# Seismic Resilience Evaluation of Urban Multi-Age Water Distribution Systems Considering Soil Corrosive Environments

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## Abstract

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## 1. Introduction

Method | Resilience Indicator Type | Resilience Indicator |
---|---|---|

Agent-based method | Proxy metric | Resilience index [16]; combination of resilience index, reliability and redundancy [17]; modified resilience index [18]; flow entropy [20]. |

Simulation method | Hydraulic metric | Satisfaction degree index [12,23,24]; pressure, demand, water serviceability, and population impacted [25]. |

Network theory method | Topological metric | Statistical and spectral metrics [27]; weighted K-level shortest-path-based topological metric [28]; edge-betweenness-based topological metric [29]; minimum cut set-based system reliability [30]; topological resilience metric (TRM) [31]; modified TRM [11,32]. |

## 2. Methodology

#### 2.1. Corrosion Model of Buried Pipelines with Different Service Ages

_{t}denotes the corrosion area of the steel pipe’s cross-section (mm

^{2}), A

_{0}is the cross-sectional area of the steel pipe before corrosion (mm

^{2}), d

_{0}represents the wall thickness of the steel pipe prior to corrosion (mm), D

_{0}is the external diameter of the steel pipe before experiencing corrosion (mm), v

_{d}indicates the corrosion velocity of the pipe in the depth direction (mm/year), T is the service age of the pipe (year), and t is the time at which corrosion occurs.

_{b}is the corrosion rate of the pipe in the radial direction (mm/year), and other parameters are identical to those in Equation (2). Furthermore, the authors of [39] presented a model describing the degradation of steel’s mechanical properties, derived from tensile tests on corroded steel:

_{w}≈ γ

_{y}, f

_{u}, δ, and E

_{s}are the yield strength, ultimate strength, elongation, and modulus of elasticity of uncorroded steel, respectively, and f

_{y}′, f

_{u}′, δ′, and E

_{s}′ represent the corresponding properties for corroded steel. D

_{w}is the mass ratio of steel lost to uncorroded steel, closely approximating the steel’s area corrosion rate. By integrating Equation (2) through (5), the correlation between the mechanical properties of steel pipes across various soil environments and their service ages can be established.

#### 2.2. Seismic Fragility of Pipelines

_{R}(x) represents the seismic vulnerability function, wherein D denotes earthquake demand, and C represents seismic capability. IM signifies the ground motion intensity parameter, m is defined as the median value of the seismic vulnerability function, and β represents the logarithmic standard deviation of the seismic vulnerability function. Considering the assumption that both structural seismic demand (D) and seismic capacity (C) follow a normal distribution, and considering the uncertainty associated with the structural vulnerability function, this leads to the derivation of the structural analytical seismic vulnerability model, as follows:

_{D}

_{|IM}and m

_{C}are the median values of structural seismic demand and seismic capacity, respectively, and β

_{D}

_{|IM}, β

_{C}, and β

_{M}are the quantified values of structural seismic demand uncertainty, seismic capacity uncertainty, and modeling uncertainty, respectively.

_{max}) as indices for ground vibration strength and structural seismic response, respectively. In establishing the finite element model of buried pipeline, this study choses the shell-equivalent spring model, using the shell unit to simulate the pipe body, while simplifying the soil body into uniform elastic–plastic springs along the axial, vertical, and transverse directions, which are connected to the nodes of each finite element to simulate the constraints on the buried pipeline in all directions of the soil body [43], and introducing the equivalent boundary spring [44] at the end of the pipeline to consider the influence of the pipeline segments outside the model. Seismic fragility curves for welded continuous steel pipes were developed across various service environments (acidic, near-neutral, and alkaline), different service ages (10, 20, 30, and 40 years), and different pipe diameters (<250 mm, ≥250 mm to <500 mm, and ≥500 mm) based on analytical vulnerability methods, as illustrated in Figure 1 and Figure 2. It is important to note that the term “service age” specifically refers to the duration of pipe usage following the onset of corrosion. The seismic fragility curves of buried pipelines enabled the determination of the likelihood that the pipelines would remain basically intact, suffer moderate damage, or experience severe damage under various seismic impacts.

#### 2.3. Hydraulic Analysis Model

_{w}is the vector of pipe roughness coefficients of the WDS, A

^{T}is the transpose of matrix A, and Q

_{N}is the vector of node flows (m

^{3}/s). Among them, the flow rate at the leakage point is calculated using the pipe leakage model based on the outflow orifice [46], and the flow rate at the pipe breakage point is calculated using the virtual reservoir model proposed by Shi [47]. Furthermore, the pressure-driven analysis (PDA) method has been widely used in the post-earthquake hydraulic simulation of WDSs to avoid negative pressure [48,49]. In this study, the node demand and pressure relationship proposed by Gupta and Bhave [50] was used:

_{i}and Q

_{i}

^{req}are the available water flows (m

^{3}/s) and the original water demand flows (m

^{3}/s) at node i, respectively. H

_{i}is the actual water pressure (m) at node i, H

_{i}

^{min}is the minimal water pressure (m) at which the node i can supply water flows, and H

_{i}

^{des}is the water pressure (m) required to fulfill the original demand flows, while n is an empirical coefficient, generally taken as 1.5~2.

#### 2.4. Seismic Resilience Assessment Method for WDS

_{0}, an earthquake occurred, causing the SP to drop to a low level (SP

_{0}), which persisted until recovery activities commenced at t

_{1}. The period from t

_{1}to t

_{2}represents the recovery stage, and by t

_{2}, SP gradually recovered to its pre-earthquake level.

_{i}(t) is the weight coefficient of node i at time t, and at any t, Σw

_{i}(t) = 1, n is the number of nodes, and NSD

_{i}(t) is the satisfaction degree of node i at time t, which is given by [23] as:

_{i}(t) and H

_{i}

^{des}(t) are the post-earthquake free water pressure (m) and demand water pressure (m) at node i at time t, respectively.

#### 2.5. Monte Carlo Simulation

_{k}is the post-earthquake SR in the kth MCS sample.

## 3. Case Study

#### 3.1. Results of Hydraulic Analysis under Different Working Conditions

_{i}

^{min}= 0 m and H

_{i}

^{des}= 15 m. Pipeline head losses were calculated using the Hazen–Williams formula. The hydraulic imbalance equation was solved using the Newton–Raphson method, with iteration accuracy set at 0.0001. For each of the 36 different working conditions previously outlined, 1000 simulations were conducted. The SP of the WDS was recorded for each simulation under varying damage states. Figure 6 displays the residual SP of the WDS across different seismic damage scenarios.

#### 3.2. Seismic Resilience Evaluation Results

#### 3.3. Results of Seismic Resilience Assessment Based on Empirical Data

_{f}

_{1}, P

_{f}

_{2}, and P

_{f}

_{3}denote the probability of the pipeline being basically intact, moderately damaged, and severely damaged, respectively, and L is the length (km) of the pipeline. R

_{f}is the pipe repair rate, expressed as repairs/km. This paper adopted the formula for calculating the R

_{f}of water supply pipelines provided by the Japan Water Works Association [54], which is expressed as follows:

_{f}= C

_{d}·C

_{p}·C

_{g}·C

_{l}·R

_{0}

_{0}= 2.88·10

^{−6}·(PGA-100)

^{1.97}

_{d}, C

_{p}, C

_{g}, and C

_{l}represent the correction factors for the pipe diameter, pipe material, topography, and liquefaction, respectively. R

_{0}is the standard pipeline repair rate.

## 4. Conclusions

- The simulation results indicated that both SP and SR remained at 100% across various service ages and soil conditions when subjected to frequent earthquakes.
- The SP and SR for the four service ages in the three soil environments did not differ significantly under the effects of a fortification earthquake. Compared with the simulation results under frequent earthquakes, there was a slight decrease, but the overall decrease was not significant.
- Under rare earthquake conditions, SP and SR varied significantly, especially in acidic environments where performance notably declined, with the worst recorded SP being 0.68 for pipelines with a service age of 40 years. Despite this, the overall performance across different conditions remained relatively high, with SP values ranging from a minimum of 83.8% to a maximum of 98.2%, and an average SP of 91%. This showed that post-earthquake restoration resources are a key factor in guaranteeing a high level of the SP.
- Holding the service age and soil environment constant, the SR diminished progressively as seismic activity intensified. Under identical service ages and high seismic intensities, the SR was lowest in acidic soil environments and highest in near-neutral soil environments. Considering a constant soil environment and high seismic activity, the SR progressively declined with the advancing service age. The assignment of multiple repair crews universally resulted in a high SR for the WDSs.
- Based on evaluations using empirical data, the SP and SR, when assessed without accounting for variations in soil corrosion conditions and the service age of the pipelines, tended to be estimated on the higher side.

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

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**Figure 1.**Seismic fragility curves for pipelines with diameters exceeding 500 mm across varied soil corrosion environments and diverse service ages.

**Figure 2.**Seismic fragility curves for pipelines across varied diameters, environments, and service ages.

**Figure 6.**SP values for WDSs with different service ages in different corrosive environments at three seismic intensities.

**Figure 7.**Functional recovery process curves of WDS with a service age of 30 years in different corrosive environments under different seismic intensities: only one repair crew was dispatched to repair the pipes sequentially.

**Figure 8.**Functional recovery process curves of WDS with a service age of 30 years in different corrosive environments under different seismic intensities: a repair team was assigned to each damaged pipe.

**Figure 9.**Functional recovery process curves of WDS with a service age of 40 years in different corrosive environments under different seismic intensities: only one repair crew was dispatched to repair the pipes sequentially.

**Figure 10.**Functional recovery process curves of WDS with a service age of 40 years in different corrosive environments under different seismic intensities: a repair team was assigned to each damaged pipe.

**Figure 11.**SR of WDSs with different service ages in different corrosive environments under different seismic intensities. Note that in order to make the figures more concise, we used the following abbreviations: acidic (AC), alkaline (AL), near-neutral (NN), frequent earthquake (FE), design earthquake (DE), rare earthquake (RE), one repair crew (ORC), and adequate repair crew (ARC).

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**MDPI and ACS Style**

Long, L.; Yang, H.; Zheng, S.; Cai, Y.
Seismic Resilience Evaluation of Urban Multi-Age Water Distribution Systems Considering Soil Corrosive Environments. *Sustainability* **2024**, *16*, 5126.
https://doi.org/10.3390/su16125126

**AMA Style**

Long L, Yang H, Zheng S, Cai Y.
Seismic Resilience Evaluation of Urban Multi-Age Water Distribution Systems Considering Soil Corrosive Environments. *Sustainability*. 2024; 16(12):5126.
https://doi.org/10.3390/su16125126

**Chicago/Turabian Style**

Long, Li, Huaping Yang, Shansuo Zheng, and Yonglong Cai.
2024. "Seismic Resilience Evaluation of Urban Multi-Age Water Distribution Systems Considering Soil Corrosive Environments" *Sustainability* 16, no. 12: 5126.
https://doi.org/10.3390/su16125126