Drainage modeling that accurately captures urban storm inundation serves as the foundation for flood warning and drainage scheduling. In this paper, we proposed a novel coupling ideology that, by integrating 2D-1D and 1D-2D unidirectional processes, overcomes the drawback of the conventional unidirectional coupling approach that fails to properly represent the rainfall surface catchment dynamics, and provides more coherent hydrological implications compared to the bidirectional coupling concept. This paper first referred to a laboratory experimental case from the literature, applied and analyzed the coupling scheme proposed in this paper and the bidirectional coupling scheme that has been widely studied in recent years, compared the two coupling solutions in terms of the resulting accuracy and applicability, and discussed their respective strengths and weaknesses to validate the reliability of the proposed method. The verified proposed coupling scheme was then applied to the modeling of a real drainage system in a region of Nanjing, China, and the results proved that the coupling mechanism proposed in this study is of practical application value.

  • A novel coupling approach by combining two unidirectional processes.

  • Comparison with bidirectional tight coupling scheme in terms of simulation accuracy and applicability.

  • The proposed coupling idea provides reliable outcomes with enhanced applicability.

The global climate change and urbanization process have contributed to an increased frequency of flooding, which has become the most frequent weather-related hazard in the last two decades, affecting 2.3 billion people worldwide, more than all other weather-related hazards combined (Wallemacq et al. 2015). At the same time, flooding is one of the world's most destructive natural disasters, not only endangering lives and property but also having a serious negative impact on society and the environment (Han & He 2021; Dong et al. 2022). It is essential to develop an efficient and trustworthy urban drainage model appropriate for various rainfall scenarios and diverse geographical regions to comprehend the development progress of urban flooding, which is the foundation for mitigating flood damage.

The sewer network flow and the overland surface flow constitute two distinct subsystems that jointly make up the urban drainage system. The simulation of overland surface flow is typically carried out with either simplified conceptual model-based hydrological methods or numerical model-based hydrodynamic methods (including one- and two-dimensional hydraulic models). Conceptual model-based hydrological and 1D hydrodynamic approaches for solving the shallow water equations (SWEs), while computationally efficient, are not as accurate as physically based 2D hydrodynamic methods (Djordjević et al. 1999; Teng et al. 2017; Sañudo et al. 2020). Modeling sewer flow is usually accomplished by solving the 1D Saint-Venant equations to examine the ability of a sewer system to cope with intense rainfall (Djordjević et al. 1999; Rossman & Simon 2022). The 1D model simulates the water hydrodynamics in the sewer network with high precision and rapid calculation, but has evident flaws when simulating urban structures (such as buildings) with obvious 2D and 3D characteristics. The 2D model deals primarily with surface water movement, but it is unable to simulate the water flow dynamics in the drainage network as effectively as the 1D model (Dong et al. 2021). Therefore, coupled 1D sewer flow and 2D overland flow models for urban inundation modeling have been extensively investigated over the last few years (Leandro & Martins 2016; Noh et al. 2016; Dong et al. 2022).

The first 1D/2D coupling was developed by Hsu et al. (2000) who used Stormwater Management Model (SWMM) to analyze the flow dynamics in a drainage network and treated its overflow as inputs to a 2D overland flow model. The process of returning overflow to the network is not accounted for with this coupling technique, and therefore, the extent and depth of flooding in the downstream area are often overestimated. In an attempt to overcome this limitation, some academic scholars incorporated the consideration of the reflux process to define the 1D/2D bidirectional interactions (Hsu et al. 2002; Seyoum et al. 2012). These studies can be further categorized into two classes according to whether or not the surface convergence process is represented in 2D. The first category applies rainfall to the 1D hydrologic model, calculating sewer flow first and then interfacing the overload with the 2D surface flow model. The second category directly adds rainfall to the 2D model, where both surface catchment and network overflow are computed from the 2D model. Flood estimation may be erroneous if the rainfall intensity exceeds the intended capacity of the sewer inlets because the hydrologic model's catchment process assumes that runoff from sub-catchments is fully collected and discharged into the sewer network. In contrast, it is a fact that in urban areas, rainfall turns into runoff immediately after it drops to the ground surface and travels across the landscape for a certain distance before it reaches a drainage inlet or manhole. A simplified hydrologic model is unable to adequately clarify this phenomenon. The findings of Chang et al. (2015) further demonstrated that the 1D/2D coupling should apply rainfall directly to the 2D model to describe the precipitation catchment process rather than employing hydrologic routing.

The previously described bidirectional coupling strategy expresses two-way interactions in terms of water exchanges between the sewer system and the surface water system, where the interaction flow is commonly a simplified calculation based on the weir or orifice equations applied to the point-source head gradient. The coefficients within these manhole flow exchange equations bear stochastic properties that inject a degree of uncertainty and potential error into the estimated flow rates (Zhang et al. 2023). And achieving coherence in simulation results necessitates the use of exceedingly small time step, invariably sacrificing the temporal smoothness of the model. Furthermore, the development of the closely coupled model involves a heightened research threshold that not only demands a profound comprehension of the 1D and 2D system dynamics but also requires substantial programming expertise. Although there exists some commercial software with strong generalization and stable performance, such as InfoWorks ICM, MIKE, and TUFLOW (Syme 1992; Innovyze 2019; DHI 2020), they are not open-source and costly to utilize. The current unidirectional coupling methods, which are relatively convenient to implement, tend to only consider the transformation of the overflow from the 1D pipe network to the 2D model, making the results comparatively imprecise. By combining two unidirectional processes – feeding the 2D confluence into the 1D sewer network, and vice versa, channeling the overflow from the 1D network back into the 2D model – we can adopt a straightforward and manageable approach that addresses the conventional unidirectional process's inability to realistically depict the rainfall-runoff process. However, to our knowledge, no studies have yet attempted to compare the effectiveness of this integrated unidirectional method with that of the bidirectional coupling strategies.

To address the above issues, this paper investigated the coupling technique to merge the outcomes of two unidirectional processes and contrasted it with the bidirectional coupling method in terms of the result correctness, calculated stability, and implementation requirements. The 2D model was applied to mimic the rainfall catchment and surface diffusion process, whereas the 1D model solely analyzes the internal flow of the sewer network. The 2D model adopted Telemac-2D and the 1D model employed SWMM (Moulinec et al. 2011; McDonnell et al. 2021), both of which are open source and hence free and accessible. The details of the two coupling approaches are covered in the section that follows.

The paper is organized as follows: first, a detailed procedural description is presented for both the new coupling scheme proposed in this study and the existing bidirectional coupling scheme; then, two case discussions are introduced to evaluate the effectiveness of the proposed methodology, where the first one cites a large-scale laboratory experiment from the literature to validate the correctness and reasonableness of the proposed model (Naves et al. 2020), and the second one applies the method to simulate a real drainage system in a specific area of Nanjing, China, to validate the applicability of the methodology to large-scale drainage systems; and finally, the major conclusions are highlighted.

Hydraulic models

2D Surface model

Numerically solving the depth-averaged shallow-water equations is the representative hydrodynamic method for 2D surface modeling, and the equations governing mass and momentum conservation take the following form:
formula
(1)
formula
(2)
formula
(3)
where h is the water depth, u and v are velocity components of the velocity vector in the x and y directions, respectively, Z is the free surface elevation, , , and are the source terms representing the wind, Coriolis force, bottom friction, and a source or a sink of momentum within the domain, and is the momentum diffusion coefficient.

Among several numerical solvers, Telemac-2D is preferred in this paper since it simulates the moving boundaries of flows on a horizontal plane by the finite element method, which offers the strength of faster speed, lower memory requirements, better robustness to absorb the drastic variation in the transition between wet and dry regions, and especially the availability of the open source (Heniche et al. 2000). Hierarchical mesh resolution is possible with Telemac-2D finite element grids (Hervouet 2007). The average vertical water depth and velocity at each point in the resolution grid are obtained by solving the SWEs on triangular or quadrilateral elements.

1D Sewer network model

SWMM has been the most commonly utilized 1D dynamic sewer network engine based on the Saint-Venant equations. A sewer network is idealized as a set of links that connect nodes and transmit flow between them. The dynamic wave routine applies the continuity and momentum equations at the pipes, while the nodes are modeled as storage components in the system and only the continuity equation is considered. The momentum equation is combined with the continuity equation to solve for the discharge rate and the water depth along each link at each time step (Rossman 2010):
formula
(4)
formula
(5)
where A is the conduit cross-sectional area, t is time, Q is the discharge in the pipe, g is the acceleration of gravity, H is the water head in the pipe, and is the friction slope, which can be expressed in terms of the Manning equation used to model steady uniform flow:
formula
(6)
where n is the Manning roughness coefficient, and R is the hydraulic radius of the flow cross-section.
While Equations (4) and (5) can calculate the flow time trajectory in pipes, another relation is required to accomplish a similar task for heads. SWMM's node-link representation of the conveyance network (Figure 1) does this by providing a continuity relationship at junction nodes that connect conduits together within a conveyance network. It is assumed that there is water surface continuity at the node and the pipes entering and leaving it. Each ‘node assembly’ contains the node itself and half the length of connecting pipe segments. Conservation of flow in the set demands that the volume variation with time be equal to the difference between the incoming and departing flows, from which the continuity equation for the node can be obtained:
formula
(7)
where H is the node water depth, t is time, is the net flow rate of the node, is the storage surface area of the node (if it is a storage node), and is the surface area contributed by the links connected to the node.
Figure 1

Node-link representation of a conveyance network in SWMM.

Figure 1

Node-link representation of a conveyance network in SWMM.

Close modal

The equations are discretized in finite difference form with spatial dispersion equal to the conduit length and are solved implicitly and iteratively using Picard's method over a given time step.

Linking methodology

The majority of surface water that contributes to urban flooding is collected in the subterranean sewer system through manholes and artificial outlets (Dong et al. 2022). When the precipitation intensity surpasses the drainage capacity of the network, it causes rainwater overflow and the emergence of surface flooding.

Combination of two unidirectional couplings (CTUC)

The two unidirectional coupling procedures consist of inflow from rainfall runoff to the nodes of the sewer network (2D-1D) and overflow from the nodes to the surface diffusive flow (1D-2D), as shown in Figure 2. The input for the 2D-1D coupling process is the rainfall time series, and the output is the discharge time series at each boundary, which is incorporated into the 1D sewer network model as the external inflow rate for each node. Manholes are defined as minor square open boundaries in the 2D model. The location of each boundary corresponds to the node position in the 1D network model, i.e., the node coordinates as the center of the squares, and the side length of the squares is specified manually. For the stability of free effluent flow calculations, TELEMAC recommends that the open boundaries should contain at least five vertices, which in this paper are all achieved to be not less than nine vertices. Moreover, considering the restricted capacity of the inlets to accommodate rainwater in actual situations, an artificial constraint was imposed on the node inflow as follows:
formula
(8)
where is the node external inflow, is the corresponding boundary outflow calculated by the 2D model, is the artificially defined maximum limit of inflow, relevant to the outlet type and dimension. If exceeds , the overage is considered to be flux remaining on the ground surface.
Figure 2

Schematic diagram of the combination of two unidirectional coupled processes.

Figure 2

Schematic diagram of the combination of two unidirectional coupled processes.

Close modal
The ultimate acquired is fed as input to the 1D-2D coupled process, i.e., the flow time series is set up in the form of a text file as a direct inflow to the SWMM nodes. The nodal overflow in the network is calculated by the 1D SWMM dynamic wave module, which incorporates the legacy quantities of the 2D-1D process to generate the point source inflow for the 2D model as follows:
formula
(9)
where is the point source flow for the 2D model, and is the nodal overflow from the 1D model. In the 1D-2D process, the point source locations are consistent with the nodes of the network, and square open boundaries no longer need to be defined. A negative point source flow in the 2D model indicates outflow from the system, and a positive flow indicates entry into the system, in the event of an overflow from the network, the point source flow should be positive in all cases.

Eventually, the 2D model results of the two processes can be aggregated with the water depths at identical spatial locations within the same time step to yield a comprehensive simulation of the entire rainfall-runoff overflow flooding procedure. The fact is that if variable time steps are adopted, it is impossible for two different 2D models to be synchronized in time, and the grid vertex locality cannot correspond accurately. Therefore, in this study, the 2D model results of the 2D-1D process are taken as the base results, with the 2D model results of the 1D-2D process as the additional results. The bathymetry values of the additional results are linearly interpolated on the time scale to correspond to the time steps of the base results, and on the spatial scale the KD-Tree algorithm (Bentley 1975) is employed to search for additional result nodes adjacent to the base result nodes at a certain distance (the mesh default size in this paper), and their average water depths are computed to be appended to the base result nodes' water depths. The treatment can be parallelized by partitioning the time series to improve computational efficiency.

The two straightforward and simple unidirectional coupling processes that make up the CTUC method's implementation eliminate the need to develop a customized 2D computational engine and coupling module and enable the direct adoption of pre-existing mature software (whether open source or not, e.g., LISFLOOD and HEC-RAS (Shustikova et al. 2019)) that focuses primarily on 2D simulation capabilities, offering a broader spectrum of application.

Bidirectional coupling (BC)

A precise characterization of the flow interaction between surface and sewer systems is important in the bidirectional coupling method. It synchronizes the simulation process of 1D and 2D models, and the water level difference between the node of the sewer network and the corresponding land surface is assessed at each time step to determine whether it is a nodal overflow or an inflow. Nodal inflow can be subdivided into free and submerged inflow according to the relation between the nodal water level and the adjacent surface elevation. Therefore, bidirectional coupling can be divided into the following three scenarios (Chen et al. 2007).

Free weir linkage
When the water level of the network node is below the corresponding elevation at the surface (i.e., ), as shown in Figure 3(a), the surface water flows into the pipe network, and the free weir equation (Equation (10)) is adopted:
formula
(10)
Figure 3

Three scenarios of flow exchange through manholes: (a) free weir linkage; (b) submerged weir linkage; and (c) orifice linkage.

Figure 3

Three scenarios of flow exchange through manholes: (a) free weir linkage; (b) submerged weir linkage; and (c) orifice linkage.

Close modal
Submerged weir linkage
When the nodal water level of the network is higher than the surface elevation but lower than the surface water level (i.e., ), as shown in Figure 3(b), the ponded water on the ground still enters into the sewer network, and the submerged weir formula (Equation (11)) is employed to calculate the influent flow here:
formula
(11)
Orifice linkage
In the case where the water level in the network exceeds the surface water level (i.e., ), as shown in Figure 3(c), the water in the network diffuses to the surface, and the orifice equation (Equation (12)) is introduced:
formula
(12)

In the above equations, is the interacting discharge, is the weir discharge coefficient, is the orifice discharge coefficient, is the weir crest width, which is the manhole perimeter in this model, is the manhole area, g is the gravitational acceleration, is the hydraulic head at the manhole, is the surface elevation, is surface water level, and is the surface water depth, thus, .

If stormwater drains from the surface into the sewer, the calculated flow exchange should be configured as a positive nodal inflow to the 1D model and a negative point source flux of the 2D model, and vice versa, for surcharge flow from the sewer toward the overland, the interchange should be set as a positive point source flow to the 2D model without a 1D nodal inflow into the network. Furthermore, a specific amount of water exchange restriction is required to ensure the model's stability, which should not exceed the cell's total water volume, i.e., Equation (13):
formula
(13)
where is the coefficient relating rainfall to topography, artificially specified according to the applied conditions, is the time step, is the element area, and is the element water depth.

A full-scale dual-drainage laboratory experimental case from the literature (Naves et al. 2020), which emulated the hydraulic and sediment transport process using a realistic rainfall simulator, was adopted in this work and the differences between the two coupling schemes concerning accuracy and implementation were analyzed and compared as the first step in verifying the dependability of the CTUC method. Then, the CTUC method was utilized to simulate the performance of a real large-scale drainage system in NJ, China during a typical local rainfall circumstance.

Model validation using a laboratory experiment

Case study description

The experiments were conducted in the Hydraulic Laboratory at the University of A Coruña (Spain), and equipped with a rainfall simulator located 2.6 m over a 36 m2 full-scale concrete street section with a sewer system (Figure 4). The street surface consists of a sidewalk with a 15-cm elevation gap and a concrete roadway. The roadway has an approximate transversal slope of 2% up to the sidewalk and a 0.5% longitudinal slope up to an outflow channel. The generated rainfall runoff is discharged into the pipe system through two gully pots (0.3 m × 0.3 m) located along the curb and a lateral outflow channel. The flow is then conveyed to a common outlet of the pipe system. The detailed surface elevation data and conduit parameter data can be obtained in WASHTREET (Naves et al. 2019b).
Figure 4

Schematic of the laboratory physical model: (a) physical facility and (b) street, drainage network, and measuring points for results validation.

Figure 4

Schematic of the laboratory physical model: (a) physical facility and (b) street, drainage network, and measuring points for results validation.

Close modal

Rainfall simulators are capable of producing rainfall with intensities of 30, 50, and 80 mm/h. The simulation employed measured rainfall intensity maps to approximate the spatial distribution of actual rainfall. To compare the results of the CTUC and BC methods with the experimental data, the discharges from the two gully pots were selected as flow indicators, the measuring point S1 as surface water depth indicator, and the measuring points S2 and S3 as inside-pipe water depth indicators (Figure 4) for reference.

Discretizing the surface domain with an unstructured triangular mesh of default size 0.05 m, the CTUC method generated a total of 29,508 elements, while the BC method created more elements irrespective of the boundary geometry, with a total of 30,186. The two gully pots and the outlet channel are set up as open boundaries for free outflow. Following previous studies conducted in this laboratory facility (Naves et al. 2019a), the Manning roughness coefficients for the surface and pipes were set at 0.016 and 0.008 s.m−1/3, respectively. For the flow exchange formula of the BC method, the weir and orifice coefficients for the inlets were set at 0.52 and 0.62 after model testing, respectively.

Results and discussions

Figure 5 compares the results of the laboratory and the numerical models applying the two coupling schemes. The first noticeable overall trend that can be observed is that the Nash–Sutcliffe efficiency (NSE) (Nash & Sutcliffe 1970) coefficients of the numerical models become better as the rainfall intensity increases. This is because the effect on the results caused by measurement uncertainty, numerical precision, and disturbance of 3D features is more pronounced under a slight rainfall impingement. Both GP1 and GP2 inlet discharges acquired satisfactory simulation results with NSE greater than 0.5, especially GP2 results with NSE around 0.9 for both. In contrast, GP1 shows a higher error due to the early raising of the runoff process with a larger peak flow value. A comparison of the results of the CTUC and the BC approach on calculated inlet flow rate reveals that the NSE values of the two techniques are in close proximity to each other, with the BC method having a slight advantage on GP2. We also compared the discharge values at steady state (Table 1) and identified that the CTUC method has a slight advancement on GP1. For surface water depth at point S1, since numerical modeling measures based on SWEs cannot accurately characterize the bathymetric oscillation caused by practical turbulence fluctuations, the values will be significantly smoother. With that premise, it can be noted that the CTUC method performs better than the BC solution in both 50 and 80 mm/h rainfall scenarios. As for the water depth measuring points of S2 and S3 in the pipes, the results are affected by the cumulative error of the 2D confluence process, yet the NSE results for both 50 and 80 mm/h events are acceptable, which proves that the current popular numerical stormwater inundation models may be inaccurate for subscale simulations under light rainfall conditions. The CTUC approach exhibits slightly worse NSE values than the BC approach in terms of pipe water depth (maximum discrepancy less than 0.05). Given the concern that urban stormwater inundation simulation is not focused on transient rheology, and the rainfall scenarios and dimensions studied tend to be sizable in scale; it can be assumed that the coupling idea proposed in this paper is dependable in terms of simulation accuracy.
Table 1

Comparison of CTUC and BC methods for the steady water discharges drained by the two gully pots

ModelsGP1
GP2
30 mm/h50 mm/h80 mm/h30 mm/h50 mm/h80 mm/h
Experiment 0.0472 0.0888 0.1414 0.1447 0.244 0.3402 
CTUC Value 0.05955 0.1027 0.1535 0.1517 0.2424 0.3465 
Error (%) 25.9534 15.6532 8.5573 4.8376 0.6557 1.8519 
BC Value 0.06052 0.1053 0.1584 0.1462 0.2429 0.3426 
Error (%) 28.2203 18.5811 12.0226 1.0366 0.4508 0.7055 
ModelsGP1
GP2
30 mm/h50 mm/h80 mm/h30 mm/h50 mm/h80 mm/h
Experiment 0.0472 0.0888 0.1414 0.1447 0.244 0.3402 
CTUC Value 0.05955 0.1027 0.1535 0.1517 0.2424 0.3465 
Error (%) 25.9534 15.6532 8.5573 4.8376 0.6557 1.8519 
BC Value 0.06052 0.1053 0.1584 0.1462 0.2429 0.3426 
Error (%) 28.2203 18.5811 12.0226 1.0366 0.4508 0.7055 
Figure 5

Comparisons between the simulated and experimental data of the CTUC and BC methods: (a) GP1; (b) GP2; (c) S1; (d) S2; (e) S3, and (f) the NSE values of all simulation scenarios.

Figure 5

Comparisons between the simulated and experimental data of the CTUC and BC methods: (a) GP1; (b) GP2; (c) S1; (d) S2; (e) S3, and (f) the NSE values of all simulation scenarios.

Close modal

Since the BC idea intervenes in the variable states of the two subsystems at each time step, it is critical to restrict the magnitude of the interactive fluxes to ensure the stability of their results. In this paper, the tuning of this limitation is manifested mostly in the coefficient α of Equation (13), where α is too large for the results to oscillate significantly, and α is too small to describe the exact amount of the interaction flow. The determination of α value is associated with the overall runoff quantity size and time step size, which is an arithmetic and time-consuming task. In addition, the BC concept cannot be implemented in a mature calculating engine without open source, which necessitates the developers to possess a profound mastery of both hydraulic mechanisms and computer programming. The CTUC approach, on the other hand, integrates two continuous hydrological processes and eliminates the necessity for additional stabilization safeguards. Moreover, the CTUC approach can be realized by utilizing mature commercial software for 2D and 1D computing that is widely available in the market, which allows for wider applicability.

Application of a real urban drainage system

Case study description

The above-validated model was applied in a coarsened real urban drainage system, which is situated in an urban center area of NJ, China, encompassing an area of approximately 18.34 km2, and its topography and network layout are shown in Figure 6. As there was insufficient field data to calibrate the model parameters, the Manning roughness coefficients for the ground surface and the network pipes were consistently specified as 0.0125 and 0.0128 s.m−1/3, respectively. The system is a stormwater network that does not receive domestic sewage and consists of 358 manholes, 357 pipes, and 76 outfalls. The pipe diameters are generally large, ranging from 500 to 2,000 mm, as the network has been simplified.
Figure 6

Topography and drainage network layout of the study area.

Figure 6

Topography and drainage network layout of the study area.

Close modal

The unstructured triangular mesh with a default resolution of 6 m was generated, and the CTUC method sets each manhole as a square open boundary. A finer grid resolution of 1 m was implemented at the boundaries, and the square sides were 2 m long. The total number of grids for the area was 1,192,979. A precipitation hydrograph with a duration of 2 h is shown in Figure 6. The rainfall distribution is spatially homogeneous. The overall simulation duration was 4 h, and the time step was set to 5 s, satisfying the Courant–Friedrichs–Lewy (CFL) stability constraint (Courant et al. 1967).

Results and discussions

We first explored the influence of different values in the CTUC method on the results (Table 2). Because of the relatively large size of the case area and the simplified drainage network model, which may not accurately reflect the positioning of the actual drainage inlets, the surface water volume rather than water depth was considered as the indicator to capture the overall inundation degree. It is clear from Table 2 that the lower the , the more pronounced restriction on the rainwater entering the network occurs, and more stagnant water remains in the surface domain, but due to the finite capacity of the network itself, the water quantity available to enter the network does not boost endlessly by increasing the . Further, the volumetric differences between the various settings were not evidently reflected over the full bathymetry map due to the large study area. Consequently, we disregarded the threshold limitation (i.e., ) for the following discussions.

Table 2

Water volume and depth statistics for the CTUC model using different

Qmax (m3/s)Final volume (m3)Rainfall (m3)Entering the network (m3)Outflow (m3)Relative errorAverage water depth at peak ponding time (m)
0.5 658475.40 1225310.89 332673.74 234225.07 9.62 × 10−5 0.05033 
605790.80 1225310.89 385598.65 233984.84 1.05 × 10−4 0.04915 
584575.60 1225310.89 406911.65 233887.05 1.08 × 10−4 0.04862 
571293.10 1225310.89 420194.15 233887.05 1.10 × 10−4 0.04851 
∞ 571236.00 1225310.89 420194.15 233944.21 1.11 × 10−4 0.04850 
Qmax (m3/s)Final volume (m3)Rainfall (m3)Entering the network (m3)Outflow (m3)Relative errorAverage water depth at peak ponding time (m)
0.5 658475.40 1225310.89 332673.74 234225.07 9.62 × 10−5 0.05033 
605790.80 1225310.89 385598.65 233984.84 1.05 × 10−4 0.04915 
584575.60 1225310.89 406911.65 233887.05 1.08 × 10−4 0.04862 
571293.10 1225310.89 420194.15 233887.05 1.10 × 10−4 0.04851 
∞ 571236.00 1225310.89 420194.15 233944.21 1.11 × 10−4 0.04850 

Figure 7 displays the spatial and temporal variations of the flood inundation process and Figure 8 shows the variation of water volume hydrograph in the study domain. Widespread apparent pooling of water (exceeds 0.1 m) commenced after t = 2,400 s. The model reached the maximum water accumulation at t = 5,100 s after the simulation started, producing the largest water volume of 874,157.61 m3, thereafter the inundation range progressively decreased in the presence of the sewer network. The surface ponding volume reduced and the reduction rate got slower with time, so did the increasing rate of volume entering the network, indicating that the sewer system gradually reached the maximum drainage capacity. After t = 12,000 s, the surface runoff around the street entrances was almost drained, so the sewer system contributed little to the water depth distribution over this period, with only partial waterlogging in the low-lying areas.
Figure 7

Spatial and temporal variations of the flood inundation process.

Figure 7

Spatial and temporal variations of the flood inundation process.

Close modal
Figure 8

Temporal variations in accumulated volumes of rain, surface, entering the network, and outflow.

Figure 8

Temporal variations in accumulated volumes of rain, surface, entering the network, and outflow.

Close modal
Figure 9 illustrates the spatial distribution of water depths at the peak flooding moment. Ponding is primarily distributed along city roadways. Part of the streets in the northwestern region are located in quite shallow terrain, but the number of manholes is comparatively small, resulting in the deepest flooding in a limited area of about 0.9 m. There is less standing water near the central arterial road (less than 0.1 m), which is in line with the design guidelines. However, the impact of buildings on the drainage system is challenging to characterize due to the lack of survey information on the buildings, and as a result, low-lying areas around roads are prone to sheet flooding.
Figure 9

Spatial distribution of flooding water depth at the highest water volume.

Figure 9

Spatial distribution of flooding water depth at the highest water volume.

Close modal
Figure 10 demonstrates the temporal variation of water depth at the four typical checkpoints in Figure 9. Point P1 is located near a dense area of manholes, which represents an area where surface stormwater can be quickly conveyed to the underground network system. Therefore, the ponding depth at point P1 is kept low throughout the simulation. P2 is located in a low-lying area far away from the manholes, where the elevation gap between the roadway and the plane in which it is located is extremely large, resulting in a continuous increase and ultimately stabilization of water depths during the simulation period. Regardless of the particular functionality of this area, this result should prompt the implementation of appropriate retrofitting measures. Points P3 and P4 are also distant from the manhole, but with the topographic variation, the ponded water will gradually flow into the network system as the rainfall intensity diminishes, and thus, the depth of standing water eventually declines.
Figure 10

Temporal distribution of water depth at checkpoints P1–P4.

Figure 10

Temporal distribution of water depth at checkpoints P1–P4.

Close modal

Among the 1D/2D coupled models that are extensively employed to simulate urban stormwater inundation developments, one generally acknowledged viewpoint is that the unidirectional coupled model from 1D network overflow to 2D surface diffusion cannot accurately characterize the rainfall surface confluence process, and hence is less dependable than the bidirectional coupled model. However, this paper proposed a new coupling scheme CTUC that combines the two unidirectional coupling processes of 2D surface confluence to 1D network and network overflow to surface flooding, and compared it with the contemporary bidirectional coupling idea BC in terms of the result accuracy and implementation requirements. This study adopted the 1D engine SWMM and the 2D engine TELEMAC, both of which are proven software that are open source and stable in operation and maintenance.

Through a laboratory experiment case designed to mimic urban street flooding, we first verified the reliability of the proposed coupling approach. The simulation results obtained by the CTUC and BC methods were compared with the experimental findings, separately, indicating that the CTUC and BC methods produced comparable results in terms of accuracy, both of which can satisfy the engineering requirements. The CTUC solution outperformed the BC marginally in the surface water depth, while it was slightly inferior to the BC strategy in the pipe water depth. The CTUC concept has a better calculation stability and applicability range than the BC method. Afterward, we applied the CTUC method to establish an actual urban drainage system model in a region of Nanjing, China, to synchronize the spatial and temporal variation of waterlogging under a 3-year rainfall condition, and the results were consistent with the physical topography and engineering characteristics of the urban area, which provided an efficient theoretical basis for the urban flood control design and renovation.

To summarize, the CTUC method proposed in this paper can generate relatively satisfactory and accurate simulation results for large-scale drainage systems, with improved computational stability and lower implementation threshold.

This work was financially supported by the National Natural Science Foundation of China (NSFC) [Grant 51978493] and [Grant 51778452].

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

The authors declare there is no conflict.

Bentley
J. L.
1975
Multidimensional binary search trees used for associative searching
.
Communications of the ACM
18
(
9
),
509
517
.
Chen
A. S.
,
Djordjević
S.
,
Leandro
J.
&
Savić
D. A.
2007
The urban inundation model with bidirectional flow interaction between 2D overland surface and 1D sewer networks
.
Courant
R.
,
Friedrichs
K.
&
Lewy
H.
1967
On the partial difference equations of mathematical physics
.
IBM Journal of Research and Development
11
(
2
),
215
234
.
DHI
2020
MIKE URBAN – 1D-2D Modelling – User Manual
.
Djordjević
S.
,
Prodanović
D.
&
Maksimović
Č.
1999
An approach to simulation of dual drainage
.
Water Science and Technology
39
(
9
),
95
103
.
Dong
B.
,
Xia
J.
,
Zhou
M.
,
Deng
S.
,
Ahmadian
R.
&
Falconer
R. A.
2021
Experimental and numerical model studies on flash flood inundation processes over a typical urban street
.
Advances in Water Resources
147
,
103824
.
Dong
B.
,
Xia
J.
,
Zhou
M.
,
Li
Q.
,
Ahmadian
R.
&
Falconer
R. A.
2022
Integrated modeling of 2D urban surface and 1D sewer hydrodynamic processes and flood risk assessment of people and vehicles
.
Science of The Total Environment
827
,
154098
.
Heniche
M.
,
Secretan
Y.
,
Boudreau
P.
&
Leclerc
M.
2000
A two-dimensional finite element drying-wetting shallow water model for rivers and estuaries
.
Advances in Water Resources
23
(
4
),
359
372
.
Hervouet
J.-M.
2007
Resolution of the Navier-Stokes Equations
. In:
Hydrodynamics of Free Surface Flows
, pp.
133
176
.
https://doi.org/10.1002/9780470319628.ch5 (accessed 17 August 2023)
.
Hsu
M. H.
,
Chen
S. H.
&
Chang
T. J.
2000
Inundation simulation for urban drainage basin with storm sewer system
.
Journal of Hydrology
234
(
1
),
21
37
.
Hsu
M.
,
Chen
S.
&
Chang
T.
2002
Dynamic inundation simulation of storm water interaction between sewer system and overland flows
.
Journal of the Chinese Institute of Engineers
25
(
2
),
171
177
.
Innovyze
2019
InfoWorks ICM Help v9.5
.
McDonnell
B.
,
Wu
J. X.
,
Ratliff
K.
,
Mullapudi
A.
&
Tryby
M.
2021
Open Water Analytics Stormwater Management Model
.
Available from: https://zenodo.org/record/5484299 (accessed 1 September 2023)
.
Moulinec
C.
,
Denis
C.
,
Pham
C.-T.
,
Rougé
D.
,
Hervouet
J.-M.
,
Razafindrakoto
E.
,
Barber
R. W.
,
Emerson
D. R.
&
Gu
X.-J.
2011
TELEMAC: An efficient hydrodynamics suite for massively parallel architectures
.
Computers & Fluids
51
(
1
),
30
34
.
Naves
J.
,
Anta
J.
,
Suárez
J.
&
Puertas
J.
2019b
WASHTREET – Hydraulic, wash-off and sediment transport experimental data obtained in an urban drainage physical model. Available from: https://zenodo.org/record/3754092/export/hx (accessed 1 October 2023)
.
Rossman
L. A.
&
Simon
M. A.
2022
Storm Water Management Model User's Manual Version 5.2
.
U.S. Environmental Protection Agency
,
Washington, DC, USA
.
EPA, 600/R–22(030)
.
Seyoum
S. D.
,
Vojinovic
Z.
,
Price
R. K.
&
Weesakul
S.
2012
Coupled 1D and noninertia 2D flood inundation model for simulation of urban flooding
.
Journal of Hydraulic Engineering
138
(
1
),
23
34
.
Shustikova
I.
,
Domeneghetti
A.
,
Neal
J. C.
,
Bates
P.
&
Castellarin
A.
2019
Comparing 2D capabilities of HEC-RAS and LISFLOOD-FP on complex topography
.
Hydrological Sciences Journal
64
(
14
),
1769
1782
.
Syme
W. J.
1992
Dynamically Linked Two-Dimensional/One-Dimensional Hydrodynamic Modelling Program for Rivers, Estuaries & Coastal Waters
.
Teng
J.
,
Jakeman
A. J.
,
Vaze
J.
,
Croke
B. F. W.
,
Dutta
D.
&
Kim
S.
2017
Flood inundation modelling: A review of methods, recent advances and uncertainty analysis
.
Environmental Modelling & Software
90
,
201
216
.
Wallemacq
P.
,
Guha-Sapir
D.
&
McClean
D.
,
CRED & UNISDR
2015
The Human Cost of Weather Related Disasters – 1995–2015
.
Zhang
D.
,
Han
D.
,
Zhong
Q.
&
Chen
Q.
2023
Classification and description of the drainage state of manholes in urban drainage systems
.
Water Science and Technology
89
(
1
),
146
159
.
This is an Open Access article distributed under the terms of the Creative Commons Attribution Licence (CC BY-NC-ND 4.0), which permits copying and redistribution for non-commercial purposes with no derivatives, provided the original work is properly cited (http://creativecommons.org/licenses/by-nc-nd/4.0/).