A Quantitative Evaluation Model for the Seismic Resilience of Water Supply Systems Based on Fragility Analysis

Abstract

A quantitative evaluation model is proposed to assess the seismic resilience of water supply systems. The water supply system is divided into three parts: water sources, aboveground infrastructures, and underground pipeline network, and importance factors for the different parts are quantified. Resilience demand is expressed as the desirable functionality loss and the recovery time of the water supply system after an earthquake. First, seismic fragility models are established for the different components of the water supply system. A water quality index is utilized to represent the impact of earthquakes on the water sources, the seismic performances of aboveground infrastructures are represented by fragility curves, and the repair rate in terms of number of repairs per kilometer is adopted for the pipeline network. Then, the post-earthquake functionality of the water supply system is quantified based on seismic fragility analysis. Changes in the water quality index are used to indicate the functionality losses related to water sources, the functionality losses of aboveground infrastructures are represented by the economic losses derived from component fragility curves, and post-earthquake functionality losses in the underground pipeline network are quantified by hydraulic simulations. The functionalities of the three parts are calculated separately, and then the overall system functionalities are obtained as the sum of the weighted functionalities of the three parts. Finally, a repair strategy is developed and the recovery time is calculated considering the system damage scenarios, system functionality analyses, and resource reserves. The proposed resilience assessment model considers all components of the water supply system, and the results are reliable when the basic information is complete and accurate.

1. Introduction

Urban water supply systems are an important part of lifeline systems and have a decisive impact on the normal operation of cities. If an urban water supply system stops working, production activities, daily life, medical treatment, and fire protection can be seriously affected, even threatening public safety. However, urban water supply systems are faced with severe earthquake threats. For example, in the 2015 Gorkha earthquake, almost all water supply pipes were damaged, and other facilities, such as pumps, wells, tanks, and water treatment plants, were also seriously damaged. More than 18 months after this devastating event, many recovery processes were still ongoing [1]. After the Kobe earthquake in Japan, more than 1.1 million people lost access to water, and the total recovery period was approximately two months [2]. Moreover, failures of water supply systems in some previous earthquakes have led to epidemics [3,4] and conflagrations [5,6,7].

A water supply system is a complex network composed of many components, and its seismic safety deserves focused attention. Some recent devastating earthquakes have shown that attention should be given not only to the seismic vulnerability and seismic risk of urban water supply systems, but also to post-earthquake recovery. The ability of urban water supply systems to cope with earthquakes should be evaluated from multiple dimensions so that they can maintain certain functions after an earthquake and recover to the expected level within an acceptable time. In the past, focus was placed on the physical protection of critical infrastructure systems, and much of the research concerning the impacts of earthquakes on urban lifeline systems relied on conventional concepts such as ‘vulnerability’, ‘reliability’, and ‘risk’ [8]. In recent years, the concept of ‘resilience’ has also attracted increasing attention. The seismic resilience of critical infrastructure systems, including urban water supply systems, is defined as the ability of these systems to withstand, adapt to, and rapidly recover from the effects of a disruptive event [9].

Subsequent to the general quantitative assessment framework offered by Bruneau et al. [10], various studies have been carried out. Many extensive review articles on resilience have been proposed, which have focused on resilience concepts and definitions [11,12], frameworks [13,14,15,16], or critical infrastructure systems [17,18,19,20]. Some studies have also been carried out specifically on the resilience of water supply systems. Based on actual seismic events, the seismic resilience of the water supply system in the Kathmandu Valley, Nepal, was assessed by Didier et al. [1] after the 2015 Gorkha earthquake, and their evaluation results demonstrated that the evolution of the community demand has a large effect on the resilience of the water supply system. Using the same earthquake event as a case study, Mostafavi et al. [21] investigated determinants of resilience in water infrastructure systems in developing countries, and the factors and their interactions affecting the resilience were delineated as exposure, sensitivity, and adaptive capacity. Some studies have focused on evaluation frameworks. Balaei et al. [3] proposed a framework for measuring the resilience of water systems based on multidimensional indicators. Zhao et al. [8] proposed a novel framework for assessing the emergency resilience of water supply networks using a newly designed performance response function based on network equilibrium theory. Different resilience indexes have also been proposed to evaluate the resilience of water supply systems. Cimellaro et al. [9] proposed a resilience index (R) for water distribution networks that is the product of demand (R1), capacity (R2), and water quality (R3). Based on a post-earthquake recovery process, Liu et al. [2] defined a resilience index, SRI, to reflect the system performance, and three pipe recovery strategies were compared using Monte Carlo simulations combined with an example. To estimate the downtime of a water supply system after an earthquake, Kammouh et al. [22] provided an empirical probabilistic model. Different from many studies, Diao et al. [23] proposed a distinctive approach that shifts the objective of seismic resilience from analyzing multiple and unknown threats to analyzing the system responses to extreme conditions. There are many uncertainties in resilience assessment, and questionnaires and interviews are often used to determine variables, factors, and indicators, and supplement the data needed in the evaluation of resilience [4,21,24,25,26,27,28].

A schematic diagram of a resilience assessment framework is shown in Figure 1. System functionality, Q(t), is usually quantified by a dimensionless percentage, which is defined as the ratio of the current available functionality to the system functionality before an earthquake event, where 100% represents no damage to the system and no degradation of the system functionality and 0 means the system is completely damaged. When an earthquake occurs at time t_EO, the internal structural and nonstructural components of the system will suffer a certain degree of damage, resulting in a decline in the system functionality. As the repair process progresses, the system components are gradually repaired, and the system functionality is gradually restored. After the repair process with a total time of T_RE, the system functionality reaches a new level, which may be equal to the pre-earthquake level or lower or higher, depending on the invested resources, the repair strategy, and other factors [10,29].

Urban water supply systems are multicomponent systems and consist of general infrastructure categories such as supply, transmission, treatment, pumping, storage, and distribution. All of these components are vulnerable to damage during earthquakes, which can cause serious disruptions to a water supply system. In most previous studies on the seismic resilience of water supply systems, pipe networks and demand nodes in these systems were taken as the objects, and the damage and repair of other components were almost ignored. Water supply system resilience is not the only result of pipe resilience, as all components should be involved in system resilience [30]. A few technical manuals and guidelines have considered a complete system. Hazus [31] described and presented the methodology for estimating the direct damage to each component of potable water systems. The NIST Guide [32,33] covers buried pipelines and aboveground infrastructures. However, how to quantitatively evaluate the functionality of each component and combine them to reflect the overall functionality of the system is unclear. In addition, unlike other lifeline systems, water quality is an important factor to be considered during resilience assessment. After an earthquake, the water quality may be affected by heavy metal substances in the strata, sediment caused by landslides, and the decay of buried animals and plants. For this reason, water quality should also be considered in seismic resilience assessments of water supply systems, and the resilience index proposed by Cimellaro et al. [9] contains this factor.

This study incorporates water quality, aboveground infrastructures, and buried pipelines into an evaluation framework that comprehensively reflects the functionality of the entire water supply system at all stages. For the quantitative evaluation of the seismic resilience of the water supply system, the two main challenges are (1) how to quantitatively evaluate the post-earthquake functionality of the water supply system, and (2) how to organize the repair process. In this study, based on the fragility curves of aboveground infrastructures, flow analyses of buried pipeline networks that consider leaks and breaks, and water quality indices, the overall functionality of the water supply system is quantified. The repair strategy of the system is determined by considering damage scenarios and the importance of each component, and on this basis, the recovery time is obtained in combination with the reserved resources. The most significant feature of this study is the defining of post-earthquake functionality losses for different parts of the water supply system based on their characteristics, and then combining them to obtain the overall loss in the system.

This paper is divided into four sections. Section 2 provides a detailed introduction to the methodology, including probabilistic seismic hazard analysis in Section 2.1, probabilistic seismic fragility analysis in Section 2.2, post-earthquake functionality analysis in Section 2.3, and repair strategy and seismic resilience evaluation in Section 2.4. Section 3 conducts and discusses a case study of a real-world water supply system in Yantian District, Shenzhen, China. Finally, Section 4 gives the conclusions and future research directions.

2. Methodology

The proposed framework for the seismic resilience of a water supply system is illustrated in Figure 2. The evaluation framework consists of four parts: (1) the probabilistic seismic hazard analysis, (2) the probabilistic seismic fragility analysis, (3) the post-earthquake functionality analysis, and (4) repair strategy and seismic resilience evaluation.

2.1. The Probabilistic Seismic Hazard Analysis

In the first step, a probabilistic seismic hazard analysis of the area where the water supply system is located is conducted. Based on the source model and attenuation model, the location, scale, mechanism, magnitude, etc., of potential earthquakes in the future can be obtained [34]. The results of a probabilistic seismic hazard analysis are seismic hazard curves, which plot the exceeding probability as a function of the ground shaking intensity [35]. The analysis results will be used as the input to the fragility analysis in Step 2.

2.2. Probabilistic Seismic Fragility Analysis

Proper seismic fragility models are the basis of resilience evaluation. In the second step of the proposed framework, a probabilistic seismic fragility analysis is performed. First, a seismic damage model is established that considers uncertainties for water sources, aboveground infrastructures, and underground pipelines. Then, using the seismic intensity obtained in the first step as the input, the extent of the damage to different parts of the entire system is assessed.

2.2.1. Seismic Fragility Analysis of Aboveground Infrastructures

The aboveground infrastructures of water supply systems typically consist of storage tanks, facilities and structures of pumping plants, facilities and structures of the water treatment plants, etc. All of these components are vulnerable to damage during an earthquake, which can cause significant disruptions to the water supply systems. According to conventional seismic engineering practices, infrastructure damage is usually divided into different states [36]. In general, fragility curves for aboveground water supply system components are modeled as lognormally distributed functions, as shown in Equation (1), which give the probability of reaching or exceeding different damage states for a given seismic intensity [31,37].

$$P(PGA)=\Phi [\frac{1}{\theta }\ln (\frac{PGA}{\beta })]$$

(1)

where PGA refers to peak ground acceleration; P(PGA) is the probability of reaching or exceeding different damage states; $\Phi [\cdot ]$ represents the standard normal cumulative distribution function; θ and β are the median and logarithmic standard deviation used to determine the fragility curves corresponding to each damage state. Porter et al. [38] introduced a set of procedures for creating fragility functions from various kinds of data. In addition, for almost all water supply system components, finite element models and nonlinear time-history analyses can be used to establish fragility curves. Engineering demand parameters for different components obtained by nonlinear time-history analyses can be used to develop probabilistic seismic demand models [39,40]. Based on probabilistic seismic demand models, fragility curves can be reached and the probability of reaching or exceeding different damage states for a selected seismic intensity can be calculated. Some technical manuals and guidelines [31,41] provide the median and standard deviation of fragility curves for different components, including aboveground water supply system components. In the resilience evaluation process, if the parameters of the fragility curves are not separately determined, these values can be referenced.

2.2.2. Seismic Fragility Analysis of Pipelines

Unlike aboveground components, the number of pipe failures or repair rates are typically used to measure the damage degree of underground pipeline networks. Some studies use empirical relationships to predict the repair rate (repairs/km) of pipe networks after earthquakes [42,43,44,45]. The data used to establish these relationships come exclusively from earthquake events in the United States and Japan. However, by comparing the results of empirical relationships with the data from the Wenchuan earthquake, these models are not suitable for China [2,46]. In addition, although these relationships have been adopted in many studies, it is worth determining whether they truly apply to other countries, especially many developing countries. Previous studies [47,48] and earthquakes [49,50,51] have indicated that seismic damage to underground pipelines is mainly caused by wave propagation. Specifications in different countries have focused on pipe stress and joint deformation caused by seismic waves [44,52,53]. In this study, a Chinese code-based pipeline fragility analysis method is adopted [2]. Buried pipelines can be either segmented or continuous. For segmented pipelines, the joint deformation determines whether they are damaged; however, for continuous pipes, pipe body stress is the decisive factor.

(1)

Pipe seismic reliability

In the proposed fragility model, the pipe reliability analysis is carried out first. For segmented pipes, joint deformation is used to measure joint seismic reliability, and the state function is expressed as:

$$Z=f(R,S)=R-S$$

(2)

where R is the allowable deformation of the joint, which follows a normal distribution $R~({\mu }_R,{\sigma }_R)$; and S is the joint deformation under the seismic load, which follows a normal distribution $S~({\mu }_S,{\sigma }_S)$. When Z < 0, damage occurs at the pipe joint. When Z > 0, no leakage occurs. According to probability theory, Z also follows a normal distribution $Z~(\mu ,\sigma )$, and $\mu ={\mu }_R-{\mu }_S$, $\sigma =\sqrt{{\sigma }_R^2+{\sigma }_S^2}$. Then, the failure probability of the joint is expressed as:

$$P_f=P(Z<0)={\displaystyle {\int }_{-\infty }^0\frac{1}{\sqrt{2\pi }\sigma }}{e}^{\frac{-(Z-\mu )}{2{\sigma }^2}} dz=\Phi (-\frac{\mu }{\sigma })$$

(3)

The joint seismic reliability, or safe probability, can be expressed as:

$$R\mathrm{e}=1-P_f=\Phi (\frac{\mu }{\sigma })$$

(4)

The above calculation results belong to the seismic reliability of a single joint. In practice, a pipeline usually contains multiple pipe joints, and it is unreasonable to assume failure independence or complete failure correlation. Ditlevsen reliability bounds [54] can be used to solve this problem [55], but the process is too complex to apply for a pipe network. Liu et al. [2] has simplified this problem: all pipe joints with the same site conditions and pipe properties are divided into a series set ${\psi }_i,i=\mathrm{1,2},3\dots n$, and n is the number of sets. Joints within the same set are assumed to have complete failure correlation, and the seismic reliability of set ${\psi }_i$ is expressed as:

$$R\mathrm{e}\left\{{\psi }_i\right\}=\min \left\{R{\mathrm{e}}_j\right\},R{\mathrm{e}}_j\in {\psi }_i$$

(5)

For different sets, failure independence is adopted, and the seismic reliability of the pipeline is given by:

$$R{\mathrm{e}}_P=\underset{i}{\Pi }\left\{{\psi }_i\right\}$$

(6)

The above analysis is based on considering the joint deformation as the parameter to control the damage state of pipelines. This analysis is suitable for pipelines with flexible or rigid joints. For continuous welded pipelines, stress should be taken as the parameter, and the analysis process is similar to the above and will not be repeated.

According to the specifications of various countries [44,52,53], the joint deformation and stress of buried pipelines can be obtained. Due to the diversity of joint and pipe materials, the means and standard deviations of the normal distributions are different. Liu et al. [2,56] and Han [57] provide corresponding values for water supply networks in China, but these values may not necessarily be suitable for other countries. However, in any case, these values can be easily obtained through testing and analysis.

(2)

Pipe damage number

In current research, it is generally assumed that the pipe damage number under seismic action follows the Poisson distribution [42,58]. The probability of n failures occurring in a pipeline with a length of L can be expressed as:

$$P(n)=\frac{{(\lambda L)}^n}{n!}{e}^{-\lambda L}$$

(7)

where n is the pipe damage number; λ is the damage rate (repairs/km); and L is the pipeline length (km). The pipe reliability Re_p can be considered as the probability of zero failure, and the following relationships can be obtained.

$$R{\mathrm{e}}_P=P(n=0)=\frac{{(\lambda L)}^0}{0!}{e}^{-\lambda L}=e^{-\lambda L}$$

(8)

Then, the pipe damage number (PDN) can be expressed as:

$$PDN=\lambda L=-\ln (R{\mathrm{e}}_P)$$

(9)

2.2.3. Impact of Earthquakes on Water Quality

Usually, urban water supply systems have strict requirements for ensuring water quality. When evaluating the seismic resilience of water supply systems, the impact of earthquakes on water quality cannot be ignored. At present, there is no globally accepted composite index to characterize water quality, and most water quality indices rely on normalizing or standardizing data according to expected concentrations and some interpretation of good versus bad concentrations [9]. Choosing which index to describe the water quality at different stages before and after an earthquake is not within the scope of earthquake engineering and will not be discussed here. Referring to previous studies [9], regardless of which index is selected, once determined, the impact of earthquakes on water quality can be defined by comparing the index values before and after earthquakes.

2.3. Post-Earthquake Functionality Analysis

Post-earthquake functionality analysis of the water supply system is conducted in the third step. Functionality models of the water supply system are established based on existing damage scenarios. Economic loss is a common indicator of the seismic resilience of an engineering system [16,59], and in this study, economic loss is used to quantify the corresponding functionality of aboveground infrastructures [60]. The economic loss of a single component, L_j, can be defined as a dimensionless factor and expressed as:

$$L_j={\displaystyle \sum _{i=1}^n{L}_{ji}}\cdot P_{D{S}_{ji}\left|IM\right.}$$

(10)

where n is the total number of damage states; $P_{D{S}_{ji}\left|IM\right.}$ is the conditional probability of component j in damage state i under a certain earthquake intensity measure; and $L_{\mathrm{ji}}$ is the economic loss ratio of component j associated with damage state i. A schematic diagram of seismic fragility curves is shown in Figure 3. The probabilities of exceedance for a specific damage state can be obtained by the seismic fragility curve, and $P_{D{S}_{ji}\left|IM\right.}$ is calculated as the difference between the exceedance probabilities of damage state i and the exceedance probabilities of damage level i + 1 [38]. The value of $L_{\mathrm{ji}}$ can be obtained from previous technical manuals and specifications [31,61]. The total loss of aboveground infrastructures at time t can be expressed as:

$$L(t)={\displaystyle \sum _{j=1}^m{\omega }_j\cdot L_j}$$

(11)

$${\omega }_j=\frac{C_j}{{\displaystyle \sum _{j=1}^m{C}_j}}$$

(12)

where m is the number of aboveground components considered in the evaluation model; ${\omega }_j$ is the importance factor and is determined by the reconstruction cost $C_j$ for each component. Then, the post-earthquake functionality of aboveground infrastructures at time t can be expressed as:

$$F_1(t)=1-L(t)$$

(13)

The post-earthquake functionality of the underground pipeline network is quantified by hydraulic simulation. Based on the fragility analysis results of the pipeline network, pipeline network damage scenarios are constructed, hydraulic simulation analyses are conducted, and the pipeline network functionality is defined based on water pressures and water demands. Referring to the methods in previous studies [2,41,56] to define the functionality of the pipeline network, the post-earthquake functionality of the pipeline network is defined as:

$$F_2(t)={\displaystyle \sum _{k=1}^n{\eta }_k}(t)N_k(t)$$

(14)

where $F_2(t)$ represents the overall functionality of the pipeline network at time t; $N_k(t)$ is the functionality of user node k at time t; and ${\eta }_k(t)$ is the importance factor of node k at time t. $N_k(t)$ is determined by the following Equation:

$$N_k(t)=\left\{\begin{array}{ll}1& h_k(t)\ge h_{k0}(t)\\ \frac{h_k(t)}{h_{k0}(t)}& h_k(t)<h_{k0}(t)\end{array}\right.$$

(15)

where $h_{k0}(t)$ is the demand head of node k at time t; and $h_k(t)$ is the actual head of node k at time t obtained by hydraulic simulation analysis. ${\eta }_k(t)$ is determined by the following Equation:

$${\eta }_k(t)=\frac{{\zeta }_k(t)d_k(t)}{{\displaystyle \sum _{k=1}^n{\zeta }_k(t)d_k(t)}}$$

(16)

where $d_k(t)$ is the water demand of user node k at time t; ${\xi }_k(t)$ is the corresponding adjustment coefficient for water demand, which can be provided by expert experience or decision-makers. In general, for key user nodes (shelters, hospitals), ${\xi }_k(t)$ should be greater than 1, and for ordinary user nodes, ${\xi }_k(t)$ can be taken as 1.

Water supply systems have strict requirements for water quality. In this study, the ratio of the selected water quality measurement indicator after earthquakes to their original value was used to represent the functionality of the water supply system related to water quality. The functionality related to water quality is defined as follows:

$$F_3(t)=\frac{q(t)}{q_0}$$

(17)

where $q(t)$ is the value of the water quality measurement index after an earthquake, and $q_0$ is the original value.

The functionalities of the three parts are calculated separately, and then summarized through the importance factors to obtain the total functional loss of the water supply system. Additionally, the importance factors are obtained by expert opinions and questionnaires, which is a common method in seismic resilience assessment [3,24,25,30,60,62]. The post-earthquake functionality of the water supply system is represented as follows:

$$Q(t)=A_1\cdot F_1(t)+A_2\cdot F_2(t)+A_3\cdot F_3(t)$$

(18)

where A₁, A₂, and A₃ are the importance factors corresponding to the aboveground infrastructure, pipeline network, and water quality, respectively. The values of A₁, A₂, and A₃ are 0.2, 0.7, and 0.1, respectively. The ratio of these values is comparable to the investment ratio of the water infrastructure system investment need [63].

2.4. Repair Strategy and Seismic Resilience Evaluation

The final step is to develop a repair strategy and complete the assessment. In this step, the repair strategy needs to be determined based on the system damage scenarios, functionality analyses, and resource reserves.

Due to differences in natural and human factors, there are obvious differences in the water supply systems of cities in different regions. For example, cold regions in China will pay attention to the anti-freezing properties of pipelines in winter, while southern regions will not consider this. Moreover, the facilities, equipment, and material reserves of cities in economically developed regions also have obvious advantages. These objective differences lead to differences in repair methods and rates, and it is necessary to evaluate the seismic resilience of water supply systems based on actual situations. When collecting the basic information of an evaluated water supply system, it should include the configuration of the water supply system employees and the reserve of emergency rescue materials. Compared with other basic information required for the resilience evaluation of the water supply system, this information is easy to obtain and accurate. In addition, when collecting basic information, interviews or questionnaire surveys can be conducted on employees of the target system to obtain the repair time of different components of the water supply system. Finally, by sorting out the investigation results, an accurate post-earthquake repair time evaluation model for specific water supply systems can be obtained.

As the repair process continues, the system damage status and system functionalities are updated, and a curve of the system function over time is obtained. Finally, based on the resilience curve, the system resilience level can be discussed.

3. Case Study

3.1. Background of the Water Supply System

The method proposed in this study has been used to assess the seismic resilience of the water supply system in Yantian District, Shenzhen, China. The water supply system consists of two water treatment plants, two pumping plants, and a large number of underground pipelines. Under normal conditions, the system serves 220,000 people in an area of 74.99 square kilometers. The water supply system buildings are all reinforced concrete frame structures, and the clean water reservoirs and water treatment reservoirs of the two water treatment plants are also reinforced concrete structures. The pipes are made of steel, ductile iron, and reinforced concrete. In this case study, a water treatment plant is simplified into a building structure, a water treatment reservoir, and a clean water reservoir, and only the main pipe with a diameter greater than 200 mm in the pipe network is considered. After simplification, 57 main pipes and 40 suer nodes are included. A simplified sketch of the water supply system is shown in Figure 4. According to Chinese code [53], the basic earthquake intensity of Yantian is VII degrees, and the rare intensity of Yantian is VIII degrees. In other words, the peak ground acceleration (PGA) of the design-basis earthquake (DBE) ground motion is 0.1 g, and the PGA of the maximum considered earthquake (MCE) ground motion is 0.2 g.

The aboveground infrastructures of the water supply system consist of two water treatment plants and two pumping plants. Water treatment plant 1 includes building structure 1, clean water reservoir 1, and water treatment reservoir 1. Water treatment plant 2 includes building structure 2, clean water reservoir 2, and water treatment reservoir 2. The two water treatment plants are assumed to be the same, and the two pumping plants are also assumed to be the same. According to Section 2.4, the importance factors for the aboveground infrastructure, water source, and pipeline network are 0.2, 0.1, and 0.7, respectively. The importance factors are determined for each component of the aboveground infrastructure according to Equation (12). For the cost of each component of the aboveground infrastructure, some relevant engineering cost models have been proposed [64,65,66], and these data can also be obtained from the design data of these projects. Two water sources are set to have the same importance factor of 0.05. The importance factors of the subsystems and components of the water supply system are shown in

Table 1. Importance factors of subsystems and components in the water supply system.

Subsystem	Importance Factor 1	Component	Importance Factor 2
Aboveground infrastructure	0.2	Building structure 1	0.027
		Clean water reservoir 1	0.018
		Water treatment reservoir 1	0.036
		Building structure 2	0.027
		Clean water reservoir 2	0.018
		Water treatment reservoir 2	0.036
		Pumping plant 1	0.019
		Pumping plant 2	0.019
Water source	0.1	Water quality 1	0.05
		Water quality 2	0.05
Pipeline network	0.7	-	-

.

3.2. Seismic Fragility and Post-Earthquake Functionality

The methods for analyzing the seismic fragility of aboveground infrastructures are mature and include establishing finite element models or using empirical fragility curves [31]. Conducting seismic fragility analyses on aboveground infrastructures and obtaining seismic fragility curves are not the focuses of this study, and some previous models have been adopted in this case study. Fragility parameters for aboveground infrastructures are shown in

Table 2. Fragility parameters for aboveground infrastructures.

Classification	Damage State	θ (g)	β
Building structure [41]	Slight	0.12	0.66
	Moderate	0.22	0.66
	Extensive	0.48	0.66
	Complete	0.79	0.66
Water treatment reservoir [67]	Slight	0.12	0.52
	Moderate	0.26	0.51
	Extensive	0.32	0.43
	Complete	0.49	0.28
Clean water reservoir [67]	Slight	0.14	0.66
	Moderate	0.36	0.59
	Extensive	0.54	0.52
	Complete	0.59	0.47
Pumping plant [31]	Slight	0.13	0.6
	Moderate	0.28	0.5
	Extensive	0.77	0.65
	Complete	1.5	0.8

. According to Equation (1) and the parameters in Table 2, the fragility curves of aboveground infrastructures have been obtained, which are shown in Figure 5. After obtaining the fragility curves, according to Equation (10) and Figure 3, the functionality losses of aboveground infrastructures under the DBE and MCE can be obtained. For water sources, this is set as a functionality decrease of 0.05 under the DBE and 0.2 under the MCE in this water supply system after discussing with experts from the water treatment plant. According to Equations (2)–(9), the amount of damage under the DBE and MCE can be calculated. For the location where the damage occurred, if a pipeline with a length of L experiences n failures, it is assumed that these n failures are evenly distributed over the length L.

After determining the quantity and locations of damage in the pipeline network, EPANET 2.2 [68] is used for the hydraulic simulation of the damaged pipeline network. Pipeline damage includes two types: leakage and breakage. According to previous studies, the leakage is modeled by adding an emitter, and the breakage is modeled by adding fictitious reservoirs at the ends of the broken pipelines and adding check valves into the broken pipeline [9,69,70]. In this case study, the same method is adopted in EPANET 2.2 [68] to simulate the hydraulic simulation of the damaged pipeline network. In addition, some studies have also conducted hydraulic simulation analysis of damaged pipeline networks through programming [2].

In an earthquake, the occurrence of pipeline damage, whether it is leakage or breakage, is stochastic. According to statistics, leakage accounts for 80% of all damage and breakage accounts for 20% [31,70]. For a pipeline network with a damage number of m, the leakage number is 0.8 m and the breakage number is 0.2 m. Note that both the number of leaks and the number of breaks should be taken as integers. The number of all possible damage scenarios N can be represented as a combination of leakage and breakage:

$$N=C_m^{0.2m}$$

(19)

All possible damage scenarios are calculated in this study, and the mean is taken as the final result. Under the action of the DBE, a total of five possible damages may occur, so the number of possible scenarios is combination number $C_5^1=5$, one of which is shown in Figure 6. Under the action of the MCE, a total of 10 possible damages may occur, so the number of possible scenarios is combination number $C_{10}^2=45$, 1 of which is shown in Figure 7. For each damage scenario, after obtaining the water pressure values of all user nodes, the post-earthquake functionality of the pipeline network can be obtained through Equations (14)–(16).

3.3. Repair Resources and Recovery Time

Repair resources such as material reserves, number of workers, and construction equipment have a significant impact on the post-earthquake recovery time of a water supply system. During the stage of collecting basic data, interviews were conducted with employees through a questionnaire survey to obtain the time for restoring the emergency water supply for each component of the water supply system. The repair times for pipeline leakage and breakage in the case study are 6 h and 12 h, respectively. Emergency restoration times for aboveground infrastructures are shown in

Table 3. Emergency restoration time for aboveground infrastructures (unit: day).

Classification	Slight	Moderate	Extensive	Complete
Building structure	0.1	1	3	14
Clean water reservoir	0.1	0.5	3	14
Water treatment reservoir	0.1	0.5	3	14
Pumping plant	0.1	0.5	2.5	13.5

. For the restoration time of water quality, due to the limited involvement of current research, it was roughly set to 2.4 h under the DBE and 7.2 h under the MCE after communication with water treatment plant staff. It should be noted that the recovery time here refers to the shortest time that the water supply system can restore its water supply capacity. For water supply systems, there is a significant difference between short-term and long-term restoration. For example, after the Wenchuan 8.0 magnitude earthquake, most city water supply systems were able to restore the water supply within a few days after the earthquake, but long-term recovery could last for several months or even more than a year. The post-earthquake repair path also has an impact on the resilience assessment of water supply systems, and some studies have compared and analyzed different repair strategies [2,60,71]. In this case study, the water supply system is repaired in the order of water sources, ground infrastructures, and underground pipeline network. For the repair priority of pipe segments in the network, a distance-based method is used [30].

3.4. Seismic Resilience Assessment

The post-earthquake functionality loss and recovery time of the water supply system were calculated following the previously proposed method. The functionality losses of the components corresponding to the DBE (PGA = 0.1 g) and MCE (PGA = 0.2 g) are presented in Figure 8. The functionality losses of components under the DBE range from 0.023 to 0.080, with the smallest being the pumping station and the largest being the building structure. The functionality losses of components under the MCE range from 0.13 to 0.309, with the smallest being the pumping station and the largest being the pipeline network. After obtaining the functionality losses of individual components, combined with their importance factors in Table 1, the functionality losses considering their contribution to the system were determined. The functionality losses of the three subsystems considering their contribution to the system are shown in Figure 9. Under the DBE, the functionality loss of the three subsystems is 0.01, 0.005, and 0.055, respectively, resulting in a total system loss of 0.07. Under the MCE, the functionality loss of the three subsystems is 0.039, 0.02, and 0.216, respectively, and the total system loss is 0.275.

There are uncertainties in the post-earthquake restoration process, such as resources and management, which are difficult to quantify, and there may be large differences between different water supply systems. In view of these uncertainties, this study takes interviews and surveys of water supply system staff as an important link in resilience assessment to determine parameters that are difficult to quantify. Based on the previously proposed repair rate and sequence, the post-earthquake restoration of the water supply system is carried out. As the repair progresses, the system’s functionality gradually recovers. After the repair is completed, the system resumes its normal water supply function.

According to the previous results, under the DBE, the number of damage scenarios is 5, and under the MCE, the number of damage scenarios is 45. The time–functionality curves of all damage scenarios under the same earthquake intensity were generated, and the mean of all curves was taken as the final result. Resilience curves for the DBE and MCE are shown in Figure 10. Under the DBE, the functionality decreased from 100% to 93%, and after 39 h of emergency repair, the water supply system resumed its normal water supply function. Under the MCE, the functionality decreased from 100% to 72.5%, and after 127 h of emergency repair, the water supply system resumed its normal water supply function. The recovery time in this case study is comparable to the recovery time of water supply systems in some cities during the Wenchuan earthquake.

In most previous studies on the seismic resilience of water supply systems, pipeline networks were taken as the objects. In this case study, from Figure 9, it can be seen that if only the pipeline network is considered, the functionality loss is 0.055 under the DBE and 0.216 under the MCE. Due to the fact that the evaluation model proposed includes a complete water supply system, the inevitable result is that the functionality loss and repair time are greater than those of models that only consider the pipeline networks. However, the model proposed in this paper is closer to the real situation.

4. Conclusions

This work proposes a quantitative evaluation method for the seismic resilience of water supply systems. In the proposed seismic resilience evaluation framework, all components in a water supply system can be considered, provided that their seismic fragility and importance factors to the system can be defined. Based on this method, the post-earthquake functionality loss and post-earthquake recovery time of the water supply system can be accurately calculated.

Water sources, aboveground infrastructures, and the pipeline network were incorporated into the evaluation framework. According to the characteristics of the different parts, different methods are used to define their fragility and post-earthquake functionality losses. Based on the functionality losses of different parts and their importance factors, the system functionality loss can be obtained. Resilience evaluation involves multiple dimensions, such as technical, organizational, social, and economic dimensions, which makes it difficult to quantify some variables and indicators. For this reason, expert questionnaires are widely used, which is also the method for obtaining the values of the importance factors for different subsystems in this study. Considering the differences in physical aspects and human factors of different water supply systems, in the evaluation framework proposed in this study, a questionnaire survey is conducted on employees of the evaluated system to obtain necessary information, such as the repair speed and recovery resources.

Using the proposed evaluation framework, the functionality loss and recovery time of a real-world water supply system under the DBE and MCE are presented. The results show that under the DBE and MCE, the functionality loss reached 7% and 27.5%, respectively. After 39 and 127 h of emergency repair, the water supply system resumes its normal water supply function. The functionality loss and repair time are greater than those of models that only consider the pipeline networks. However, the model proposed in this paper is closer to the real situation.

The proposed resilience evaluation model can consider all components of the water supply system, and the results are reliable when the basic information is complete and accurate. In addition, some issues in the evaluation framework are worth further exploration, such as the seismic fragility of water sources and the values of the importance factors for different components and subsystems, which are determined by the uncertainty of resilience. The evaluation framework proposed in this study is reliable and innovative.