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Evaluating the system reliability of corroding pipelines based on inspection data a
M. Al-Amin & Wenxing Zhou a
b
TransCanada Corporation , 450-1st Street SW, Calgary , AB , Canada T2P 5H1
b
Western University , 1151 Richmond Street, London , ON , Canada N6A 5B9 Published online: 09 May 2013.
To cite this article: M. Al-Amin & Wenxing Zhou (2013): Evaluating the system reliability of corroding pipelines based on inspection data, Structure and Infrastructure Engineering: Maintenance, Management, Life-Cycle Design and Performance, DOI:10.1080/15732479.2013.793725 To link to this article: http://dx.doi.org/10.1080/15732479.2013.793725
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Structure and Infrastructure Engineering, 2013 http://dx.doi.org/10.1080/15732479.2013.793725
Evaluating the system reliability of corroding pipelines based on inspection data M. Al-Amina1 and Wenxing Zhoub* a
TransCanada Corporation, 450-1st Street SW, Calgary, AB, Canada T2P 5H1; bWestern University, 1151 Richmond Street, London, ON, Canada N6A 5B9
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(Received 15 October 2012; final version received 9 January 2013; accepted 24 February 2013) This paper presents a methodology for evaluating the time-dependent system reliability of a pressurised gas pipeline segment containing multiple active metal-loss corrosion defects. The methodology incorporates three distinctive failure modes of the pipe segment due to corrosion, namely small leak, large leak and rupture. The growth of the depth of individual corrosion defect is assumed to follow a power-law function of time. The Bayesian updating and Markov Chain Monte Carlo (MCMC) simulation techniques are used to quantify the parameters of the power-law growth model based on data obtained from multiple inspections carried out at different times. The simple Monte Carlo and MCMC techniques are combined to evaluate the system reliability. A numerical example involving an in-service gas pipeline located in Alberta, Canada, is used to illustrate the proposed methodology. Results of the sensitivity analysis suggest that the use of a defect-specific or segment-specific growth model for the defect depth has a marked impact on the evaluated system reliability. The proposed methodology can be incorporated in reliability-based pipeline corrosion management programmes to assist integrity engineers in making informed decisions about defect repair and mitigation. Keywords: pipeline; metal-loss corrosion; system reliability; inspection; Bayesian updating and MCMC simulation
1.
Introduction
Metal-loss corrosion is one of the most common attributing factors to failures of underground natural gas pipelines, which are highly pressurised with typical operating pressures ranging from 4 to 10 MPa (Nessim, Dawson, Mora, & Hassanein, 2008; Pipeline and Hazardous Materials Safety Administration [PHMSA], 2012). Corrosion may occur on both the outside surface and the inside surface of a pipe: the former is external corrosion whereas the latter is internal corrosion. The reliability-based corrosion management programme has received increasing attention from pipeline operators (Kariyawasam & Peterson, 2008) over the past decade. Such a programme typically consists of three cyclic steps: first, detecting and sizing external and internal corrosion defects on a pipeline using the in-line inspection (ILI) technology; second, evaluating the failure probability of the pipeline as a result of the corrosion defects and finally, mitigating the defects if the failure probability exceeds a certain allowable level. To this end, implementation of the reliability-based corrosion management programme requires accurate evaluation of the failure probability of pipelines due to corrosion defects so that defect repairs can be scheduled to meet the required safety levels while optimising the allocation of limited resources for repair and mitigation. The failure mechanisms of a pressurised gas pipeline containing an active external or internal corrosion defect
*Corresponding author. Email:
[email protected] q 2013 Taylor & Francis
can be broadly classified into two categories: small leak and burst (Canadian Standard Association [CSA], 2007). Small leak occurs if the defect penetrates the pipe wall and burst occurs if the internal pressure exceeds the burst resistance at the corrosion defect, resulting in plastic collapse of the pipe wall. A burst can be further categorised as a rupture or a large leak due to the relatively low velocity of the decompression wave during the release of the gas (Qiu, Gong, & Zhao, 2011). Rupture occurs if the through-wall defect resulting from the burst extends unstably in the longitudinal direction of the pipeline, whereas large leak is the plastic collapse of the pipe wall without unstable axial extension of the defect (CSA, 2007). It is important to distinguish different failure modes in that their corresponding consequences differ significantly (Nessim, Zhou, Zhou, & Rothwell, 2009; Rothwell & Stephens, 2006; Zhou, 2011), with the consequences of ruptures generally being the most severe and those of small leaks being the least severe. Different allowable failure probabilities (or target reliability levels) have been proposed for different failure modes of gas pipelines (CSA, 2007): more stringent allowable failure probabilities for ruptures and large leaks, and less stringent values for small leaks (CSA, 2007). Corrosion growth modelling plays an important role in forecasting the failure probability of a corroding pipeline (Kariyawasam & Peterson, 2010; Nessim et al., 2008).
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M. Al-Amin and W. Zhou
Due to its simplicity, the linear growth model (Coleman & Miller, 2010; Fenyvesi & Dumalski, 2005; Huyse & Roodselaar, 2010; Nessim et al., 2008) is the most commonly used corrosion growth model in practice, whereby the defect is assumed to grow at a constant rate over time. It is also recognised (Ahmed & Melchers, 1997; Caleyo, Vela´zquez, Valor, & Hallen, 2009; Maes, Faber, & Dann, 2009; Melchers, 2003a, 2003b; Romanoff, 1989; Soares & Garbatov, 1999) that the growth of metal-loss corrosion can be better characterised by non-linear models than by the linear model. Furthermore, two different approaches have been reported in the literature (Kariyawasam & Peterson, 2010; Nessim et al., 2008) to characterise the growth of a group of corrosion defects on a given pipe segment, namely the defect-specific and segment-specific approaches. In the former approach, the growth of individual defect is evaluated, whereas in the latter approach, the average growth of the group of defects within the segment is calculated. The defect-specific approach is considered to be more advantageous than the segment-specific approach because the growth paths for different defects can vary markedly (Ahammed, 1998; Southwell & Bultman, 1975). Extensive research has been carried out in the past to evaluate the reliability of pressurised pipeline containing active metal-loss corrosion defects (Ahammed, 1998; Ahammed & Melchers, 1997; Caleyo, Gonza´lez, & Hallen, 2002; Hong, 1997; Stephens & Nessim, 2006; Zhou, 2010). The corrosion defects were often assumed to grow in a linear fashion in previous studies. Furthermore, the same (uncertain) growth rate was typically assumed for different defects considered in the reliability analysis. To the authors’ best knowledge, the consideration of defect-specific growth models in the reliability analysis of corroding pipelines has not been reported in the literature. The main objective of the work reported in this paper was to develop a methodology that can be used to evaluate the time-dependent system reliability of a segment of a pressurised gas pipeline containing multiple active corrosion defects by incorporating a non-linear defect-specific corrosion growth model. The depth of a given corrosion defect was assumed to follow a power-law growth path over time. The parameters of the power-law model were evaluated through the Bayesian updating based on data obtained from inspections. The failure probabilities associated with three distinctive failure modes, namely small leak, large leak and rupture, were evaluated using a simulation-based approach that consists of both the simple Monte Carlo simulation for generating random samples of the pipe geometric and material properties as well as the defect length and the Markov Chain Monte Carlo (MCMC) simulation for generating random samples of the defect depth. The methodology is illustrated using a numerical example that involves a natural gas pipeline segment located in Alberta, Canada.
This paper is organised in seven sections. The limit state functions associated with small leak, large leak and rupture are presented in Section 2. This is followed by the capacity models for burst and rupture in Section 3. Section 4 describes the Bayesian power-law growth model for the depths of corrosion defects. Section 5 includes the basic assumptions adopted in the reliability analysis as well as the procedure of evaluating the system reliability using a combination of the simple Monte Carlo simulation and MCMC simulation techniques. A numerical example is given in Section 6 to illustrate the proposed methodology. Results of the sensitivity analysis with respect to the autocorrelation between the MCMC samples of the defect depths, spatial variability of the length growth rates of different defects and corrosion growth model assumptions are also presented in Section 6. The main findings of the study are summarised in Section 7.
2.
Limit state functions
Metal-loss corrosion on pipeline causes volumetric loss of metal in the pipe wall. The geometry of a typical metalloss corrosion defect on a pipeline is illustrated in Figure 1. The length, width and depth of the defect are measured in the longitudinal, circumferential and through-wall thickness directions, respectively, of the pipeline. Based on the above-defined defect dimensions, the limit state functions for a pipe containing a single active corrosion defect under internal pressure are developed in the following. The limit state function, g1(t), for the corrosion defect penetrating the pipe wall at a given time t is g1 ðtÞ ¼ 0:8wt 2 dðtÞ;
ð1Þ
where wt is the wall thickness of the pipeline and d(t) is the maximum depth of the corrosion defect (see Figure 1) at time t. The use of 0.8wt (as opposed to wt) is based on practical experience (S. Kariyawasam, private communication, 2012), which suggests that once the defect depth reaches 0.8wt, the remaining ligament of the pipe wall is prone to developing cracks that could lead to a leak. Note that Caleyo et al. (2002) also used 0.8wt in the leakage limit state function to carry out the reliability assessment of corroding pipelines. The limit state function, g2(t), for plastic collapse under internal pressure at the defect at time t is given by g2 ðtÞ ¼ r b ðtÞ 2 p;
ð2Þ
where rb(t) denotes the burst pressure resistance of the pipe at the defect at time t and p is the internal pressure of the pipeline. The internal pressure varies with time and should ideally be treated as time dependent. A previous study (Zhou, 2011) suggests that ignoring the time dependency of the internal pressure has a negligible impact on the
3
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Structure and Infrastructure Engineering
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“River bottom”path of contour map of defect
Width (W)
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Length (L)
Maximum depth (d)
Wall thickness (wt)
Longitudinal axis of pipeline
Figure 1.
Dimensions of a typical corrosion defect on pipeline.
reliability of corroding pipelines if the coefficient of variation (COV) of the annual maximum internal pressure is about 2%. Information in the literature (CSA, 2007; Jiao, Sotberg, & Igland, 1995) indicates that the COV of the annual maximum internal pressure of gas pipelines is in general small (ranging from 1% to 3.4%); therefore, we assumed the internal pressure to be time independent in this study. The burst pressure resistance is a function of geometric and material properties of the pipeline and the defect depth and length (see Figure 1). Since the defect size monotonically increases over time, the burst pressure resistance monotonically decreases over time. Given a burst, the unstable axial extension of the through-wall defect that results from the burst is defined as a rupture and is governed by the limit state function g3(t) as follows:
at time t. A burst is classified as a rupture if g3(t) # 0; otherwise, it is defined as a large leak. Based on the limit state functions defined by Equations (1) –(3), failure of a gas pipeline can be categorised into three modes, namely small leak, large leak and rupture. Because these limit state functions involve monotonically increasing defect geometry and monotonically decreasing pipe resistance, and because the internal pressure is assumed to be time independent, the probabilities of small leak, large leak and rupture within a time interval [0, t ], Psl(t), Pll(t) and Prp(t), respectively, are defined as follows:
g3 ðtÞ ¼ r rp ðtÞ 2 p;
where ‘ > ’ represents the intersection (i.e. joint event). In estimating the probabilities of small leak and burst, it is assumed that the occurrences of burst and small leak at a given defect are mutually exclusive (Zhou, 2011).
ð3Þ
where rrp(t) is the pressure resistance of the pipeline at the location of the through-wall defect resulting from the burst
Psl ðtÞ ¼ Prob½g1 ðtÞ # 0 > g2 ðtÞ . 0;
ð4aÞ
Pll ðtÞ ¼ Prob½g1 ðtÞ . 0 > g2 ðtÞ # 0 > g3 ðtÞ . 0; ð4bÞ Prp ðtÞ ¼ Prob½g1 ðtÞ . 0 > g2 ðtÞ # 0 > g3 ðtÞ ¼ 0; ð4cÞ
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M. Al-Amin and W. Zhou
3. Burst and rupture pressure models In this study, the B31G modified criterion developed by Kiefner and Vieth (1989) was selected to evaluate the burst pressure resistance of a pipe containing a single corrosion defect. According to this model, the pressure resistance rb is calculated as follows: 2wtsf 1 2 0:85ðd=wtÞ d # 0:8; ð5aÞ rb ¼ e 1 2 0:85ðd=MwtÞ wt D 8 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi l2 l4 l2 > 2 0:003375 ðDwtÞ < 1 þ 0:6275 Dwt 2 Dwt # 50 M¼ ; > l2 l2 : 3:293 þ 0:032 Dwt Dwt . 50
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ð5bÞ
sf ¼ sy þ 68:95 MPað10 ksiÞ;
ð5cÞ
where sf and sy are the flow stress and yield strength of the pipe material, respectively; e is a multiplicative model error and can be characterised based on the test-topredicted burst pressure ratios for a set of full-scale tests of corroded pipes (Zhou & Huang, 2012); D is the outside diameter of the pipeline; d is the maximum depth of the defect (see Figure 1); l is the defect length and M is the Folias factor or bulging factor. The rupture pressure resistance model recommended in Annex O of the Canadian pipeline standard CSA Z66207 (CSA, 2007) was employed in this study. This model was developed by Kiefner and Vieth (1989) based on the flow stress-dependent failure criterion for pressurised pipelines containing through-wall flaws. The rupture pressure resistance, rrp, is calculated as follows: r rp ¼
2wtsf : MD
ð6Þ
The Folias factor M in Equation (6) can be calculated using Equation (5b). The model error for Equation (6) was ignored in the analysis due to a lack of relevant information.
4.
Corrosion growth models
Consider m active corrosion defects on a pipe segment. The maximum depth of corrosion defect i (i ¼ 1, 2, . . . , m), di(t), was assumed to follow a power-law growth path given by di ðtÞ ¼ ai ðt 2 toi Þbi þ hi ;
ð7Þ
where t (years) is the time elapsed since the time of installation; hi represents the model error of the power-law growth model associated with defect i, which is assumed to follow a normal distribution with a zero mean and a variance of s2h ; the model errors for different defects are assumed to be independent and identically distributed, and
ai, bi and toi define the growth path for defect i. The parameter ai (ai . 0) is indicative of the growth of the defect depth within 1 year from the defect initiation; bi (bi . 0) defines the rate of change of the growth path; that is, bi ¼ 1, bi . 1 and 0 , bi , 1 characterise a linear, an accelerating and a decelerating growth path, respectively, and toi (years) represents the corrosion initiation time (e.g. the time interval between the installation and the time at which defect i starts to grow). Further consider that the m defects have been subjected to n ILIs. The parameters of the growth models, i.e. ai, bi, toi and s2h , can be evaluated using a hierarchical Bayesian method (Congdon, 2010) based on the ILI data. This method assumes that the measured maximum depth of defect i at the jth ( j ¼ 1, 2, . . . , n) inspection, yij, is related to the corresponding actual maximum depth dij as follows: yij ¼ aj þ bj dij þ 1ij ;
ð8Þ
where aj and bj are the calibration parameters of the ILI tool employed in the jth inspection, which characterise the bias of the tool (i.e. if aj ¼ 0 and bj ¼ 0, the tool is unbiased; if aj – 0 and bj ¼ 0, the tool has a constant bias and if aj – 0 and bj – 0, the tool has both constant and non-constant bias); 1ij represents the random scattering errors of the measured maximum depth of the ith defect at the jth inspection, and dij is evaluated from Equation (7) with the parameter t replaced by tj, where tj is the time interval between the installation of the pipeline and jth ILI. Assuming ai, bi, toi, hi and s2h to be uncertain quantities, one can combine the prior distributions of these parameters and ILI data to evaluate the corresponding posterior distributions. The exchangeability condition (Bernardo & Smith, 2007) was also assumed in the Bayesian updating; that is, the growth paths of different defects are independent conditional on the uncertain parameters. Because it is not feasible to develop the posterior distributions analytically, the MCMC simulation is employed to numerically evaluate the posterior distributions. Details of the hierarchical Bayesian method are given in AlAmin, Zhou, Zhang, Kariyawasam, and Wang (2012). A linear growth was assumed for the defect length; that is, the length of a corrosion defect was assumed to grow at a constant (but uncertain) rate over time (Caleyo et al., 2002; Hong, 1997; Zhou, 2011). Therefore, the length of defect i can be predicted as follows: li ðtÞ ¼ r li ðt 2 toi Þ;
ð9Þ
where li(t) is the length of defect i, t years after installation and rli denotes the length growth rate for defect i. In practice, the time-dependent reliability of a given pipe segment often needs to be evaluated or forecast starting from the time of the most recent inspection. Let t denote the time elapsed since the last inspection (i.e. the forecasting year). It follows that the length of defect i at
Structure and Infrastructure Engineering the forecasting year t; li(t), can be written as li ðtÞ ¼ l0i þ r li t;
following analysis procedure was employed: ð10Þ
where loi is the initial length (i.e. at the time of last inspection) of defect i.
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5.
System reliability analysis
5.1 Basic assumptions In this study, the system reliability of a pipeline was evaluated on a joint-by-joint basis; that is, a pipe joint, with a typical length of 10 – 20 m, containing multiple active corrosion defects was considered as a system. Because failure of any defect on a pipe joint implies failure of the joint, it follows that the pipe joint is a series system. The internal operating pressure of the pipeline was assumed to be a time-independent random variable. The internal pressure, pipe geometry and material properties (i.e. diameter, wall thickness and yield strength), length growth rate and the model error associated with the burst pressure model were assumed to be mutually independent for a given defect. Each of these parameters except the length growth rate was further assumed to be fully correlated for all the defects in a given joint, whereas the length growth rate was assumed to be independent for different defects. 5.2
5
Analysis procedure
A combination of the simple Monte Carlo simulation and MCMC simulation techniques was used to evaluate the system reliability of a given pipe joint containing multiple active corrosion defects. Because the parameters of the growth model for the defect depth were obtained from the Bayesian updating using the MCMC technique, it is advantageous to retain the random samples of these parameters generated from MCMC and incorporate the samples in the reliability analysis. Furthermore, the correlations among these parameters are fully preserved by directly using the MCMC samples in the reliability analysis. The simple Monte Carlo technique was used to generate random samples of the other parameters in the reliability analysis such as the pipe wall thickness, yield strength and model error. Samples of di(t) were obtained by substituting MCMC samples of ai, bi, toi and hi into Equation (7) for different t values corresponding to the forecasting years. Because the model error, hi, associated with the power-law model is normally distributed, the random samples of di(t) may be less than zero or greater than 100%wt, which are impossible in reality. To address this, the distribution of di(t) was truncated at the lower bound of zero and upper bound of 100%wt. To calculate the system reliability of a pipe joint containing m active corrosion defects over a forecasting period of t years since the last inspection, the
(1) Generate N random samples of the maximum depth for each of the m defects at each year within the forecasting period t using the procedure described in the previous paragraph. (2) Set sl(t), ll(t) and rp(t) ¼ 0, where sl(t), ll(t) and rp(t) denote the counters of small leaks, large leaks and ruptures, respectively, that occur in a given forecasting year t (t ¼ 1, 2, . . . , T). (3) For a given simulation trial k (k ¼ 1, 2, . . . , N), check if the system has failed and determine the corresponding failure mode within the forecasting period T as follows (a) Generate samples of the material properties (e.g. sy) and geometric properties (e.g. wt and D) of the pipeline, initial lengths loi (i ¼ 1, 2, . . . , m) and length growth rates rli of the defects, the internal operating pressure p and the model error e; (b) Start from the forecasting year t ¼ 1 and carry out the following: i) obtain a set of m random samples of the maximum defect depth, di (i ¼ 1, 2, . . . , m) at t, one for each of the m defects. Because the B31G modified criterion is only valid for di # 80%wt, set di ¼ 80%wt if di . 80%wt; ii) calculate the lengths of the defects li at t using Equation (10); iii) calculate g1 ¼ 0:8wt 2 maxfd i }; i iv) substitute the values of wt, D, e, sy, li and di into Equation (5); calculate g2 ¼ min fr b;i } 2p; i v) if g1 . 0 and g2 . 0, set t ¼ t þ 1 and repeat steps (3.b.i) through (3.b.iv); if g1 ¼ 0 and g2 . 0, set sl(t) ¼ sl(t) þ 1; if g2 # 0, , , calculate g3 ¼ r rp 2 p, where r rp is the rupture pressure of the defect with the lowest burst pressure at t; and set ll(t) ¼ ll(t) þ 1 if g2 # 0 and g3 . 0; set rp(t) ¼ rp(t) þ 1 if g2 # 0 and g3 # 0 and (4) Repeat steps (3.a)–(3.b) for N simulation trials. Once the counts of sl(t), ll(t) and rp(t) are obtained for the N simulation trials, the cumulative probabilities of small leak, large leak and rupture up to a given forecasting year t, Psl(t), Pll(t) and Prp(t), are evaluated as follows: t 1X Psl ðtÞ < slðjÞ; ð11aÞ N j¼1 Pll ðtÞ
