Inverse Gaussian process-based corrosion growth modeling and its application in the reliability analysis for energy pipelines

Abstract

This paper describes an inverse Gaussian process-based model to characterize the growth of metal-loss corrosion defects on energy pipelines. The model parameters are evaluated using the Bayesian methodology by combining the inspection data obtained from multiple inspections with the prior distributions. The Markov Chain Monte Carlo (MCMC) simulation techniques are employed to numerically evaluate the posterior marginal distribution of each individual parameter. The measurement errors associated with the ILI tools are considered in the Bayesian inference. The application of the growth model is illustrated using an example involving real inspection data collected from an in-service pipeline in Alberta, Canada. The results indicate that the model in general can predict the growth of corrosion defects reasonably well. Parametric analyses associated with the growth model as well as reliability assessment of the pipeline based on the growth model are also included in the example. The proposed model can be used to facilitate the development and application of reliability-based pipeline corrosion management.

Keywords

pipeline / metal-loss corrosion / inverse Gaussian process / measurement error / hierarchical Bayesian / Markov Chain Monte Carlo (MCMC)

Introduction

Metal-loss corrosion is a major threat to the structural integrity of steel energy pipelines world-wide, the majority of which are highly pressurized underground oil and natural gas pipelines. The reliability-based corrosion management programs have been increasingly employed by the pipeline operators. The corrosion growth modeling plays a key role in the various tasks involved in the pipeline corrosion management program such as determination of the inspection interval, evaluation of the failure probability and development of phased mitigation actions to satisfy the safety constraint and optimize the allocation of limited resources.

The corrosion process is subjected to significant uncertainties of both temporal and spatial variability. The Markov process and Gamma process are two stochastic process models that have been commonly applied in corrosion growth modeling [1-3]. For example, Hong [1] proposed a non-homogenous Markov process to model the corrosion growth based on data obtained from corrosion experiments. Maes et al. [2] and Zhang et al. [3] reported non-homogenous Gamma process- (NHGP-) based models formulated in the hierarchical Bayesian framework to characterize the growth of defects on pipelines based on data obtained from multiple inline inspections (ILI). The study in [3] differs from that in [2] as the former considers the initiation time of each individual defects and incorporates a general form of the measurement errors associated with ILI data, which includes the biases and random scattering error as well as the correlations between the random scattering errors associated with the ILI tools used in different inspections.

A recent study [4] reported in the literature suggests that the inverse Gaussian process (IGP) is a viable alternative to characterize the corrosion process. The mathematical tractability of IGP [4] also facilitates incorporating the IGP-based corrosion growth model in a Bayesian framework to evaluate and update the model parameters based on the corrosion inspection data. However, reports on the IGP-based models to characterize the growth of corrosion defects on pipelines are scarce in the literature.

In this study, we aim to develop an IGP-based growth model for the depths of corrosion defects on energy pipelines by incorporating the ILI-reported depths collected from multiple ILI runs. The growth model is formulated in a hierarchical Bayesian framework to deal with uncertainties in the growth model and inspection data. The Markov Chain Monte Carlo (MCMC) simulation techniques are employed to carry out the Bayesian inference and numerically evaluate the probabilistic characteristics of the model parameters. The proposed growth model is illustrated and validated through an example involving a set of real ILI data collected from a natural gas pipeline that is currently in service in Alberta, Canada. The use of the developed growth model for evaluating the time-dependent system reliability of a joint of the above mentioned pipeline is also included in the example.

Inverse Gaussian process

Consider that X is a random variable and follows an inverse Gaussian distribution with the probability density function (PDF), f_X(x|μ,θ), parameterized as [5]

(1)f_X((x|\mu ,\theta ))=\sqrt{{\displaystyle \frac{\theta }{\mathrm{2}\mathrm{\pi }}}}{x}^{-\mathrm{2}/\mathrm{3}}\exp \left(-{\displaystyle \frac{\theta {(x-\mu )}^{\mathrm{2}}}{\mathrm{2}{\mu }^{\mathrm{2}}x}}\right)I_{(\mathrm{0},\infty )}(x),

with μ (μ>0) and θ (θ>0) denoting the mean and shape parameters of X, respectively, and I_{(0, ∞)}(x) denoting an indicator function that equals unity for x>0 and zero otherwise. The mean and variance of X are μ and μ³/θ, respectively.

Let {X(t); t≥0} denote an inverse Gaussian process (IGP) with a PDF given by [4]

(2)f_{X(t)}(x(t)|\mathit{\Theta }(t),\xi {(\mathit{\Theta }(t))}^{\mathrm{2}})=\sqrt{\xi /\mathrm{2}\mathrm{\pi }}\mathit{\Theta }(t)x{(t)}^{-\mathrm{3}/\mathrm{2}}\exp \left(-{\displaystyle \frac{\xi {(x(t)-\mathit{\Theta }(t))}^{\mathrm{2}}}{\mathrm{2}x(t)}}\right)I_{(\mathrm{0},\infty )}(x(t)),

where Θ(t) and ξ(Θ(t))² denote the mean and shape functions, respectively, with ξ denoting a scale parameter. The IGP has three properties: 1) X(t) equals zero at t = 0 with probability one; 2) X(t) has independent increments; and 3) for all t₂>t₁≥0, X(t₂) - X(t₁) follows an inverse Gaussian distribution with a PDF given by Eq. (1), where the mean and shape parameters equal Θ(t₂) - Θ(t₁) and ξ(Θ(t₂) - Θ(t₁))², respectively.

Based on Eq. (1), the mean and variance of X(t), denoted by E[X(t)] and Var[X(t)], respectively, are given by

(3a)E[X(t)]=\mathit{\Theta }(t),

(3b)Var[X(t)]=\mathit{\Theta }(t)/\xi .

Inspection data and growth models for multiple defects

The inspection data reported by the ILI tools are always associated with certain sizing errors including biases and random scattering error that will impact the estimation of model parameters. Suppose m active corrosion defects on a particular pipe segment have been inspected and sized n times over a certain period of time. The measured depth of the i^th defect at the j^th inspection, y_ij, (i = 1, 2, …, m; j = 1, 2, …, n) can be related to the corresponding actual depth, x_ij, through the following equation [6,7]:

(4)y_{ij}=a_j+b_j{x}_{ij}+{ϵ}_{ij},

where a_j and b_j denote the constant and non-constant bias associated with the ILI tool used in the j^th inspection, and ϵ_ij denotes the random scattering error associated with the ILI-reported depth of the i^th defect at the j^th inspection, and is assumed to be normally distributed with a zero mean and known standard deviation [8]. It is further assumed that for a given inspection j, ϵ_ij and ϵ_lj (i ≠ l) are independent, whereas for a given defect i, ϵ_ij and ϵ_ik (j ≠ k) are correlated with a correlation coefficient of ρ_jk (i.e., the correlation coefficient between the random scattering errors associated with the j^th and k^th inspections) [8]. Denote BoldItalic_i = (E_i1, E_i2 … E_in)′ as the vector of random scattering errors associated with the n inspections for defect i, with “′” representing transposition. It follows from the above assumption that BoldItalic_i is multivariate normal-distributed and has a PDF given by

(5)f_{{\mathit{E}}_i}({\mathit{ϵ}}_i)={\displaystyle \frac{\mathrm{1}}{{(\sqrt{\mathrm{2}\mathrm{\pi }})}^{n/\mathrm{2}}{\left|{\sum }_{{\mathit{E}}_i}\right|}^{\mathrm{1}/\mathrm{2}}}}\exp \left(-{\displaystyle \frac{\mathrm{1}}{\mathrm{2}}}{{\mathit{ϵ}}^{\prime }}_i{\sum }_{{\mathit{E}}_i}^{-\mathrm{1}}{\mathit{ϵ}}_i\right),

where ∑_BoldItalici denotes the n × n variance-covariance matrix of BoldItalic_i with each of elements equal to ρ_jkσ_jσ_k (j = 1, 2, …, n; k = 1, 2, …, n), and σ_j denotes the standard deviation of the random scattering error associated with the tool used at the j^th inspection. A Bayesian methodology has been reported in [8] to evaluate a_j, b_j, σ_j and ρ_jk by comparing the ILI-reported and corresponding field-measured depths for a set of static (i.e., repaired) corrosion defects.

The growth of the depth associated with each individual defect is assumed to follow an inverse Gaussian process given by Eq. (2), where Θ(t) is assumed to be a power-law function of time, i.e., Θ(t) = α(t – t₀)^δ with t₀ denoting the initiation time of the corrosion defect, α denoting the average growth over the first unit time interval since t₀, and δ indicating the trend of the mean growth path. The mean growth (i.e., Θ(t)) is an accelerating, decelerating and linear growth path for δ>1, δ<1 and δ = 1, respectively.

In this study, the four parameters of the IGP-based growth model, namely α, t₀, δ and ξ, were assumed to be uncertain parameters. Furthermore, we assumed α and t₀ to be defect-specific, and δ and ξ to be common for all defects. It then follows from Eq. (2) that the growth of the i^th defect between the (j - 1)^th and j^th inspections, ΔX_ij, is inverse Gaussian distributed and has a PDF of f_ΔXij(Δx_ij|ΔΘ_ij, ξ(ΔΘ_ij)²). The mean value of ΔX_ij, ΔΘ_ij, equals α_i(t_j – t_i0)^δ for j = 1 and α_i(t_j – t_i0)^δ - α_i(t_j-1 – t_i0)^δ for j = 2, 3, …, n, where α_i denotes the average growth of the i^th defect over the first unit time interval since t_i0; t_j denotes the time of the j^th inspection, and t_i0 denotes the initiation time of the i^th defect. The actual depth of the i^th defect at the time of the j^th inspection, x_ij, is then obtained from x_ij = x_i,_j-1 + Δx_ij, with x_i0 assumed to be zero.

Bayesian updating of the growth model

Overview the Bayesian updating

The Bayesian updating is an approach to evaluate the probability distributions of a set of unknown parameters of a given model by combining the existing knowledge of these parameters with the new information contained in the data [9]. The mechanism for combining the information is Bayes’ theorem. The existing knowledge of model parameters is reflected by the prior distribution. The new information in the data is incorporated in the Bayesian updating through the so-called likelihood function, and the probability distribution obtained from the updating is known as the posterior distribution. The likelihood function, prior and posterior distributions are described in the following sections.

Likelihood function

Likelihood function of the inspection data

Consider defect i and denote BoldItalic_i = (y_i1, y_i2,…, y_ij, …, y_in)′ and ΔBoldItalic_i = (Δx_i1, Δx_i2, …, Δx_in)′. It follows from Eqs. (4) and (5) as well as the assumptions described in Section 3 that the likelihood of the inspection data, BoldItalic_i, conditional on the actual depth increments, ΔBoldItalic_i, is given by

(6)L(y_i|\mathrm{\Delta }{x}_i)={(\mathrm{2}\mathrm{\pi })}^{-n/\mathrm{2}}{\left|{\sum }_{ϵ}\right|}^{-\mathrm{1}/\mathrm{2}}\exp \left[-{\displaystyle \frac{\mathrm{1}}{\mathrm{2}}}(y_i-a-b{x}_i){\sum }_{{\mathit{E}}_i}^{-\mathrm{1}}(y_i-a-b{x}_i)\right],

where BoldItalic_i = (x_i1, x_i2, …, x_ij, …, x_in)′ with the j^th element, x_ij = {\sum }_{k=\mathrm{1}}^j\mathrm{\Delta }{x}_{ik}; BoldItalic = (a₁, a₂, …, a_j, …, a_n)′, and BoldItalic is an n × n diagonal matrix with the j^th element equal to b_j.

Likelihood functions of the growth of depth

Assume that Δx_ij and Δx_lj (i ≠ l) are conditionally independent of each other for inspection j; in other words, the exchangeability condition [9] is applicable to Δx_ij (i = 1, 2,…, m) conditional on α_i, δ, ξ and t_i0 for inspection j. The joint PDF of ΔBoldItalic_i can be written as

(7)$$\begin{gathered} f_{\mathrm{\Delta }{X}_i}(\mathrm{\Delta }{x}_i|a_i,\delta ,\xi ,t_{\mathrm{0}i})={\mathrm{\Pi }}_{j=\mathrm{1}}^n{f}_{\mathrm{\Delta }{X}_{ij}}(\mathrm{\Delta }{x}_{ij}|a_i,\delta ,\xi ,t_{\mathrm{0}i}) \\ ={\mathrm{\Pi }}_{j=\mathrm{1}}^n\sqrt{\xi /\mathrm{2}\mathrm{\pi }}\mathrm{\Delta }{\mathit{\Theta }}_{ij}\mathrm{\Delta }{x}_{ij}^{-\mathrm{3}/\mathrm{2}}\exp \left(-{\displaystyle \frac{\xi {(\mathrm{\Delta }{x}_{ij}-\mathrm{\Delta }{\mathit{\Theta }}_{ij})}^{\mathrm{2}}}{\mathrm{2}\mathrm{\Delta }{x}_{ij}}}\right). \end{gathered}$$

Prior distribution

For m active corrosion defects, the growth model described in Section 3 involves a total of 2m+ 2 basic parameters, namely m defect-specific parameters α_i and initiation times t_i0 (i= 1, 2,…, m) and two common parameters δ and ξ. In this study, the gamma distribution was selected as the prior distributions of α_i, δ and ξ (i= 1, 2,…, m) based on the consideration that the gamma distribution ensures α_i and ξ to be positive quantities and can be conveniently made as a non-informative distribution. The prior distribution of the initiation time t_i0 was assigned a uniform distribution with a lower bound of zero and an upper bound equal to the time interval between the installation of the pipeline and the first detection of the i^th defect. The prior distributions of α_i and t_i0 associated with all defects were further assumed to be identical and mutually independent (iid). Details of the prior distributions of α_i, t_i0 (i= 1, 2,…, m), ξ and δ are given as follows:

(8a){\alpha }_i~\mathrm{\pi }G(p,q),

(8b)t_{i\mathrm{0}}~{\mathrm{\pi }}_{\mathrm{U}}(\mathrm{0},u),

(8c)\xi ~\mathrm{\pi }\mathrm{G}(r,s),

(8d)\delta ~{\mathrm{\pi }}_{\mathrm{G}}(v,w),

where π_G(∙) and π_U(∙) denote the PDF of the gamma and uniform distributions, respectively, and p, q, u, r, s, v, and w are known distribution parameters of the prior distributions. The selection of the values of these parameters is discussed in Section 5.

Posterior distribution

Let BoldItalic denote the unknown model parameters and BoldItalic denote the inspection data. The joint posterior distribution of BoldItalic, p(BoldItalic|BoldItalic) can be obtained by combining the joint prior distribution of BoldItalic, π(BoldItalic), with the likelihood, L(BoldItalic|BoldItalic), of the inspection data according to Bayes theorem [9,10]:

(9a)p(\mathit{\theta }|\mathit{D})={\displaystyle \frac{L(\mathit{D}|\mathit{\theta })\mathrm{\pi }(\mathit{\theta })}{{\displaystyle \int L(\mathit{D}|\mathit{\theta })\mathrm{\pi }(\mathit{\theta })\mathrm{d}\mathit{\theta }}}}\propto L(\mathit{D}|\mathit{\theta })\mathrm{\pi }(\mathit{\theta }),

where “∝” represents proportionality. If the distribution parameters (denoted by BoldItalic) of the prior distribution of BoldItalic are also considered to be uncertain, the joint posterior distribution of the parameters BoldItalic and BoldItalic, denoted by p (BoldItalic, BoldItalic), can be formulated as the same manner as Eq. (9a) based on the hierarchical Bayesian theorem, i.e.,

(9b)p(\mathit{\theta }\mathit{,}\lambda |\mathit{D})={\displaystyle \frac{L(\mathit{D}|\mathit{\theta })\mathrm{\pi }(\mathit{\theta }|\lambda )\mathrm{\pi }(\lambda )}{{\displaystyle \int \mathrm{...}}{\displaystyle \int L(\mathit{D}|\mathit{\theta })\mathrm{\pi }(\mathit{\theta }|\lambda )\mathrm{\pi }(\lambda )\mathrm{d}\mathit{\theta }\mathrm{d}\lambda }}}\propto L(\mathit{D}|\mathit{\theta })\mathrm{\pi }(\mathit{\theta }|\lambda )\mathrm{\pi }(\lambda ),

It is not possible to analytically derive the joint posterior distributions of the parameters of the growth model; therefore, the MCMC simulation techniques were used to numerically evaluate the posterior distribution. In this study, the software OpenBUGS [11] was employed to carry out the MCMC simulation and Bayesian inference of the probabilistic characteristics of the model parameters. The generic distribution option was used to define the inverse Gaussian distribution in OpenBUGS [11].

Numerical example

Model validation

This section presents the growth models developed for 62 external corrosion defects identified on a natural gas pipeline that is currently in service in Alberta, Canada. This pipeline was constructed in 1972 and has been inspected by the magnetic flux leakage (MFL) tools in 2000, 2004 and 2007. The 62 defects, which spread over a pipe segment with a length of about 82 km, were excavated, field measured and mitigated in 2010. The field-measured depths were assumed to be free of measurement errors [8]; that is, the actual depths of the 62 defects in 2010 are known. A summary of the ILI-reported and field-measured depths for the defects is given in Table 1, where the symbol wt denotes pipe wall thickness and the symbol %wt denotes the unit of defect depth measured by MFL tools. The apparent growth paths indicated by the ILI-reported and field-measured depths for the 62 defects are illustrated in Fig. 1. Note that the non-monotonic trend of the growth paths for some defects is attributed to the measurement errors involved in the inspection data. The three sets of ILI data obtained in 2000, 2004 and 2007 were used to carry out the Bayesian updating and evaluate the probabilistic characteristics of the parameters of the growth models for each of the 62 defects. The growth model was then validated by comparing the predicted depths in 2010 with the corresponding actual depths.

The calibrated biases and random scattering errors associated with individual ILI tools as well as correlations between the random scattering errors of different ILI tools are as follows [8]: a₁ = a₂ = 2.04 (%wt) and a₃ = -15.28 (%wt); b₁ = b₂ = 0.97 and b₃ = 1.4; σ₁ = σ₂ = 5.97 (%wt) and σ₃ = 9.05 (%wt); ρ₁₂ = 0.82 and ρ₁₃ = ρ₂₃ = 0.7, with the subscripts “1,” “2” and “3” denoting the parameters associated with the ILI-reported data in 2000, 2004 and 2007, respectively.

The distribution parameters of the prior distributions given by Eq. (8) were selected as follows: p = 1, q = 1, u = 28 (year), r = 1, s = 1, v = 1 and w = 1. These values were observed to lead to a good convergence of the MCMC simulation. The impact of the distribution parameters of the prior distributions on the posterior distributions as well as the predictive accuracy of the growth model is discussed in the following section.

A total of 20000 sequences of samples were generated from the MCMC simulation and the first 3000 sequences of samples were treated as the burn-in period and therefore discarded. The samples in the remaining sequences were used to make inference of the statistics of the model parameters. The PDFs of the posterior marginal distributions of ξ and δ, which are common for all defects, are plotted in Figs. 2(a) and (b), respectively. For comparison, the prior distributions are also plotted in the same figure.

For brevity, the posterior distributions of α_i and t_i0 associated with each of the 62 defects are depicted by the box-plots instead of PDF curves and shown in Figs. 3(a) and 3(b), respectively. In Fig. 3, the mid-line, left and right sides of a given box indicate the median, 25- and 75-percentile values of the posterior marginal distribution of each parameter. The left and right ends of the arms of each box indicate the 2.5- and 97.5-percentile values of the corresponding marginal distribution. The bracketed number at the right end of the arm represents the defect number. The vertical line indicates the overall mean of this parameter for all the defects. The overall mean of α and t₀ for the 62 defects are 1.2%wt/year and 10.4 years, respectively.

Figures 2 and 3 indicate that most of the marginal distributions of model parameters are not symmetric and even highly skewed for certain defects. Therefore, the median values of the marginal distributions were selected as the point estimates of model parameters and used to predict the corrosion growth based on Eq. (2).

The predictive quality of the growth model was demonstrated by a comparison of the predicted depths in 2010 with the corresponding field-measured depths for the 62 defects (see Fig. 4). The predicted depths were obtained by substituting the median values of α_i, t_i0 and δ associated with each of the 62 defects evaluated from the MCMC simulation into Eq. (3a). Figure 4 suggests that the proposed model can predict the growth of corrosion defects reasonably well for majority of the defects considered in that the predicted depths for 90% of the 62 defects fall between the two bounding lines representing actual depth±10%wt. The range of±10%wt is commonly used in the pipeline industry as a confidence interval for the calibration of the ILI tools and was used in this study as a metric of the predictive accuracy of the growth model.

The mean, 10- and 90-percentile values of the growth paths for four arbitrarily selected defects, Defects #13, #15, #18 and #19, are plotted in Fig. 5(a) through 5(d), respectively. The mean and standard deviation of the defect depth at a given time were evaluated using Eqs. (3a) and (3b), respectively, based on the median values of the model parameters associated with this particular defect. The 10- and 90-percentile values were then approximately evaluated assuming that the depth at a given time is Gaussian-distributed according to the central limit theorem. For comparison, the corresponding ILI-reported depths in 2000, 2004 and 2007 as well as the field-measured depth in 2010 are also plotted in the same figure. The results in Fig. 5 indicate that the predicted average growth rate differs from defect to defect, which is expected because the parameter α is defect-specific. For example, the average growth rate of Defect #19 is the highest among the four defects plotted and equals 1.9%wt/year, followed by Defects #15, #13 and #18 with average growth rates equal to 1.6, 1.3 and 1.1%wt/year, respectively.

Parametric analysis

To investigate the impact of the assumptions associated with the model parameters on the predictions, two scenarios, designated by Scenarios I and II respectively, were further considered. For brevity, the case corresponding to the results shown in Section 5.1 is referred to as the baseline case, i.e., α and t₀ are both defect-specific and δ and ξ are common for all defects. Scenario I assumes that only t₀ is defect-specific, whereas scenario II assumes that only ξ is common for all the defects. The predicted depths corresponding to Scenarios I and II were obtained in the same manner as that for the baseline case. The results indicate that the percentage of the predictions for the 62 defects falling within the region of actual depth±10%wt is 72% for Scenario I and 48% for Scenario II. The fact that Scenario I results in a poorer prediction than the baseline case suggests that it is more reasonable to assume α to be defect-specific (as in the baseline case) than common for all the defects (as in Scenario I). This is because α represents the mean of the growth within the first unit time interval since initiation and is expected to differ from defect to defect due to the spatial variability of the corrosion defects. Furthermore, the fact that the predictions corresponding to Scenario II are markedly worse than those corresponding to the baseline case implies that δ does not vary markedly from defect to defect. Therefore, assuming δ to be common for all the defects (as in the baseline case) allows the Bayesian updating to assimilate the growth trends associated with all the defects and therefore make a reasonable inference of the statistics of δ. Our investigation indicates that assuming ξ to be either defect-specific or common for all defects, in general, has a negligible impact on the predictions.

To investigate the influence of the prior distributions of model parameters on the predictive quality, we considered two additional scenarios, designated by Scenarios III and IV respectively, both of which are the same as the baseline case except that the shape parameter (i.e., p) of the prior distribution of α_i equals 10 in Scenarios III and 0.1 in Scenarios IV (as opposed to 1.0 in the baseline case). The analysis results indicate that about 21% (61%) of the predictions of the 62 defects fall within the region of actual depth±10%wt for Scenario III (IV). This is mainly because p = 10 and 0.1 in conjunction with q = 1 in Eq. (8a) implies that the means of the growth in the first unit time interval since t₀ equal 10%wt/year and 0.1%wt/year, respectively. This prior knowledge about the growth rate might be too distant from the realistic scenario to lead to a good convergence of the MCMC simulation as well as good estimate of the posterior distributions. The prior distribution of α_i in the baseline case (i.e., p = q = 1) implies that the mean and coefficient of variation (COV) of the growth in the first unit time interval since t₀ equal 1%wt/year and 100%, respectively, which are considered reasonably representative of the reality. From this perspective, the prior distribution of α_i specified in the baseline case can be regarded, to certain extent, as an informative distribution. The comparison between the baseline case and Scenarios III and IV highlights the importance of properly selecting the prior distributions for the Bayesian updating.

Application in the system reliability assessment

In this section, we apply the developed growth model to the system reliability assessment of the corroding gas pipeline considering three distinctive failure modes, namely small leak, large leak and rupture. Details of the mechanisms of small leak, large leak and rupture can be found in [12]. The failure of a pipeline segment due to small leak, large leak and rupture can be formulated based on the following three limit state functions [13].

The limit state function for a corrosion defect penetrating the pipe wall at a given time t, g₁(t), is

(10a)g_{\mathrm{1}}(t)=\mathrm{0.8}wt-d(t),

with wt denoting the wall thickness of the pipeline, and d(t) denoting the depth of the corrosion defect at time t, i.e., x(t) used in the growth model. The use of 0.8wt as opposed to wt in the above equation is consistent with typical industry practice, where a remaining ligament thinner than 0.2wt is considered prone to developing cracks that could lead to leaks.

The limit state function for plastic collapse under internal pressure at the defect at time t, g₂(t), is given by

(10b)g_{\mathrm{2}}(t)=r_b(t)-p(t),

where r_b(t) denotes the burst pressure resistance of the pipe at the defect at time t and p(t) is the internal pressure of the pipeline at time t.

The limit state function for the unstable axial extension of the through-wall defect that results from the burst, g₃(t), is given by

(10c)g_{\mathrm{3}}(t)-r_{rp}(t)-p(t),

where r_rp(t) is the pressure resistance of the pipeline at the location of the through-wall defect resulting from the burst at time t. Note that we use g₁(t), g₂(t) and g₃(t) to emphasize that g₁, g₂ and g₃ are all time-dependent as a result of the deterioration of pipe resistance over time and time-dependency of the internal pressure.

Given the above, the probabilities of small leak, large leak and rupture within a time interval [0, t], denoted by P_sl(t), P_ll(t) and P_rp(t) respectively, are defined as follows [14]:

(11a)Psl(t)=\mathrm{P}\mathrm{r}\mathrm{o}\mathrm{b}[t_s\prec t_b\cap t_s\le t],

(11b)

(11c)

where Prob[·] denotes the probability of an event; t_s and t_b denote the times at which the defect depth just reaches 0.8wt and p(t) outcrosses r_b(t) for the first time, respectively, and “∩” represents the intersection (i.e., joint event). Note that the probability of small leak occurring first then followed by a burst is ignored in the analysis. The probabilities given by Eq. (11) were evaluated through the Monte Carlo (MC) simulation technique following the procedures described in [13]. The PCORRC model [15] as given by Eq. (12) was selected to evaluate r_b(t)

(12)r_b(t)=\zeta {\displaystyle \frac{\mathrm{2}{\sigma }_u}{D}}[\mathrm{1}-{\displaystyle \frac{d(t)}{wt}}(\mathrm{1}-\exp (-{\displaystyle \frac{\mathrm{0.571}L(t)}{\sqrt{D(wt-d(t))}}})\left)\right]

where σ_u is the pipe tensile strength; D is the pipe diameter; L(t) is the defect length (i.e., in the longitudinal direction of the pipeline) at time t and ζ is the model error.

The flow stress-dependent failure criterion for a through-wall flaw developed by Kiefner et al. [16] was employed in this study to calculate r_rp(t) as follows:

(13)r_{rp}(t)={\displaystyle \frac{\mathrm{2}wt{\sigma }_f}{M(t)D}},

where σ_f is the flow stress and equals 0.9σ_u [12] and M(t) is the Folias factor given by (the notation M(t) is used to emphasize that M is in general a function of time because the defect length can grow over time)

(14)M(t)=\{\begin{array}{ll}\sqrt{\mathrm{1}+\mathrm{0.6275}{\displaystyle \frac{L{(t)}^{\mathrm{2}}}{Dwt}}-\mathrm{0.003375}{\displaystyle \frac{L{(t)}^{\mathrm{4}}}{{(Dwt)}^{\mathrm{2}}}}},\hfill & {\displaystyle \frac{L{(t)}^{\mathrm{2}}}{Dwt}}\le \mathrm{50};\hfill \\ \mathrm{3.293}+\mathrm{0.032}{\displaystyle \frac{L{(t)}^{\mathrm{2}}}{Dwt}},.\hfill & {\displaystyle \frac{L{(t)}^{\mathrm{2}}}{Dwt}}>\mathrm{50}\hfill \end{array},

To carry out the reliability analysis, we selected nine defects located on one particular pipe joint (approximately 17.5 m long) from the 62 defects considered in Section 5.1. Although the defects have been repaired and ceased growing since 2010, they were hypothetically assumed to be still active and follow the growth path predicted from the growth model. This assumption allows evaluation of the time-dependent failure probability of the pipeline under the threat of the nine active defects for the sole purpose of illustrating the application of the developed growth model in the reliability analysis. The pipe joint selected was made from API 5L X52 steel with a specified minimum yield strength (SMYS) of 359 MPa and a specified minimum tensile strength (SMTS) of 456 MPa, and has an outside diameter of 508 mm, an operating pressure of 5.66 MPa and a nominal wall thickness of 5.56 mm. Due to lack of information of the defect lengths associated with the nine defects in the ILI report of 2007, we assumed that the defect lengths are static and follow an identical and independent lognormal distribution with a mean of 30 mm and a COV of 50% based on the information provided in [12]. The internal pressures of different years were assumed to be identical and independent Gumbel-distributed random variables [17]; at a given year, the internal pressures associated with all the defects were assumed to be fully correlated. The statistics associated with the parameters involved in the reliability analysis are summarized in Table 2. The spatial variability of diameter, wall thickness, tensile strength, defect depth, defect length and model error associated with different defects was ignored and all these parameters were assumed to be mutually independent.

A total of 100,000 simulation trials were used to evaluate the probabilities of small leak, large leak and rupture over a 10-year forecasting period since the most recent inspection (i.e., from years 2008 to 2017). For each simulation trial, the growth paths of the nine defects over the period from years 2008 to 2017 (i.e., corresponding to t = 36 to 45 in Eq. (2)) were generated from Eq. (2) with the values of α_i, t_i0, δ and ξ equal to the corresponding medians of their posterior distributions evaluated from the MCMC simulation. The probabilities of small leak, large leak and rupture over a ten-year forecasting period are shown in Fig. 6. Figure 6 indicates that the probability of small leak at a given time is the highest over the entire forecasting period, for this example, followed by the probabilities of large leak and rupture in a descending order. The probabilities of small leak, large leak and rupture gradually increase over time with a similar trend. For example, the failure probability at the end of the forecasting period is about twice as high as that at the beginning of the forecasting period for each of the three failure modes considered.

Conclusions

We propose an inverse Gaussian process- (IGP-) based model to characterize the growth of depths of metal-loss corrosion defects on oil and gas steel pipelines. The model parameters (i.e., α, t₀, δ and ξ) were assumed to be uncertain and evaluated using the hierarchical Bayesian methodology based on the imperfect inspection data obtained from multiple ILI runs. The parameters α and t₀ were assumed to be defect-specific, whereas δ and ξ were assumed to be common for all defects. The biases, random scattering errors as well as correlations between the random scattering errors associated with the ILI tools used in different inspections were accounted for in the Bayesian inference. The Markov Chain Monte Carlo simulation techniques were employed to carry out the Bayesian updating and numerically evaluate the probabilistic characteristics of the parameters in the growth model.

An example involving real ILI data collected from a natural gas pipeline that is currently in service in Alberta, Canada was used to illustrate the proposed growth model and its application in evaluating the time-dependent failure probability of the pipeline due to corrosion. The growth models were developed for 62 external corrosion defects that have been subjected to multiple ILI runs, and were subsequently excavated, field measured and repaired. The ILI data were used to carry out the Bayesian updating and evaluate the parameters in the growth models corresponding to each of the 62 corrosion defects considered. The median values of the posterior distributions of the model parameters were then used to predict the depths of the defects at the time of the field measurements. A comparison of the predicted and field-measured depths indicates that the proposed model can predict the corrosion growth reasonably well: the absolute differences between the predicted depths and the actual depths are less than and equal to 10%wt for 90% of the 62 defects.

To illustrate the application of the growth model in the reliability analysis, nine defects located on the same pipe joint were selected from the 62 defects. The failure probability of the pipe joint due to corrosion and internal pressure was then evaluated for three distinctive failure modes, i.e., small leak, large leak and rupture over a forecasting period of 10 years since the most recent inspection. The simple Monte Carlo technique was employed in the reliability analysis by taking into consideration uncertainties in the corrosion growth, material and pipe geometric properties as well as internal pressure. The growth model and procedure for the reliability analysis described in this study can be relatively easily implemented in a pipeline corrosion management framework to facilitate the structural integrity management of pipelines. It is noted that the growth model developed in this study ignores the spatial correlation between the growths of different defects, and is a phenomenological model that does not attempt to address the mechanism of metal-loss corrosion.

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