# Where to Find Water Pipes and Sewers?—On the Correlation of Infrastructure Networks in the Urban Environment

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Materials and Methods

#### 2.1. Geometric Analysis and Parameter Sensitivity

#### 2.2. Graph Analysis

#### 2.2.1. Cycle Indicator (CI)

^{2}) is used for generating an MWG (|E| and |V| are the number of edges and vertices). Based on the MWG the cycle indicator (CI) can be evaluated. The overall runtime complexity of the MWG algorithm (Steiner tree approximation) is the sum of the MST algorithm complexity O(|E| log(|V|)) (Using Kruskal’s algorithm [21]) and TRIM algorithm complexity O(|V|

^{2}) (check each vertex if it is part of one unique edge and, if yes, deleting it). Additionally, more accurate, but also more computationally intense, approximations have been established [22]. In this work all analyses regarding the CI indicator are performed on datasets of water supply and sewer network graphs. Based on these graphs the generalized minimum Steiner tree is equivalent to a minimum spanning tree minus all non-terminal leafs. Hence, the choice of the approximation function has no impact on the results of CI.

#### 2.2.2. Leaf Indicator (LI)

#### 2.3. Description of Case Studies (CS)

## 3. Results and Discussion

#### 3.1. Area of Interest

^{2}. CS2 and CS3 have nearly identical sizes resulting in close population densities when comparing these values with the population values in Table 1. Due to clipping the water supply and sewer network data to a common area of interest, some data is not regarded. The amount of clipped data is shown by comparing the total network length of the original input dataset (Table 1: rows 6 and 8) with that of the clipped data (Table 2: rows 7 and 9). Disregarded data of sewer networks is marginal for all three case studies. Neglected data of the water supply networks for the CS1 is also marginal. CS2 and CS3 show a higher deviation, with 24% and 15% of the data being clipped, respectively. This clipping results from the lower service area of the sewer networks. These clipped network zones are usually those parts of the sewer network in which not all inhabitants are connected to the sewer system, or in which long main trunks occur in the water supply network. Comparing the widest diameter of pipes occurring within the clipped (Table 2: row 3) and original (Table 1: row 2) water supply dataset, CS2 shows that many portions of the main trunks are not considered in this case. This is because of missing information of the sewer network in the identical area and long main trunks from the reservoir to the valley within the water supply network. Generally speaking, the service area of water supply systems is more likely to be larger when compared to the service area of the sewer system within the identical case study.

#### 3.2. Parameter Sensitivity

#### 3.3. Geometric Analysis

#### 3.4. Graph Analysis

#### 3.5. Applications of the Results Obtained from the Graph and Geometric Analyses

## 4. Summary, Conclusions, and Outlook

## Acknowledgments

## Author Contributions

## Conflicts of Interest

## Appendix A

Pipe Type According to Case Study | Pipes below Different Street Types (%) | |||||
---|---|---|---|---|---|---|

Motorway | Primary | Secondary | Tertiary | Other | All | |

CS1: service pipes | 0.00 | 0.01 | 0.00 | 0.00 | 0.44 | 0.43 |

CS1: distribution mains | 0.40 | 5.29 | 1.31 | 8.37 | 49.37 | 57.79 |

CS1: secondary mains | 0.12 | 3.54 | 1.51 | 7.87 | 16.66 | 25.38 |

CS1: trunk mains | 0.25 | 1.35 | 1.08 | 2.16 | 6.42 | 9.67 |

CS1: All | 0.77 | 10.19 | 3.90 | 18.39 | 72.89 | |

CS2: service pipes | 0.00 | 0.00 | 0.00 | 0.04 | 1.70 | 1.73 |

CS2: distribution mains | 0.00 | 3.21 | 1.18 | 8.82 | 47.59 | 56.93 |

CS2: secondary mains | 0.00 | 1.91 | 2.61 | 0.62 | 12.01 | 15.74 |

CS2: trunk mains | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 |

CS2: All | 0.00 | 5.12 | 3.78 | 9.48 | 61.31 | |

CS3: service pipes | 0.00 | 0.00 | 0.00 | 0.02 | 0.19 | 0.20 |

CS3: distribution mains | 0.08 | 7.91 | 5.73 | 5.45 | 44.27 | 59.84 |

CS3: secondary mains | 0.08 | 0.09 | 2.75 | 0.17 | 4.67 | 6.82 |

CS3: trunk mains | 0.00 | 0.00 | 0.00 | 0.00 | 0.01 | 0.02 |

CS3: All | 0.16 | 8.00 | 8.49 | 5.65 | 49.14 |

Conduit Diameter d (cm) According to Case Study | Conduits below Different Street Types (%) | |||||
---|---|---|---|---|---|---|

Motorway | Primary | Secondary | Tertiary | Other | All | |

CS1: $\mathit{d}<25$ (%) | 0.04 | 0.06 | 0.01 | 0.12 | 2.97 | 2.88 |

CS1: $25\le \mathit{d}<100$ (%) | 0.24 | 5.42 | 2.37 | 8.75 | 62.16 | 65.60 |

CS1: $\mathit{d}\ge 100$ (%) | 0.13 | 4.17 | 1.25 | 6.84 | 19.58 | 23.90 |

CS1: $\mathit{d}\ge 0$ (%) | 0.41 | 9.64 | 3.63 | 15.70 | 84.71 | |

CS2: $\mathit{d}<25$ (%) | 0.00 | 0.12 | 0.16 | 0.92 | 14.11 | 14.35 |

CS2: $25\le \mathit{d}<100$ (%) | 0.00 | 2.79 | 3.90 | 8.57 | 50.43 | 60.74 |

CS2: $\mathit{d}\ge 1000$ (%) | 0.00 | 1.30 | 0.33 | 0.12 | 7.23 | 8.21 |

CS2: $\mathit{d}\ge 0$ (%) | 0.00 | 4.20 | 4.39 | 9.62 | 71.76 | |

CS3: $\mathit{d}<25$ (%) | 0.00 | 0.01 | 0.77 | 0.41 | 8.43 | 9.24 |

CS3: $25\le \mathit{d}<100$ (%) | 0.09 | 4.91 | 5.96 | 5.48 | 51.25 | 63.39 |

CS3: $\mathit{d}\ge 100$ (%) | 0.00 | 0.11 | 3.03 | 0.53 | 2.72 | 5.81 |

CS3: $\mathit{d}\ge 0$ (%) | 0.09 | 5.03 | 9.75 | 6.41 | 62.40 |

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**Figure 4.**Sensitivity analysis of additional street width and covered network length below the streets for three alpine case studies.

Row | CS1 | CS2 | CS3 | |
---|---|---|---|---|

Population (-) | 1 | 120,000 | 13,000 | 11,000 |

Widest pipe diameter (mm) | 2 | 600 | 500 | 250 |

Widest conduit diameter (mm) | 3 | 2444 | 2130 | 1800 |

Lowest pipe diameter (mm) | 4 | 25 | 31 | 30 |

Lowest conduit diameter (mm) | 5 | 150 | 100 | 150 |

Total water supply network length (km) | 6 | 217.67 | 64.2 | 61.9 |

Pipe length/inhabitant (m) | 7 | 1.81 | 4.93 | 5.63 |

Total Sewer network length (km) | 8 | 202.88 | 41.8 | 47.1 |

Conduit length/inhabitant (m) | 9 | 1.69 | 3.22 | 4.28 |

Row | CS1 | CS2 | CS3 | |
---|---|---|---|---|

Area of interest—AI (km^{2}) | 1 | 25.14 | 5.88 | 5.35 |

Population density (Pop/ km^{2}) | 2 | 4774 | 2210 | 2056 |

Widest pipe diameter (mm) | 3 | 600 | 305 | 250 |

Widest conduit diameter (mm) | 4 | 2,440 | 2130 | 1800 |

Lowest pipe diameter (mm) | 5 | 25 | 31 | 30 |

Lowest conduit diameter (mm) | 6 | 150 | 100 | 150 |

Total water supply network length (km) | 7 | 216.75 | 48.7 | 52.20 |

Pipe length/inhabitant (m) | 8 | 1.81 | 3.74 | 4.75 |

Total sewer network length (km) | 9 | 201.93 | 41.4 | 46.82 |

Conduit length/inhabitant (m) | 10 | 1.68 | 3.19 | 4.26 |

Total street network length (km) | 11 | 419.48 | 78.1 | 67.87 |

Type | Motorway | Primary | Secondary | Tertiary | Other |
---|---|---|---|---|---|

Assumed width (m) | 21 | 14 | 14 | 13 | 2 |

Corrected width (m) | 28 | 21 | 21 | 20 | 9 |

Case Study | Motorway | Primary | Secondary | Tertiary | Other |
---|---|---|---|---|---|

CS1 | 6.12 | 7.67 | 2.60 | 7.81 | 75.80 |

CS2 | 0.35 | 6.89 | 2.62 | 8.37 | 81.77 |

CS3 | 2.97 | 5.48 | 7.88 | 3.78 | 79.88 |

CS1 | CS2 | CS3 | Average | |
---|---|---|---|---|

#Nodes (-) | 7128 | 790 | 452 | |

#Edges (-) | 7600 | 878 | 602 | |

85% percentile of CI (%) | 49.84 | 58.53 | 60.67 | 56.35 |

LI (%) | 11.18 | 12.81 | 11.89 | 11.96 |

CS1 | CS2 | CS3 | Average | |
---|---|---|---|---|

#Nodes (-) | 5395 | 1232 | 1700 | |

#Edges (-) | 5756 | 1248 | 1710 | |

85% percentile of CI (%) | 34.98 | 19.25 | 29.29 | 27.84 |

LI (%) | 28.62 | 80.83 | 64.68 | 58.04 |

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**MDPI and ACS Style**

Mair, M.; Zischg, J.; Rauch, W.; Sitzenfrei, R.
Where to Find Water Pipes and Sewers?—On the Correlation of Infrastructure Networks in the Urban Environment. *Water* **2017**, *9*, 146.
https://doi.org/10.3390/w9020146

**AMA Style**

Mair M, Zischg J, Rauch W, Sitzenfrei R.
Where to Find Water Pipes and Sewers?—On the Correlation of Infrastructure Networks in the Urban Environment. *Water*. 2017; 9(2):146.
https://doi.org/10.3390/w9020146

**Chicago/Turabian Style**

Mair, Michael, Jonatan Zischg, Wolfgang Rauch, and Robert Sitzenfrei.
2017. "Where to Find Water Pipes and Sewers?—On the Correlation of Infrastructure Networks in the Urban Environment" *Water* 9, no. 2: 146.
https://doi.org/10.3390/w9020146