This paper proposes a framework for a reliabilitybased flow analysis for a water pipe network after an earthquake. For the first part of the framework, we propose to use a modeling procedure for multiple leaks and breaks in the water pipe segments of a network that has been damaged by an earthquake. For the second part, we propose an efficient systemlevel probabilistic flow analysis process that integrates the matrixbased system reliability (MSR) formulation and the branchandbound method. This process probabilistically predicts flow quantities by considering systemlevel damage scenarios consisting of combinations of leaks and breaks in network pipes and significantly reduces the computational cost by sequentially prioritizing the system states according to their likelihoods and by using the branchandbound method to select their partial sets. The proposed framework is illustrated and demonstrated by examining two example water pipe networks that have been subjected to a seismic event. These two examples consist of 11 and 20 pipe segments, respectively, and are computationally modeled considering their available topological, material, and mechanical properties. Considering different earthquake scenarios and the resulting multiple leaks and breaks in the water pipe segments, the water flows in the segments are estimated in a computationally efficient manner.
Water pipe networks constitute one of the largest of our essential infrastructure assets and make a substantial contribution to economic services, industrial activities, quality of life, and the environment. Their primary duty is to provide ample amounts of water at a pressure that is sufficient to satisfy the demands of all consumers. A reliable water supply is essential to communities and is one of the socalled “services of general interest” that are vital to general welfare, public health, and the collective security of populations, as well as economic activities [
Many research efforts have attempted to estimate the seismic performance or probabilistic flow quantity of water supply networks using Monte Carlo simulation (MCS) or nonsimulationbased methods. Recently, several researchers [
This paper proposes a framework for a systemlevel probabilistic water network flow analysis after a seismic event. The first part of the proposed framework involves the computational modeling of a water pipe network considering the components that are leaking and broken components due to earthquake shaking. In this study, multiple software packages including ArcGIS [
The proposed framework includes a computational network modeling procedure for hydraulic analysis, a damage modeling procedure for leaks from and breaks in pipe segments, and a systemlevel probabilistic flow analysis procedure combining the MSR method and the branchandbound method. The modeling part of the framework can be further improved by introducing more realistic assumptions and precise modeling schemes.
The first procedure of the proposed framework consists of network modeling for hydraulic analysis, which includes the following steps. (i) The topology of the water pipe network of interest is prepared. An initial drawing of the model is overlaid on a base map obtained from Google Earth, originally developed by Keyhole Inc. in 2001 and later acquired by Google in 2004. Using Google Earth, the locations of the junctions and the pipes of the network are roughly determined. The other parameters, including the pipe diameter, roughness, and the locations of pumps, reservoirs, and tanks, are not included in the model at this stage. The model is stored as a keyhole markup language (KML) file containing the network’s coordinate information for pipelines and junctions. (ii) The KML information from Google Earth is decompressed using GPS Visualizer, an Internetbased publicdomain software product developed in 2002. Through the use of GPS Visualizer, the coordinates are extracted from the Google Earth KML file. A new KML file is created using GPS Visualizer which contains the accurate coordinates and ground profile of the water pipeline network. (iii) The KML file created in GPS Visualizer is imported into the ArcGIS software to undertake further modeling of the water pipeline network. ArcGIS is a software product based on the Geographic Information System (GIS) and developed by ESRI. In ArcGIS, additional parameter information such as the pipe diameter, pipe roughness, elevation of the junctions, and water demand locations and levels is added. Graphically unsnapped junctions in the Google Earth model are fully snapped using the snapping tool in ArcGIS with 0.001 decimal degrees, for example. A drawing interchange format (DXF) file is created in ArcGIS and then imported into EPANET (a hydraulic analysis software product developed by the US Environmental Protection Agency). Since EPANET cannot read DXF files directly, EPACAD is used to convert the DXF file into an input file for EPANET. (iv) In EPANET, the remaining parameters are defined, including the locations of reservoirs, tanks, and pumps, the required demand at junctions, and other flowrelated parameters. Once the network modeling is complete, a hydraulic analysis can be carried out with EPANET to estimate the water flows in the pipeline segments and the head pressures of the pipe junctions. EPANET uses a gradientbased algorithm developed by Todlini and Pilati [
To model leaks and breaks in multiple pipe segments after a seismic event, a modeling scheme in GIRAFFE [
Hydraulic model of pipe break.
Hydraulic model of pipe leak.
Figure
In the proposed framework, the MSRbased uncertainty quantification method [
For each system state, any corresponding system quantity such as the posthazard flow of each pipeline can be estimated by using the water flow analysis model proposed in Sections
Using the probability vector
The systemlevel probabilistic flow analysis described in Section
The branchandbound method is used to sort the system state probability vector in (
For each process, we check whether the element with the maximum value contains all
The proposed framework is applied to the water pipe network shown in Figure
Example water pipe network with 11 pipe segments.
If an earthquake occurs, it is assumed that each pipeline in the network will assume one of the following three damage states: undamaged, leaking, or broken. The failure probabilities of the pipes are estimated using the equation below for repair rates, given as a function of the peak ground velocity (PGV) in the HAZUS technical manual [
The failure probability of each pipe is computed using a Poisson process along a dimension of length. This paper deals only with failures induced by ground shaking and uses the repair rate model given by (
The PGV is computed from the following attenuation relationship [
When a pipe segment is damaged by the ground shaking of an earthquake, it can assume either one of two states: leaking or broken. The likelihoods of these occurring are assumed to be 0.8 and 0.2, according to the probability distribution used in Zolfaghari and Niari [
Using the proposed flowbased reliability analysis framework, the mean, standard deviation, and coefficient of variation (c.o.v.) of the flow rate in pipe 5 for given earthquake magnitudes were estimated. These are listed in Table
Mean, standard deviation, and c.o.v. of water flow from node 1 to pipe 5.
Mean (m^{3}/day)  Standard deviation (m^{3}/day)  c.o.v.  


8219.4  11736  1.4278 

9164  17396  1.8983 

9182.7  23725  2.5837 
Uncertain 
8934.4  16461  1.8335 
Probability of water flow in pipe 5.
The results for pipe 6 are listed in Table
Mean, standard deviation, and c.o.v. of water flow in pipe 6.
Mean (m^{3}/day)  Standard deviation (m^{3}/day)  c.o.v.  


−3746  8224.4  2.1955 

−2907.3  14166  4.8725 

−2654.3  20539  7.738 
Uncertain 
−3100.8  13107  4.3982 
Probability of water flow in pipe 6.
The computational cost incurred for the analysis of the full system states is about 148 hours using MATLAB on a computer with an Intel I7 CPU (2.80 GHz) and 3 GB of RAM, and it can be considered as unaffordable. However, it should be noted that, for all the results and plots, the quantity vector
To further reduce the computational cost, the branchandbound method introduced in Section
Comparison of results obtained with proposed branchandbound approach and by Monte Carlo simulation for pipe 5.
Comparison of results obtained with proposed branchandbound approach and by Monte Carlo simulation for pipe 6.
Comparison of the results obtained with proposed branchandbound approach and by Monte Carlo simulation for pipe 1.
Comparison of results obtained with proposed branchandbound approach and by Monte Carlo simulation for pipe 2.
Figure
In Figure
Figure
Figures
Figure
Number of identified branches in branchandbound analysis, normalized by number of full system states.
Figure
Computation time for branchandbound analysis normalized by computational time required to evaluate overall system state.
Note that the branchandbound method in the proposed framework is based on the order of the probabilities of the system states without considering the system quantities, although we are interested in the prediction of flow rate quantities in terms of their partial descriptors which are the functions of both the probabilities and the associated quantities as shown in (
The proposed framework was extensively applied to the 20pipe water supply network shown in Figure
Link and demand node data of water distribution network [
Link number  Length (km)  Pipe type  Diameter (mm)  Probability of failure  Node number  Water demand (m^{3}/day) 

1  1.183  CIP  400  0.2844  2  2237 
2  0.818  DIP  400  0.2084  3  1678 
3  2.193  DIP  800  0.3474  4  3356 
4  1.472  DIP  400  0.5721  5  4475 
5  0.973  CIP  400  0.4211  6  2797 
6  0.474  DIP  400  0.2126  7  2237 
7  0.726  DIP  400  0.4049  8  3356 
8  0.673  DIP  400  0.2717  9  3356 
9  0.248  CIP  800  0.0855  10  5593 
10  0.440  CIP  800  0.1514  11  2237 
11  0.616  CIP  400  0.1751  12  5593 
12  0.522  DIP  400  0.1621  13  3915 
13  0.562  CIP  800  0.1677  14  3915 
14  1.559  CIP  400  0.3930  15  5034 
15  0.411  DIP  400  0.1155  
16  1.230  DIP  400  0.3099  
17  1.409  DIP  400  0.3429  
18  0.524  CIP  400  0.1492  
19  2.033  CIP  400  0.8942  
20  0.437  DIP  400  0.1490 
DIP: ductile iron pipe; CIP: castiron pipe.
A 20pipe water supply network in Kobe city.
Modeling of Kobe city network in Google Earth.
Hydraulic model of Kobe city network in EPANET.
Figure
Predicted flows (m^{3}/day) for pipes (a) 4, (b) 13, (c) 16, and (d) 20 using the branchandbound method.
However, Figure
Predicted flows (m^{3}/day) for pipes (a) 1, (b) 2, (c) 3, and (d) 9 using the branchandbound method.
Figure
Number of identified branches in branchandbound analysis normalized by number of full system states.
Figure
Computation time required for branchandbound analysis.
This study has proposed an efficient systemlevel probabilistic flow analysis framework for estimating the postevent performance of a water pipe network in a probabilistic way. The framework consists of two parts: (
The authors declare that there is no conflicts of interest regarding the publication of this paper.
The authors thank Ms. Seungyeon Ji and Mr. Abdul Mauzir of Western Sydney University for their contribution to discussions about and improvement of this manuscript. This work was supported by the Australian Research Council (ARC) under its Linkage project (Project no. LP140100030). This work was also supported by the 2017 Research Fund (1.170013.01) of UNIST (Ulsan National Institute of Science and Technology).