# Seismic Reliability Assessment of Water Supply Systems Considering Critical Paths
## 3 Application and Results
### Method of Earthquake Simulation and Pipeline Damage Probability Calculation  
Generally, wide-area waterworks systems are built to sustain earthquakes of magnitude \(\sim\)6.5. Therefore, we chose to simulate magnitude 7 earthquakes in order to assess water supply capability before and after an earthquake. Earthquake location was determined by dividing the target area into six regions (2 \(\times\) 2 squares) and setting earthquakes to occur at the nine intersections of these regions. When setting the earthquake locations, a uniform distribution was assumed to account for uncertainties such as the depth of the earthquake. Using Monte Carlo Simulations, a total of 100,000 earthquakes \begin{table}
\begin{tabular}{l c c c c c c c} \hline \hline \multirow{2}{*}{**Critical Path**} & \multirow{2}{*}{**Net Inflow (m\({}^{3}\)/d)**} & \multirow{2}{*}{**Number of**} & \multirow{2}{*}{**Total Pipe**} & \multicolumn{2}{c}{**Average Diameter (mm)**} & \multicolumn{2}{c}{**Min/Max**} \\  & & & & **Length (m)** & & **Diameter (mm)** \\ \cline{5-8}  & & & & **(Rank)** & & **Total Length/No.** & **Length** & \\ \cline{5-8}  & & & **(Rank)** & & **of Pipe** & **Weighted** & **Min** & **Max** \\ \hline RES1 Pipeline & 316,645 (2) & 108 (6) & 38,782 (2) & 1487 (1) & 1762 & 500 & 2400 \\ RES2 Pipeline & 732,455 (1) & 140 (4) & 69,369 (1) & 1424 (4) & 1989 & 100 & 3300 \\ RES3 Pipeline & 316,645 (2) & 137 (5) & 37,198 (4) & 1436 (2) & 1438 & 100 & 2200 \\ RES4 Pipeline & 194,551 (5) & 36 (8) & 13,954 (7) & 1432 (3) & 1566 & 250 & 2200 \\ RES5 Pipeline & 34,582 (8) & 34 (9) & 10,633 (9) & 1285 (6) & 1168 & 800 & 2000 \\ RES6 Pipeline & 34,582 (8) & 41 (7) & 10,736 (8) & 1340 (5) & 1671 & 100 & 2400 \\ RES7 Pipeline & 96,905 (7) & 291 (2) & 33,925 (5) & 417 (9) & 546 & 40 & 1200 \\ RES8 Pipeline & 211,363 (4) & 311 (1) & 38,105 (3) & 452 (8) & 553 & 40 & 1350 \\ RES9 Pipeline & 102,036 (6) & 202 (3) & 26,562 (6) & 619 (7) & 534 & 50 & 1200 \\ Total (Entire & & & & & & & \\ System) & 2,039,765 & 1300 & 279,263 & 1099 & 1315 & 40 & 3300 \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Water Supply and Pipe Characteristics of Each Critical Path. Figure 5: Distribution of Pipe Diameter in Each Critical Path. were simulated to achieve statistically appropriate convergence. A Ground Motion Prediction Equation (GMPE) was applied to calculate the probability of failures of individual pipes due to seismic wave attenuation after an earthquake, as in a previous study [31]. Equation (3) of [31] which was shown to be suitable based on earthquake data from Korea, was applied. \[\log(y)=a+bM+clog\sqrt{R_{epi}^{2}+h^{2}}+dR_{epi}+es \tag{3}\]  
Here, \(a\), \(b\), \(c\), \(d\), and \(e\) are coefficients calculated from the regression analysis, and values of \(-1\), \(0\), and \(1\) are used for the modification coefficients given for each observation. The coefficient \(h\) was added to prevent unrealistic results that may occur within a \(10\) km radius of the station. \(y\) corresponds to \(PGV\) (Peak Ground Velocity, m/s), \(M\) is the magnitude of the earthquake, and \(R_{epi}\) is the distance to the epicenter of the earthquake. Finally, the pipeline repair rate (\(RR\)) was calculated using the \(PGV\) computed by GMPE and the characteristics of the pipeline. In this study, as suggested in Equation (4), the repair rate was obtained by applying the corresponding coefficient in accordance with the pipe type (C1), pipe diameter (C2), buried topography (C3), and liquefaction condition (C4). The modification factors \(C1\)-\(C3\) of the target area--the A-City water pipe network system--were applied based on the basic data presented in [32]. For C4, information on liquefaction was not available, so it was assumed to be \"none\" and a value of \(1.0\) was uniformly applied to all pipelines.