# Topological Analysis and Application of Urban Drainage Network

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## Abstract

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## 1. Introduction

- A data sorting method for urban drainage networks is proposed, emphasizing the flow direction in the drainage network system, and the basic principle that is needed to follow in topology building of urban drainage networks is discussed.
- According to the connection relationship of the network topology relationship, we trace the potential drainage outfall for each drainage node and use the control area of the node with the same potential drainage information as a subcatchment.
- The allocation ratio of the rainwater between subcatchments is calibrated dynamically with a hydrodynamic model and then applied to further hydrological analysis.

## 2. Methodology

#### 2.1. Drainage Network Data Combing Method Based on Potential Outfall

#### 2.1.1. Correction of Flow Direction

- $V$, a set of vertices;
- $E\subseteq \left\{\left(x,y\right)\mid \left(x,y\right)\in {V}^{2}\mathrm{and}x\ne y\right\}$, a set of edges, which are ordered pairs of vertices.

- We name structure (a) as an “All-up” node, which is defined as a point upstream of several pipeline but not downstream of any pipeline. For such points, we should warn against it and only retain it after confirming that it is actually correct.
- We name structure (b) as “ALL-down” node. The definition of such a node is that the point is downstream of several pipelines, but not upstream of any pipeline. When such a node appears, an error should be prompted, because such a node will not appear in reality, and the inflow of the node needs to have a reasonable place to go.
- We name structure (c) “Cycle”, which is the same as the definition of cycle in graph theory. Starting from a node in this structure, it can return to itself according to the direction of the pipelines. Such structures make no sense when only gravity is driving them.

#### 2.1.2. Topology Analysis

- Analyze the in-degree and out-degree of each node.
- The node in-degree = 0 (no upstream), defined as type = 0, which is usually rain grate or household pipe entrance;
- The node in-degree > 0 and out degree > 0 (not the starting point or end point of the drainage network), defined as type = 1, which is generally a manhole;
- The node out-degree = 0 (without downstream), defined as type = 2, which is generally an outlet mouth or an inlet of a sewage plant.

- Mark potential outfall information.
- If a point type is 2, then its outfall information is marked by its own number;
- If a point type is not 2, the downstream of the point is retrieved according to the topological relationship, and the drainage outfall information of the downstream node is returned for recursion;

- Classification—Nodes with the same outfall information are defined as homogeneous nodes.

#### 2.2. Subcatchment Based on Network Connectivity

#### 2.3. Dynamic Split Ratio at Branchpoint of Bifurcation Tube

- $x$ = distance (ft);
- $t$ = time (sec);
- $A$ = flow cross-sectional area (ft
^{2}); - $Q$ = flow rate (cfs);
- $H$ = hydraulic head of water in the conduit ($Z+Y$) (ft);
- $Z$ = conduit invert elevation (ft);
- $Y$ = conduit water depth (ft);
- ${S}_{f}$= friction slope (head loss per unit length);
- $g$ = acceleration of gravity (ft/sec
^{2}).

- ${R}_{split}$ = percentage of a pipe to the total discharge of a branch;
- ${Q}_{a,b}$ = flow rate of the two pipes at the inlet.

#### 2.4. SBPO Modeling Method Details

- $d$= reservoir water depth (ft);
- $i$ = rate of rainfall (ft/s);
- $e$ = surface evaporation rate (ft/s);
- $f$ = infiltration rate (ft/s);
- $q$ = runoff rate (ft/s).

- $S$ = cross sectional area (m
^{2}); - $h$ = the water level (m);
- ${q}_{r}$ = runoff rate (m
^{3}/s); - ${q}_{d}$ = discharge rate (m
^{3}/s).

## 3. Study Area and Scenario Design

#### 3.1. Study Area

#### 3.2. Scenario Design

## 4. Results

#### 4.1. Results of Drainage System Topology Combing and Subcatchment Based on Potential Outfall

#### 4.2. Result of Outfalls’ Total Inflow

^{6}LTR, and the relative error is 2.18%, which indicates that the two simulation methods of SWMM and SBPO have a high degree of consistency in the discharge volume at the outfall, and the water allocation results of the two methods are the same. However, the SBPO method can significantly reduce the computational burden of the model and allow us to view the start and end points of the urban water flow movement from a realistic and meaningful perspective, rather than simply applying the subcatchment only as the calculation units of the service stormwater model. The results also prove the feasibility of using the network combing method based on the outfall to confirm the internal independent relationship of the drainage system.

^{3}/s. In order to observe the specific situation at the outfall, we further checked the flow process lines of outfall 688 with return periods of “1a”, “10a”, “50a”, “100a” and“6·28” (Figure 12):

## 5. Discussion

#### 5.1. Basic Drainage Network Data Processing

#### 5.2. Application of Potential Outfall Information Based on Network Node

## 6. Conclusions

- The point-line relationship of the drainage network system is generalized to a directed graph, and the topological relationship of the drainage network can be quickly and effectively sorted into the DAG diagram of the outfall leading to the river or sewage by changing and correcting the partial structures such as “ALL-DOWN”, “ALL-UP” and “CYCLE” defined by the in-out degree. This can greatly simplify the difficulty of sorting out the intricate underground drainage network data in urban areas.
- By tracing the network upstream from the outfall with the correct representation of the flow direction, the researchers can retrieve the potential outfall at each node of the network system. The relative independence of the drainage network system can be explained by clustering the drainage network system from this perspective. The branchpoint points defined by the potential outfall information are the key to water exchange between subcatchment areas and the keynote of overflow analysis in the combined sewer area.
- The SBPO modeling method is highly consistent with the SWMM of a high-resolution drainage network in terms of drainage outfall volume. Under the SBPO method, the time of computing the peak discharge and discharge duration can significantly reduce with the buffer effect of the storage on the drainage network system.

## Author Contributions

## Funding

## Data Availability Statement

## Conflicts of Interest

## Appendix A

Parameter Set | SPBO | SWMM | ||
---|---|---|---|---|

Ruoff-generation | Sub-catchment | Area, Width, Slope, Imperv, N-inperv, N-perv, Dstore-Imperv, Dstore-Perv, OUTLET | ||

Runoff-conveying | Drainage Network | Faraway Branchpoint | Near Branchpoint | Node: Invert EI MAX. Depth Initial Depth Surcharge Depth Ponded Area |

Topological information: Inlet Node, Outlet Node | Node: Invert EI MAX. Depth Initial Depth Surcharge Depth Ponded Area | |||

Conduit: Inlet Node Outlet Node Shape MAX. depth Length Roughless InletOffset OutletOffset InitialFlow | Conduit: Inlet Node Outlet Node Shape MAX. depth Length Roughless InletOffset OutletOffset InitialFlow |

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**Figure 2.**Suspicious topological structure in drainage network. (

**a**) All-up; (

**b**) ALL-down; (

**c**) Cycle.

**Figure 3.**Pipeline with flow direction opposite to slope. Where, A and B represent nodes; a, b and c represent pipeline.

**Figure 5.**(

**a**) Two pipelines with a diameter of 1 m. The inlet offset of one bifurcation pipe is constantly raised. (

**b**) The diameter of one pipeline is 1 m, and the diameter of the other pipeline is increased above it.

**Figure 7.**Location of the study area and related display. (

**a**) Study area; (

**b**) DEM; (

**c**) land use; (

**d**) distribution of network.

**Figure 13.**Branchpoint to outfall pipelines side profile. (

**a**) branchpoint”2160” to outfall “76”; (

**b**) branchpoint”2160” to outfall “1214”.

Design Frequency (a) | The Total Rainfall (mm) | The Biggest Rain Density (mm/5 min) |
---|---|---|

1 | 24.953 | 5.805 |

2 | 33.778 | 7.858 |

3 | 38.940 | 9.059 |

5 | 45.442 | 10.571 |

10 | 54.267 | 12.624 |

20 | 63.091 | 14.677 |

30 | 68.252 | 15.878 |

50 | 74.755 | 17.39 |

100 | 83.579 | 19.443 |

Potential Outfall | Area (ha) | Length of Conduits (m) | Number of Nodes | Number of Potential Outfalls | Combined System |
---|---|---|---|---|---|

(688) | 143.5 | 19,697.853 | 1146 | 1 | YES |

(76, 1214) | 103.9 | 14,906.404 | 1020 | 2 | YES |

(2998) | 20.01 | 2775.734 | 86 | 1 | YES |

(688, 860, 4193) | 19.04 | 1061.384 | 69 | 3 | YES |

(4916) | 16.13 | 2223.359 | 164 | 1 | NO |

(51) | 15.08 | 3090.572 | 235 | 1 | NO |

(76, 688, 1214) | 14.82 | 2725.267 | 219 | 3 | YES |

(4015) | 11.02 | 1117.715 | 95 | 1 | NO |

(50) | 7.202 | 994.165 | 122 | 1 | NO |

(1214) | 5.414 | 800.299 | 55 | 1 | YES |

(4016) | 5.067 | 783.716 | 32 | 1 | NO |

(688, 3284) | 3.903 | 49.525 | 59 | 2 | YES |

(4017) | 3.815 | 563.841 | 26 | 1 | NO |

(1094) | 1.681 | 656.529 | 16 | 1 | NO |

(4018) | 1.153 | 145.865 | 14 | 1 | NO |

(4020) | 0.897 | 81.848 | 9 | 1 | NO |

Computing Time (s) | Design Frequency | |||||||||
---|---|---|---|---|---|---|---|---|---|---|

1a | 2a | 3a | 5a | 10a | 20a | 30a | 50a | 100a | MEAN | |

SWMM | 273 | 280 | 288 | 290 | 291 | 296 | 290 | 302 | 311 | 291.22 |

SBPO | 62 | 60 | 65 | 78 | 80 | 86 | 91 | 93 | 98 | 79.22 |

Improve Efficiency (%) | 77.29 | 78.57 | 77.43 | 73.10 | 72.51 | 70.95 | 68.62 | 69.21 | 68.49 | 72.91 |

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## Share and Cite

**MDPI and ACS Style**

Ren, H.; Liu, S.; Li, M.; Zhang, H.; Wang, H.; Hao, X.; Cui, J. Topological Analysis and Application of Urban Drainage Network. *Water* **2022**, *14*, 3732.
https://doi.org/10.3390/w14223732

**AMA Style**

Ren H, Liu S, Li M, Zhang H, Wang H, Hao X, Cui J. Topological Analysis and Application of Urban Drainage Network. *Water*. 2022; 14(22):3732.
https://doi.org/10.3390/w14223732

**Chicago/Turabian Style**

Ren, Hancheng, Shu Liu, Min Li, Hongping Zhang, Huiying Wang, Xiaoli Hao, and Jie Cui. 2022. "Topological Analysis and Application of Urban Drainage Network" *Water* 14, no. 22: 3732.
https://doi.org/10.3390/w14223732