Probabilistic failure investigation of small diameter cast iron pipelines for water distribution

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Probabilistic failure investigation of small diameter cast iron pipelines for water distribution ⁎

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Jian Jia,b, , Jia Hong Laib, Guoyang Fub, Chunshun Zhangb, , Jayantha Kodikarab a b

Key Lab of Ministry of Education for Geomechanics and Embankment Engineering, Hohai University, Nanjing, China Department of Civil Engineering, Monash University, Clayton, Australia

A R T IC LE I N F O

ABS TRA CT

Keywords: Cast-iron pipes Corrosion Failure prediction Probabilistic inverse analysis (PPM) Probabilistic physical modelling

At present, a significant proportion of Australia’s ageing water infrastructure is composed of castiron pipes. The corrosion of cast-iron pipes is the main triggering factor for Australia’s water industry. It has been found that the failure mode of small diameter water mains (< 300 mm external diameters) from Yarra Valley Water is mainly the longitudinal failure (hoop-stressed) instead of broken back failure, which is unexpected. Therefore, the purpose of this paper is to examine the failure mechanism of longitudinal failure of small diameter cast-iron pipes. The observational failure database provided by Yarra Valley Water show limited data availability. This has stimulated the focus of this paper: to examine the corrosion mechanism by utilising probabilistic inverse analysis for unknown physical parameters. Focusing on the likelihood of failure, this paper investigates the performance of small diameter cast-iron pipes by using statistical analysis based on the observational failure lifetime data within pipe cohorts. Background studies on the deterioration of cast-iron pipes due to corrosion are made to help set up the probabilistic physical modelling (PPM). Using the updated corrosion parameters, the lifetime probabilities of failure and the hazard rate of the cast iron model are derived from PPM. Last, it is found that the modelling results agree reasonably well with prediction curves of statistical data, indicating that the proposed method for pipe lifetime prediction is promising.

1. Introduction Cast iron was widely used as a common material for water mains more than 50 years ago. At present, there are still many ageing cast-iron pipes underground to service the community. As these pipes further deteriorate, the potential failure of such pipes is a major concern to both water utilities and the public. Thus, there are increasing demands for failure prediction of these pipes. Having the ability to predict the remaining service life of cast-iron water mains would allow water utility companies to manage those ageing assets more efficiently. In order to predict a water mains remaining service life, fundamental knowledge is required to understand the current state of the pipe and the key physical properties that contribute to pipe failure. Probabilistic physical modelling (PPM) is an extension of the deterministic modelling [1,2], and it was introduced in this study for failure prediction of small diameter cast-iron pipes. PPM is adopted because the physical properties of cast iron and the soil, as well as the corrosion parameters, are subject to significant variations; thus, deterministic modelling is not appropriate as it requires all the variables to be deterministic (identified). Prior to PPM, statistical analysis of the observational failure data will be conducted to determine the statistical information for key parameters of the pipes. The statistical analysis of the observational failure data was

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Corresponding authors at: Department of Civil Engineering, Monash University, Clayton, Australia (J. Ji). E-mail addresses: [email protected] (J. Ji), [email protected] (C. Zhang).

https://doi.org/10.1016/j.engfailanal.2019.104239 Received 13 June 2019; Received in revised form 9 October 2019; Accepted 14 October 2019 1350-6307/ © 2019 Elsevier Ltd. All rights reserved.

Please cite this article as: Jian Ji, et al., Engineering Failure Analysis, https://doi.org/10.1016/j.engfailanal.2019.104239

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Nomenclature g(x, t) SCF(t) e

List of symbols a, b τ(t) cs, rs t0 D R d de d' σy σnom v ξ P x

the major and minor axis of an elliptical corrosion patch corroded depth when the pipe age is t years the corrosion parameters which represent the interception and corrosion rate transit point of time in corrosion model external diameter of the pipe external radius of the pipe initial wall thickness before pitting corrosion occurred effective wall thickness remaining wall thickness in the centre of a corrosion pit tensile failure stress of cast iron material nominal hoop stress of uniformly corroded pipe Poisson ratio uniform corrosion coefficient internal operating water pressure vector containing all physical parameters

σe PPM F(t) f(t) R(t) λ(t) f(ti; θ) g(·) gmes(·) g (x ;̂ t j )

x̂ xi Tl To α β

governing the pipe behaviour time-dependent limit state function at lifetime t stress concentration factor (at time t) corresponding error between measurement value and model prediction standard deviation of errors probabilistic physical modelling cumulative distribution function of failure data probability mass/distribution function (PMF/PDF) reliability function of failure data hazard rate function of failure data probability density function at lifetime ti, with unknown parameters vector θ response quantity of a physical modelling measurement value of the response at time t model prediction of the response at a specific time t attribute vector containing n input parameters the ith random value time to left truncation location parameter of Weibull function shape parameter of Weibull function scale parameter of Weibull function

conducted within the concept of pipe cohorts, which will classify pipes into a few groups. Generally speaking, for Australian water utilities, water mains can be broadly divided into small diameter (< 300 mm external diameters) and large diameter (> 300 mm external diameters). 2. Background study 2.1. Grey cast-iron pipes Understanding the grey cast-iron pipe is essential before we proceed to identify the problems. According to a study by the American Water Works Association (AWWA) in the US during 1992, more than two thirds of existing water mains were metallic (about 48% cast iron and 19% ductile iron) [3]. A survey conducted by ref. [4] among 21 Canadian cities also revealed a similar distribution of pipe material types. These studies show that cast-iron pipes were commonly used back then, and there are many ageing cast-iron pipes still buried underground to service the community. From the 1880s to the early 1930s, grey cast-iron pipes were manufactured by pouring molten cast iron into a series of upright sand molds created in a pit. The molds were then removed and the pipes rolled free, cleaned, examined and tested [5]. These pipes are commonly known as Pit Cast Iron. Spun Cast Iron was then introduced commercially during the 1920s/1930s and widely used

Fig. 1. Timeline of ferrous pipe material in eastern Australia (Modified from [7;8]). 2

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over the next two decades. Spun Cast Iron pipes are manufactured by using molds made of sand and metal, cast horizontally, and cooled externally with water to prevent damage from the molten metal that was poured into molds to form the pipe [5]. Spun Cast Iron was known to have better material uniformity in comparison with Pit Cast Iron, with corresponding improvements in material properties [6]. In 1948, ductile iron pipe was introduced by changing the composition of iron, which makes it more ductile and less prone to graphitisation. Graphitisation is a term used to describe the selective dissolution of iron from the iron/carbon matrix (leeching out of iron components) [6]. Ductile iron pipe was not industrially produced until 1960 and all new iron pipes were ductile iron by 1980s. Fig. 1 shows the summary of the manufacturing timeline of grey cast-iron pipes.

2.2. Observational pipe failure data The observational failure of cast-iron pipes data provided by Yarra Valley Water consist of 33,963 failure data records. The observation data contains some important information such as water mains pipe diameter, pipe age, pipe length, and also failure modes which will be analysed in this study. It is worth noting that the observational failure data shows most of the pipes were installed before the 1980s, which indicates that they are less likely to be ductile cast iron. Therefore, the two major classes of cast-iron pipe that are relevant to this study are Pit Cast Iron and Spun Cast Iron. Fig. 2 shows the general pipe external diameter of the failure data from Yarra Valley Water. Two major groups of failure data were identified to be 100 mm and 150 mm external diameters. The failure mode of water mains data collected from Yarra Valley Water were categorised and shown in Fig. 3. The data collected from Yarra Water Valley indicate that the small diameter cast-iron pipes mainly failed by longitudinal failure of the hoop-stressed type (53%), followed by circumference failure of the axial-stressed type (21%). In normal circumstances, small diameter cast-iron pipes will fail by the circumference failure mode due to soil ground movement, which causes high bending stress [9], and the longitudinal failure is more commonly observed in larger diameter pipes [10–12], which is mainly caused by highly concentrated stresses in the vicinity of corrosion areas subjected to high water pressures. However, it was not the case for the water mains in Yarra Valley Water. Therefore, the purpose of this study is to investigate the fundamental failure mechanism of longitudinal failure and to explain why this happened. In order to investigate the failure mechanism of longitudinal failure, it is essential to understand the key physical properties contributing to pipe failure. According to Refs. [6,13–15], they include the pipe structural properties, material types, pipe soil interactions, internal and external loads, and material deterioration due to electro-chemical or microbiological corrosion. Fig. 4 shows typical corrosion patterns that were observed from field pipe samples. In addition, Ref. [6] also stated that the main deterioration mechanism on the exterior of cast-iron pipes is electro-chemical corrosion with the damage occurring in the form of corrosion pits. The statement earlier was also justified by the filed evidence obtained from five different Australia utilities, whereas longitudinal failure was often found to be caused by internal water creating pressure and corrosion [10]. The damage of grey cast iron is often correlated to the presence of ‘graphitisation’. In addition, the physical environment of the pipe also has a significant impact on the deterioration rate. Soil characteristics such as moisture content, chemical and microbiological content, and aeration, are factors that might accelerate the corrosion of metallic pipes [6]. In short, pipe failure is likely to occur when the structural integrity of the pipes has been compromised by operational stresses or environmental factors such as corrosion, degradation or inadequate installation [6]. In most circumstances, data are unavailable or it is very costly to acquire this information from water supply industries. For instance, the observational failure data provided by Yarra Valley Water does not specify the loadings subjected internally and externally to the cast-iron pipes or the operating water pressure of the cast-iron pipes. The lack of data availability has narrowed the focus of this paper to only examine the corrosion failure mode.

Fig. 2. Failure data from Yarra Valley Water. 3

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Fig. 3. Failure Mode of Water Mains in Yarra Valley Water.

Fig. 4. Field observations of typical corrosion patterns and longitudinal failure in water mains.

2.3. Deterioration of pipe wall thickness due to corrosion According to the observational failure data obtained from Yarra Valley Water, it was found that most pipes were installed before the 1980s, in which case they are most likely grey cast iron subjected to various patterns of corrosion. The corrosion activity of castiron pipes generally involves either wall thinning or pitting, which ultimately lead to wall thickness reduction. A case study conducted by several geotechnical experts also revealed there were two types of thickness loss due to corrosion: the uniform corrosion and pitting corrosion [16]. Uniform corrosion occurs when the pipe wall thickness experiences an all-round reduction, whereas pitting corrosion only occurs when there is localised corrosion patch or pits. In general, corrosion pits would occur first, and some of them may expand into uniform corrosion over time. The corrosion of cast-iron pipes is a time-dependent process that follows a monatomic bi-modal trend with time [17]. The rate of corrosion has been the interest of many experts to help understand the corrosion behaviour, but ultimately it was found that the rate changes with time while featuring an initially high value. In normal circumstances, the unknown parameters in the corrosion model are evaluated by site observation data or experience for simplicity. In this paper, the main objective is to investigate the long-term corrosion behaviour of cast-iron pipe, of which the exponential model was given as follows [6]:

Fig. 5. Schematic illustration of bilinear corrosion model [18]. 4

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τ (t ) = rs t + cs [1 − exp(−t / to)]

(1)

where τ (t ) is the corroded depth when the pipe age is t years; t0 is the transit point of time where the corrosion rate approaches a steady state in equilibrium with the surrounding environment after an initially high value. This model is a simplification of a multiple phase model for practical purposes [18]. Meanwhile, cs and rs are the corrosion parameters which represent the interception and corrosion rate of that phase, respectively, as shown in Fig. 5. These parameters are important because most of the ageing buried pipelines are now in the steady state phase. The simplified exponential model can describe the development of corrosion with time directly, where the corrosion progress is at relatively quicker rate until t reaches a certain lifetime, after which the corrosion rate will become constant. It is also worth noting that there are some surface protection techniques to prevent cast-iron pipes from earlier corrosion; for example, by using the cathodic protection system. To the best of the authors’ knowledge, the cathodic protection technique, which is a very effective corrosion reduction technique, might not have been widely used more than 50 years ago in the Australian pipeline industry. In this study, we tried to investigate the failure of Australian cast-iron pipes. These pipes were mainly installed during the period between the 1930s and the 1980s. Hence, we cannot clarify with confidence the specific corrosion protection techniques adopted at that time. Consequently, the effect of corrosion protection on cast-iron pipe wall deterioration will not be discussed in this study. 2.4. Stress prediction of pipes subjected to general corrosion In general, it is expected that corrosion in cast-iron pipelines will cause a steady reduction in wall thickness. The corroded depth with respect to pipe age for pitting corrosion was shown earlier by Eq. (1). The remaining wall thickness with corrosion pits, d’, is defined by:

d' = d − τ (t )

(2)

where d is the initial wall thickness before pitting corrosion occurred and τ (t ) is the reduction in wall thickness[18]. On the other hand, uniform corrosion should also be considered for pipe stress analysis. Effective wall thickness, de, will be introduced to describe the uniform deterioration, which is given by:

de = d − ξτ (t )

(3)

where ξ represents the uniform corrosion rate as a ratio to the pitting corrosion rate; e.g. taken as 0.1 in this study due to the absence of other supporting data [18]. The nominal stress of uniformly corroded pipes, σnom, for simplification purposes, is defined based on the thin walled assumption:

σnom =

PD 2d e

(4)

where P is the internal operating water pressure, and D is the mean external diameter of the cast-iron pipe. 2.5. Stress prediction of pipes subjected to corroded pits/patches In addition to the ongoing uniform corrosion deterioration acting on cast-iron pipes, the presence of localised defects will increase the principal stresses significantly in the event of high internal and external loadings. The sudden increase in working stress is related to the stress concentration factor (SCF), which is defined as the ratio of concentrated stress of localised corrosion pit to uniformly corroded pipe stress [18–21]. As the corrosion process is a time-dependent process, the working stress at time t subjected to pitting or patch corrosion is defined below

σ (x , t ) = σnom × SCF (t )

(5)

where SCF (t) is the stress concentration factor at time t. Comprehensive studies have been conducted to investigate the SCF for through wall and remaining wall corroded pipes. In most cases, the remaining wall model will be more critical in comparison with the through wall model in relation to the corrosion failure mode. Therefore, only the SCF model for remaining wall corrosion was used in this study. Specifically, the pipeline research project from Monash University conducted a series of 3D finite element analysis of corroded cast-iron pipes, and the equation below is the proposed SCF regression model for remaining wall (SCFRWC ) corrosion patterns [18,22–24]. Table 1 Coefficients of stress concentration factor prediction equation. α1

α2

α3

α4

α5

α6

α7

2.80E−05 β1 1.071

3.00E−05 β2 2.09

7.096 β3 11.677

3.00E−06 β4 0.733

3.00E−05 β5 1.348

0.011 β6 5.755

0.797 β7 0.84

5

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4

SCF = 1 +

⎧ 3(1 − ν 2) ⎪ α1 ⎨ 2 ⎪ α4 ⎩

a Rd

β1

a Rd

β4

( ) ( )

+ α2

b Rd

β2 β5

5

b Rd

( ) +α ( )

β3

( ) +α ( )

+ α3

τ Rd

6

τ Rd

⎫ β ⎪ τ ⎞ , α ⎛ β6 ⎬ 7 d − τ ⎝ ⎠ ⎪ ⎭

0⩽b⩽

R2 − (R − d )2 (6)

where R is the pipe radius, a and b are respectively the major and minor axis of an elliptical corrosion patch, Poisson ratio v is 0.3, and the regression parameters are listed in Table 1. 3. Methodology 3.1. Statistics of pipe failure data Statistical analysis is required to obtain statistical information such as the basic physical parameters of the observational failure data. The result obtained was then used for comparison with the lifetime failure prediction of cast-iron pipelines using probabilistic physical modelling. Statistical analysis in this paper involved several techniques, mainly by the histogram, the probability mass/ distribution function (PMF/PDF) f(t), and the hazard rate function λ(t). It is noted that the hazard rate function, λ (t ) , also known as the failure rate function, provides an alternative way of describing a failure distribution and it provides an instantaneous (at time t) rate of failure [25]. The conditional probability of failure per unit of time is defined below:

λ (t) =

f(t) R(t)

(7)

where R(t) is the reliability function which can be expressed as shown below: (8)

R (t ) = 1 − F (t )

where F(t) is the cumulative distribution function of failure data. Furthermore, it was found that the observational failure lifetime from Yarra Valley Water is not complete in terms of data availability. The cast-iron pipe failure data do not specify what types of cast-iron pipe has been installed. Therefore, a manual screening process is needed to separate the failure data into two main groups, namely Pit Cast Iron and Spun Cast Iron. For analysis consistence, the observational data were divided into their respective cohorts as listed in Table 2. Meanwhile, the pipe thickness for Spun Cast Iron was found using Fig. 6, where a linear relationship equation has been defined for Class-C type Cast Iron. On the other hand, the linear equation in Fig. 6 is not applicable for Pit Cast Iron. This is because Pit Cast Iron existed before Australian Iron and Steel Standards had been established. Therefore, the British Standard Specification [26] was included in this study to find the pipe thickness for Pit Cast Iron. Table 2 summarises the pipe thickness for each cohort. 3.2. Operating pressure failure data For pressurised pipe failure investigation, the operating water pressure is an essential factor. Unfortunately, such important data was not clearly provided by Yarra Valley Water in the observational failure database. Therefore, attempts have been made to obtain prior knowledge of operational water pressure indirectly through their respective distribution zones. Distribution zones of cast-iron water mains for four different cohorts as mentioned in Table 1 have been filtered and the column charts were analysed. The results will not be presented in this paper due to dimension constraints. However, notable results were observed showing that there was a huge similarity of the water distribution zone for each material cohort. It was found that Pit Cast Iron mainly failed in Preston Residential and Surrey Hills Residential. On the other hand, it was found that Spun Cast Iron generally failed in Mitcham-Morang and Olinda-Mitcham zone. Since the operating pressure plays an important role in the physical modelling analysis, failure in obtaining such data would indicate it is hard to achieve the primary objective of the study, which is to investigate why small diameter cast-iron pipe in Yarra Valley Water primarily failed by longitudinal failure. Based on the observational failure from other water utility company, South East Water, the static wall pressure for small diameter cast-iron pipe varies between 700 kPa and 720 kPa. However, the static water pressure mentioned above did not take the transient behaviour of water into consideration, which could add up to 30% to the operational pressure in water pipe. Transient events can be initiated when large pressure forces and rapid fluid accelerations are introduced into water distribution systems, which could cause disturbance in the water during a change in mean flow conditions [27]. In most cases, transient events are most severe in highTable 2 Summary of pipe thickness for classified corroded pipe cohorts (longitudinal failure mode). Pipe Cohorts

Number of Failures

Thickness (mm)

100 mm 150 mm 100 mm 150 mm

10,747 2751 3978 208

9.39 10.64 10.16 12.45

Spun Cast Iron Spun Cast Iron Pit Cast Iron Pit Cast Iron

6

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Fig. 6. Relationship between wall thickness and pipe diameter for spun cast iron pipes [18].

elevations areas or locations with low static pressures [28]. Therefore, the assumed prior knowledge of operating pressure in back figuring the corrosion parameter are 800 and 1000 kPa. 3.3. Maximum likelihood estimate for observed failure data The Weibull distribution is an important distribution, especially for reliability analysis [29]. Lifetime failure predictions of the cast-iron pipelines could serve as valuable information for future research. The lifetime prediction of the cast-iron pipe was done by using the updated corrosion parameters obtained earlier and fitting them into the Weibull distribution, of which the cumulative distribution function is written as α

F (t: α, β ) = 1 − e−{(t − T0)/ β )}

(9)

where To is the location parameter, also known as the honeymoon-period (HMP) for buried water pipes. Meanwhile α is the shape parameter and β is the scale parameter. Based on these expressions above, the hazard rate function can also be deduced by using the formula shown below

H (t: α, β ) =

dF (t: α, β )/ dt α t − T0 ⎞ = ⎜⎛ ⎟ 1 − F (t: α, β ) β⎝ β ⎠

α−1

α

e−{(t − T0)/ β )}

(10)

There are many methods for estimating the shape and scale parameters of Weibull functions, which are also known as Weibull parameters. Analytical method (maximum likelihood estimate, MLE) was chosen herein to estimate the Weibull parameters because the traditional graphical methods usually involve a greater probability of error, which is undesirable. In general, the MLE method let t1, t2, …, tn be a random sample of size n drawn from a probability density function ft (ti; θ) where θ is the vector of unknown parameters. Then, the joint (likelihood) probability mass function is known as n

L=

∏ ft (ti, θ)

(11)

i=1

On the other hand, the MLE for θ is realised by maximising L, or equivalently, the logarithm of L. Essentially, the solution will have the form as shown below

dlogL =0 dθ

(12)

MLE was utilised in dealing with left-truncated Weibull data and normal Weibull fitting, which will be introduced in the subsequent studies. For a truncated Weibull distribution with probability distribution function f(t: α, β), the likelihood function reads as follows: n

log L =

∑ log f (ti: α, β ) − n·log[f (T1: α, β ) − f (T0: α, β )]

(13)

i=1

where T1 is the time to left truncation, and T0 is the location parameter. 3.4. Probabilistic physical modelling of corroded pipelines Probabilistic Physical Modelling (PPM) is always known as an advanced approach to predict the failure probability of pipelines. On the subject of deteriorated pipelines, a time-dependent limit state function (t-LSF) can be defined as shown in the equation below:

g (x , t ) = σy − σ (x , t )

(14)

where x is the vector containing all physical parameters that were involved in the time-dependent working stress as defined earlier 7

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and σy is the tensile failure stress. The tensile failure stress for Pit and Spun Cast Iron was assumed as time-invariant parameter in this paper, which is 100 MPa and 130 MPa respectively. This assumption is valid because corrosion has been taken into consideration in the calculation and the strength varies with the age of the pipelines originating from corrosion or graphitisation of the metal [30]. In short, for a given range of time series with different sets of parameters, the pipe will not fail when g(x,t) > 0, but it fails when g (x,t) < 0, and it is at the limit state when g(x,t) = 0. From a probabilistic point of view, the limit state function involves many parameters, which, subject to significant variations, makes it difficult to calculate analytically. Therefore, Monte Carlo simulation was introduced herein to solve the t-LSF, based on the physical parameter uncertainties as previously reported in [1] and summarised in Table 3. Note that the computer package software @RISK by Palisade [31] was adopted to run the Monte Carlo simulations. Last, it should be known that the PPM is only limited to pipelines that experience a first-time failure system (non-repairable). 3.5. Bayesian inverse analysis of corrosion parameters Corrosion parameters are important factors commonly derived by studying the soil moisture and soil environment surrounding the pipes [32]. Based on Fig. 5, corrosion parameter cs can be described as the corrosion depth at the initial stage of the corrosion process; T0 is the transit point of time where the corrosion rate approaches a steady state value in equilibrium with the surrounding environment after an initially high rate and rs is the corrosion rate growing steadily after T0. The corrosion parameters are unknown by nature, and subject to a large amount of uncertainties. Back figuring corrosion parameters can be realised by utilising the stress-based physical failure model that was mentioned earlier. In addition to the corrosion mechanism, the model involves other physical parameters; e.g. the internal water pressure at failure, the tensile strength of cast iron, and the corrosion patch geometries, etc. The above-mentioned physical parameters together constitute the unknown random variables (as shown in Table 2) that govern the pipe failure mechanism. In this study, we employed the Bayesian inverse analysis to back figure them by using the observational data in hand. The Bayesian method is briefly introduced below. A time-dependent response of any physical process can be described by a function or model (output, e.g. the t-LSF defining the onset of pipe failure) of several attributes (inputs, e.g. the pipe physical parameters), such as:

g (x ;̂ t ) = g (x1, x2 , ...,x n ; t )

(15)

where g(·) is the response quantity of a physical modelling, x ̂ = (x1, x2 , ...,x n ) is the attribute vector containing n input parameters, and t denotes the time. Practically, the model predictions may not be very accurate and are always subject to some errors due to various sources of uncertainties. For example, the input parameters are usually not well known and the model function may not be so perfectly defined. If measurements are taken on the response quantity at some discrete points of time, we have the following generic relationship:

gmes (x ;̂ t ) = g (x ;̂ t ) + e

(16)

where gmes(·) is the measurement value of the response at time t, and e is the corresponding error between measurement value and model prediction. In the probabilistic point of view, the error e is a realisation of a measurement error if there are a large number of measurements being conducted. A simple assumption is that the measurement error is a Gaussian process with zero-mean and variance σe2. In other words, e can be regarded as a random variable following normal distribution with mean value = 0 and standard deviation = σe. As a result, the likelihood function of measurement error by considering N observations or measurements of the response, gmes, j = gmes (x ;̂ t j ) at discrete points of time tj (j = 1 to N), is given by [33] N

L [gmes,1, ...,gmes, N |x ]̂ =

∏ j=1

2 ⎡ 1 [gmes, j − g (x ;̂ t j )] ⎤ exp ⎢− 2 ⎥ σe σe 2π ⎣ 2 ⎦

1

(17)

where g (x ;̂ t j ) is the model prediction of the response at a specific time tj. In most cases, the PDF of model parameters x ̂ is not well known. From Bayesian theory, we can estimate the posterior PDF of x ,̂ f X ̂ (x )̂ by use of the prior PDF p X ̂ (x )̂ and the likelihood function of response observations [34], such that Table 3 Empirically statistical information of physical parameters for pipe failure analysis. Physical parameter

Statistical distribution

Mean value

COV

Tensile strength Water pressure Corrosion rate rs Corrosion cs Corrosion T0 Patch factor ξ1 Patch factor ξ2

Lognormal Normal Lognormal Lognormal Lognormal Normal Normal

130 MPa 1000 kPa 0.1 mm/year 5 mm 10 years 1 1

0.2 0.3 0.3 0.2 0.2 0.25 0.25

8

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f X ̂ (x )̂ = f X ̂ [x ̂ |gmes,1, ...,gmes, N ] = cp X ̂ (x )̂ L [gmes,1, ...,gmes, N |x ]̂

(18)

where c is a normalisation constant. Note that the above equation is referred to as indirect Bayesian updating because the likelihood depends on observations of the response g(·) instead of on the parameter x ̂ by itself [35]. It is noted that the normalisation constant c is very difficult to be explicitly solved, hence limiting the Bayesian updating of the posterior PDF in the practice. Alternatively, this problem can be solved using the Markov Chain Monte Carlo (MCMC) simulations, which has revolutionised the application of Bayesian statistics. MCMC is an efficient method for the simulation of unknown probability distributions based on the concept of the Markov chain. In brief, considering an unknown PDF, we can construct a Markov chain by sequentially drawing samples xi’s, starting from an arbitrary value x0, in such a way that xi+1 is independent from xi−1, xi−2, … , x0. The stationary distribution of such a chain is our target PDF. Some algorithms available for the implementation of MCMC include Metropolis-Hasting algorithm, Cascade Metropolis-Hasting algorithm, Gibbs sampling, Slice sampling, and Metropolis-within-Gibbs algorithm, etc. [33].

4. Result and discussion 4.1. Statistical analysis of pipe failure lifetime According to the pipe cohort information as detailed in Table 1, statistical analyses were carried out on the failure lifetime data. Fig. 7 shows their histogram plots. For convenience of mathematical description, Weibull distributions were fitted to the histograms. Taking the first cohort failure data as an example, MLE on Weibull distribution is shown in Fig. 8. It was found that the Weibull distribution can generally represent the histogram. However, the hazard rate would be substantially overestimated by this method. As discussed previously, the discrepancy could be the result of the incompleteness of the observational period, which only commenced two decades ago. Hence, MLE on left-truncated Weibull distribution was conducted on the same cohort data. With truncation lifetime = 25 y, the corresponding results are shown in Fig. 9. By comparison, it is clearly indicated that the latter approach is more rational for statistically explaining the pipe failure data. In order to consider the no-observation period of the pipe failure data, we introduced the left truncation MLE for Weibull regression, where the truncation lifetimes are 25 years and 67 years for Spun Cast Iron and Pit Cast Iron pipes, respectively. The same statistical analysis is conducted on the other three cohort data. Fig. 10 shows the histograms and hazard rates resulted from truncated MLE regressions, where Spun Cast Iron pipes are subjected to 25 years truncated value and Pit Cast Iron to 65 years.

Fig. 7. Histogram plots of observed failure data of four different pipe cohorts. 9

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Fig. 8. MLE matching a Weibull distribution to the observed failure data of 100 mm Spun Cast Iron pipe cohort.

Fig. 9. MLE matching a left-truncated Weibull distribution to the observed failure data of 100 mm Spun Cast Iron pipe cohort.

4.2. Back figure results It is noted that without loss of generality, the case study that follows will focus on the pipe cohort of the largest amount of failure data; i.e. the 100 mm Spun Cast Iron pipes. Based on the Bayesian framework that was mentioned in Section 3.5, the MCMC simulation for statistically back figuring physical parameters involves constructing an e-LSF based likelihood function on observed pipe failure data. The following probabilistic assessment was performed on the 100 mm Spun Cast Iron pipe cohort. MCMC trace and histogram plots are shown in Figs. 11 and 12 with 10,000 simulations in total. A burn-in of 5000 simulations was executed and Table 4 summarises the back-figured statistics of the physical parameters, which are regarded as posterior distributions as a result of the MCMC simulations.

4.3. Comparison between modelling results and observation data The probability of pipe failure analysis conducted earlier is usually known as the top-down approach, which is solely based on the probabilistic investigation of the observational failure data. On the other hand, the probabilistic physical modelling on failure mechanism (bottom-up approach) can also be used to analyse the lifetime probability of pipe failure [1]. The success of the latter will heavily depend on the statistical information of physical parameters. Unfortunately, data collection work is seldom directed to these physical parameters in the pipeline industry. Thus, characterisation of the parametric uncertainties is not readily implementable from field evidence. Alternatively, we can use the Bayesian back-figured statistics as shown in Table 4 for our probabilistic physical modelling. The instantaneous probability of failure is directly computed by modelling the failure mechanism, which is equivalent to the hazard rate curve as shown in Fig. 13. In addition, the hazard rate curve can be converted to the cumulative distribution probability curve using Eqs. (7) and (8). Fig. 14 shows the comparison of lifetime prediction using probabilistic physical modelling method with the MLE estimation on observed failure data for 100 mm Spun Cast Iron. It was obvious that lifetime probability of failure results obtained from the topdown method and the bottom-up method showed similar hazard rate predictions with some discrepancy due to modelling errors. By comparing the result from Fig. 14, it is also worth noting that the bottom-up method tends to underestimate the failure probability at pipe early lifetime and overestimate at later lifetime. One possible explanation for this phenomenon could be due to the observational failure data consisting of different sources of errors, whereas the ‘first time to fail’ record was not recorded accurately. According to the observational failure data, it was found that many cast-iron pipes were installed before the 1990s; the earliest was installed in 1864. However, the recorded ‘first time to fail’ year was more than 100 years after the installation year, which is technically 10

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Fig. 10. MLE matching left-truncated Weibull distributions to the observed failure data of (a) 150 mm Spun Cast Iron pipe cohort; (b) 100 mm Pit Cast Iron pipe cohort; and (c) 150 mm Pit Cast Iron pipe cohort.

impossible by any means. Consequently, our failure mechanism involves model errors in terms of the SCF prediction. These combined error effects on the discrepancy need to be investigated thoroughly in future studies. 5. Conclusion In this paper, the main focus was to investigate the reasons contributing to small diameter cast-iron pipe failures in Yarra Valley Water that mainly fail by longitudinal failure. Background studies of key physical properties contributing to pipe failure have been made to better understand the dominant failure mechanism of cast-iron pipelines. Also, extensive studies regarding the deterioration of pipe structural capacity due to corrosion have been conducted to help set up the long-term corrosion behaviour model by utilising the probabilistic physical modelling (PPM) method. Prior to PPM, statistical analysis was conducted and statistical information of the basic physical parameters of the observational failure data were obtained using Bayesian inverse analysis. The back-figured results have provided an updated list of corrosion parameters, cs and rs which could be useful as future reference for other researchers if they decide to adapt the simplified exponential model when studying the corrosion behaviour of small 11

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Fig. 11. Trace plots of random variables by MCMC simulations.

Fig. 12. Histogram plots of random variables by MCMC simulations. Table 4 Bayesian back-calculated statistics of random variables. Random variable

Mean value (posterior)

Standard deviation (posterior)

Corrosion rate, rs (mm/y) Corrosion rate, cs (mm) Corrosion coefficient, 1/T0 (1/y) Breakage water pressure (kPa) Yield strength (MPa) Longitudinal patch ratio Circumferential patch ratio

0.02 6.07 0.18 2834.32 118.60 4.28 1.03

0.0044 0.44 0.055 296.43 11.19 1.09 0.34

diameter cast-iron pipes. In addition, the lifetime failure prediction and the hazard rate function were computed by utilising the updated corrosion parameters. Finally, comparison of the PPM was made with results obtained from statistical analysis. It was found that results obtained from the top-down method (idealised through field evidence) and the bottom-up method (utilising PPM) show similar results, with some discrepancy at the early lifetime stage, and after the pipe age exceeded 50 years. The possible reason is attributed to the incompleteness of failure data and stress-based physical modelling errors.

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Fig. 13. Probability of pipe failure curves predicted by probabilistic physical modelling with Bayesian updated statistics.

0.4 MLE on truncated Weibull

0.35

Probabilistic physical modelling

Hazard rate

0.3 0.25 0.2 0.15 0.1 0.05 0 5

10

15

20

25

30

35 40 45 50 Pipe age (Years)

55

60

65

70

75

80

Fig. 14. Comparison between MLE-observational data and proposed modelling prediction values for 100 mm Spun Cast Iron pipes.

Declaration of Competing Interest The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

Acknowledgements Research student Hong-Zhi Cui of Hohai University helped compiling the list of symbols. This publication is an outcome of the “Advanced Condition Assessment & Pipeline Failure Prediction Project (ACA&PFPP)” funded by the Sydney Water Corporation, The Water Research Foundation (USA), Melbourne Water, Water Corporation (WA), UK Water Industry Research Ltd, South Australia Water Corporation, South East Water, Hunter Water Corporation, City West Water, Monash University, the University of Technology Sydney, and the University of Newcastle. The research partners are Monash University (lead), the University of Technology Sydney, and the University of Newcastle. The first author would also like to thank the financial support from the National Natural Science Foundation of China (51609072, 51879091), and the Fundamental Research Funds for the Central Universities (2018B01214, 2019B44114).

Appendix A. Supplementary material Supplementary data to this article can be found online at https://doi.org/10.1016/j.engfailanal.2019.104239. 13

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