A comparison of three dual drainage models: shallow water vs local inertial vs diffusive wave

Pluvial floods can cause huge economic damage. Physical scale models require large data extraction instrumentation and installations. Numerical models on the contrary allow a cost efficient solution to simulate floods.

To simulate flood dynamics in urban areas, the drainage network is often divided into two sub-systems: the minor drainage system composed of buried pipes and manholes and the major drainage system that includes watercourses and flow pathways along the surface. An accurate flood model should not only replicate the flow dynamics in the sewer system but also on the surface as well as the interaction between both systems. The flow in the minor system is commonly modelled by one-dimensional (1D) sewer models, while either a 1D or two-dimensional (2D) approach (Leandro et al. 2009) can be used to simulate flooding in the major system. 1D/1D models were the first type of urban flood models adopting the dual drainage (DD) concept (Leandro et al. 2011). One of the first proposals for DD was made by Ellis et al. (1982) using SWMM (Storm Water Management Model) to design DD 1D/1D Systems, where the two systems remained separate although they were calibrated as if they worked together. More recently the approach has seen some significant improvements by Prodanović et al. (1998), Djordjević et al. (1999), Hsu et al. (2002), Nasello & Tucciarelli (2005), Chen et al. (2007), Leandro et al. (2009), Mahdizadeh et al. (2011, 2012), Seyoum et al. (2012), Borsche & Klar (2014), Fraga et al. (2015), Chen et al. (2015), Russo et al. (2015), Martins et al. (2016c), and Leandro et al. (2016).

Djordjević et al. (1999) proposed an approach to the DD concept to better replicate the urban flood conditions with an integrated sewer and surface hydraulic model. This concept included the use of energy and mass conservation equations to compute the linkage. Nasello & Tucciarelli (2005) applied a similar concept to a 1D/2D model with linkage through weir and orifice equations. The weir and orifice equations are usually calibrated with parameters obtained from experimental studies or using computational fluid dynamics simulations that can account for the full complexity of the linkage structure (Lopes et al. 2015; Martins et al. 2014; Leandro et al. 2014b).

Mahdizadeh et al. (2011, 2012) introduced a novel model for the shallow water equations (SWEs) where inflows and outflows are computed using the approach they termed a flux differencing methodology. Abdullah et al. (2012) developed a methodology to account for elevated roads and bridges. Seyoum et al. (2012) coupled SWMM to a 2D SWE local inertial model. Borsche & Klar (2014) derived a system of ordinary differential equations governing the flow in the manholes. Fraga et al. (2015) used an experiment in order to validate a 1D/2D coupled model where the rainfall-runoff transformation in urban environments has been validated using a real-scale street section.

Modelling of flow in the major system is usually done with the 2D non-linear SWEs. These equations can be obtained from the Navier-Stokes equations by assuming an incompressible and inviscid fluid, by neglecting the influence of the vertical acceleration and assuming hydrostatic pressure distribution. They are often used to describe a wide range of free surface flows (Chertock et al. 2015). Simplified versions have been developed (Ponce 1990; Aronica et al. 1998; López-Barrera et al. 2012; Chen et al. 2015; Martins et al. 2015; Fernández-Pato & García-Navarro 2016) in order to reduce the computational cost whilst maintaining an acceptable level of accuracy. Examples can be seen in the applications of local inertial equations (Aronica et al. 1998; Martins et al. 2016a, 2016b), which neglect the convective acceleration, and diffusive wave equations without inertia (Lal 1998; Cea et al. 2010; Wang et al. 2011; López-Barrera et al. 2012; Chen et al. 2015; Fernández-Pato & García-Navarro 2016) that neglect the convective and local accelerations. The spatial and time discretization used to model surface flows varies in each model.

This paper focuses on the wave propagation of three hydrodynamic models under DD conditions. We compared the models using a novel methodology in a real test case scenario with surcharge from a sewer network. The importance of knowing the limitation of each simplified model in such a complex situation is important, as the correct assessment of the flood depths is critical for analysis of flood impacts.

Three 2D overland flow models were studied: full dynamic model (SWE), gravity wave model (GWM) and parallel diffusive wave model (PDWAVE). The sequential version of PDWAVE was used in order to be comparable to the other models.

Herein the equations are discretized in an unstaggered structured grid. A finite volume explicit scheme with cell centred control volumes is used. The time step increment is controlled by CFL condition, which is proportional to (Leandro et al. 2014a). PDWAVE uses a prediction correction Wet-Dry front treatment that restricts the movement of water in case of negative depths, obtaining absolute mass conservation (Leandro et al. 2014a).

Although the proximity search is effective, usually there is disagreement between the terrain elevation data obtained from the DTM and the corresponding manhole cover elevation from the sewer network data (Chen et al. 2007). If the crest is lower than the surface the maximum is used as the value for both the surface and the crest. The assumption that the crest elevation is defined differently from the surface elevation is also viable since the cell area might be high and ponding occurs; the ponding can be simulated by elevating the crest or lowering the surface, thus allowing retention.

is the head in the manhole (1D sewer model), the depth in the overlapping 2D mesh call of the surface model, t the previous time step, the time step intended to be updated, the sewer model head estimation after time-steps, the discharge calculated through Equation (19), the surface model time step, the sewer model time step, the area of the Manhole, and the area of the cell connected.

The three modelling approaches and the linkage performance were compared using three different indicators, which rely mainly on the depths, heads and discharges in the linkages and on the surface. The first measure is the total water volume exchanged between surface and sewer system, which allows a global comparison between all three models. The second comparison is based on the hydraulic head and flow rates at linked manholes, allowing detailed observation of the flow interaction between the sewer and the surface systems. The third indicator analyses the inundation extent data to indicate the main differences in overland flow propagation. Average used for GWM was 0.141, for SWE 0.111 and for PDWAVE 0.01. In PDWAVE was set as 0.01 s. The CFL coefficient and Manning's roughness coefficient were kept constant as 1 and 0.03, respectively.

The third indicator is based upon flood depths and extents in the overland flow models.

The comparison of the three overland flow models under the DD framework will be made using three different sets of parameters. The computational time was approximately 188 min, 183 min, and 284 min for PDWAVE, GWM and SWE, respectively.

The left vertical axis represents the value for each overland flow model whilst the right vertical axis indicates the differences between the models. The results in Figure 4 show close agreement. Note that the difference between models is presented in absolute value. Since GWM has the higher volume on the surface, the difference between GWM and PDWAVE is higher than between PDWAVE and SWE. The peaks obtained in the first half hour are mainly due to PDWAVE's smaller wave speed propagation when compared to GWM and SWE. As such there is a higher depth on top of the manhole and therefore a smaller exchange. The values of the accumulated volume on the left image are almost superimposed and the differences are small. The difference at the end of the simulation is below 0.16% for all surface models, which is a good agreement. The difference between SWE and GWM is smaller than the one between PDWAVE and the other two models.

The image on the right shows the discharged volume every 60 s. The differences are very small and the volume is almost coincident. In detail, the difference between the models is shown in the right vertical axis. The major difference, found at t ≈1 h, are relatively small: 4 m³ difference for a discharged volume of 300 m³. Figure 4 also shows that the cumulative discharge is higher in SWE and GWM than in PDWAVE. When the discharge is above 500 m³ GWM tends to have a slightly bigger discharge, as seen by the negative values of volume difference.

In Figure 5, PDWAVE has higher hydraulic head for most of the manholes among all models. This trend can be seen in PBIAS Manhole Head since most of the coefficients for PDWAVE are positive on the x and y axes. This is reinforced by the negative values for GWM and SWE when compared with PDWAVE (⁠ axis). As a consequence of higher head and due to the surcharging flow, PDWAVE also has higher discharges for the majority of the manholes when compared to GWM and SWE. It should be noted that the differences are kept well within reasonable limits (<5%) for the vast majority of the manholes.

provides an insight into the relative global difference for the manhole head as a percentage. Most of the manhole heads are in very good agreement and are kept well below 0.03 (3% difference) with GWM and SWE relation below 0.01. Points with notable differences are always below 0.12. In discharge the discrepancies are larger. Values are usualy kept below 0.08. There is a tendency of PDWAVE to distance itself from both GWM and SWE as seen by an almost line formed by the markers present in the discharge and head and, except for some outliers, the relation between GWM and SWE is usually less than 0.02.

values are with a very good agreement. Manhole heads are kept within a correlation higher than 0.975 with the exception of only a few points. The same holds for the discharges, showing a very good correlation between all the datasets. It should be noticed that the GWM vs. SWE and SWE vs. GWM relations have considerably higher correlations than the ones with PDWAVE involved with the lower value of 0.995 for the head and 0.98 for the discharge.

Node 2190 has some of the major differences between models with regard to discharges. The maximum difference between PDWAVE and the other models is 0.1 m and the discharge 0.04 m³/s. The manhole head is solely changed by the sewer system and discharges are from the surface to the manhole as seen in the right figure. The major differences are explained by a 0.04 m difference (see Table 1) on the surface between PDWAVE and the other models. Between GWM and SWE the difference is considerably smaller and is in full scale due to the surface flow. The maximum difference in discharge is small and lower than 0.005 m³/s, which is a good agreement.

Node 3056 shows the worst correlation regarding manhole heads. The difference is relatively small, 0.05 m and can be easily explained by a difference in the surface elevation as shown in Table 1. The difference is of 0.1 m. The discharge is controlled not only by the manhole head but also the differences between the elevation on the surface and the manhole head. This shows that the surface flow elevation is smaller in GWM and SWE.

Node 3004 has 0.08 m difference in the surface elevation between PDWAVE and the other models. Besides this difference, the surface crest is in between these values. These differences force a higher water level in PDWAVE as seen in the beginning of the manhole head figure and a lower level in GWM and SWE. The head of the manhole is therefore very close to the manhole crest elevation in GWM and SWE and to the surface elevation in PDWAVE. PDWAVE has some oscillations as a small part of the flow is retained in the surface and directly influences the manhole head. GWM and SWE have a small difference in elevations between the crest and the surface that allows the flow to leave the cell without disturbing the manhole. Therefore, the head inside the manhole is always the crest elevation and all the flow that would increase the manhole head is expelled to the surface. This situation increases the discharge as seen in the right figure during the first hour. After the first hour, the discharge diminishes and the head increases as a direct influence of the surface flow inverting the flow from a surcharge to a discharge.

Node 3079 is one of the few nodes where the flow is truly bidirectional. The manhole crest is at 86.38 m whilst the surface is at 86.42 m for all models. The discharge only happens when the manhole hydraulic head exceeds the surface water level and the behaviour of all models is similar. The results almost overlap except for the descending branch where GWM and SWE have a smoother transition. This transition is due to the influence of the sewer network as there is a faster decrease of head for GWM and SWE. Eventually the models return to similar values since all the flow that arrives at the cell is drained to the sewer. Some discharge oscillations are seen in GWM and SWE after 3 h. The unsteady nature of both models can explain this as some oscillations occur in the surface flow. The discharge in GWM and SWE is also slightly higher than in PDWAVE.

Node 1912 is controlled by the manhole head and is therefore a good indicator of the global changes until 30 min. Afterwards the differences reflect the flow over the manhole. PDWAVE achieves a faster peak whilst GWM and SWE achieve a higher head peak. The discharges are almost overlapping with some small differences that are a reflection of the head in the manhole.

Node 3067 also shows consistency between data as shown in Table 1 for of Average Balance and Head. The elevations are very similar, with only a small divergence for 3067 when the head becomes lower than the crest resulting from a global sewer system response. The oscillations seen are due to the proximity of the crest to the surface elevation in the SWE and GWM whilst for PDWAVE the manhole crest is 0.08 m higher than the surface.

Overall the results are in very good agreement with some major differences explained by the different discretization and some small differences attributed to the models. The manhole's head and discharges are very similar between models. The results also show that an equal representation of the surface and the linkage is of the utmost importance since some major differences can occur due to different representations. As expected GWM has closer results to SWE than PDWAVE. This is due to the differences in the governing equations but also on the discretization. Indeed GWM and SWE share the same irregular mesh, and therefore the models share the same exact elevations at the surface (Table 1).

PDWAVE on the contrary uses a regular grid, and therefore the surface node elevations inevitably differ from the other two models.

Figure 7 shows that there is no dominant model as the distribution has roughly the same area for all the models. PDWAVE has predominance upstream. This is also seen near the manhole with the highest discharges showing that PDWAVE tends to ‘retain’ more water upstream as opposed to GWM that shows higher depths downstream due to a higher but slower wave front.

Figure 8 shows the direct comparison between all the models tested for the three time steps mentioned along with the maximum extent. The figure is arranged so that the time steps are presented in columns and the comparison in rows. In each figure, the horizontal axis represents model 1 and the vertical axis for model 2. This is assessed by the first and second model in the row's header. Two functions based on Equation (30) where plotted. These two functions imply that between them there is a 95% confidence that any value of model 1 is within 2 of model 2. Each dot represents a cell depth for the compared models and the colour of the area where the point is.

The analysis from Figure 8 shows a high correlation for GWM vs. SWE. For 50th minute the red area shows a small difference in depth as GWM has a higher depth than SWE. The vast majority of points are near the origin and the values are very similar between themselves. The points then spread through the line showing a very good relation. The 80th minute is when the differences between the models are larger. The confidence interval is larger than except for the maximum extent and some areas show slight disagreement, namely the area shown in cyan. This area has a flow with up to 30 cm difference and the difference is due to no loss in the momentum when the flow goes over a small bump in the south-east of the area. The red and light green area also shows some differences, in the magnitude of 10 cm once again due to a slower but higher depth of propagation by GWM. Minute 110, with the exception of some scattered points shows a slightly higher spread than minute 50 but with a larger concentration of wet cells. Red, cyan, blue and orange stand out as the areas with the most differences. This difference is however of small magnitude with higher values of about 10 cm.

GWM vs. PDWAVE (Figure 8) has a larger difference than GWM vs. SWE. At 50 min the results, although more scattered than SWE vs. GWM show a good agreement and the spread is kept within 15 cm. The most noticeable areas are the red and dark green. The red area is kept within reasonable limits except for some points that lie outside the confidence interval, for the dark green area, some points with water show difference up to 20 cm. GWM predicts some flooded areas that PDWAVE does not that can have differences up to 80 cm in the green and orange area. For the 80th minute the differences increase. Light blue areas are now predicted higher in PDWAVE than GWM and blue and green areas higher in GWM than PDWAVE. This is a direct outcome of the mesh discretization. Flow is restricted in some areas due to the structured mesh of PDAWAVE thus routing the flow through the cyan area to the light blue whilst for GWM the flow runs through cyan to dark green. The red area is stable and with a very high resemblance. At 110th min the scatter is diminished and the depth starts to converge as flow arrives. GWM flooded point that had no depth in PDWAVE starts now to get flooded in dark green area and the points reflect that as they tend to decrease the difference somewhat linearly.

SWE vs. PDWAVE has a similar trend as GWM vs. PDWAVE. The 50th and 110th minutes are very similar with the same zones flooded. The main differences reside in the red area, that is more similar between SWE and PDWAVE than GWM and PDWAVE due to fewer nodes outside the confidence interval. For minute 80, the light blue area shows a different tendency than GWM vs. PDWAVE. With still some differences, the light blue area shows an increase linearly whilst in GWM vs. PDWAVE there was a complete scatter. This could point to a place where the PDWAVE flow is more similar to SWE than GWM is to SWE.

The results on Figure 10 show that the coefficient in Equation (34) for SWE and GWM is always very high. The comparisons with PDWAVE for both the GWM and SWE show a lower coefficient with the number of agreed cells always larger than the disagreed ones. After 30 min the coefficient surpasses 0.33 which shows that at least twice the number of the disagreeing cells are in agreement. This is in line with the previous results as all of the models are very similar in extent as shown in Figure 9.

In this paper three overland flow models (SWE, GWM and PDWAVE) with different levels of complexity in DD modelling were evaluated and compared using the Keighley test case. To enable a fair overland flow comparison, all three models were coupled with the same fully dynamic sewer network model (SIPSON). A coefficient was used to evaluate the models without assuming either as correct. The linkage is obtained by solving sequentially the sewer and the overland flow models with interpolated discharge values from the sewer model used as internal boundary condition of the surface model. This was based on an exchange formulation that allows for the connection between two models with different time steps by interpolating the discharges and heads.

Three indicators were adopted to compare and verify the agreement between models, namely: (a) a volume exchange between sewer and overland flow models; (b) the hydraulic head and discharge rates at the linked manholes; and (c) the inundation extents and model predominance. A novel methodology based on RGB scale was used to interpret the results, which was particularly useful for the situation where none of the three sets of model results is considered as a benchmark (‘accurate’) solution.

For this case study (and given the predominance of relatively low slopes), the results are consistent between all coupled models in terms of the extents of flooding, depth and volume exchanged. Nonetheless, it was also shown for the test conducted that the PDWAVE has higher maximum flood depth and has a slower propagation of the flood front. The GWM showed higher depths downstream as the wave front is slower and therefore higher than SWE. Overall the results show a better agreement between GWM and SWE than between PDWAVE and the other models, and the simplified overland models can produce comparable results to fully dynamic models with less computational cost.

We are grateful to Bradford City Council and John Blanksby for the access to the Keighley pipe network data. Thanks are also due to the UK Geomatics Group, Environment Agency, and Ordinance Survey for the provision of LiDAR and digital map data, respectively. The data used were obtained from the cited references. This research is partially funded by the FCT (Portuguese Foundation for Science and Technology) through the Doctoral Grant SFRH/BD/81869/2011 financed through the POPH/FSE program (Programa Operacional Potencial Humano/Fundo Social Europeu). This study had the support of the Portuguese Foundation for Science and Technology (FCT) Project UID/MAR/04292/2013. This research is part of the SINATRA project which is supported by the United Kingdom NERC Flooding from Intense Rainfall programme (grant NE/K00896X/1).