# A New Rapid Simplified Model for Urban Rainstorm Inundation with Low Data Requirements

^{1}

^{2}

^{3}

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## Abstract

**:**

## 1. Introduction

## 2. Materials and Methods

#### 2.1. Overview

- (1)
- Because there is a one-to-one correspondence between drainage basins and multiple flood sources, the fill/spilling processes of inundation from each flood source can be simulated in a distributed way.
- (2)
- Compared with irregular storage cells in existing models, drainage basins with larger sizes can reduce the number of calculation cells. The fewer the calculation cells in the model, the fewer are the calculation steps that need to be carried out, thus raising the computation efficiency of the inundation model.
- (3)
- By simplifying the inundation spaces of drainage basins to regular spatial geometries, inundation modeling with a good balance between accuracy and cost-efficiency can be achieved when detailed topographic features cannot be represented explicitly in input data.

#### 2.2. Construction of Calculation Cells

- (1)
- Area. The area of each calculation cell (m
^{2}) can be calculated according to the number and resolution of DEM cells inside each calculation cell. - (2)
- Mean elevation. The mean elevation of all DEM cells located in each calculation cell (m) can be obtained based on the statistical analysis tools in ESRI ArcGIS.
- (3)
- Mean slope. The average value of the slopes at all DEM cells located in the calculation cell (%). The slope of each DEM cell can be calculated by the surface analysis tools in ESRI ArcGIS.
- (4)
- Storm gate capacity. To simplify the interactions between the surface and the drainage sewer system, this study assumes that the capacity of the storm gate inside each calculation cell is equal to the full design capacity of the downstream drain pipe (Q
_{f}, m^{3}/s), which can be computed by [33]:$${Q}_{f}=\frac{1}{{n}_{c}}{{A}_{f}{R}_{f}}^{2/3}{{S}_{f}}^{1/2}$$_{c}is the Manning’s roughness coefficient of the drain pipe, A_{f}(m^{2}) is the full cross-section area of the drain pipe, R_{f}(m) is the hydraulic radius for full drain pipe flow, and S_{f}is the friction slope of the drain pile, which can be defined as:$${S}_{f}=\frac{{e}_{s}-{e}_{e}}{l}$$_{s}and e_{e}(m) are elevations of the start node and end node of the drain pipe, respectively, and l (m) is the length of the drain pipe. - (5)
- Flood Volume. The flood volume q (m
^{3}) of each calculation cell is expressed as follows:$$q={q}_{s}+{q}_{e}$$_{s}(m^{3}) is the overflow volume from the storm gates when the corresponding drain pipes reach their capacities and q_{e}(m^{3}) is the volume of water that cannot be conveyed in time by the drainage system after the storm begins. Both q_{s}and q_{e}can be obtained by establishing the Storm Water Management Model (SWMM) developed by the U.S. Environmental Protection Agency (EPA), which can provide reliable simulation for the water flow in drainage sewer systems and surcharged flow at storm gates.

#### 2.3. Filling Processes inside Each Calculation Cell

^{3}), based on the filling/spilling assumption of the inundation scenario, Q can be represented by:

^{i}is the flood volume flowing in from the adjacent cells, and q

^{o}is the volume of residual water spilling out to adjacent cells. q

^{i}and q

^{o}can be obtained using the method outlined in Section 2.4.

^{3}) is the volume of stormwater distributed in each calculation cell. In TRSFM, Equation (2) can be solved by establishing a volume-level function [17] or flood routing algorithm [23,24] for water inside each impact zone according to the DEM data. For the NRSFM proposed in this study, the interior inundation space of each cell is simplified to a regular spatial geometry according to its topographic features, and the volume-level function can then be easily derived in a geometric way, which reduces the computational complexity. The rainstorm inundation always first appears near the storm gate, which is generally located in the middle region with a low elevation for each cell, and then expands to the boundary. The storm gate thereby serves as the vertex, and the inundation space of each cell is approximated as an upside-down circular cone with the same topographic features, called the drainage basin cone. The surface of each cell is approximated by the bottom circle of the drainage basin cone with the same value of area (see Figure 2).

^{o}to be zero, based on the mass balance principles in the drainage basin cone, Equation (2) can be rewritten as:

^{o}> 0. In this case, the residual water will spread across the boundary of the cell and flow to other cells. Let A

_{i}and h

_{i}be the inundation area and depth; they can be computed as follows:

#### 2.4. Spilling Processes between Calculation Cells

- (1)
- For a current calculation cell in an urban setting, let E be its mean elevation; the downward slope from the current cell to one of its adjacent cells can then be written as:$$(E-{E}^{\prime})/d$$$$d=\sqrt{{\left(p.x-{p}^{\prime}.x\right)}^{2}+{\left(p.y-{p}^{\prime}.y\right)}^{2}}$$
- (2)
- For each calculation cell adjacent to the current one, the downward slope is calculated and its value is recorded. Then, the adjacent cell with the maximum downward slope can be defined as the destination of the residual water.
- (3)
- For each normal cell in the study area, Steps (1) and (2) are used to find the flow direction of the spreading water.
- (4)
- For the current cell, if all of the adjacent cells satisfy $e-{e}^{\prime}<0$, the current cell is a depression cell. When a depression cell has been entirely flooded, the residual water will not immediately spread to the other cells but will accumulate above the inundation cone until the water level is higher than the mean elevation of any neighbour cell that has not yet been entirely flooded.

^{o}can be computed as:

_{c}is the volume of the drainage basin cone and can be calculated as follows:

^{o}is the height of the cylinder; based on the mass conservation principles, h

^{o}can be written as:

^{o}be the mean elevation of the lowest neighbourhood cell that has not yet been entirely flooded, and e

^{s}be the elevation of storm gate. If ${e}^{s}+{h}^{o}+h\le {e}^{o}$, there will be no water spilling out from the current cell, and the maximum inundation depth can be expressed as h

^{o}+ h. If e

^{s}+ h

^{o}+ h > e

^{o}, the residual water will finally flow to the neighbour cell with a mean elevation of e

^{o}, and q

^{o}can be rewritten as:

^{o}to q

^{i}of the next cell according to its flow direction. Combined with its own q and q

^{o}, the inundation scenario inside the next calculation cell can be then simulated using the filling model introduced in Section 2.3. The simulation procedures above are repeated until the flood volume from every flood source has been assigned and the results of the inundation scenarios over the entire study urban setting are then obtained.

## 3. Case Study

#### 3.1. Study Area

^{2}, mainly of residential land use with some institutional and educational portions and some open areas with forest cover and shrub cover. However, no significant water system can be found in the satellite image. According to the topographical information derived from a 30-m resolution DEM dataset (provided by the NASA ASTER global digital elevation map (GDEM)), the study area is mainly composed of flat topography with a slight gradient in the south, hills in the north, and a range of elevation between 24 m and 147 m. The campus has suffered from flood inundation disasters for years because of the high frequency of severe rainstorm events in Wuhan and the deficiencies in the drainage system. In 13 July 2013, Wuhan City was hit by a six-year-frequency rainstorm event that caused serious flood inundation in the middle-west of the campus and severely threatened the safety and property of students. As a result, there is obvious practical value in applying the NRSIM and other inundation models to this area.

^{3}and 2081 m

^{3}).

^{2}, is surrounded by East Lake to the north, South Lake to the west, Tang Xun Lake to the south, and hills to the east. Using the method in Section 2.2, the floodplain was divided into 2992 drainage basins depending on the spatial distribution of 2992 major storm gates. Based on the simulation results for the six-year, 2-h design rainstorm provided by the SWMM, 216 storm gates were identified as flood sources in the large-scale floodplain (see Figure 10b).

#### 3.2. Model Validation

- (1)
- Fit indicator of inundation extent (FIE).$$FIE=\frac{{N}_{o}}{{N}_{t}+{N}_{o}+{N}_{l}}$$
_{o}is the number of overlapping monitoring points that were simulated as flooded points by both models, N_{t}is the number of monitoring points flooded in the testing model (NRSIM or TRFSM) but dry in the LISFLOOD model, and N_{l}is the number of monitoring points that were dry in the testing model but flooded in the LISFLOOD model. Apparently, the value range of FIE is from 0 to 1, and a higher FIE represents a better performance of inundation extent simulation. - (2)
- Mean depth deviation (MDD) is the mean value of depth deviations (DD) for all overlapping monitoring points. For each point, DD was represented as the relative error between simulated depth and reference depth provided by LISFLOOD:$$DD=\frac{\left|{D}_{t}-{D}_{l}\right|}{{D}_{l}}\times 100\%$$
_{t}is the maximum depth value simulated by the testing model (NRSIM or TRFSM) and D_{l}is the maximum depth value simulated by LISFLOOD. A smaller MDD represents a better performance of the maximum inundation depth simulation.

## 4. Results and Discussion

#### 4.1. Inundation Simulation for the Campus of HUST

#### 4.2. Inundation Simulation for a Large-Scale Floodplain

#### 4.3. Validation for Time Efficiency

## 5. Conclusions

## Acknowledgments

## Author Contributions

## Conflicts of Interest

## References

- Shuster, W.D.; Bonta, J.; Thurston, H.; Warnemuende, E.; Smith, D.R. Impacts of impervious surface on watershed hydrology: A review. Urban Water J.
**2005**, 4, 263–275. [Google Scholar] [CrossRef] - Kubal, C.; Haase, D.; Meyer, V.; Scheuer, S. Integrated urban flood risk assessment—Adapting a multicriteria approach to a city. Nat. Hazards Earth Syst. Sci.
**2009**, 9, 1881–1895. [Google Scholar] [CrossRef] - Phillips, B.C.; Yu, S.; Thompson, G.R.; de Silva, N. 1D and 2D Modelling of Urban Drainage Systems using XP-SWMM and TUFLOW. In Proceedings of the 10th International Conference on Urban Drainage, Copenhagen, Denmark, 21–26 August 2005.
- Hunter, N.M.; Horritt, M.S.; Bates, P.D.; Wilson, M.D.; Werner, M.G.F. An adaptive time step solution for raster-based storage cell modelling of floodplain inundation. Adv. Water Resour.
**2005**, 28, 975–991. [Google Scholar] [CrossRef] - Shih, D.; Yeh, G. Identified model parameterization, calibration, and validation of the physically distributed hydrological model WASH123D in Taiwan. J. Hydraul. Eng.
**2011**, 16, 126–136. [Google Scholar] [CrossRef] - Shih, D.; Chen, C.H.; Yeh, G. Improving our understanding of flood forecasting using earlier hydro-meteorological intelligence. J. Hydrol.
**2014**, 512, 470–481. [Google Scholar] [CrossRef] - O’brien, J.S.; Julien, P.Y.; Fullerton, W.T. Two-Dimensional water flood and mudflow simulation. J. Hydraul. Eng.
**1993**, 119, 244–261. [Google Scholar] [CrossRef] - Sampson, C.C.; Smith, A.M.; Bates, P.B.; Neal, J.C.; Alfieri, L.; Freer, J.E. A high-resolution global flood hazard model. Water Resour. Res.
**2015**, 51, 7358–7381. [Google Scholar] [CrossRef] [PubMed] - Bates, P.D.; De Roo, A.P.J. A simple raster-based model for flood inundation simulation. J. Hydrol.
**2000**, 236, 54–77. [Google Scholar] [CrossRef] - Pietro, P. Suitability of the diffusive model for dam break simulation: Application to a CADAM experiment. J. Hydrol.
**2008**, 361, 172–185. [Google Scholar] - Bates, P.B.; Horritt, M.S.; Fewtrell, T.J. A simple inertial formulation of the shallow water equations for efficient two-dimensional flood inundation modeling. J. Hydrol.
**2010**, 387, 33–45. [Google Scholar] [CrossRef] - Da Paz, A.R.; Collischonn, W.; Tucci, C.E.M.; Padovani, C.R. Large-scale modelling of channel flow and floodplain inundation dynamics and its application to the Pantanal (Brazil). Hydrol. Process.
**2011**, 25, 1498–1516. [Google Scholar] [CrossRef] - Neal, J.C.; Schumann, G.; Bates, P.B. A subgrid channel model for simulating river hydraulics and floodplain inundation over large and data sparse areas. Water Resour. Res.
**2012**, 48, 328. [Google Scholar] [CrossRef] - Néelz, S.; Pender, G. Benchmarking of 2D Hydraulic Modeling Packages; Environment Agency: Bristol, UK, 2010.
- Neal, J.; Villanueva, I.; Wright, N.; Willis, T.; Fewtrell, T.; Bates, P. How much physical complexity is needed to model flood inundation? Hydrol. Process.
**2012**, 15, 2264–2282. [Google Scholar] [CrossRef] - Gouldby, B.; Sayers, P.; Mulet-Marti, J.; Hassan, M.; Benwell, D. A methodology for regional-scale flood risk assessment. Proc. Inst. Civ. Eng. Water Manag.
**2008**, 161, 169–182. [Google Scholar] [CrossRef] - Lhomme, J.; Sayers, P.B.; Gouldby, B.P.; Samuels, P.G.; Wills, M.; Mulet-Marti, J. Recent development and application of a rapid flood spreading method. In Proceedings of the Flood Risk 2008 Conference, Oxford, UK, 30 September–2 October 2008; Taylor and Francis Group: London, UK, 2008. [Google Scholar]
- Zerger, A.; Smith, D.I.; Hunter, G.J.; Jones, S.D. Riding the storm: A comparison of uncertainty modeling techniques for storm surge risk management. Appl. Geogr.
**2002**, 22, 307–330. [Google Scholar] [CrossRef] - Falter, D.; Vorogushyn, S.; Lhomme, J.; Apel, H.; Gouldby, B.; Merz, B. Hydraulic model evaluation for large-scale flood risk assessments. Hydrol. Process.
**2013**, 27, 1331–1340. [Google Scholar] [CrossRef] - Liu, Y.; Pender, G. A new rapid flood inundation model. In Proceedings of the First IAHR European Congress, Edinburgh, UK, 4–6 May 2010.
- Krupka, M.; Wallis, S.; Pender, S.; Neélz, S. Some practical aspects of flood inundation modeling. Publ. Inst. Geophys. Pol. Acad. Sci.
**2007**, E-7, 129–135. [Google Scholar] - Krupka, M. A Rapid Inundation Flood Cell Model for Flood Risk Analysis. Ph.D. Thesis, Department of Built Environment, Heriot-Watt University, Edinburgh, UK, 2008. [Google Scholar]
- Yang, T.H.; Chen, Y.C.; Chang, Y.C.; Yang, S.C.; Ho, J.Y. Comparison of Different Grid Cell Ordering Approaches in a Simplified Inundation Model. Water
**2015**, 7, 438–454. [Google Scholar] [CrossRef] - Chen, J.; Hill, A.A.; Urbano, L.D. A gis-based model for urban flood inundation. J. Hydrol.
**2009**, 373, 184–192. [Google Scholar] [CrossRef] - Turner, A.B.; Colby, J.D.; Csontos, R.M.; Batten, M. Flood Modeling Using a Synthesis of Multi-Platform LiDAR Data. Water
**2013**, 5, 1533–1560. [Google Scholar] [CrossRef] - Jung, Y.H.; Kim, D.Y.; Kim, D.W.; Kim, M.M.; Lee, S.O. Simplified Flood Inundation Mapping Based On Flood Elevation-Discharge Rating Curves Using Satellite Images in Gauged Watersheds. Water
**2014**, 6, 1280–1299. [Google Scholar] [CrossRef] - Blumensaat, F.; Wolfram, M.; Krebs, P. Sewer model development under minimum data requirements. Environ. Earth Sci.
**2012**, 65, 1427–1437. [Google Scholar] [CrossRef] - Chen, A.S.; Evans, B.; Djordjevic, S.; Savic, D.A. Multi-layered coarse grid modelling in 2D urban flood simulations. J. Hydrol.
**2012**, 470, 1–11. [Google Scholar] [CrossRef][Green Version] - Li, Z.F.; Wu, L.X.; Zhu, W.; Hou, M.L.; Yang, Y.Z.; Zheng, J.C. A New Method for Urban Storm Flood Inundation Simulation with Fine CD-TIN Surface. Water
**2014**, 6, 1151–1171. [Google Scholar] [CrossRef] - Zhao, D.Q.; Chen, J.N.; Wang, H.Z. GIS-based urban rainfall-runoff modeling using an automatic catchment-discretization approach: A case study in Macau. Environ. Earth Sci.
**2009**, 56, 465–472. [Google Scholar] - Lhomme, J.; Bouvier, C.; Perrin, J.L. Applying a GIS-based geomorphological routing model in urban catchments. J. Hydrol.
**2004**, 299, 203–216. [Google Scholar] [CrossRef] - Schmitt, T.G.; Thomas, M.; Ettrich, N. Analysis and modeling of flooding in urban drainage systems. J. Hydrol.
**2004**, 299, 300–311. [Google Scholar] [CrossRef] - Hsu, M.H.; Chen, S.H.; Chang, T.J. Inundation simulation for urban drainage basin with storm sewer system. J. Hydrol.
**2000**, 234, 21–37. [Google Scholar] [CrossRef][Green Version] - O’Callaghan, J.; Mark, D. The extraction of drainage networks from digital elevation data. Comput. Vis. Graph. Image Process.
**1984**, 28, 323–344. [Google Scholar] [CrossRef] - Ishigaki, T.; Kawanaka, R.; Onishi, Y.; Shimada, H.; Toda, K.; Baba, Y. Assessment of safety on evacuating route during underground flooding. In Advances in Water Resources and Hydraulic Engineering; Springer: Berlin/Heidelberg, Germany, 2009; pp. 141–146. [Google Scholar]

**Figure 1.**Model structure of new rapid simplified inundation model (NRSIM): (1) Filling model: for each flood source, the volume of inundation water is first distributed over the corresponding drainage basin by using the water spreading algorithm; and (2) Spilling model: water spilling between flooded drainage basins is computed by using an improved D8 algorithm.

**Figure 3.**Spatial distribution of a central digital elevation model (DEM) cell and its eight adjacent cells in the drainage basin cone.

**Figure 6.**The section diagram of inundation cylinders above the inundation cone of a depression cell.

**Figure 7.**Study area: main campus of Huazhong University of Science and Technology (HUST), Wuhan, China.

**Figure 10.**(

**a**) Study area of a large-scale floodplain in the city of Wuhan; and (

**b**) distribution of drainage basins and flood sources in the large-scale floodplain.

**Figure 11.**Comparison of inundation extent between LISFLOOD and: NRSIM (

**a**); and traditional rapid flood spreading model (TRFSM) (

**b**) using a 30-m resolution DEM.

**Figure 12.**Relationship between thresholds of flooded depth and fit indicator of inundation extent (FIE) of TRFSM and NRSIM by using a 30-m resolution DEM.

**Figure 13.**Comparison of inundation extent between LISFLOOD and: NRSIM (

**a**); and TRFSM (

**b**) using a 90-m resolution DEM.

**Figure 14.**Comparison of fit indicator (

**a**); and mean depth deviation (MMD) (

**b**) between NRSIM and TRFSM by using different resolution DEMs.

**Figure 15.**Comparison of inundation extents in a large-scale floodplain between LISFLOOD and: NRSIM (

**a**); and TRFSM (

**b**) using a 30-m resolution DEM. For a 90-m resolution DEM, the results for NRSIM and TRFSM are given in (

**c**,

**d**), respectively.

Drainage Basin | Flood Volume (m^{3}) | Drainage Basin | Flood Volume (m^{3}) |
---|---|---|---|

1 | 363 | 38 | 1974 |

8 | 851 | 40 | 540 |

15 | 2199 | 48 | 515 |

16 | 2103 | 61 | 457 |

28 | 169 | 62 | 2081 |

35 | 761 | 65 | 1168 |

**Table 2.**Comparison of computational efficiency between NRSIM, TRFSM and LISFLOOD using the computational times when applied to the study area and a large-scale urban floodplain.

Models | Study Area | Large-Scale Floodplain | ||
---|---|---|---|---|

30-m DEM | 90-m DEM | 30-m DEM | 90-m DEM | |

TRFSM | 55.8 s | 12.6 s | 27 min 52 s | 5 min 20 s |

NRSIM | 2.1 s | <1 s | 53.9 s | 17.5 s |

LISFLOOD | 4 min 3 s | 51.8 s | 2 h 10 min 25 s | 25 min 16 s |

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## Share and Cite

**MDPI and ACS Style**

Shen, J.; Tong, Z.; Zhu, J.; Liu, X.; Yan, F. A New Rapid Simplified Model for Urban Rainstorm Inundation with Low Data Requirements. *Water* **2016**, *8*, 512.
https://doi.org/10.3390/w8110512

**AMA Style**

Shen J, Tong Z, Zhu J, Liu X, Yan F. A New Rapid Simplified Model for Urban Rainstorm Inundation with Low Data Requirements. *Water*. 2016; 8(11):512.
https://doi.org/10.3390/w8110512

**Chicago/Turabian Style**

Shen, Ji, Zhong Tong, Jianfeng Zhu, Xiaofei Liu, and Fei Yan. 2016. "A New Rapid Simplified Model for Urban Rainstorm Inundation with Low Data Requirements" *Water* 8, no. 11: 512.
https://doi.org/10.3390/w8110512