The drainage network is the key infrastructure that drains storm water runoff on urban areas. Inundation in urban surfaces is always caused by the surcharge, which means the capacity of the drainage network cannot support the urban surface runoff. Therefore, drainage network models are often used for simulating urban storm-water runoff when detailed pipe-network data are available (Lee and An, 2019). The main goals of such models are to simulate the streamflow in the underground drainage network and to provide the flow hydrograph at the outlets of urban catchments or sub-catchments.

In a confined channel or in a pipe, the flow is generally considered to be one dimensional. Drainage network models solve equations derived by ensuring mass and momentum conservation between two cross sections a distance x apart, which yields the well-known one-dimensional Saint-Venant equations. In the urban drainage network, as the pipeline discharge increases or decreases, the pipeline flow may change from open-channel flow to pressurised flow back and forth. Therein, the open-channel flow has a free surface, and the pressure can be approximated as satisfying the assumption of hydrostatic pressure. But the open-channel flow control equation is used for the simulation of open-channel drainage flow (Eqs. 1 and 2). However, the pressurised flow does not have a free surface, and the pressure no longer meets the hydrostatic pressure assumption. Thus, a set of modified equations (Eqs. 3 and 4) are used. These models were also called “0-term” models (Néelz and Pender, 2013) corresponding to the full-term shallow-water-based models.

The mass and momentum conservation equations for open-channel flows with water level Z and flow discharge Q as variables are as follows. 1Conservationofmass∂Z∂t+1B∂Q∂x=02Conservationofmomentum∂Q∂t+∂∂xQ2A+gA∂Z∂x+gASf=0 Here Q represents the flow discharge, Z represents the water level, Sf represents the friction slope, A is the flow cross-section area, B is the width of water surface, t is time, and g is the gravitational acceleration.

The mass and momentum conservation equations for pressurised flows with piezometric head H and flow discharge Q as variables are as follows: 3∂H∂t+a2gA∂Q∂x=0,4∂Q∂t+∂∂xQ2A+gA∂Z∂x+gASf=0, where H is the piezometric head, and a is the wave velocity.

A drainage network model is generally coupled with an urban hydrological or hydraulic model to quantify surface runoff on the urban surface. For example, urban hydrological models, such as the Urban Drainage and Sewer Model (MOUSE) (DHI, 2004), EPA SWMM (Rossman, 2010), and other research models (Schmitt et al., 2004; Simões et al., 2010), are frequently implemented or redeveloped to simulate urban rainfall–runoff and flood overcharge from drainage maintenance holes. However, these models cannot reproduce the dynamic processes of urban inundation. To simulate urban surface water flooding, dual drainage models that combine drainage networks with urban street networks have been developed (Djordjevic et al., 2005; Simões et al., 2010). During flood events, drainage pipe flows generally produce surcharge at urban drainage manholes, resulting in flooding on the street surface. Dual drainage models take into account both flows within the drainage system and surcharge over the streets during intense rainfall. This type of model has two interactive modules: (1) an underground module that consists of a sewer system with known manholes, inlets, and control structures and (2) a surface module including flow paths, retention basins in local depressions, or other artificial control structures (brinks, ponds) made of channels (Mark et al., 2004; Djordjevic et al., 2005). In essence, the 1D model represents the surface flow path (mainly streets) on top of a 1D-pipe-flow model, with exchanges through gully port, catch basin, or other coupling junctures. Then, urban flood risks are assessed based on a dual-drainage-modelled output hydrograph, the water remaining in the drainage network, and water depths on street surfaces with the aid of the analytical capabilities of Geographic Information System (GIS). However, a 1D model cannot be used to reproduce the rainfall–flood process as it cannot be one-way flow in the whole urban area. It is considered over-assumed when treating surface flow as 1D channelled flow, which clearly limits its application.

In recent decades, high-resolution digital elevation model (DEM) and digital surface model (DSM) data with more detailed spatial information are increasingly available. Hydrodynamic models built upon SWEs have demonstrated strong capabilities in providing more detailed flood information in urban areas, such as distributed floodwater depths and velocities. SWE-based models have been frequently applied to fluvial and coastal flooding but were recently refined for urban surface water flooding (Gómez et al., 2011; Xia et al., 2017).

Urban surface floods generally do not have a fixed path or direction and are generally widespread in two-dimensional or even three-dimensional space in the case of large water depth. The motion of an incompressible, viscous fluid can be described by the Navier–Stokes equation. Due to the heavy calculation burden and complicated algorithm design, it is obvious that traditional hydrological methods and one-dimensional hydrodynamic models cannot effectively deal with such problems. In practical applications, the equations are generally simplified appropriately according to the specific characteristics of the flow conditions. The urban surface runoff is generally shallow-water flow; that is, the movement of shallow water with free surface under the action of gravity. The two-dimensional shallow-water equations can realise the accurate calculation of urban flood inundation, which can be obtained by simplifying the Navier–Stokes equations in the vertical direction. It can meet most of the application requirements in engineering practices and has been widely used in shallow-water research. Based on comprehensive reviews of SWE-based models for both fluvial flows and urban surface water, we summarise the governing equations as below (Audusse et al., 2004; Liang and Marche, 2009; Toro, 2013). 5Conservationofmass∂h∂t+∂qx∂x+∂qy∂y=R+E.6Conservationofmomentum∂qx∂t+∂∂xuqx+∂∂yvqx=Sbx+Sfx.7∂qy∂t︸(1)+∂∂xuqy︸(2)+∂∂yvqy︸(3)=Sby+Sfy︸(4), where x and y are the two Cartesian directions, t is time, h is the water depth, qx and qy are the x and y components of the discharge per unit width, u and v are the x and y components of the flow velocity, z is the bed elevation, g is the gravitational acceleration, R is the source or sink term representing net rainfall intensity (runoff term) (rainfall intensity is accumulated into the mass equation as a mass input), and E is a pipe–surface exchange flow term to connect the flow between 2D surface runoff and 1D drainage network flow. Sb and Sf are the bed slope source term vectors and friction effect source term vectors, respectively. The numbers below the equation represent the different terms of the shallow-water equations: (1) local acceleration, (2) convective acceleration, (3) pressure + bed gradients, and (4) friction.

Apart from the models reviewed in Sect. 3.1 and 3.2, an emerging method, the hydrogeomorphic approach, has been recently developed for flood hazard management and mapping (Nardi et al., 2013; Zheng et al., 2018). Unlike the physical-based models above, hydrogeomorphic approaches are based on the concept of fractal river basins or hydrogeomorphic theories, and floodplains are identified as unique morphologic landscape features. The inundation area is clearly recognisable by using terrain data analyses (Di Baldassarre et al., 2020). It is not necessary to estimate the synthetic flood hydrograph, and the floodplain can be determined consistently in different regions. Specifically, a simplified hydrologic analysis can provide the elevation thresholds of the potential inundated grid under different discharge conditions and then identify the floodplain cells for a different return period (Nardi et al., 2018).

More recently, the increasing availability of high-accuracy digital terrain models (DTMs) provides new opportunities for morphometric analysis of floodplain mapping. And this floodplain delineation method has been of significant interest because of the low requirements of time-series data and high computational efficiency. It is being developed to apply in global and large-scale catchment fluvial flood mapping and has also been applied in urban areas in recent years (Brown de Colstoun et al., 2017; Nardi et al., 2018). However, its model concept implies that such a method can predict an approximate inundation area but cannot simulate the dynamic processes of flooding (time series of water depths, velocities, etc.) that are vital for risk assessment.

Drainage network models, particularly coupled with hydrological methods, are highly computationally efficient, owing to their simple structures; thus, they show merits in applications that focus on flood-related hydrological analysis in an urban catchment and have low requirements for the representation of full hydrodynamics. The fast operation of this model type makes it suitable for large-scale simulations with various temporal scale evaluations that require thousands of longer-run simulations with little detail of the flow behaviours. Inclusion of pipe network and manholes in models can help to estimate potential surcharge, which can be used as a reference for other inundation models, e.g. 1D street network models or 2D surface water models. The surface runoff simulation of these models is mainly based on the conceptual hydrological models or grey-box models. Although urban rainfall–runoff processes can be properly estimated by these models through linking a drainage network, the over-assumption of flow dynamics does not allow for the simulation of surface flow inundation, which is a key indicator for risk assessment. Thus, there will be a lack of detailed spatial dynamic information of urban surface waters, such as flood depth and velocity. Moreover, this type of model usually solves the 1D equations for drainage flows. At the city scale, the drainage network system is often complex with a large number of drainage nodes and pipes. Although automatic GIS procedures can improve efficiency to a certain extent, it is still time-consuming data-demanding to set up and verify this type of model. For operational flood management, there is currently a high demand for thorough, detailed spatial information, such as the depth and velocity of surface flow in each street and accurate data for each residence. Drainage network models cannot meet the above requirements for urban surface water flooding.

A power of work has been devoted to developing 2D shallow-water-based numerical models in recent decades. These include simplified and full SWEs models, models coupled with drainage networks, and even coupled hydrological and hydrodynamic models. In the above overview, SWE-based models have proven to be capable of reproducing surface flow reasonably well for flooding in urban areas, accurately predicting velocity, flood extent, and water level.

High-resolution topographical data, such as lidar, DEM, or DSM data, are now becoming available and can have a fine resolution of about 20 cm or even less. Several studies have developed applications of SWE-based models in a variety of scales by developing new algorithms or new model framework and utilisation of high-resolution topography datasets. These models are advancing current practices to make practical simulations of urban surface water inundation driven by extreme rainfall. Experimental and field in situ data have always been considered an essential supporting source for flood model validation. In this concern, some recent studies have specifically developed physical models and approaches to gather detailed experimental or field data to support flood model verification (Rubinato et al., 2013; Lopes et al., 2015; Gómez et al., 2019; Martins et al., 2018). For example, Rubinato et al. (2013) and Lopes et al. (2015) collected hydraulic data form a physical model and applied the data to verify the performance of a numerical model. Although there are some, good-quality benchmark datasets are still lacking in the research community for urban flood model validation.

Compared to full SWE-based models, simplified SWE-based models have been typically used in larger-scale urban flood modelling with coarser grids and simplified treatment of urban features (e.g. city scale, even continental scale) because of the relatively low computational costs and the capability to simulate surface water dynamics (e.g. Yu et al., 2016). However, the inherent model conceptualisation implies that these models cannot capture the shock wave adequately. They may not be an ideal choice when simulating detailed urban flood dynamics with infrastructure.

Recently, sophisticated full SWE-based models have also been applied to city-scale flood modelling with the advances of accelerated algorithms (e.g. Glenis et al., 2018; Xia et al., 2019; Sanders and Schubert, 2019). Such models are capable of simulating the full surface water dynamics. However, they require high-quality data, which could result in high uncertainty depending on the data source (Willis et al., 2019). Both simplified and full SWE-based models have weaknesses, and further improvements are the following current challenges:

The concentration of buildings in urban areas plays a vital role in the magnitude of inundation and its hydrodynamics; however, current SWE-based models either ignore its effects or make over-assumptions. Some attempts with the inclusion of urban features mainly focus on small-scale validations of numerical schemes, whereas larger-scale applications have been hardly studied.

Hydrogeomorphic approaches can identify the inundation area directly from the topography. Thus, it requires much less computation time and no time-series data. However, there are still many limitations for the application of these.

Only approximated inundation area can be obtained. Flow dynamic information (water depth, velocity, etc.) cannot be provided.

In view of the current advances of urban flood models, there are still deficiencies in improving model reliability and efficiency. Urban areas have many complex underlying surface characteristics, and in reality, when the capacity of drainage networks is insufficient, the pipe flow will overcharge to the ground surface. At present, simulation methods of the exchange of pipe flows and surface water have only focused on local-scale modelling. Some existing numerical models often directly use empirical formulas or simplified methods that are still lacking in stability and accuracy. Therefore, mechanisms between pipe flow and surface water and their modelling approaches would help simulate drainage flooding in urban areas. This could be accomplished by integrating drainage network models with overland flow routing models as some studies have done, but further refinement is needed. Moreover, the question of which model conceptualisation is more appropriate for urban surface water flooding is still unanswered and in need of further investigation with the support of high-quality data.

The spatial heterogeneity of urban catchments is typically more profound, and surface water depths are generally shallower. Both pose numerical challenges in solving SWEs over frictional and extreme, irregular terrain. Although the application of parallel computing technology to improve the computational efficiency of the model has become a trend, efficient urban flood simulation and even real-time flood prediction with better resolutions are still difficult to achieve. In the light of the high computational costs for large-scale modelling at high resolution, more accurate and faster model algorithms are urgently needed. This is critical for achieving a city-scale urban flood prediction in real time.

With the continuous improvement of remote sensing technology, data become more readily available. A data-rich environment also encourages model calibration, validation, and assimilation. In practical terms, the accuracy of terrain data obtained from modern lidar system has met the requirements for surface flow simulation, but it is necessary to fuse such dataset with digital map data of buildings and land use to realise the maximum development and utilisation of contained information. Furthermore, these terrain data are readily available, and topographical data with a grid scale of roughly 30 m at best hardly meets accuracy requirements for urban flood models.

Model calibration is an essential way to reduce uncertainty over model parameters, but to this day, such data have been scarce for urban areas. Despite the frequency of urban floods, field observations during urban flooding are rarely available for model calibration and validation. Calibration data will be the key factor constraining the future development of urban flood inundation models. Some effective methods or tools are therefore urgently needed to infer from these limited data sources, extending the quantity and range of typically available calibration–validation data. Development of physical urban flood models is an option to gather benchmark data for urban flood modelling as some researchers have done (e.g. Rubinato et al., 2017). Nowadays, the application of social media for both collection and dissemination of flood information is increasingly recognised and thus provides an important basis for flood inundation estimation (Fohringer et al., 2015; Smith et al., 2017). The flood related information provided by the general public through social media such as Twitter or Weibo is also effective and a valuable calibration–validation data source as an addition to the hardly available traditional monitoring data. Besides, studies (e.g. Waller et al., 2018; Ziliani et al., 2019) have verified that the combination of data assimilation and numerical model is used operationally to improve model performance and reduce uncertainties in flood prediction. Ziliani et al. (2019) assimilated field data into the flood model, and the prediction result was improved up to 90 %. Among the relevant literature, this method is mostly applied to fluvial floods with the support of satellite-based data or field water-level measurements but rarely applied to urban pluvial floods. Moreover, as it is recognised that current 2D hydrodynamic models are still computationally demanding and challenging for real-time applications at large-scale, recent innovative modelling exploration has focused on the machine-learning approach for fluvial flooding; for example, a deep convolutional neural network (CNN) method has been developed by Kabir et al. (2020). Such an approach is presented for rapid prediction of fluvial flood inundation. However, a good model training still requires good-quality inundation data and/or a robust hydrodynamic model. Application of the machine-learning approach in urban flood modelling is promising but still very challenging.

Inter-model and interdisciplinary approaches can help to develop the strengths of the various approaches while avoiding shortcomings. Facing the knowledge gap among urban flood risk management, innovative use of computer-based visualisation and virtual reality (VR) technology has been shown to encourage greater engagement amongst diverse participants. A combined simulation–visualisation platform can become an important shared learning tool and there are good prospects for developing an interactive model through the use of computer-based visualisation and virtual reality technology (Wang et al., 2019; Zhi et al., 2020; Yang et al., 2021). The innovation will be helpful for practitioners to communicate and perceive an extreme flood event. With the help of interactive 3D visualisation tools, the extent of inundation and other features such as water depth and floodwater velocity can be better viewed and understood. A combined simulation–visualisation approach can enhance decision support by incorporating 2D inundation modelling and 3D data visualisation. Besides, as mentioned in the previous part, multi-source data such as social media and remote sensing provides an excellent source of model calibration–validation data during and after flood events. Its application may be further enhanced when coupled with accelerated real-time urban flood modelling. In other words, the combination of social media data and an efficient simulation model provides a strong support to build a real-time surface water flood warning system. The astute combination of models is promising, and it will be successfully developed and applied in the future.