International Conference on Applications of Statistics and Probability in Civil Engineering (ICASP) (12th : 2015)

Bayesian approach to estimate corrosion growth from a limited set of matched features Dann, Markus R.; Huyse, Luc Jul 31, 2015

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12th International Conference on Applications of Statistics and Probability in Civil Engineering, ICASP12 Vancouver, Canada, July 12-15, 2015 Bayesian Approach To Estimate Corrosion Growth From A Limited Set Of Matched Features Markus R. Dann Assistant Professor, Department of Civil Engineering, University of Calgary, Calgary, Canada Luc Huyse Facilities Engineering Functional Authority Program Manager, Chevron ETC, Houston, USA   ABSTRACT: Corrosion is a time-dependent hazard for pipelines that gradually decreases the resistance of pressure containment. The corrosion growth can be inferred from a set of matched corrosion features observed in two successive in-line inspections. Experience shows that the measured corrosion growth between two inspections has often a mean value around zero and, usually, nearly half of the matched features have negative measured growth. Negative growth values are physically impossible as the underlying true corrosion process has strictly non-negative corrosion growth increments. In this paper, a Bayesian probability model is presented to estimate the actual corrosion growth conditional on the observed growth from a set of matched corrosion anomalies. The model assumes independence between sizing error and true feature depth and produces a strictly non-negative corrosion growth process that explicitly accounts for non-growing features. An Empirical Bayes approach is used to determine the prior distribution of the corrosion growth. The key findings in this paper are (1) the variance of the actual corrosion growth process is less than the observed variance of the direct measurements, (2) the upper percentiles of the posterior corrosion growth distribution may be lower than the direct measurements, and (3) the posterior distribution of the corrosion growth is non-negative. A numerical example is provided. 1. INTRODUCTION Corrosion is a time-dependent hazard for carbon steel pipelines. The metal loss gradually decreases the pipeline resistance with respect to pressure containment. Corrosion can ultimately lead to pipeline failures, such as leakage and rupture, with possibly severe consequences for safety, the economy, and the environment. The objective of a pipeline integrity management (PIM) program is to ensure safe and reliable pipeline operations and to prevent pipeline failures and unnecessary shutdowns. PIM typically relies on a 3-step approach: inspection, analytical assessment, and, if necessary, repair to avoid pipeline failures due to corrosion. In-line inspections (ILIs) can be performed periodically to identify and size corrosion anomalies in pipelines. Corrosion growth analysis and a fitness-for-service (FFS) assessment of the reported corrosion features determine the safe remaining lifetime of the system and the re-inspection interval (Bubenik et al. 2014). Required maintenance and repair actions are executed as the final step of the PIM process. The corrosion growth analysis is often based on matched features, which are reported anomalies that are identified in the same location during at least two successive ILIs. The measured maximum pit depths of such matched features are used to forecast future corrosion growth. While the measured corrosion growth can – and often does – take negative values (Dawson and Kariyawasam 2009; Dann and Huyse 2014), the corrosion growth process is strictly non-negative  1 12th International Conference on Applications of Statistics and Probability in Civil Engineering, ICASP12 Vancouver, Canada, July 12-15, 2015 with zero and positive growth increments at discrete points in time. Negative measured corrosion growth is caused by two reasons: a) Feature mismatch: two truly different features are falsely matched as the same corrosion anomaly.  b) Sizing error: two features are matched correctly, but the reported depths are subject to uncertainty. To minimize the influence of falsely matched features on the FFS assessment and the subsequent decision making process, only features that are matched with high confidence are considered for the corrosion growth analysis in the proposed model. The subset of correctly matched features will still show negative measured corrosion growth due to the effect of sizing errors.  The objective of this paper is to support the integrity assessment of pipelines by determining the actual corrosion growth conditional on the observed growth from a (limited) set of matched anomalies. Special emphasis is given to feature pairs with a negative measured corrosion growth. Section 2 outlines the basic sizing error model that establishes the relationship between actual and measured depth for an individual corrosion feature. Section 3 demonstrates why matched features often have a measured growth that is negative. Section 4 introduces the Bayesian probability model (Gelman et al. 2004) to estimate the true corrosion growth from a set of matched features. A numerical example of a subsea pipeline in service is provided in Section 5 and the conclusions are presented in Section 6.  2. SIZING ERROR MODEL An additive sizing error model (Maes and Salama 2008) is commonly used to define the relationship between the measured and actual depth of a corrosion anomaly i at an inspection at time t:  )()()( ttXtY iii ε+=       for i = 1, …, n (1) where Yi is the measured depth of the corrosion anomaly, Xi is the true depth of the corrosion anomaly and εi is the sizing error, which is independent of the true feature depth. The exact values of Xi and εi remain unknown and the ILI only provides information about their sum Yi. The lower and upper bounds for both quantities Xi and Yi are 0 and 100% nwt (= nominal wall thickness). The Bayesian network (Jensen 2001) of equation (1) is shown in Figure 1. The elements of vector αε are the parameters for specifying the distribution of the sizing error εi.    Figure 1: Bayesian network for the measured depth of an individual corrosion feature.    Vendors of ILI tools are not required to provide the complete probability distribution of the sizing error (POF 2009). Instead they usually provide a confidence interval without specifying the full sizing error distribution. For example, standard high resolution magnetic flux leakage (MFL) tools, which are often used for corrosion detection (i.e. wall loss) inspections, have a depth sizing accuracy of ±10-15% nwt at an 80% confidence level, depending on the geometry of the feature (POF 2009). A common assumption about the sizing error εi is to treat it as an independent random variable following a normal distribution with a zero mean and a standard deviation σε, which fits the normal distribution to the given confidence interval:  εi(t)|σε(t) ~ N(mean = 0, var = σε2)     (2) Figure 2 shows the probability density function (= pdf) of the measured depth Yi in (1) given the true depth Xi and assuming the normally distributed sizing error in (2). The standard deviation σε is 7.8% nwt, which corresponds to a ±10% nwt confidence interval for εi at an 80% confidence Xi εi Yi αε Actual depth Sizing error Measured depth Parameter sizing error  2 12th International Conference on Applications of Statistics and Probability in Civil Engineering, ICASP12 Vancouver, Canada, July 12-15, 2015 level. The normally distributed sizing error is unbounded, which leads to an unbounded measurable feature depth Yi. For example, if the true depth Xi is small, a significant portion of the pdf of the measureable depth is located in the impossible negative domain (Figure 2).   Figure 2: Conditional pdf of measured depth assuming normally distributed sizing error with a standard deviation of 7.8% nwt.   Figure 3: Comparison of pdfs that have 80% confidence between 40% nwt and 60% nwt depth.   To overcome the limitation of the unbounded normal distribution in (2) and Figure 2, Figure 3 and Figure 4 compare different distributions (e.g. normal, log-normal, beta, and gamma). All displayed pdfs have a probability content of 80% between the lower and upper confidence bounds. While all pdfs in Figure 3 have a similar shape, their shapes differ considerably if the true feature depth is small (Figure 4). Verifying what type of distribution best describes the sizing error is outside the scope of this paper. Although a normally distributed sizing error is applied in this paper, the model is not restricted to this choice.   Figure 4: Comparison of pdfs that have 80% confidence between 5% nwt and 25% nwt depth.       3. WHY IS THE MEASURED CORROSION GROWTH OFTEN NEGATIVE? Considering two ILIs at times t1 and t2 > t1, the relationship between the measured and actual corrosion growth of feature i can be inferred from equation (1):  iii XY ε∆+∆=∆        for i = 1, …, n (3) where ΔYi = Yi(t2) – Yi(t1) is the measured corrosion growth, ΔXi = Xi(t2) – Xi(t1) is the true corrosion growth and Δεi = εi(t2) – εi(t1) is the difference in sizing errors. Since features either increase in size (X(t2) > X(t1)) over time or do not grow (X(t2) = X(t1)), the actual corrosion growth increment in equation (3) can only take zero and positive values (ΔXi ≥ 0). If the sizing error difference is zero (Δεi = 0), the actual corrosion growth is equal to the measured growth (ΔXi = ΔYi).  To further investigate equation (3), it is assumed the sizing error follows a normal distribution with a mean value of zero and known variance σε2(t). Hence, the difference in sizing errors Δεi = εi(t2) – εi(t1) in equation (3) is also normally distributed (Benjamin and Cornell 1970): )2,0(normal~ 212221 σσσσε ci −+∆        (4) 0.000.010.020.030.040.050.06-20 -10 0 10 20 30 40 50 60 70 80 90 100Conditional PDF f(y i|xi)Measured feature depth Yi [% nwt]True size = 10%nwt True size = 30%nwtTrue size = 50%nwt True size = 70%nwt0.000.010.020.030.040.050.060 10 20 30 40 50 60 70 80PDF [-]Feature depth [% nwt]Normal PDFLog-normal PDFBeta PDFGamma PDFCI bounds0.000.010.020.030.040.050.060.070 10 20 30 40 50 60 70 80PDF [-]Feature depth [% nwt]Normal PDFLog-normal PDFBeta PDFGamma PDFCI bounds 3 12th International Conference on Applications of Statistics and Probability in Civil Engineering, ICASP12 Vancouver, Canada, July 12-15, 2015 where σ1 and σ2 are the standard deviations of the sizing errors for ILI 1 and ILI 2, respectively, and c is the correlation coefficient of the sizing errors.  The probability Pr(ΔYi < 0|ΔXi = Δxi) that the measured corrosion growth ΔYi is negative given the true corrosion growth ΔXi = Δxi is presented in Figure 5 for different correlation coefficients c.    Figure 5: Probability that the measured corrosion growth of a single feature is less than zero as a function of the actual corrosion growth. Normally distributed sizing errors are used with varying degrees of correlation.  If the actual corrosion growth approaches zero, the likelihood of measuring a negative corrosion growth approaches 50%. The dependence of the probability Pr(ΔYi < 0 | ΔXi = Δxi) on the correlation coefficient between the sizing errors is pronounced when the correlation between subsequent sizing errors is high. 4. CORROSION GROWTH MODEL ASSUMING INDEPENDENCE BETWEEN TRUE FEATURE SIZE AND SIZING ERROR The corrosion growth model is based on equation (3) and the corresponding Bayesian network is given in Figure 6. The observable quantity in the model is the measured corrosion growth ΔYi of feature i (i = 1, …, n). It is the sum of the actual corrosion growth ΔXi ≥ 0 and the sizing error difference Δεi. The sizing errors εi(t2) and εi(t1) belong to the two inspections ILI 1 and ILI 2. They follow a normal distribution with a given parameterization (2). The correlation coefficient c as applied in (4) can be treated as fixed (as shown in Figure 6) or as a known random variable to include the uncertainties on the correlation of the sizing errors into the corrosion growth analysis.   Figure 6: Bayesian network for inferring the actual corrosion growth of a feature observed during two inspections and assuming independence between true feature depths and sizing error.  The random variable of interest in Figure 6 is the true corrosion growth ΔXi. It can be determined conditional on the measured Δyi using the Bayesian inference for i = 1, …, n:  fi(Δxi|Δyi) = kiLi(Δxi|Δyi)fi(Δxi)   (5) where fi(Δxi|Δyi) is the posterior pdf of actual corrosion growth, Li(Δxi|Δyi) is the likelihood function of the actual corrosion growth as specified in the model in Figure 6, and fi(Δxi) is the prior pdf of Δxi. The multiplier ki is used in (5) to achieve a posterior pdf that integrates to 1. Assuming a uniform prior pdf in (5), the posterior pdf fi(Δxi|Δyi) is directly proportional to the likelihood function. If the fixed correlation coefficient in Figure 6 is used, the posterior pdf of the local corrosion growth Δxi becomes a truncated normal pdf with the lower and upper bounds of 0 and 100% nwt, respectively:   Δxi|Δyi, αε ~ trunc-N(mean = μi, var =  σi2) (6) where μi and σi are the mean value and the standard deviation of the local corrosion growth, respectively. Both quantities are functions of the measured corrosion growth Δyi and the vector αε 0%5%10%15%20%25%30%35%40%45%50%0 5 10 15 20 25 30Pr[measured corrosion growth  ΔY i< 0 | ΔXi= Δx i]Actual corrosion growth Δxi [% nwt]c = 0c = 40%c = 80%ΔXi Δεi ΔYi αε(t1) Actual corrosion growth Sizing       error difference Measured corrosion growth Parameter sizing error  αε(t2) Correlation coefficient ILI 1 ILI 2 c   4 12th International Conference on Applications of Statistics and Probability in Civil Engineering, ICASP12 Vancouver, Canada, July 12-15, 2015 of parameters defining the two distributions of the sizing errors including the correlation coefficient. For details on determining the truncated normal pdf fi(Δxi|Δyi), μi and σi in (6) see Cohen (1991).  After determining the local corrosion growth of all matched features, an estimate of the overall corrosion growth is required for the set of matched features as well as the features that have not been matched. Assuming the matched features are a good representation of all the features in the pipeline, the overall corrosion growth can be determined as a mixture of the local corrosion growths (Hogg et al. 2005):  )()(),...,( 1 jjjn yfyxfyyxf ∆∆∆=∆∆∆ ∑  (7) where f(Δx|Δy1,…,Δyn) is the posterior pdf of the actual corrosion growth of the set of matched features,  fj(Δx|Δyj) is the posterior pdf of the local corrosion growth of a feature with measured corrosion growth Δyj, and f(Δyj) is the empirical pdf of the measured corrosion growth. The summation in (7) goes over all non-zero elements of f(Δyj). Equation (7) ignores the statistical uncertainties in the prediction of the overall (population-based) corrosion growth, but it can be included by adjusting the empirical pdf f(Δyj). 5. NUMERICAL EXAMPLE A 16” subsea pipeline with a length of 5 km is considered (the same example is in Dann and Huyse 2014). Two inspections were performed; the second ILI (ILI 2) was performed four years after the first one (ILI 1). Each inspection detected around 2,000 corrosion anomalies with the most severe corrosion in the first 1.7 km of the pipeline.  After successfully matching all girth weld locations, almost all reported clusters from ILI 2 are matched with high confidence to the clusters from ILI 1. These n = 213 clusters and the deepest features are analyzed with regard to their growth over the course of four years. In this example, the very few unmatched elements that have been removed from the analysis are only shallow features which neither make a noticeable impact on the subsequent results nor raise immediate concern from an integrity perspective. Figure 7 shows the measured depth yi of the matched features at ILI 1 and ILI 2. The reporting threshold of 10% nwt is clearly visible for ILI 2. The maximum reported depth is 43% nwt and 50% nwt at ILI 1 and ILI 2, respectively.   Figure 7: Measured depth of the matched features in the first 1.7 km.   Figure 8: Measured depth change between the two ILIs of the matched features.  The change in maximum corrosion pit depth between the two inspections is given in Figure 8 as a function of the log-distance (i.e. location along the pipeline). The maximum and minimum observed depth changes are 22% nwt and                            -18% nwt, respectively, which both occur near a log-distance of 500 m. In this example, the distribution of the observed change in depth is not very dependent on the log- distance and remains fairly constant over the entire first mile (1.6 km). Figure 8 shows that the positive and negative 051015202530354045500 200 400 600 800 1000 1200 1400 1600 1800Reported depth y i[% nwt]Log-distance [m]ILI 1ILI 2-20-15-10-505101520250 200 400 600 800 1000 1200 1400 1600 1800Measured depth change Δy i[% nwt]Log-distance [m] 5 12th International Conference on Applications of Statistics and Probability in Civil Engineering, ICASP12 Vancouver, Canada, July 12-15, 2015 changes in depth are also fairly evenly distributed around zero. Figure 9 shows the normal probability plot of the measured depth change. The median is located at -1% nwt and the standard deviation is 7.1% nwt. The shape of the graph suggests a distribution close to a Gaussian distribution, which indicates the sizing errors associated with the two ILIs have a large influence on the observed corrosion growth. This large influence of the sizing errors illustrates the concept that the observed growth can differ (significantly) from the true depth growth (see Section 2).   Figure 9: Normal probability plot of the measured depth change of the matched features.  The 213 matched features are analyzed using the Bayesian model developed in Section 4. The prior distribution of the actual corrosion growth Δxi in (5) is assumed to be mixture of 1. Δxi = 0 with probability p to represent the possibility that the feature is not growing between the two inspections. 2. Δxi|λ ~ exponential(λ) with probability 1 – p where λ is the mean corrosion growth.  The prior distribution for the unknown probability p is assumed to be 50% based on Figure 9 where around half of the matched features have negative observed growth. The parameter λ describing the Δxi > 0 portion in the prior distribution is estimated according to the Empirical Bayes (EB) theory (Carlin and Louis 2009) using the reported growth data Δyi > 0. The EB point estimate of λ is 5.4% nwt. Under these assumptions, the posterior distributions for the real time-averaged growth, given the observed growth between two successive inspections, are given in Figure 10 to Figure 14. Figure 10 to Figure 12 show the posterior cumulative distribution function (cdf) of the actual corrosion growth for individual observations of pairs of features with measured growth values of -18% nwt (minimum measured growth), zero, and +22% nwt (maximum measured growth), respectively. The prior distribution shifts the majority of the posterior corrosion growth to the left of the observed growth in Figure 12. This tendency of oversizing in the upper percentiles has been theoretically justified (Huyse and van Roodselaar 2010; Dann and Huyse 2014) and confirmed in practice as well (Huyse et al. 2011).   Figure 10: Posterior and prior cdf of the actual corrosion growth of a single feature with a measured corrosion growth of -18% nwt.   Figure 11: Posterior and prior cdf of the actual corrosion growth of a single feature with a measured corrosion growth of zero. -3-2-10123-20 -15 -10 -5 0 5 10 15 20 25Normal probability scale [-]Measured depth change ΔYi [% nwt]ILI dataTrendline0%10%20%30%40%50%60%70%80%90%100%0 5 10 15 20 25 30CDFActual corrosion growth [% nwt]Prior CDFPosterior CDF (c = 80%)Posterior CDF (c = 40%)Posterior CDF (c= 0)0%10%20%30%40%50%60%70%80%90%100%0 5 10 15 20 25 30CDFActual corrosion growth [% nwt]Observed growthPrior CDFPosterior CDF (c = 80%)Posterior CDF (c = 40%)Posterior CDF (c= 0) 6 12th International Conference on Applications of Statistics and Probability in Civil Engineering, ICASP12 Vancouver, Canada, July 12-15, 2015  Figure 12: Posterior and prior cdf of the actual corrosion growth of a single feature with a measured corrosion growth of +22% nwt.  Figure 13 and Figure 14 show the posterior cdf and exceedance probability plot of the overall corrosion growth, respectively. While the influence of the correlation between sizing errors on the actual corrosion growth is evident for some of the matched pairs (e.g. Figure 12), its influence on the overall corrosion growth distribution is very low (Figure 13).  Although nearly half of the matched pairs have negative measured growth, the Bayesian model correctly reflects that the true corrosion growth is strictly non-negative. Figure 14 shows in the log-exceedance probability plot that the posterior corrosion growth distribution exhibits a lower likelihood for large, actual corrosion growth than the observed data would suggest, irrespective of the correlation between sizing errors. This observation reflects the lower variance of the true corrosion growth relative to the ILI-measured growth. Figure 15 shows the posterior probability p of zero corrosion growth for an individual feature as a function of the observed growth. The likelihood of zero growth exceeds 50% until the measured growth reaches +4% nwt. In some respects, Figure 15 represents the complement of Figure 5 and shows the likelihood of non-zero corrosion growth as function of the ILI data matching.  Considering the entire set of matched features in Figure 13, the posterior probability p slightly increases compared to the prior value.  Figure 13: Posterior and prior cdf of the actual corrosion growth of the entire set of matched features for varying correlation coefficients.   Figure 14: Log-exceedance probability plot of the corrosion growth of the entire set of matched features.   Figure 15: Prior and posterior probability of zero actual corrosion growth as a function of the observed growth. 6. CONCLUSIONS Experience shows the measured corrosion growth of matched features from two or more ILIs can be 0%10%20%30%40%50%60%70%80%90%100%0 5 10 15 20 25 30CDFActual corrosion growth [% nwt]Observed growthPrior CDFPosterior CDF (c = 80%)Posterior CDF (c = 40%)Posterior CDF (c= 0)0%10%20%30%40%50%60%70%80%90%100%-20 -15 -10 -5 0 5 10 15 20 25 30CDFActual corrosion growth [% nwt]Observed growthPrior CDFPosterior CDF (c = 80%)Posterior CDF (c = 40%)Posterior CDF (c= 0)1.0E-031.0E-021.0E-011.0E+000 5 10 15 20 25Exceedance ProbabilityActual corrosion growth [% nwt]Observed growthPrior CDFPosterior CDF (c = 80%)Posterior CDF (c = 40%)Posterior CDF (c= 0)0%10%20%30%40%50%60%70%80%90%-30 -25 -20 -15 -10 -5 0 5 10 15 20 25 30Pr(ΔX = 0 | Δy i)Measured corrosion growth Δyi [% nwt]PriorPosterior (c = 80%)Posterior (c = 60%)Posterior (c = 40%)Posterior (c = 20%)Posterior (c = 0) 7 12th International Conference on Applications of Statistics and Probability in Civil Engineering, ICASP12 Vancouver, Canada, July 12-15, 2015 negative. The analysis, limited to only perfectly matched features, demonstrates that standard sizing accuracies of MFL inspection tools produce negative measured growth although the actual corrosion growth process is strictly non-negative. Furthermore, the common assumption of normally distributed sizing errors seems appropriate for medium-sized features, but it causes inadmissible values for very small and large features. A simple Bayesian probability model, which relies on a non-parametric corrosion growth process is developed for analyzing a limited set of matched features from two successive inspections. The assumed corrosion growth process explicitly includes the possibility that features do not grow between inspections. The local corrosion growths of the matched features are mixed together to obtain an estimate of the system-wide corrosion growth. The Bayesian corrosion growth model exhibits the following characteristics: 1. The distribution of the corrosion growth is non-negative. 2. The variance of the actual corrosion growth process is less than the variance of the direct measurements. 3. The upper percentiles of the posterior corrosion growth distribution are lower than the direct measurements. The latter characteristic has important ramifications when planning the next inspection interval or scheduling necessary repairs. ACKNOWLEGEMENT This work was partially funded through the Pipeline Integrity Technology Development Program at Chevron Energy Technology Company. The support from the Facilities, Operations & Reliability Focus Area manager and the members of the Technical Management Team is gratefully acknowledged. The first author is thankful for the grant on pipeline integrity and risk assessment received from the University of Calgary. REFERENCES Benjamin, J. R., and Cornell, C. A. (1970). Probability, Statistics, and Decision for Civil Engineers, MacGraw-Hill Book Company, New York, US. Bubenik, T., Harper, W. V., Moreno, P., and Polasik, S. (2014). "Determining reassessment intervals from successive in-line inspections." 10th International Pipeline Conference, ASME, Calgary, Canada. Carlin, B. P., and Louis, T. A. (2009). Bayesian Methods for Data Analysis, CRC Press, Boca Raton, US. Cohen, A. C. (1991). Truncated and censored samples: Theory and applications, Marcel Dekker, Inc., New York, US. Dann, M. R., and Huyse, L. (2014). "Pragmatic approach to estimate the corrosion rates for pipelines subject to complex corrosion." 10th International Pipeline Conference, ASME, Calgary, Canada. Dawson, J. S., and Kariyawasam, S. (2009). "Understanding and accounting for pipeline corrosion growth rates." 17th Joint Technical Meeting on Pipeline Research, EPRG - PRCI - APIA, Milan, Italy. Gelman, A., Carlin, B. P., Stern, H. S., and Rubin, D. B. (2004). Bayesian Data Analysis, CRC Press, Boca Raton, US. Hogg, R. V., McKean, J. W., and Craig, A. T. (2005). Introduction to Mathematical Statistics, Pearson Prentice Hall, Upper Saddle River, US. Huyse, L., Monroe, J., and van Roodselaar, A. (2011). "Effects of In-Line Inspection Sizing Uncertainties on In-The-Ditch Validation: Review of Results." Rio Pipeline Conference & Exposition, ASME, Rio de Janeiro, Brazil. Huyse, L., and van Roodselaar, A. (2010). "Effects of inline inspection sizing uncertainties on the accuracy of the largest features and corrosion rate statistics." 8th International Pipeline Conference, ASME, Calgary, Canada. Jensen, F. V. (2001). Bayesian Networks and Decision Graphs, Springer, New York, US. Maes, M. A., and Salama, M. M. (2008). "Managing ILI Inspection Uncertainties." PipeLine and Gas Technology, 7(5), 34-41. POF (2009). "Specifications and requirements for intelligent pig inspection of pipelines." Pipeline Operators Forum.  8 

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