4.1.1. Results for Scenario 1: Single-Pipe Failure

Figure 4 and Table 4 show the correlation between reliability indices and hydraulic measures according to the application of Scenario 1. In Figure 4, the points in the scatter plot represent the reliability index values and hydraulic measures derived from each of the 17 networks (*x*-axis—Reliability index; *y*-axis—Hydraulic measure), and Table 4 summarizes the correlation coefficients calculated from each scatter plot. The plots and table values indicated in yellow are index-measure combinations with relatively high correlation coefficients of 0.95 or higher.

**Figure 4.** Scatter plots of Scenario 1 (*x*-axis—Reliability index; *y*-axis—Hydraulic measures). The plots highlighted in yellow show high correlation coefficient of 0.95 or higher.


**Table 4.** Correlation coefficients for Scenario 1.

Note: The values highlighted in yellow indicate high correlation coefficient of 0.95 or higher.

In Scenario 1, a broken pipe causes increment of flow and head loss to substitute paths. The hydraulic indices (RI, MRI, and API) are proportional to excessive nodal pressure in normal water demand conditions. When the pipe is broken, the network which has bigger excessive pressure remains more backup pressure after the event, thus the redundancy measures (Red1 and Red2) have relatively high correlation with the hydraulic indices (*R* values are 0.98 and 0.97, respectively).

The topological indices (AD and NE) quantify the diversity of water paths. For the pipe failure scenario, therefore, these indices show relatively high correlation with all measures indicating good representation of system performance under the specific condition (*R* values are higher than 0.95 for Red2, Rob2, and Ser2).

The entropic index (FE) quantifies the diversity and relative uniformity of flow distribution. When a pipe is isolated, the higher flow diversity leads to better adaptation against rapid increase of

water flow. However, increased flow occurs in limited local paths, thus the overall correlation with the measures is relatively low compared to other indices.

Among the mixed indices (PHRI, RRI, and NRI), PHRI shows good correlation, especially with the serviceability measures (*R* values are 0.98 for Ser1 and 0.97 for Ser2). For a pipe isolation condition, a network with a lower hydraulic gradient gets better service. RRI and NRI are calculated in a similar way to RI, thus, they also show high correlation with the redundancy measures, but have a relatively low correlation with other measures.
