the discharge of non-standard thin-plate weirs in urban combined sewer overflow systems: a case study

Since the 21st century, global warming and human activities have directly affected the spatial and temporal distribution of urban water circulation (Qin 2015), and the situation of urban water security is becoming more and more serious. Although local governments have intensified the rectification of protecting urban drainage systems and the water environment, some of the municipal drainage system construction is still lagging, leading to frequent overflow pollution issues during heavy rainfall events (Zhao et al. 2020).

Municipal drainage systems include rainwater drainage systems and sewage drainage systems, usually implemented in the form of combined sewer systems or separate sewer systems for the discharge of rainwater and sewage. Compared to the relatively reasonable but difficult to construct separate sewer systems, the interception systems in combined sewer systems have the characteristics of a short construction period, quick effect, and strong applicability (Zhou et al. 2015). Therefore, in order to improve the water quality of urban water bodies, many cities in China have carried out the construction of intercepted drainage systems. Such as Fuzhou, Shenzhen, Guangzhou and Nanchang, have adopted intercepted drainage system construction in some areas, and achieved good performance (Gao 2013; Li et al. 2020, 2022).

As the connection system that links the combined sewer system and receiving water body, UCSOS undertakes the function of collecting and discharging domestic sewage, industrial wastewater, and rainwater. When it's sunny or lightly raining, combined sewage is transported to the sewage treatment plant as normal. However, during heavy rain, when the volume of transported water exceeds the sewage system's capacity, it can lead to overflow pollution, impacting the ecological environment of the city's rivers and lakes. According to statistics, CSO discharges contribute to 10–80% of the annual load for different contaminants including suspended solids, Escherichia coli, intestinal enterococci, total nitrogen and total phosphorus in receiving urban rivers. In addition, CSO is also a source of emerging contaminants including stimulants, analgesics, NSAIDs, beta-blockers and so on. Thus, how to balance the capacity of the sewage treatment plant with the volume of sewage has become a common problem in these cities.

At present, the overflow frequency and overflow volume of CSO are the most common indicators used in overflow pollution control (Huang et al. 2021). However, due to the late start of CSO pollution research and lack of basic monitoring data such as overflow discharge, overflow frequency, and water quality, it is difficult to evaluate CSO pollution effectively, and CSO pollution has become a major problem for Chinese cities to eliminate black-odorous water and improve the overall urban water environment. In particular, affected by the shortage of urban land resources, there exists a large amount of NTPWs. Meanwhile, due to the restriction of engineering structure, it is hard to setup conventional flow monitor facilities to monitor CSO, which makes it more difficult to control CSO pollution. Therefore, reliable and convenient discharge monitoring methods are urgently needed. many scholars have also done a lot of research on water quality, frequency, and discharge of CSO pollution. Viviano et al. (2017) used caffeine as a tracer to identify CSO contributions to phosphorus in river water and found that more than half of the total phosphorus loads transported by the river are due to the CSO discharges. Daan Bertels conducted a statistical analysis on the overflow volume and frequency occurring in the WPC Kluizen area, revealing that about 80% of the total overflow volume is attributed to around 20% of the CSO events. Hofer et al. (2018) presented a novel surrogate method to detect CSO events by using two low-cost temperature sensors. However, discharge measurement techniques are intrinsically complicated to deal with (Melching 2006). The main challenge focuses on the monitoring of CSOs as overflow structures were not originally built for monitoring purposes. As a result, they often exhibit complex hydrodynamics and uncertainties associated with traditional measurement processes, if estimated, usually are considerably high. Therefore, Maté Marín et al. (2018) designed a pre-calibrated device that can be installed downstream from the existing CSO structure to measure the overflow volumes by water level gauge in the device. To sum up, most research focuses on concentration and overflow frequency monitoring, and the monitoring of overflow discharge lacks accuracy because the discharge measurement techniques are intrinsically complicated.

Although it is a convenient method to indirectly measure discharge through its relationship with water level, how to obtain a credible relationship between water head and discharge in a thin-plate weir is still a difficult problem in fluid mechanics calculation. In 1920s, Rehcock proposed his own discharge and discharge coefficient calculation formula based on numerous experiments. Later, Sarginson modified the Reibock empirical formula into an expression containing the Weber number (We) (Peng et al. 2004). Arvanaghi & Oskuei (2013) carried out the numerical simulation of the sharp-crested weir and established the corresponding relationship between H/W (static head over the weir divided by weir height), Froude number and Reynolds number on the weir discharge coefficient under different conditions. Hadano et al. (2015) deduced the dimensionless parameter to characterize the flow variation characteristics through the law of momentum conservation and deduced the formulas for both discharge and upstream head through the obtained unique relationship between the dimensionless critical depth and the upstream head from the crest in subsequent studies (Hadano et al. 2020). Liu et al. (2021) obtained the relationship between the modified inundating coefficient and the value of the upstream head and weir height by conducting tests on water-retaining weir. Through a series of physical model tests in different discharges, Tang et al. (2022) established the discharge coefficient calculation formula of contracted weir considering D/d (where D is the thickness of the retaining wall, d is the thickness of the weir) and h/P (where h is static head over the weir, P is weir height). Although there are many scholars have made achievements in the research of standard thin-plate weir flow, the theoretical research of NTPW is still insufficient. During heavy rain events, water levels typically rise, causing backflow in the municipal drainage system, resulting in overflow or surcharging of weirs, combined with the influence of irregular structures on the flow, leading to complex hydraulic characteristics. Monitoring the discharge of NTPW under these conditions poses a significant challenge in CSO pollution control practices.

Nanchang City, situated in the central northern region of Jiangxi Province, China, serves as the central hub of the Poyang Lake eco-economic region, encompassing a 393 km² urban built-up area. The topography of Nanchang City is characterized by predominantly flat terrain, complemented by rolling hills in the northwest. The city boasts a well-developed water system, featuring a river network density of 0.57 km². Positioned within the subtropical humid monsoon climate zone, Nanchang experiences warm and humid weather conditions. The city receives an annual average precipitation of 1,611.8 mm, with 143.9 days of rainfall per year and 5.7 days annually reporting precipitation exceeding 100 mm.

The original overflow device uses cement block stones with a roughness of 0.015–0.02 as the surface, while the transition section of the physical model is made of organic glass, and the downstream channel is built with bricks and covered with cement mortar. After polishing with sandpaper, the overall roughness of the model is reduced to 0.008–0.012, which meets the requirements of roughness similarity.

The experiment adopts a self-circulating water supply system, the water used in the experiment is delivered into the water tank from an underground water reservoir through a water supply pipe and then flows through the culvert, transition section, and ultimately returns to the reservoir through a drainage channel. In practice, the water level in the tank is adjusted by controlling the opening of valve I and valve II (valve II is connected to the drainage pipe, which can return the water to the underground water reservoir), and the water level in the river is adjusted by controlling the downstream baffles. In order to reduce the impact of turbulent flow inside the culvert on the flow through the weir, three flow stabilizing plates are installed at the junction of the tank and the culvert to buffer the flow. Additionally, two piezometric pipes are placed inside the culvert to measure the upstream water head and total head. Three piezometric pipes are installed in the river to measure the downstream water level. One piezometric pipe is placed upstream of the triangular weir to measure the water level and calculate discharge through the triangular weir.

Figure 5(b) shows a free flow condition in the physical model, where the upstream water level is 0.133 m and the downstream water level is 0.8 m.

As can be seen from Equation (5), m is related to and ⁠. In addition, according to the test conditions and monitored discharge of the physical model, the values of m, ⁠, and ⁠, are calculated by Equations (2), (6), and (7), respectively. In this way, the fitting method can be used to obtain the functional expressions of m, 1/Re, and 1/We.

Figure 6(b) shows a submerged flow condition in the physical model, where the upstream head is 0.13 m and the downstream water level is 0.12 m.

Therefore, by conducting numerous physical model tests under different working conditions, the values of and H can be obtained. Additionally, by combining with formula (9), the value of can be calculated. Finally, the fitting method can be used to obtain the functional relationship between ⁠, and H.

Figure 7(b) shows a pressurized flow condition in the physical model, where the upstream head is 0.17 m and the downstream water level is 0.165 m.

According to the working conditions of the physical model test and the corresponding flow monitored, the value of and can be obtained, and the can be calculated by formula (13). Finally, the fitting algorithm is used to obtain the functional expressions of ⁠, and ⁠.

In this study, we need to obtain the DCMM through the discharge calculation formulas under different flow conditions, but the function expression of the coefficient in the discharge calculation formula is difficult to obtain, so we introduce the fitting method to solve the expression of coefficient. There are many common function-fitting algorithms, such as GA, PSO, SA. GA is one of the evolutionary algorithms, which finds the optimal solution by imitating the mechanism of selection and heredity in nature, and it has been widely used in the field of fitting which is not limited by function form (Tang et al. 2007). Thus, we use GA to fit the relationship of three types of coefficients with related physical quantities.

Subsequently, a large number of functions are randomly generated as the parent generation using the aforementioned approach. They are then sorted according to their fitness, and a group of parents with higher fitness is selected through the method of roulette wheel selection. The selected parents then undergo genetic operations, including crossover and mutation, based on a predefined ratio (as shown in Figure 9(b)–9(d)). This process results in the creation of the initial child generation. These child functions subsequently become the new parent functions, and the entire process is repeated. Ultimately, after a finite number of iterations, the child function with the highest fitness, represents the best-fitted function (Ren et al. 2012).

In order to verify the accuracy of the DCMM, two typical working conditions are selected under three flow conditions (free flow, submerged flow, and pressurized flow), and the difference between the simulated flow of DCMM and the measured discharge of the physical model is compared. The MRE, RMSE, and NSE are selected to evaluate the accuracy of the DCMM. The calculation formula is the same as above.

Under free flow conditions, the influence of viscosity and gravity on liquid flow is greater than surface tension. Therefore, the Reynolds number and Froude number have a greater impact on the discharge coefficient than the Weber number (Negm et al. 2002). Table 1 shows the contribution of the Reynolds number term and the Weber number term in Equation (10) to the discharge coefficient under various free flow working conditions, confirming that the Reynolds number has a greater impact on the flow coefficient compared to the Weber number.

The free flow condition is a normal designed working state of overflow weir. The discharge coefficient of the standard thin-plate weir can be calculated using the empirical coefficient formula, Raybock discharge coefficient formula and Sarginson discharge coefficient formula (Peng et al. 2004) (Table 2).

Due to the combined impact of the shortage of urban land resources and engineering costs, irregular large diffusion angles often appear in overflow devices, resulting in poor flow dynamics and affecting the overflow efficiency of NTPWs. The DCMM of NTPW in this study is an application of data-driven and scenario-based modeling in urban drainage engineering. The possible range of the discharge coefficient under free flow conditions can be calculated to be between 0.252 and 0.362. It is difficult for NTPWs to reach the discharge capacity of standard thin-plate weirs, with their discharge efficiency in free flow conditions being 16–42% lower than that of standard thin-plate weirs.

During rainstorm events, the water levels of urban rivers and lakes generally rise, causing surcharging in the municipal drainage network. According to observations by the urban drainage management department, during heavy rainstorms (100–200 mm/day), there is a higher frequency of submerged flow in the overflow devices, with a water level difference of 0.20–0.40 cm between the upstream and downstream. During extreme rainstorms (≥200 mm/day), the culvert becomes completely filled with water, resulting in pressurized flow inside the culvert, which can cause significant surcharging in the upstream municipal drainage network. At this point, there is a significant difference in water level between the upstream and downstream of the overflow device, which increases with the intensity of rainfall, even exceeding 0.5 m. The overflow capacity of the overflow device under the above-mentioned conditions is shown in Table 3.

By observing the upstream and downstream water level of the overflow device during urban rainstorms, the DCMM can be used to evaluate its discharge capacity under different rainstorm events, which provides a mathematical basis for the review and improvement of the overflow device.

Combined sewer systems and separate sewer systems are two kinds of drainage systems which coexist in urban drainage, which are all reasonable drainage systems and have their own advantages, and combined sewer systems will exist for a long time. Therefore, accurate monitoring and control of CSO is urgently needed in these cities. According to statistics, there are 101,100 km of confluence pipelines in China, accounting for 12.59% of the total drainage pipelines in 2020. Although the proportion of confluence pipe networks is relatively low, there are still problems with the co-existing of combined municipal effluent systems and separate sewer systems in urban drainage systems, and the exist of incomplete intercepted drainage systems. In the past decades, overflow pollution has been very serious, and has been criticized by environmental inspectors.

The DCMM of NTPW constructed in this study can realize reliable and convenient discharge monitoring with the help of the water level-discharge relationship in the overflow device, thus providing an effective measure to obtain the basic monitoring data of CSO, and providing technical support for local CSO pollution control.

• (1)

  The DCMM for NTPW effectively addresses the challenge of evaluating CSO pollution, especially in monitoring overflow from NTPWs. The verification results demonstrate the relatively accurate construction of the DCMM in this study.

• (2)

  The DCMM modeling method applies data-driven and scenario-based modeling in urban drainage engineering. It can assess discharge capacity during different rainstorm events and provide reliable and convenient discharge monitoring, offering technical support for local CSO pollution control.

• (3)

  While the DCMM constructed in this particular case may not be universally applicable and its accuracy may be limited by the conditions of building the physical model, the method of combining physical models with GA to evaluate NTPW discharge capacity proves to be an effective approach under complex hydraulic conditions. This approach also has value for implementation in other cities.

M.T. investigated, organized the framework of the article, assited in writing – review and editing, and performed data analysis, and discussed the results. Y.W. assisted in original draft preparation, collected the experimental data, designed the algorithm, data analysis, and prepared figures and tables. Q.X. collected the experimental data, data analysis, and prepared figures. H.C. and W.X. organized experiments and provided ideas for improvement. All authors reviewed the manuscript.

This study was funded by the research project of Nanchang Urban Planning & Design Institute of China (NO. ky2021-06), the Joint Major Project of ‘Science and Technology + Water Conservancy’ in Jiangxi Province (2022KSG01007).

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

The authors declare there is no conflict.