Corps de l’article

1. Introduction

The structure of aging pipes becomes increasingly fragile largely due to corrosion of their inner surfaces or linings (SCHOCK, 1990) and poses major financial challenges for municipalities and concerns regarding the quality of services for citizens. Subject to the various forces acting on them, weakened pipes are susceptible to breaks (EISENBEIS, 1994). This, in turn, can cause a leak, which once detected, can be repaired. Despite a certain number of interventions (repairs and replacements of pipes), pipe breaks continue to increase in several North American cities (DUCHESNE et al., 2011). The history of these interventions, if recorded, provides a good indication of the structural state and aging of the pipe network. In recent years, certain water network managers have recorded these interventions, thus creating an historical database of pipe breakages. In order to better plan their investments, municipalities should be equipped with tools enabling them to track and predict the evolution of pipe aging (DUCKSTEIN and PARENT, 1994; GERMANOPOULOS et al., 1986).

Two significant reviews of the different models used to evaluate the structural deterioration of pipes were conducted in KLEINER and RAJANI (2001) and RAJANI and KLEINER (2001). The authors classified these models into two categories: deterministic and probabilistic. Furthermore, they underlined the difficulty in gathering the amount of data required to establish a deterministic model capable of simulating the aging of pipes throughout an entire pipe network (KLEINER and RAJANI, 2001). WALSKI and PELICCIA (1982) also underlined the difficulty in gathering the amount of data required to establish a deterministic model capable of simulating the aging of pipes throughout an entire distribution system. Additionally, BERARDI et al. (2008) argued that efforts to introduce complex analytical techniques become futile when data are either not available or are of poor quality. Hence, to model the aging of pipes in water distribution networks, probabilistic models should be used, except in the case where sufficient data are available to use a deterministic approach for predicting asset failure.

Comprehensive reviews of the literature concerning probabilistic models predicting the rate of water main pipe breaks can be found in VILLENEUVE et al. (1998), PELLETIER (2000), MAILHOT et al. (2000), GOULTER and KAZEMI (1988 and 1989), TSITSIFLI and KANAKOUDIS (2010), TSITSIFLI et al. (2011) and JOWITT and XU (1993). VILLENEUVE et al. (1998), used survival analysis to model the time elapsed between successive breaks. To use survival analysis in a classical manner (calibration by the maximum likelihood) however, one must know the complete history of the breakages, the time elapsed between pipe installation and the first break, the time between the first and second break, and so forth. In addition, studies have shown that the time between two successive breaks is different from the amount of time between installation and the first break (CLARK and GOODRICH, 1989; CLARK et al., 1982; ANDREOU et al., 1987). Thus, a different statistical distribution should be used to represent each of these intervals. The most frequently used statistical distributions to represent the time between pipe installation and the first pipe break or the time between two successive breaks are the Weibull and exponential distributions (ANDREOU et al., 1987; EINSENBEIS, 1994). PELLETIER (2000) and MAILHOT et al. (2000) developed a statistical modeling approach that takes into consideration the periods when pipe failures were and were not recorded. This was the first appearance of this methodological approach in the literature to model water main pipe breaks. Case studies were conducted demonstrating the pertinence and validity of this methodology (PELLETIER, 2000; PELLETIER et al., 2003). This approach was also used to establish an optimal process to replace pipes (MAILHOT et al., 2003). KANAKOUDIS and TOLIKAS (2001) also introduced a methodology that calculated the optimum replacement time for the pipes of a water network. This was done by performing a techno-economic analysis that took into account the costs related to the repair or replacement of trouble-causing parts of a network. GOULTER et al. (1993), JOWITT and XU (1993), KANAKOUDIS (2004), and KANAKOUDIS and TOLIKAS (2004) also provide examples of methodologies for optimal replacement.

TOUMBOU et al. (2014), inspired by the approach of MAILHOT et al. (2000), developed a general model which permits the use of explanatory covariates which take into account factors which can lead to pipe breaks (pipe age, pressure, temperature, soil humidity, pipe diameter and length, type of pipe material, etc.). The study by TOUMBOU et al. (2014) was primarily focused on the theoretical development of the general model, which advantageously allows for combinations of distributions permitting better predictions of pipe breaks over time. If the last distribution used in the combination is exponential, one can calculate analytically the average annual number of failures in a network since an equation can be developed to estimate the probability of failure of each of the pipes during a given year. Several other models using explanatory covariates have been developed (LE GAT and EISENBEIS, 2000; KLEINER and RAJANI, 2001; RAJANI and KLEINER, 2001; KRETZMANN and VAN ZYL, 2004; VANRENTERGHEM-RAVEN, 2008; DAVIS et al., 2008; BERARDI et al., 2008; YAMIJALA et al., 2009; WANG et al., 2009; KLEINER et al., 2009; ALVISI and FRANCHINI, 2010).

In the present study, we propose and compare three models for the simulation of pipe breaks in a water main network: a linear regression model, the Weibull-Exponential-Exponential (WEE) model and the Weibull-Exponential-Exponential-Exponential (WEEE) model, each of which are described in detail. The objectives were to evaluate the capability of the three models to predict future pipe breaks, to identify the best suited model as a function of the modeling objectives and to determine the most appropriate calibration method for the WEE and WEEE models, according to the available data. All these evaluations were performed using a database of recorded breaks in a real water main network of a municipality in the province of Quebec.

2. Case Study

The database of the municipality under study includes breaks that were observed and recorded from 1976 to 2007, on the pipes of its water distribution network that were in place in 2008. In addition to the number of breaks on each pipe by year, the database contains the pipe installation dates (from 1944 to 2006) and the diameter and length of each pipe. The network, in 2008, contained 10 258 pipe segments for a total pipe length of 396 km. Pipe segments, that we refer to simply as “pipes” throughout the text, are defined as portions of the water distribution network that are located between two adjacent street junctions, with constant slope, diameter, and material. The discretization of the network into pipes was carried out by the City’s staff. The database used for model development and calibration was constructed by the authors, from data provided by the City. For the breaks that occurred from 1997 to 2007, the City provided two geo-referenced databases: one for the pipes and one for the breaks. The spatial joint of these two databases was performed by the authors. For the breaks that occurred from 1976 to 1996, the City provided, along with a geo-referenced database of the network, a list of the break dates with the civic address of the nearest building. Many operations were conducted by the authors to locate these breaks on the corresponding pipes, including spatial joints, corrections of street names and manual localizations. The numerical operations involved in the construction of the final database were performed rigorously, in an attempt to include as much information as possible and to make sure this information reflected what was provided by the City. However, it is possible that not all of the pipe breaks in the water network, from 1976 to 2007, were recorded properly by the City. In the absence of other information, we can only assume that the provided databases accurately represented the state of the pipe network. Furthermore, the increasing trend in the annual total pipe breaks and their interannual variability suggest that the breaks were properly recorded.

3. Presentation of the Models

In this study, only time (t) was used as an explanatory variable to model the evolution of breaks in the system. As presented in TOUMBOU et al. (2014), other explanatory variables (e.g. pipe diameter or material) can be integrated in the models. However, it was observed that the presence or absence of the pipe diameter and material in the model did not significantly change the results when computing the total annual number of breaks for the whole network (when the models integrating explanatory variables were simulated and compared to those without explanatory variables for several networks in the province of Quebec; see DUCHESNE et al., 2011). Indeed, the incorporation of explanatory variables other than time into the model can significantly affect the probability that a specific pipe will break during a specific year, but the total number of breaks for the entire network during a year might not be affected. For this reason the only explanatory variable considered in this paper is time (t). Three models are used and compared in this paper. Each of them is described briefly below.

If the sole objective of a municipality is to evaluate the evolution of pipe breaks over time the simplest method is to fit a linear regression. However, the use of a linear regression does not allow to take into account different explanatory factors or pipe replacement scenarios. In this case, the model which is used is a linear regression of the form:

where y represents the annual number of pipe breaks, x is the number of years since the reference year, and a and b are parameters which value is determined during calibration.

To better reflect the evolution of pipe breaks over time as well as to account for pipe replacements, models that take these aspects into account should be used. Water distribution pipe breaks can be likened to a failure rate process that is represented by survival functions as amply shown in the literature (VILLENEUVE et al., 1998; PELLETIER, 2000; MAILHOT et al., 2000; MAILHOT et al., 2003; KLEINER and RAJANI, 2001; RAJANI and KLEINER, 2001; KRETZMANN and VANZYL, 2004; VANRENTERGHEM-RAVEN, 2008; BERARDI et al., 2008; DAVIS et al., 2008; KLEINER et al., 2009; WANG et al., 2009; YAMIJALA et al., 2009; TOUMBOU et al., 2014). Here, we use the exponential and Weibull distributions (KALBFLEISCH and PRENTICE, 2003) to build the two models under study: WEE and WEEE. The corresponding survival functions, that represent the distribution of the time elapsed between two successive breakages or between the first break and pipe installation, are written respectively:

where FE is the exponential survival function, FW is the Weibull survival function, t represents time, and k and p are scalar parameters.

The WEE model represents the time elapsed from installation to the first break using a Weibull distribution, the time elapsed between the first and second break using an exponential distribution, and the time between subsequent breaks using another exponential distribution. A similar definition holds for the WEEE model.

4. Calibration of the Models

For the linear model, the value of the parameters a and b were obtained by using the least squares (LS) method consisting of minimizing the sum of squared deviations between the annual numbers of observed and simulated breaks in the network for each year of the calibration period. For the other models, calibration was more complex, as we explain in the section which follows.

Two methods are considered for the calibration of the WEE and WEEE models, namely the least square (LS) and the maximum likelihood (ML) methods. To apply the LS method, it is required to compute the total number of pipe breaks for each year. To apply the ML method, the likelihood function must be developed. These two developments are detailed in the following sections.

4.1 Calculation of the average annual number of breakages

The average number of breakages for a pipe between T and T + DT can be found using:

with P(a, b) the probability of having a breaks between 0 and T, and b breaks between T and T + DT.

The general equation to estimate the average number of breakages, given in Equation 3, was developed in TOUMBOU et al. (2014) for the WEE model and the following equation was obtained:

with

and

where k2 and k3 are calibration parameters.

An equation to calculate the average number of breaks can also be obtained for the WEEE model using the general model presented in TOUMBOU et al. (2014) as follows:

with

and

where k4 is a calibration parameter.

Thus, the cumulative number of breaks in the whole network (N pipes), for a given year A, can be calculated using the following equation:

In this paper, the cumulative number of breaks given in Equation 6 was used for model calibration with the LS method and also for the simulation of the total annual number of breakages.

4.2 Calculation of the likelihood function

To establish the likelihood function for the WEE and WEEE models, one must first calculate P(n), n ≥ 0, which is the probability of having n breakages between Tb and Ta regardless of the number of breaks between 0 and Tb. The likelihood function is written as follows:

where par is the vector of parameters ((kj, pj) where j = 1, 2, …m + n), bi is the number of recorded breaks for pipe i, N is the total number of pipes in the network and Pi(bi) is the probability for pipe i to have bi breaks between Tb and Ta, the starting and ending time of data collection period, regardless of the number of breaks between 0 and Tb.

To determine the value of the model parameters for WEE, the following function, namely the natural logarithm of the likelihood function, was maximized:

where

For the WEEE model, the logarithmic expression of the likelihood function L is shown in equations 10 and 11.

5. Results

In order to simulate the evolution of the annual number of breakages in the network, the parameter values in each of the models were first evaluated. Since we sought to verify the capability of the models to predict future breaks, the models were first calibrated using the period from 1976 to 1996. This then allowed comparing the model-based predictions with the actual observations which were recorded from 1997 to 2007.

To demonstrate the ability of simulating replacement scenarios, the WEEE model, as an example, was calibrated using the data from 1976 to 2007. The annual number of breaks simulated with this model from 1997 to 2007 was then compared to the number of breaks simulated with the WEEE model calibrated using the data from 1976 to 1996, and assuming a pipe annual replacement rate equal to 0.5% of the total network length (which is the average annual replacement rate that the municipality actually applied from 1997 to 2007, that can vary between pipe material and diameter). Finally, the WEEE model was used to simulate the impact on the annual number of breaks of four pipe replacement scenarios: 0.5%, 1.0%, 1.5%, and 2.0% replacement of the total network length.

where

5.1 Calibration for the period 1976 to 1996

Calibration of the WEE and WEEE models was carried out using the LS and ML methods while considering the state of the network in 1996. Values of the parameters for this calibration are presented in Table 1. Note that this calibration was carried out under constraints to meet the assumption that successive breaks occur more and more frequently (k1 < k2 < k3 < …< kn). The calibration of the linear model was conducted using the LS method. The obtained parameters were: a = 4.2416 and b = -231.99.

Table 1

Calibration parameter values for the period 1976 to 1996

Valeur des paramètres de calage pour la période 1976 à 1996

Calibration parameter values for the period 1976 to 1996
a

LS: least squares; ML: maximum likelihood; 96: calibrations on the 1976-1996 period.

-> Voir la liste des tableaux

Then these parameters were used to simulate the average number of annual breaks. Simulated break curves are presented in Figure 1. In this figure, one can see that the calibration by the LS method overestimates the predicted number of breaks compared to the observed breaks from 1997 to 2007. It is also evident that the ML method seems the most realistic in terms of prediction of pipe breaks. This shortcoming of the LS method is likely linked to the small amount of available data to determine the values of the parameters as demonstrated by BAYRAK and AKKAYA (2010) and TSURU and HIROSE (2009). In fact, for the LS method, there were only 21 data points, representing the annual number of breaks for the period from 1976 to 1996. In contrast, using the ML method, we were able to incorporate information for each of the 9 233 pipes that were in the network in 1996. This included each break, its date of occurrence, the age of the pipe at the time of breakage, and the age of the remaining unbroken pipes. This leads us to conclude that if one has all of the information for each of the pipes, then it is more appropriate to use the ML method for short observation periods. If all of the necessary information to apply this method is not available, then a simple trend line can provide satisfactory results as seen in Figure 1.

Figure 1

Results of the calibrated models for the observation period from 1976 to 1996 and observed breaks (the WEE_ML_96 and WEEE_ML_96 curves overlap). LS: least squares; ML: maximum likelihood; 96: calibrations on the 1976-1996 period

Résultats des modèles calés avec les observations de la période 1976 à 1996 et nombre annuel de bris observés (les courbes WEE_ML_96 et WEEE_ML_96 sont superposées)

Results of the calibrated models for the observation period from 1976 to 1996 and observed breaks (the WEE_ML_96 and WEEE_ML_96 curves overlap). LS: least squares; ML: maximum likelihood; 96: calibrations on the 1976-1996 period

-> Voir la liste des figures

Figure 1 also shows that the results of the simulations for the two models WEE and WEEE are very similar for each of the two calibration methods, as their respective curves are overlapping. For the remainder of the study we thus employed the WEEE model. Although the number of simulated pipe failures by both the WEE and WEEE models were similar, the WEEE model seems best suited for the case study. Indeed, as shown in Table 2, the number of third or more breaks is relatively important at 4.07% of pipes in the studied network, and the WEEE model uses different distributions for the first, second, third, fourth, and subsequent breaks, while only three distributions are used in the WEE model (for the first, second, third, and subsequent breaks).

Table 2

Percentage of pipes according to the total recorded number of breaks

Pourcentage de conduites selon le nombre total de bris enregistrés

Percentage of pipes according to the total recorded number of breaks

-> Voir la liste des tableaux

5.2 Calibration for the period 1976 to 2007

The WEEE model was calibrated with the observed data for the period from 1976 to 2007, using the LS and ML methods. The calibration values are presented in Table 3. Once again, this calibration was carried out under the assumption that successive breaks occur more and more frequently.

Table 3

Calibration parameter values for the period 1976 to 2007

Valeurs des paramètres de calage pour la période 1976 à 2007

Calibration parameter values for the period 1976 to 2007
a

LS: least squares; ML: maximum likelihood.

-> Voir la liste des tableaux

Considering the observations from 1976 to 2007, consisting of 32 data points representing the average annual number of breaks and 10 258 pipes in the network in 2008, the number of failures simulated by the LS method was very similar to that simulated by the ML method (Figure 2). This can be explained by the fact that there are more observed data than in the previous assessments, allowing for a better calibration with the LS method. It should be noted that here, the ML method used information from 10 258 pipes in the network in 2008, while the LS method only required 32 observations to generate similar results.

Figure 2

Results of the calibrated models for the observation period from 1976 to 2007 and observed breaks. LS: least squares; ML: maximum likelihood

Résultats des modèles calés avec les observations de la période 1976 à 2007 et nombre annuel de bris observés

Results of the calibrated models for the observation period from 1976 to 2007 and observed breaks. LS: least squares; ML: maximum likelihood

-> Voir la liste des figures

The linear regression, with parameter values a = 2.7945 and b = -118.65, also gave very good results. It is important to note that the linear regression model allows for the simulation of the evolution of pipe breaks, as shown in figures 1 and 2, however, it does not take into account certain factors such as the diameter, length, or type of material of the pipes. Additionally, this model does not allow pipe replacement scenarios to be taken into consideration when simulating the annual number of pipe breaks. As a result, to account for these important factors, it is necessary to resort to other statistical models such as the WEE and WEEE models.

5.3 Capability of the models to predict future pipe breaks

To validate the predicted values generated by the models, it was necessary to compare them with the recorded observations. However, the values simulated with the models are averages of the annual numbers of pipe breaks. Given the dispersion in the observed values, it was difficult to directly compare them with those provided by the models. In order to make this comparison, we determined the observed average annual values from the cumulative break curve. In order to do this, a second degree polynomial was fit to the cumulative observed break curve. For all practical purposes these curves are overlapping (Figure 3). We then differentiated the resulting cumulative model, which gives what we have designated as the average observed annual breaks (AOAB). The line representing these average values is slightly above the trend line and the curve simulated by the WEEE_ML_07 model which were previously obtained (Figure 4).

Figure 3

Cumulative breaks from 1976 to 2007, observed and simulated with a second degree polynomial

Bris cumulés de 1976 à 2007, observés et simulés avec un polynôme du second ordre

Cumulative breaks from 1976 to 2007, observed and simulated with a second degree polynomial

-> Voir la liste des figures

Figure 4

Results for the derived model (AOAB), the WEEE_ML_07 model and the linear regression, compared to the observations from 1976 to 2007. ML: maximum likelihood

Résultats du modèle dérivé (AOAB), du modèle WEEE_ML_07 et de la régression linéaire, comparés avec les observations pour la période de 1976 à 2007

Results for the derived model (AOAB), the WEEE_ML_07 model and the linear regression, compared to the observations from 1976 to 2007. ML: maximum likelihood

-> Voir la liste des figures

Thus, we used this average to evaluate the predictive abilities of the WEE and WEEE models. In Table 4, we present the pipe breaks for different years, for the period 1997 to 2007, which were predicted by the WEEE_ML_96, WEEE_LS_96, and AOAB models.

Table 4

Predicted number of average annual breakages with the WEEE and AOAB models

Nombre annuel moyen de bris prédits avec les modèles EEE et AOAB

Predicted number of average annual breakages with the WEEE and AOAB models
a

ML: maximum likelihood; LS: least squares; 96: calibrations on the 1976-1996 period.

-> Voir la liste des tableaux

As compared to the AOAB model, the best predictive model to use, assuming the network was in the same state as it was in 1996, is the WEEE calibrated with the ML method. The maximum annual difference between the results for both models (3.5%) was reached in 2007 (173 breaks for the AOAB model and 179 breaks for the WEEE_ML_96 model), which was an excellent result given both the short observation period and the dispersion of the observed breaks (Table 4).

During the period from 1997 to 2007, the city carried out an average pipe replacement rate of 0.5% of the total network length per year. To validate the model, the simulated results of the WEEE_ML_96 model (calibrated with data from 1976 to 1996) assuming a pipe replacement scenario of 0.5% over the period 1997 to 2007, were thus compared to those obtained from the WEEE_ML_07 model (calibrated with data from 1976 to 2007). Results of these simulations are presented in Table 5.

Table 5

Predicted number of average annual breakages with the WEEE model for different scenarios

Nombre annuel moyen de bris prédits avec le modèle WEEE pour différents scénarios

Predicted number of average annual breakages with the WEEE model for different scenarios
a

ML: maximum likelihood; 96: calibrations on the 1976-1996 period.

-> Voir la liste des tableaux

Results obtained from the WEEE_ML_96 model with a 0.5% replacement scenario and WEEE_ML_07 model were similar from 2003 onward. Figure 5 illustrates the data which are presented in Table 5 and, again, shows the ability of the model to simulate the evolution of pipe breaks and the ability to model pipe replacement scenarios.

Figure 5

Simulation of the 0.5% pipe replacement scenario on the 1996 network and comparison with the WEEE_ML_96 model results, the WEEE_ ML_07 model results, and the observed breaks. ML: maximum likelihood

Simulation du scénario annuel de remplacement de 0,5 % de la longueur totale du réseau de 1996 et comparaison avec les résultats du modèle WEEE_ML_96, les résultats du modèle WEEE_ML_07 et les bris observés

Simulation of the 0.5% pipe replacement scenario on the 1996 network and comparison with the WEEE_ML_96 model results, the WEEE_ ML_07 model results, and the observed breaks. ML: maximum likelihood

-> Voir la liste des figures

5.4 Implementation of replacement scenarios

As previously stated, the importance of the WEE and WEEE models is their ability, when calibrated, to simulate replacement scenarios in a water main network. These scenarios indicate what will happen to the evolution of annual breaks after replacement of pipes in the network. This information is essential for municipalities which would like to both plan pipe replacements and know what effect they will have on the number of annual breaks. Independent of social costs, municipalities can evaluate the costs of each scenario and compare them with the costs of damage repair. Furthermore, they can establish maintenance and repair policies knowing the short and long term impacts of their efforts. However, although models simulating the evolution of the number of pipe breaks over time are quite helpful in assessing the global costs associated with the renewal of water pipes, the actual decision to replace a pipe is often based on a variety of other factors, including inspection data, hydraulic performance and required work on other pipes located in the same trench. In the following, we assume that the oldest pipes take priority for replacement.

The WEEE_ML_07 was used to simulate four pipe replacement scenarios. The average annual replacement rates of 0.5%, 1.0%, 1.5%, and 2.0% of the total network length were used as example scenarios. The results of these simulations are presented in Figure 6 and Table 6. These results demonstrate that, in order to maintain the same average annual rate of pipe breakage as in 2009 (162 breaks), it is necessary to replace 1.5% of the length of the network annually. If the municipality under study maintains an average rate of 0.5% replacement, as was done over the course of the last ten years, the average annual number of breakage will increase from 162 in 2009 to 196 in 2027.

Figure 6

Replacement scenarios simulated with the WEEE_ML_07 model

Scénarios de remplacement simulés avec le modèle WEEE_ML_07

Replacement scenarios simulated with the WEEE_ML_07 model

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Table 6

Predicted annual number of breaks according to various pipe replacement scenarios (the indicated percentages correspond to the percentage of total length of the network)

Nombre annuel de bris simulés selon différents scénarios de remplacement de conduites (les pourcentages indiqués correspondent à des pourcentages de la longueur totale du réseau)

Predicted annual number of breaks according to various pipe replacement scenarios (the indicated percentages correspond to the percentage of total length of the network)

-> Voir la liste des tableaux

6. Conclusion

Calibrations and simulations with the WEE and WEEE models in this study were made possible by the models generalized by TOUMBOU et al. (2014). We first demonstrated the ability of the aforementioned models to predict annual pipe breaks over time. We also evaluated the capacity of these models to incorporate specific pipe replacement scenarios in a water main network. This was achieved by using replacement and break data provided by the municipality under study. If the period of observation is short (less than 25 observations), we recommend using the ML method for calibration, however this is only possible when one has access to all of the information for each of the pipes of the network. If this is not the case, a trend line will be sufficient to predict the number of breaks over time, if no changes are made to the network; though this type of curve does not allow one to account for pipe replacement scenarios. If information for each of the pipes is not available and replacement scenarios need to be accounted for, then the models could be calibrated with the LS method. These results were obtained using only pipe age as an explanatory variable to model the occurrence of pipe breaks. Using covariates such as pipe diameter, length, or material could be possible with the WEE and WEEE models but would most probably lead to the same conclusions.