# Determination of Optimal Meshness of Sewer Network Based on a Cost—Benefit Analysis

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Materials and Methods

#### 2.1. Case Study

^{2}, of which 42% corresponds to impervious surfaces. Furthermore, the local environmental, agricultural and geological agency has identified 28 different land uses in the area (see Figure A1 [13]). In order to simplify the flood damage analyses, these land uses are reclassified to match the categories used in the depth–damage curves proposed by Huizinga et al. [14]. These curves are used in this study for analyzing flood damage. Further information about this is provided in Section 2.5. and Appendix A. The study area is reclassified into six major land uses: Agriculture, Industry, Commerce, Residential, Infrastructure and No Damage. This last one corresponds to green areas where flooding is not expected to cause any economic damage. Information regarding the reclassification process can be found in Appendix A. The structure of the sewer systems is quantified using the Meshness coefficient proposed by Reyes-Silva et al. [11], which indicates whether the structure of a sewer network is predominantly branched or meshed. This coefficient ranges between 0% and 100%, representing a network structure with the minimum or maximum possible number connected pipes per node. The results show that the analyzed subnetwork has a predominantly branched structure, with a Meshness value of 25%.

#### 2.2. Scenario Construction

#### 2.3. Hydrodynamic Simulations and Node Flooding

#### 2.4. Flood Depth and Flooded Area Determination

^{2}digital surface model obtained from the state service for geoinformation and geodesy Saxony (Staatsbetrieb Geobasisinformation und Vermessung Sachsen (GeoSN)). Each individual cell acts as storage and the cells located above a flooding node are considered as source cells. These cells are initially filled with the corresponding flood volume reported in the EPA SWMM results. Then, if topography allows, the algorithm distributes the flooding water of each source cell to its eight adjacent dry cells. Flood water is allocated into a nearby cell only if the height of the source cell is higher than the nearby one, thus resembling gravitational surface flow. Based on the distributed volume and the number of cells with water, a flood depth is calculated. If this depth is higher than a given threshold, the new cells filled with water act as new source cells and the process continues iteratively until a depth threshold is reached. When the process is over, it is possible to analyze the extent of the flooded area and the flood depth associated with each flooded node in the study area. Although this approach neglects several hydrodynamic processes of surface water diffusion [20], its results are considered appropriate enough to analyze the effects of different layouts on node flooding.

#### 2.5. Maximum Flood Damage Costs

- Max. damage
_{2019}= maximum damage for price level 2019; - Max. damage
_{year of issue}= maximum damage in the year of issue; - CPI
_{2019}= CPI for 2019; - CPI
_{year of issue}= CPI for the year of issue;

- D represents the damage in terms of €/m
^{2}(year 2019) - Subscripts a, id, c, r and if indicate the land use
- H corresponds to the flood depth in meters

#### 2.6. Pipe Related Costs

- C represents the pipe installation cost in terms of €/m
- D corresponds to the pipe diameter in mm

#### 2.7. Cost–Benefit Analysis

## 3. Results

#### 3.1. Influence of Meshness in Flood Volumes and Flood Damage

#### 3.2. Cost–Benefit Analysis

^{6}) for the year 2019. For each Meshness scenario, the flood damage results for the different return periods are combined to obtain an expected annual flood damage, measured in millions of Euros per year. For more information on how these values are combined to obtain an expected annual flood damage values, see Appendix B. As seen before, an increase in Meshness is associated with a decrease in flood damage. For instance, by comparing the scenarios with the lowest (0%) and highest Meshness possible (45%), it can be seen that in the last case, the expected annual damages are reduced by almost 20%. Furthermore, the present values of the expected damages for the entire pipe life cycle are calculated by multiplying the expected annual values by 50 years (life expectancy of pipes) without considering inflation. Lastly, the incremental benefits of each scenario are calculated as the difference between the total expected damage of the 0% Meshness case and the total expected damages for the corresponding scenario. Benefits here represent the amount of euros that are not lost due to flood damage by increasing the Meshness in the system. As an example, increasing Meshness from 0% to 5% leads only to a reduction in the expected flood damages for the entire life expectancy of the sewer system of 2.03 million Euros, while increasing Meshness up to 45% leads to a saving of 32 million Euros due to avoiding flood damage.

^{6}). The total investment for each scenario corresponds to the sum of the installation cost at the beginning of the project and the O&M cost during the life expectancy of the system. This value is obtained by multiplying the annual O&M fees by the pipe life expectancy (50 years) without considering inflation. Since an increase in Meshness can only be obtained by adding more pipes into the network, the total investment costs increase with higher Meshness degrees. In this context, costs are calculated as the difference between the total investments for the 0% Meshness case (i.e., the base conditions) and the total investments for the corresponding scenario. Costs thus represent the amount of additional money that needs to cover the installation and O&M of the additional pipes that lead to Meshness increments. As an example, for the 25% Meshness scenario, it is necessary to invest 7.33 million more for the initial installation and O&M of the system in comparison to the base scenario (0% Meshness).

^{6}€ per 1% increase in Meshness. On the other hand, increments of costs as a function of Meshness can be expressed with an exponential function (Figure 4B). Hence, for low Meshness values, costs increase at a slower rate than the benefits. As a consequence, the ratio between benefits and costs is higher for low Meshness values. This explains why the optimal B/C occurs in the 10% Meshness scenario and then decreases as Meshness increases. Considering this, it is hypothesized that there is a possible critical Meshness value which is economically unfeasible (B/C ratio below 1). This critical value should represent a network layout which is predominantly meshed, i.e., high Meshness and it is a cost-ineffective measure. Nevertheless, this remains to be tested.

## 4. Discussion

## 5. Summary and Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## Appendix A

Original Land Use | Reclassified Land Use |
---|---|

Mixed area | 50% Residential area + 50% Commercial |

Agriculture | Agriculture |

Special crops | |

Orchard | |

Economic grassland | |

Commercial area/technical infrastructure | Commercial |

Anthropogenic used special areas (excavation area, storage...) | Industry |

Field wood/group of trees (dense/closed), 100 m to 1 ha | Infrastructure |

Bushes | |

Green and open spaces | |

Traffic areas | |

Bedrocks | No Damage |

Afforestation | |

Damp forest | |

Running water | |

Vegetation accompanying the water | |

Mixed deciduous forest | |

Foliage-coniferous mixed forest | |

Deciduous forest (pure stock) | |

Mixed coniferous forest | |

Coniferous forest (pure stock) | |

Open areas | |

Ruderal corridor, perennial corridor | |

Still waters | |

Forest edge areas/pre-forests | |

Residential area | Residential area |

**Figure A1.**Land use classes in the study area (adapted from [13]).

## Appendix B

^{6}EUR/year. Similar procedures were done to obtain the expected annual damage values for the other Meshness scenarios.

Interval | Return Period [Year] | Probability [-] | Δ Probability [-] | Damage [10^{6} EUR] | Mean Damage [10^{6} EUR] | Expected Damage per Year [10^{6} EUR/Year] |
---|---|---|---|---|---|---|

2 | 0.5 | 0 | ||||

1 | 0.4 | 5.55 | 2.22 | |||

10 | 0.1 | 11.09 | ||||

2 | 0.05 | 12.20 | 0.61 | |||

20 | 0.05 | 13.30 | ||||

3 | 0.03 | 14.89 | 0.45 | |||

50 | 0.02 | 16.48 | ||||

4 | 0.01 | 17.66 | 0.18 | |||

100 | 0.01 | 18.84 | ||||

5 | 0.01 | 18.84 | 0.19 | |||

∞ | 0 | 18.84 |

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**Figure 2.**Flood volumes (

**A**) and flood damages (

**B**) for the different rain events as a function of Meshness, including the total storage capacity of the system for each analyzed Meshness degree.

**Figure 4.**Regressions between benefits (

**A**) and costs (

**B**) as a function of Meshness. Including fitted lines, their 95% prediction interval (red and magenta lines, respectively).

**Table 1.**Benefits of increasing Meshness in the UDN (Present value in euros 2019). Negative values represent savings due to avoiding flood damage.

Meshness [%] | Expected Flood Damage per Year [ 106 €/Year] | Flood Damages in Life Expectancy of 50 Years [106 €] | Total Incremental Benefits [106 €] |
---|---|---|---|

0 | 3.64 | 182.00 | - |

5 | 3.60 | 179.97 | −2.03 |

10 | 3.49 | 174.60 | −7.39 |

15 | 3.46 | 173.06 | −8.94 |

20 | 3.42 | 171.17 | −10.82 |

25 | 3.27 | 163.72 | −18.28 |

30 | 3.29 | 164.66 | −17.34 |

35 | 3.25 | 162.54 | −19.45 |

40 | 3.23 | 161.64 | −20.35 |

45 | 2.99 | 149.63 | −32.37 |

Meshness [%] | Installation [10^{6} €] | O&M [10 ^{6} €/Year] | O&M in Life Expectancy of 50 Years [10 ^{6} €] | Total Investment [10 ^{6} €] | Total Incremental Costs [10 ^{6} €] |
---|---|---|---|---|---|

0 | 81.05 | 0.227 | 11.35 | 92.41 | - |

5 | 81.69 | 0.230 | 11.48 | 93.17 | 0.76 |

10 | 82.78 | 0.233 | 11.64 | 94.42 | 2.01 |

15 | 83.66 | 0.236 | 11.78 | 95.44 | 3.03 |

20 | 85.10 | 0.241 | 12.03 | 97.13 | 4.72 |

25 | 87.36 | 0.247 | 12.37 | 99.73 | 7.33 |

30 | 88.28 | 0.250 | 12.51 | 100.79 | 8.38 |

35 | 92.68 | 0.261 | 13.06 | 105.74 | 13.33 |

40 | 94.33 | 0.267 | 13.37 | 107.71 | 15.30 |

45 | 96.90 | 0.275 | 13.74 | 110.64 | 18.24 |

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**MDPI and ACS Style**

Reyes-Silva, J.D.; Frauches, A.C.N.B.; Rojas-Gómez, K.L.; Helm, B.; Krebs, P. Determination of Optimal Meshness of Sewer Network Based on a Cost—Benefit Analysis. *Water* **2021**, *13*, 1090.
https://doi.org/10.3390/w13081090

**AMA Style**

Reyes-Silva JD, Frauches ACNB, Rojas-Gómez KL, Helm B, Krebs P. Determination of Optimal Meshness of Sewer Network Based on a Cost—Benefit Analysis. *Water*. 2021; 13(8):1090.
https://doi.org/10.3390/w13081090

**Chicago/Turabian Style**

Reyes-Silva, Julian D., Ana C.N.B. Frauches, Karen L. Rojas-Gómez, Björn Helm, and Peter Krebs. 2021. "Determination of Optimal Meshness of Sewer Network Based on a Cost—Benefit Analysis" *Water* 13, no. 8: 1090.
https://doi.org/10.3390/w13081090