Modeling of failure rate of water-pipe networks Małgorzata Kutyłowska *
Institute of Environment Protection Engineering, Wroclaw University of Technology, pl.
Grunwaldzki 9, 50-377 Wroclaw, Poland.
Phone: 0048 713204084, Fax: 0048 713282980, E-mail: malgorzata.kutylowska@pwr.edu.pl
* - corresponding author Abstract
The paper describes the reliability of selected water-pipe networks in the Polish median size city X and Z. The main goal of the paper is to compare the results of modeling of failure rate with the experimental data. On average in the analyzed time the main conduits, distribution and service pipes were characterized by failure rates (fail./(km·a)) equaled to 0.27, 0.40 and 0.59 in the city X, respectively. On average in the city Z the failure rates of main conduits, distribution and service pipes were equaled to 0.30, 0.32 and 0.78, respectively. The model, described in the literature, was slightly modified by author (model M3 was created) to achieve better convergence with experimental data. The results of modeling could be useful for water utilities because they can predict (using the model M3) in next years the reliability of the water-pipe network. Such prediction is necessary when the schedule of renovation is created.
Keywords: failure rate, mathematical models, water-pipe networks 1. Introduction
Water supply network is a very important part of buried infrastructure. Water should be delivered to the consumers under required pressure, with required amount and with proper quality. To achieve these goals it is necessary to monitor the technical state of water pipes continuously. Nowadays a proper maintenance and operation is crucial to increase the reliability and safety of water-pipe network. Among other things the pressure control inside the pipes is one of the activity which leads to decrease the number of failures and unreliability of the whole system [4]. This is directly connected with the operation of the water system.
Another way of optimization is mathematical modeling and forecasting, using typical models and artificial intelligence, some reliability indicators. The modeling allow us to assess the condition of water pipes quite quickly [20]. According to many authors [3, 15, 17] modeling the reliability indicators as well as the technical condition and afterwards using the modeling results by water utilities can lead to increase of water supply network efficiency and water quality. The management improvement could be also observed. Moreover, the modeling enables to estimate properly the costs of pipe reconstruction [2].
2. Aim and range of studies
The main goal of the paper is to compare the values of failure rates obtained on the basis of proposed mathematical model [19, 20] and results of own investigations which include the operational data from two water utilities [14]. In this paper the model proposed by Shamir and Howard [19] was modified to achieve better convergence with experimental data and to show relationships between way of exploitation and water-pipe network reliability. Comparative analysis was carried out for one reliability indicator (failure rate, λ) which was calculated for
water mains, distribution and service pipes in years 1999-2012 and 2001-2012 in the Polish city X and Z, respectively. Two analyzed water-pipe networks are the part of buried infrastructure of two median size cities. The city X is situated at the mining area and systematic modernization and renovation of old water pipes made of steel and grey cast iron is carried out. In the city Z the renovation is done when the water pipe is characterized by high failure frequency, mainly house connections are modernized or replaced. The length of main conduits is stable in two cities. The significant increase (about 14% and 32% which means ca. 16 km and 29 km) of distribution conduits is observed, respectively in the city X and Z. The increment of service pipes’ length is similar in the city X and Z and equals to about 21%. The material structure (at the end of 2012) of main and distribution conduits is displayed in the figure 1 and 2. In two systems over the half of total length of main and distribution pipes is made of grey cast iron. This fact has an influence on the higher failure frequency. In the systems where dominate new materials the situation is opposite. The studies carried out in Poland and abroad pointed out that pipes made of PVC or PE were characterized by lower failure rates, even below 0.10 fail./(km·a) [5, 11-13, 15, 18].
Fig. 1. Material structure of main and distribution pipes in the city X
Fig. 2. Material structure of main and distribution pipes in the city Z
56.6
17.2 16.0
8.4
1.8 0.0
10.0 20.0 30.0 40.0 50.0 60.0
Grey cast iron PE Steel PVC AC
Length, %
Material
64.1
24.0
7.4
2.0 1.9 0.6
0.0 10.0 20.0 30.0 40.0 50.0 60.0 70.0
Grey cast iron
PE PVC Ductile cast iron
Steel AC
Length, %
Material
3. Materials and methods
There are a lot of mathematical models which are used to forecast the technical condition, number of failures and reliability of water-pipe networks. The detailed description and some critique of existed statistical and physically based models was carried out by Kleiner and Rajani [8, 16]. Nowadays sometimes some modeling approaches are done without necessity of using operational data [6, 9], which theoretically should have an influence on the decreasing time of analysis. But the quality of modeling in such cases could be lower, because artificial not real data (from existing water utilities) were taken into consideration during the forecasting process. Of course the majority of modeling approaches were based on the operational information [1, 7, 21]. That is why it seems to be significant to check and compare, using data from two Polish water utilities, the results of modeling with experimental data.
The forecast of failure rate (equation (1)) was carried out on the basis of proposed by Shamir and Howard [19] deterministic model which was also described in the literature [9, 21].
(1)
where
λ(t) – failure rate at time t, fail./(km·a) λ(t0) – failure rate at time t0, fail./(km·a) t – current time of analysis, year
t0 – initial time of analysis, year
a – coefficient depended on the diameter and material of the pipe, -
As it was mentioned, for instance by Kleiner [8], this model is simple but also has some limitations. The main assumption of this model is exponential increment of failure rate in analyzed period of time. The pipe’s replacement or renovation as well as changing condition inside or outside of the conduit (pressure, soil and air temperature, water demand) are not taken into consideration. Coefficient “a” varies in the range 0.05-0.15 [19, 20]. Three values of this coefficient (0.05, 0.10 and 0.15) were used to predict failure rate in two water networks in the city X and Z. The model described by the relation (1) was chosen particularly to the comparative analysis because at such simple example it is possible to indicate that during the prediction of technical condition and failure rate a lot of changing parameters should be taken into consideration (e.g. the pressure traffic [3]). Unfortunately, some detailed information which theoretically should be included to the model are difficult to obtain from water utilities in Poland and also abroad. That is the reason why generally the level of failure rate has been assessed on the basis of relationship (2) where all the variables should be and are known [5, 10, 11, 22].
(2)
where
N – number of failures, unit Δt – analyzed period of time, year L – length, km
To achieve better convergence and include operational conditions of existed water pipes, the modification of model (1) was proposed by author. New proposition of failure rate
modeling is described by relation (3). It was indicated that the difference between model (1) and second-degree polynomial gave good results of failure rate modeling in two Polish cities X and Z. The attempt of using linear function, instead of polynomial, was done, but the agreement between modeling results and experimental data was not satisfactory from engineering point of view. As it was mentioned above water utilities do not register all necessary information about the operation of the system. Very important data such as: pipe- laying depth, kind of soil, pressure traffic, age of pipes and others are not available for modeling purposes. That is why simple formula is necessary to predict the failure rate. In the proposed modification (second-degree polynomial) of model described previously in the literature, only time is a variable which should be known. Constant coefficients D, C and E should be established on the basis of way of exploitation and maintenance in the selected water-pipe network.
(3)
4. Results and discussion
The experimental values, from two Polish cities, of failure rate are described as solid line.
The results obtained using model (1) and (3) with three values of coefficient “a” (0.05, 0.10 and 0.15) are displayed in the figures as “M1, a” and “M3, a”, respectively.
The failure rate of main conduits during 14 and 12 years of exploitation was decreasing (fig. 3 and 4) in the city X and Z, respectively. The maintenance of water mains is improving and year by year less number of damages is registered. On average, in the analyzed time, the main conduits were characterized by similar values of failure rate which were equal to 0.27 fail./(km·a) and 0.30 fail./(km·a) in the city X and Z.
The analysis of figures 3 and 4 shows that values obtained on the basis of the model M1 are significantly different than results of investigation results obtained on the basis of operational (experimental) data as well as on the basis of proposed model M3. As it was mentioned before, the model M1 included only time which theoretically influenced the deterioration of water pipes. The time increase causes the increase of failure rate. But from engineering and exploitation point of view the failure rate is rather random phenomenon which depends on the external factors, e.g. temperature, pressure, pipe-laying depth, pipe material and diameter [5]. Of course, in the reality, the deterioration of water-pipe network increase in time due to many factors (e.g. workmanship, kind of material and pressure inside the pipe), but not so significantly as it is shown in the figure 3 (line M1, a=0.15). According to Shamir and Howard [19], the coefficient “a” depends on the kind of material, but there is lack of information concerning the values of this coefficient if we consider pipe made of grey cast iron, steel or another type.
The proposed modification of failure rate modeling (equation (3)) also include only time as a variable but the constant coefficients D, C and E had an great influence on the convergence with experimental data. The proposition of values of these coefficients, established by the author during the analysis of way of exploitation of water-pipe network in the city X and Z, is displayed in the table 1.
Fig. 3. Experimental and forecasted failure rate of water mains in the city X
Fig. 4. Experimental and forecasted failure rate of water mains in the city Z
The values of coefficients D, C and E depend on such parameters, which are changing in time and way of exploitation: frequency of pipe section renovation, kind of renovation (what kind of materials were used), influence of mining area, frequency of pipe inspection, general way of maintenance, diameter and the pipe function (main, distribution or service pipes).
Each water-pipe network is different and should be considered independently. The proposed model M3 could be used for predicting the failure rate in other water systems but the values of constant coefficients D, C and E should be established after deep analysis of way of maintenance of water-pipe network, because they depend on various factors which should be taken into consideration during the analysis of water-pipe network in each city. Also the knowledge about the history of changing in the past values of reliability indicator as e.g.
failure rate is very important and should be taken into consideration during the modeling. The constant D is very similar for main conduits in the city X and Z. Other values of coefficients D, C and E are different in each city. It means that operational conditions and way of maintenance are not similar. The deterioration of water pipes in the city X is caused, among other things, by mining exploitation. The ground movement caused by mining has an influence on the faults occurred in the pipes. But deteriorated pipes are replaced more often.
0.00 0.50 1.00 1.50 2.00 2.50 3.00
1999 2000 2001 2002 2003 2004 2005 2006 2007 2008 2009 2010 2011 2012 Failure rate, fail./(km·a)
Year
Experimental M1, a=0.05 M1, a=0.10 M1, a=0.15
M3, a=0.05 M3, a=0.10 M3, a=0.15
0.00 0.20 0.40 0.60 0.80 1.00 1.20 1.40
2001 2002 2003 2004 2005 2006 2007 2008 2009 2010 2011 2012
Failure rate, fail./(km·a)
Year
Experimental M1, a=0.05 M1, a=0.10 M1, a=0.15
M3, a=0.05 M3, a=0.10 M3, a=0.15
On another hand, the city Z is not situated at the mining area, but the renovation of damaged pipe section is not planned and only very vulnerable parts are replaced.
Table 1. Values of constant coefficients D, C, E in the city X and Z
a City X City Z
D C E D C E
Main conduits
0.05 0.003 0.000 0.017 0.006 -0.033 -0.148
0.10 0.006 0.003 0.019 0.008 -0.030 -0.149
0.15 0.014 -0.029 0.077 0.012 -0.041 -0.128 Distribution
pipes
0.05 -0.003 0.108 -0.084 0.004 0.002 -0.029
0.10 0.003 0.113 -0.078 0.007 0.010 -0.034
0.15 0.018 0.051 0.035 0.014 -0.009 -0.004
House connections
0.05 0.004 -0.006 -0.061 0.014 -0.025 -0.118 0.10 0.009 -0.001 -0.058 0.021 -0.006 -0.131 0.15 0.021 -0.054 0.038 0.039 -0.052 -0.058 The analysis of failure rate of distribution conduits in the city X (fig. 5) shows convergence between model M3 and experimental results. According to the investigations, slight decrease of failure rate was observed from value 0.67 fail./(km·a) in 1999 to 0.42 fail./(km·a) in 2012. The model proposed in this paper (equation (3)) better predicts the failure rate of distribution pipes in two Polish cities than typical deterministic model. In the reality the failure rate does not increase in the time so quickly as it could be observed in the figures 5 and 6 (see lines M1). Even the decrease is observed due to renovation of deteriorated pipe sections. Some pipe sections (characterized by high failure frequency) are replaced by completely new and failure rate modeling should include information about the real way of maintenance, e.g. kinds and time of modernization of selected pipe sections. On average experimental failure rate of distribution pipes was equal to 0.40 and 0.32 fail./(km·a) in the city X and Z, respectively. For example average failure rates predicted by model M3 for different coefficient “a” were equal to 0.38, 0.42 and 0.50 fail./(km·a) in the city Z.
Fig. 5. Experimental and forecasted failure rate of distribution pipes in the city X
0.00 1.00 2.00 3.00 4.00 5.00
1999 2000 2001 2002 2003 2004 2005 2006 2007 2008 2009 2010 2011 2012
Failure rate, fail./(km·a)
Year
Experimental M1, a=0.05 M1, a=0.10 M1, a=0.15
M3, a=0.05 M3, a=0.10 M3, a=0.15
The climate and the depth of frost penetration are random factors which depend on the region as well as on the year. They are also very important indicators which have an influence on the changes of failure rate year by year. The constant coefficients in the proposed model M3 depends on the local conditions of the temperature changes. The relative errors are listed in the table 2.
Fig. 6. Experimental and forecasted failure rate of distribution pipes in the city Z The indicator λ for house connections in the city X and Z is displayed in the figure 7 and 8, respectively. The failure rates in the city Z were predicted with higher relative errors than in another city. But still the obtained results using model M3 are better than M1. On average experimental failure rates of service pipes were equal to 0.59 and 0.78 fail./(km·a) in the city X and Z, respectively. Average failure rates predicted by model M3 for different coefficient
“a” were equal to 0.66, 0.70 and 0.86 fail./(km·a) in the city X.
Table 2. Relative errors of model M3 in the city X and Z
a Relative error, %
City X City Z
Main conduits
0.05 3.64 6.17
0.10 7.00 7.08
0.15 13.21 10.92
Distribution pipes
0.05 9.79 4.33
0.10 15.14 7.92
0.15 29.29 16.25
House connections
0.05 6.14 13.00
0.10 10.50 24.33
0.15 26.00 42.75
0.00 0.50 1.00 1.50 2.00 2.50
2001 2002 2003 2004 2005 2006 2007 2008 2009 2010 2011 2012
Failure rate, fail./(km·a)
Year
Experimental M1, a=0.05 M1, a=0.10 M1, a=0.15
M3, a=0.05 M3, a=0.10 M3, a=0.15
Fig. 7. Experimental and forecasted failure rate of service pipes in the city X
From engineering point of view the relative errors for main and distribution pipes are acceptable. Only for house connections in the city Z the divergences between experimental data and the model M3 are quite high. Proposed model M3 is the first attempt of modeling failure rate of water-pipe network on the basis of experimental data done by the author. In the future the model M3 will be checked using operational data from other water utilities. The model M3 will be developed to establish universal values of constant coefficients D, C and E.
Fig. 8. Experimental and forecasted failure rate of service pipes in the city Z
The better (than described above) convergence between experimental data and results of modeling was obtained by Tabesh et al. [20] because during the modeling of failure rate using artificial neural network the coefficient R2 was equal to 0.989. Generally, the artificial intelligence predicts some operational parameters slightly better than typical models [7, 21].
0.00 0.50 1.00 1.50 2.00 2.50 3.00 3.50 4.00 4.50
1999 2000 2001 2002 2003 2004 2005 2006 2007 2008 2009 2010 2011 2012
Failure rate, fail./(km·a)
Year
Experimental M1, a=0.05 M1, a=0.10 M1, a=0.15
M3, a=0.05 M3, a=0.10 M3, a=0.15
0.00 0.50 1.00 1.50 2.00 2.50 3.00 3.50 4.00 4.50 5.00 5.50
2001 2002 2003 2004 2005 2006 2007 2008 2009 2010 2011 2012
Failure rate, fail./(km·a)
Year
Experimental M1, a=0.05 M1, a=0.10 M1, a=0.15
M3, a=0.05 M3, a=0.10 M3, a=0.15
The problem is only in the availability of operational data used for modeling purposes. On the basis of information obtained from two water utilities in the city X and Z it was not possible to use artificial intelligence in modeling. Simple model was proposed to predict indicator λ.
The results of modeling could be useful for water utilities because using the model M3 they can predict in some years the reliability of the water-pipe network. Such prediction is necessary when the schedule of renovation is created.
5. Conclusions
The following conclusions can be drawn:
- Reliability of the water-pipe network was established on the basis of one important indicator – failure rate. According to own investigations the reliability of analyzed water networks in two cities is a little bit higher than in another Polish median size city [10], but similar to that obtained in Canada [15].
- The forecasting of failure rate, using proposed mathematical model M3, was characterized by acceptable, from engineering point of view, level of convergence with experimental data. The relative errors varied in the ranges 3.64-29.29% and 4.33- 42.75% in the city X and Z, respectively. The divergences between model M1 and experimental data were higher because the model assumptions were based on the thesis that the deterioration was simply connected with the age. This is proper assumption only if there is lack of pipes’ modernization. The model M1 did not include the information about other factors influencing the deterioration (pressure, water demand, pipe material, kind of soil, pipe-laying depth, frost penetration, climate) as well as the renovation or replacement of pipe sections. The restoration of deteriorated parts of the water supply system is very important factor which should be included to the models.
- Water-pipe networks differ between each other and their maintenance should depend on many factors like: the size of the city, the pipe-laying depth, the climate, the age of the pipes and others. That is the reason why models should also include the information about changeable conditions of maintenance and operation. The model M3 is the first attempt carried out by author to modify existed models and predict failure rate in selected Polish cities.
- Data collection is very important factor during the assessment of the reliability level of water-pipe network. The operational data should be precise and without any scarcity.
In the reality, the data collected by water utilities are sometimes incomplete and in such cases the modeling using operational data is difficult or even impossible. That is why the proposed model M3 include only time as a variable to simplify the information needed for modeling purposes. The values of coefficients D, C and E in the model M3 depend on: frequency and kind of pipe section renovation, influence of mining area, frequency of pipe inspection, diameter. Each water-pipe network is different and should be considered independently. In the future the model M3 will be verified using operational data from other water utilities. The model M3 will be developed to establish universal values of constant coefficients D, C and E.
- The results of modeling could be useful for water utilities because using the model M3 they can predict in next years the reliability of the water-pipe network. Such prediction is necessary when the schedule of renovation is planned.
Acknowledgement
Author thanks Water Utilities in the city X and Z for giving the possibility of using and analyzing their operational data.
Fellowship co-financed by European Union within European Social Fund References
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