Highlighting the role of the model user and physical interpretation in urban flooding simulation

The diffusion of two-dimensional (2D) models is a current trend and they are substituting or complementing one-dimensional (1D) application when facing engineering problems. However, the increased demand for information to use and calibrate these 2D models became the main constraint for their consistent application. On the other hand, the Light Detection and Ranging (LiDAR) technology is becoming usual to survey terrain surface and it can offer terrain details to the 2D model representation, allowing the refinement of the modelling mesh with high-resolution information (Neal et al. 2011). However, this kind of detailed information is also (usually) more expensive and more difficult to process.

Abdullah et al. (2017) compared many LiDAR filtering algorithms and found that, even if producing detailed maps, none of them is fully reliable in capturing some important urban features. According to Mishra et al. (2022), there are some challenges related to the representation and description of urban landscapes for atmospheric, hydrologic, and hydraulic modelling. These challenges are related to the scales involved, to assumptions needed to couple these processes, and to the type of models being used to represent these features.

Other problems related to 2D models in urban regions are pointed out by Abily et al. (2013), who highlighted the rapid changes in the flow regime and numerical problems with the drying and flooding elements of the modelled mesh throughout the simulation. Almeida et al. (2016), on the other hand, found out that small changes in the representation of the urban landscape are not observed in resolutions greater than 1 m. Similarly, according to Mark et al. (2004), to represent the characteristics of an urban landscape, the mesh must have elements of less than 5 m. In practice, however, large-scale models would not support such a refined mesh (Jamieson et al. 2012). Besides, the difficulty related to processing such a great dataset is that it is difficult to imagine a situation in which the flooding results should be known (in useful terms) at each square metre of the urban landscape. In general, the water depths in the streets (and, consequently, in the buildings) and the flow velocities in the main directions of flow (along the streets, for example, and not transversally from sidewalks to roadways) are the main required results. Additionally, the urban environment can produce discontinuities in the flow surface that cannot be captured by 2D equations.

In urban areas, however, the search for an increasingly refined resolution is a trend to many modellers, largely due to the widespread use of LiDAR technology and also to the computational development that allows a greater refinement of the modelling mesh (Neal et al. 2011). In fact, during the last decade, topographic datasets created based on LiDAR and photogrammetry technologies have become widely used by consulting companies for various applied study purposes including flood risk studies (Abily 2015).

In the case of the 2D urban models, the resolution of urban 2D models was usually limited to 1-m grids (for example, Mark et al. 2004; Aronica & Lanza 2005; Fang & Su 2006; Gallegos et al. 2009; Leandro et al. 2009; Gallien et al. 2011; Almeida et al. 2012). Advances in computational resources and the recent availability of high-resolution LiDAR data have allowed the first 2D simulations of urban floods with lower resolutions, reaching the order of up to 10 cm (Ozdemir et al. 2013).

Ozdemir et al. (2013) simulated the same urban region with different resolutions and observed the elevation of the depths and maximum water velocities in 37 and 32%, respectively, with the reduction of the resolution from 10 cm to 1 m, while the extension of the flooded areas decreases by about 6%. However, it is important to highlight that a complementary discussion should be considered. When refining the mesh, gradual transitions may be lost and one of the main hypotheses assumed by the Saint-Venant equations is also consequently lost. This is something that should be investigated, since the results may be affected by the chosen scale of representation. For example, a formidable change in bottom slope could be introduced in a mesh of 10 cm if the water is crossing the sidewalk edge. In this situation, a slope of nearly zero (considering an almost flat sidewalk) would rise to something about 100% if the water goes down 10 cm (from sidewalk to roadway) in a 10 cm mesh. These small meshes can be a source of significant numerical instabilities. Besides, this velocity down the sidewalk is not important in the representation of the flooded street.

Leandro et al. (2016) emphasizes that unless the flow remains confined within street limits and channel networks, 1D flow models are not applicable and 2D flow models should be chosen. However, the same authors (Leandro et al. 2016) recognize that surface runoff in urban areas is highly complex due to interaction with artificial structures that interfere with flow paths, requiring a precise pre-defined set-up for 1D models, which is difficult, but also makes it difficult (and eventually improper) to apply 2D models, since a real 2D surface is not defined throughout the modelled space, due to urban structures interactions. When flow paths are well known, 1D flow models can be used in an agile and efficient way, and it is an economical alternative to 2D models (Spry & Zhang 2006).

On the other hand, a different (and coherent) situation occurs in an unoccupied floodplain with the formation of a continuous surface of water, allowing an adequate use of 2D models. Chang et al. (2015) point out that high imperviousness rates and short concentration times are common elements of urban hydrology and, under these conditions, drainage networks play a major role in modern cities by transporting surface runoff during storm events.

After confronting numerical problems in calculating flow velocities and suggesting more detailed meshes as a possible solution, Néelz & Pender (2013) affirmed that the simple mesh refinement is not a viable option, since it can make modelling impracticable in computational terms, overcoming the ability to perform multiple simulations, quantify uncertainties, perform risk studies, calibrate models, and so on.

Cunge (2014) pointed out that except for very simple situations, the model user must be aware of basic hypotheses and physical laws that supported the software construction. Only then will the user be able to distinguish coherent modelling results from results completely incompatible with the physical reality of the system.

Modelling implies creating an image of the reality. However, images are only the initial modelling input, and their use in a model can generate positive or negative results. It is important to know the limitations of the chosen equations and what questions of interest are to be modelled to correctly evaluate the process and adequately translate the image in mathematical terms.

Flood maps obtained with hydraulic simulation models are subject to multiple uncertainties, which are often ignored or misinterpreted (Dimitriadis et al. 2016). The resulting differences are mostly assigned to the quality of topographic and input data; to the high uncertainty caused by several input variables; to the treatment of wetting and drying elements; to the hypothesis in the mathematical description of the natural phenomena; to the lack of knowledge in the input parameters; and to the natural phenomena inherent randomness (Neal et al. 2011; Abily 2015; Abily et al. 2016). Friction and source terms, inflows and normal depth boundaries may also differ subtly between codes, both in terms of approach and parameterization. All of these factors may alter simulation results before any consideration of numerical scheme choices or physical complexity is taken into account, adding significant uncertainty to any discussion (Neal et al. 2011).

Considering this, some works focused their analysis on the uncertainty produced by the model parameters and its structure, raising the question of how much physical complexity is needed to simulate flood inundation (Horritt & Bates 2002; Pappenberger et al. 2006).

In this context, this article offers a counterpoint to the current trend of indiscriminately moving towards more sophisticated models, especially without understanding its limitations, while not giving the proper attention to the physical interpretation that governs the modelled phenomena. In addition, this work seeks to stress the importance of the modeller as a key element in the modelling process, highlighting the physical interpretation role and the importance of knowing about model potential and limits.

In a certain way, the ‘novelty’ of this article falls on recovering the modelling spirit of former model developers that needed to work with the representativeness of the important physical details to soundly and efficiently equate the problem, showing that the current trend can dangerously lead to inadequate modelling choices (with unnecessary details, or high computational costs or even lacking physical representativeness), including the possibility of making ‘mechanical’ decisions that are only driven by model outputs and not by a deeper engineering reasoning. In this sense, our contribution refers to raising a discussion about how much a detailed model is really needed, recognizing that mathematical models are always limited simplifications of the reality and modellers should be aware about what responses are required from the model simulations (one model cannot give all of the possible answers to any case). In this discussion, we sustain that the physical interpretation provided by the modeller is the core element of any successful modelling process.

To support this discussion, we will demonstrate that a Quasi-2D model, associated with a modelling procedure of systematically interpreting the physical reality, can represent a real system with the same acceptable reliability of a complete 2D model using a huge dataset, but with much less computational efforts and data needs. We emphasize that the absolute results obtained in this study refers only to the chosen particular model; however, in relative terms, these results will be the basis for our argumentation towards a more consistent and immersive modelling procedure as key elements to obtain adequate results. This proposal opens the door for a deeper discussion alerting to the risks of being overshadowed by technological facilities.

The model chosen to fulfil this aim was MODCEL, a Quasi-2D model (Miguez et al. 2017) that was used in two different ways: as a raster model that represents the region of interest in an automatic way, using a high-resolution grid, like 2D models; and as in its original conception, using irregular cells to represent the watershed surface and urban structures in a way that is completely dependent on physical interpretation and cannot function as a blind automatic application.

These two modelling alternatives were used to reproduce a test proposed by the British Environment Agency (Néelz & Pender 2013) for model comparison and benchmarking purposes. Additionally, we also investigate one real case study in Canal do Mangue basin, comparing the modelling results with real data.

MODCEL was originally developed by Miguez (2001). It is a Quasi-2D model (CUNGE et al. 1980) that represents the urban space using homogeneous compartments, called cells. The concept of flow cells was initially developed by Zanobetti & Lorgeré (1968) and enshrined by Cunge et al. (1980).

The cells may present an irregular form, composing an adjustable mesh that covers the whole space of the basin in an interconnected way, forming a flow network, linked in pairs by 1D equations. Thus, MODCEL can represent the two-dimensional characteristics of the river basin in a simplified way, only using 1D equations. The model can describe natural and artificial watercourses and also detailed elements of the urban fabric (streets, squares, parks, roofs, etc.), the flow in the underground storm drains, and the mutual connections between the upper and lower layers formed by such elements.

Since its conception, MODCEL is in continuous improvement and its last version can be found in detail in Miguez et al. (2017). Some previous applications of this model can be found in literature (see, for instance, Miguez et al. 2015; Nardini & Miguez 2016; Barbedo et al. 2015).

Different hydraulic equations can represent the connections between cells, ranging from the classical Saint-Venant dynamic equation to broad crested weirs, orifices, pumps, flap gates, operated gates, etc. These different hydraulic links can overcome the possible discontinuities of the flooding surface solution. By rendering the flow through 1D equations written for the possible pre-defined flow paths (as a result of the modeller interpretation), the model preserves simplicity and saves computational time.

The key concept of MODCEL lies in the topographic and hydraulic representations. In this process, the modeller is responsible for interpreting the geometric and hydraulic characteristics of the terrain that characterizes each cell as well as to connect all the cells in a functional and representative flow network. In this context, MODCEL does not need a Digital Terrain Model (DTM) in the strict sense, since it will receive information interpreted (and reduced) from the available topography, as defined by the modeller.

Another version of MODCEL was created to work as a Raster model, aiming to represent the region of interest in an automatic way, using a mesh of cells of regular formats and identical dimensions, simulating an usual 2D model application.

Raster models are those that transform each point of a DTM in a ‘pixel’. In this case, the structure of MODCEL was adapted so each pixel represents a flow cell that is connected to the other cells in its surroundings by the Saint-Venant dynamic equation, without the inertia terms and without interpreting the terrain, in a similar way done by 2D models.

The benchmarking proposed by the British Environment Agency exercise (Néelz & Pender 2013) involves 10 test cases and one of the objectives of this research is to provide a set of data against which a model can be evaluated by its developer. In recent years, several authors have used these tests to validate their models (Jamieson et al. 2012; Coulthard et al. 2013; Leandro et al. 2014; Beevers et al. 2016; Guidolin et al. 2016).

Test 8, in particular, was designed to compare urban flood models and is divided into two parts, 8A and 8B. In this paper, we used test 8A as reference. This test assumes that the flood arises from two sources: a uniformly distributed rainfall event applied to the modelled area; and a point discharge source (simulating a storm drain overflow), reaching a peak of 5 m³/s, lasting 15 min, 35 min after the start of the rainfall event.

The second application refers to the use of the interpretative model in a real urban watershed, the Canal do Mangue basin. This is an urban watershed of 45.4 km², located in the north zone of the city of Rio de Janeiro, Brazil, which outflows into the Guanabara Bay, suffering tide effects.

Tide levels in Guanabara Bay, which are able to cause backwater effects, were used as downstream boundary conditions. This information was obtained from reports made by Rio-Águas³ and it can be seen in Figure 3(b).

Firstly, the two modelling approaches (MODCEL – RASTER and MODCEL – Quasi-2D) were used to reproduce the benchmarking test proposed by the British Environment Agency (Néelz & Pender 2013) for comparison purposes with other models, especially 2D models, which had their results already published in the same reference (Néelz & Pender 2013). This phase aimed to evaluate the differences found between the two modelling alternatives, since the same area was simulated under the same controlled conditions, which were established by the British test set-up. This standardization intends to highlight the different results obtained with the different models, when using the same reference conditions.

After that, the Canal do Mangue case study was used to demonstrate that the physical interpretative modelling approach used together with limited datasets could also produce reliable results (compared to real measured data) in a real urban flooding case.

In the benchmarking case, both modelling approaches followed the test requirements pre-established for the Manning coefficient values, where 0.02 was set to the paved areas, while the green areas received the value of 0.05. Runoff coefficients were taken as 1, since the effective rainfall was directly informed to the model. Also, both simulations used the same time-step of 1 s.

There was a significant difference in the runtime of the two modelling approaches. The MODCEL – RASTER provided the results in 3 days, while the MODCEL – Quasi-2D finished its simulation in 190 seconds (more than 1,300 times less). The computer used in both applications was a standard desktop, with an Intel processor i7-5500 U of 2.4 GHz, with 16 GB of RAM.

Considering the maximum flood water levels, it is possible to observe that the MODCEL – RASTER is more precise in the spatial flooding representation (due to the refined mesh), while the MODCEL – Quasi-2D model offers an envelope of responses (showing an apparently greater flooded area, due to the greater cell sizes, which makes a given water depth representative of a larger area). Taking the maximum flood water level of <0.06 m as an example, it is observed that in the MODCEL – RASTER the flooded area is about 80,388 m² while with the MODCEL – Quasi-2D the flooded area is about 73,662 m².

The results obtained with MODCEL were also compared with the results obtained with several other models published in the benchmarking document (Néelz & Pender 2013), as shown in the brief resume of Table 1.

The application of MikeFlood and Infoworks followed the same rules established by The British Environmental Agency. Therefore, the value of the Manning coefficients used in the MODCEL applications were the same as the original test. According to the benchmarking document, the MikeFlood simulation used 8 CPUs, a time-stepping of 1.5 s, and reached a total runtime of 367 s. In another way, the Inforworks simulation used a multi-processing GPU, a time-stepping of 30 s, and reached a total runtime of 66 s. More details of the MikeFlood and Infoworks application can be found in Néelz & Pender (2013).

The Canal do Mangue watershed covers an area of 45.4 km² and it is a dense and consolidated urban area usually affected by floods. Following the previous standards, if this watershed was discretized using a mesh of cells, each one with 9 m², the modelled area would require more than 5 million elements, which makes the MODCEL – RASTER use rather unfeasible.

In such cases, Quasi-2D models based in physical interpretation may be useful, as they can produce reliable results with much less computational efforts and data needs. For this reason, only the MODCEL – Quasi-2D alternative was applied for representing the Canal do Mangue case and its results are compared to real measured data.

This study intended to discuss the importance of the modeller role and to highlight the possibility of reaching adequate results using simple approaches, by highlighting the physical interpretation in the modelling process. To cope with this aim, we used a Quasi-2D model – MODCEL – as a discussion supporting tool in two different ways: as a raster model, simulating a detailed spatial grid, like the usual 2D model approach; and highlighting physical interpreted decisions that are translated into larger surface model elements (cells) and links (hydraulic relations). These two modelling approaches were used to reproduce a test proposed by the British Environment Agency (Néelz & Pender 2013) for comparison and benchmarking purposes.

The results for both configurations were consistent with the other models tested in the prediction of water levels, where all the obtained answers remained within the expected range of variation – this first impression validates the possibility of simplifying model choices, if sound physical-based interpretations are backing these choices. Regarding the velocity results, MODCEL in the Raster configuration presented velocities within the range of the other models, while the Quasi-2D configuration seemed to slightly overestimate the velocities at the reference points. However, 2D models were more susceptible to greater velocity variations, due to greater changes of bottom slopes in short distances, while the Quasi-2D representation was more stable.

It is possible to observe that the surface representation of the maximum water levels obtained by the two modelling alternatives presented very similar results. The Raster alternative produced a more detailed spatial response, since spatial modelling is more precise in this alternative, but it is really reliable and representative only if the terrain model is the result of a detailed field survey (with LIDAR technology, for example, and controlled by field surveys). On the contrary, it can be only an illustrative representation, in case the DTM was obtained from the interpolation of a non-detailed source. On the other side, MODCEL – Quasi-2D used traditional contour curves as a reference, producing greater cell sizes that tend to also show a greater flooded area (since the water level is referred to the cell centre, which usually occupies the lowest terrain level in the cell). However, it is just a visual effect resulting from a representation feature. The calculations are adherent to the expected results.

In commenting on the results of other models to which the British test was applied, Néelz & Pender (2013) concluded that differences in topography approach suggest that a 2 m grid may be insufficiently precise for high-resolution modelling of shallow urban flows, particularly if accuracy in predictions are expected. However, Néelz & Pender (2013) also emphasize that it should not be concluded at this stage that the refinement of the grid is a path to be followed since it is likely to have counterproductive effects in terms of computational efficiency.

Since the results obtained in the British test showed that the alternative with physical interpreted decisions could represent a real system with similar fidelity in comparison with a complete 2D model, but with much less computational efforts and data needs, this approach was also used in the Canal do Mangue case study. It is important to emphasize that in the Canal do Mangue case the 2D modelling could result in an unfeasible simulation in regular computers due to its dimension.

The Canal do Mangue basin is an urban watershed characterized by a complex hydraulic system. This place is also known for its recurring cases of flooding in the city of Rio de Janeiro. Due to the existence of rainfall and fluviometric data in almost all major channels in this area, it was possible to build and test a model using real information.

The results showed that the behaviour of the maximum levels mapped by the MODCEL – Quasi-2D are in accordance with the data records made by the Municipality. This approach proved to obtain good results, with relatively low computational costs, providing an interesting tool for planning actions to mitigate flood problems.

As MODCEL in its original and interpretative form is completely dependent on physical interpretation and cannot function as a blind automatic application, it obliges the modeller to investigate and understand how the real system works. By saying this, we emphasize that the elaboration time of this kind of model might be time-consuming, because it is important to reach a prior understanding of the modelled phenomena.

However, even if this apparent difficulty could be seen as a weakness, in a first moment, in fact it turns into the model main strength, since this process inevitably leads the user to better know the investigated problem and to gain sensibility to criticize the model answers. The understanding of the system responses to causal factors can be of great value in understanding model hypothesis, limitations and potentials, helping in interpreting model results and projecting solutions to hydraulic problems.

The hydrodynamic modelling process has been experiencing a rapid evolution in terms of computer processing capacity and topographic data surveying, leading to the widespread use of 2D models, with refined meshes and new perspectives that were not possible some time ago. However, as highlighted by Abbott & Vojinovic (2009) and Cunge (2014), it is important to use these new technologies and possibilities maintaining the modeller knowledge about model hypothesis and consequent limitations. If this aspect is not carefully considered, it can lead to dangerous consequences when model users are not aware of model limitations and use mathematical models as ‘black boxes’. The technological facilities are not able to solely respond for an adequate model response.

This manuscript intends to discuss the importance of the model user being aware of model limitations and to propose that valuing the physical interpretation in the mathematical modelling process can lead to simpler and efficient model approaches, not necessarily requiring the most advanced technologies or the more sophisticated models. The results obtained in the cases simulated here support our statement that the adequate physical interpretation and the awareness of the modeller role are in the core position of any reliable modelling process.

The benchmarking test proposed by the British Environment Agency to assess model behaviours was used as a first case. In the first case shown in our work, we simulated the urban watershed test using a regular detailed grid raster approach, similar to 2D model uses, and an irregular mesh of cells, representing greater portions of the watershed, using a Quasi-2D model, settled on a physical-based interpretation. When evaluating the time spent to obtain the results, the MODCEL – Quasi-2D simulation had a much smaller runtime than that of MODCEL – RASTER. However, the elaboration time of MODCEL – Quasi-2D was more time-consuming. In the end, both applications were able to find equivalent results, but the physical-based interpretation led to a significant lower computational effort and using much less data, when compared to the 2D approach. This result supported the second case shown in this work, where the Quasi-2D model was applied in a real case, in which the use of a 2D model would have been unfeasible. This simulation was successfully compared with measured data, giving adequate responses based on the interpretative process.

It was observed that more sophisticated approaches become extremely demanding in terms of data and computational resources, thus imposing substantial barriers to their usage, especially in developing countries where detailed topographic datasets are not available and investments are limited. On the other hand, extremely simple models introduce several assumptions and uncertainties to the modelling. Finding an acceptable compromise between accuracy and parsimony is the key for a successful hydrodynamic modelling.

In the cases presented in this work, the use of a complete 2D model might have been unnecessarily complex, and simpler schemes, like a Quasi-2D model, would be able to perform just as well. In this sense, we intend to shed light on the importance of modelling decisions based in physical interpretation. In certain cases, this approach can be more effective for obtaining adequate results than simply selecting a more complex model demanding more complex topographic data.

The results presented here, however, should be treated with caution. It is expected that results can change from application to application and there is no general response. However, the relative results seem sound and helped to support the proposed research hypothesis considering that a physical-based process conducted by a conscient modeller is a key point for model reliability and optimization. That is, this work does not intend to define a ‘better’ modelling approach. It is not possible (nor desirable). The results obtained in this work are a warning against blind model uses. No model should be previously enabled to any general situation. Besides, in a complementary (and crucial) way, after analysing the model hypothesis and limitations and considering the physical interpretations, it is always important to remember that calibrating and validating a mathematical model against real observed data is necessarily and intrinsically associated with its reliability.

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

The authors declare there is no conflict.