Corrosion Growth Modeling Based on Mass In-Line Inspection Data Using Variational Inference

This paper establishes a dynamic Bayesian network to model the growth of corrosion defects on energy pipelines. The integrated model characterizes the growth of defect depth by a homogeneous gamma process and considers the biases and random errors associated with the in-line inspection (ILI) tools. The distributions of the mean value and coefficient of variation of the annual growth of defect depth are learned from multiple ILI data using the parameter learning technique of Bayesian networks. With the same technique, the distributions of the biases and standard deviation of random errors associated with ILI tools are learned from ILI data and their corresponding field measurements. An example with real corrosion management data is used to illustrate the process of developing the model structure, learning model parameters and predicting the corrosion growth and time-dependent failure probability. The results indicate that the model can in general predict the growth of corrosion defects with reasonable accuracy and the ILI-reported and field-measured depth can be used to update the time-dependent failure probability in a near-real-time manner. In comparison with existing growth models, the graphical feature of Bayesian networks makes it more intuitive and transparent to users. The employment of parameter learning provides a semi-automated and convenient approach to elicit the probabilistic information from ILI and field measurement data. The above advantages will facilitate the application of the model in the practice of corrosion management in pipeline industry.

Inverse Gaussian process-based corrosion growth model for energy pipelines considering the sizing error in inspection data

To manage the integrity of corroded pipelines reliable estimates of the current and future corrosion growth process are required. They are often obtained from in-line inspection data by matching defects from two or more inspections and determining corrosion growth rates from the observed growth paths. In practice only a (small) subset of the observed defects are often reliably matched and used in the subsequent corrosion growth analysis. The information from the remaining unmatched defects on the corrosion growth process are typically ignored. Hence, all decisions that depend on the corrosion growth process such as maintenance and repair requirements and re-inspection intervals, are based on the information obtained from the (small) set of matched defects rather than all observed corrosion anomalies. A new probabilistic approach for estimating corrosion growth from in-line inspection data is introduced. It does not depend on defect matching and the associated defect matching uncertainties. The reported defects of an inspection are considered from a population perspective and the corrosion growth is determined from two or more defect populations. The distribution of the reported defect sizes is transformed into the distribution of the actual defect sizes by adjusting it for detectability, false calls, and sizing uncertainties. The obtained distribution is then used to determine the parameters of the assumed gamma-distributed corrosion growth process in order to forecast future metal loss in the pipeline. As defect matching is not required all reported corrosion defects are used in the probabilistic analysis rather than the truncated set of matched defects. A numerical example is provided where two in-line inspections are analyzed.

SUMMARY & CONCLUSIONSModeling corrosion growth for complex systems such as the oil refinery system is a major challenge since the corrosion process of oil and gas pipelines are inherently stochastic and depends on many factors including exposures to environmental conditions, operating conditions, and electrochemical reactions. Moreover, the number of sensors is usually limited, and sensor data are incomplete and scattering, which hinders the capability of capturing the corrosion growth behaviors. Therefore, this paper proposes Multi-sensor Corrosion Growth Model with Latent Variables to predict the corrosion growth process in oil refinery piping. The proposed model is a combination of the hierarchical clustering algorithm and the vector autoregression (VAR) model. The clustering algorithm aims to find the hidden (i.e., latent) data clusters of the measured time series data, from which the time series from the same cluster will be included in the VAR model to predict the corrosion depth from multiple sensors. The model can capture the relationship between sensor time series data and identify latent variables. A real case study of an oil refinery system, in which in-line inspection (ILI) data were collected, was utilized to validate model. Regarding corrosion growth prediction, the paper compared the prediction accuracy of VAR model with other three forms of power law model, which is widely accepted to expect the time-dependent depth of corrosion such as power function (PF), PF with initiation time of corrosion (PFIT), and PF with initiation time of corrosion and covariates (PFCOV). The results showed that VAR model has the lowest prediction error based on the mean absolute percentage error (MAPE) evaluation for test data. Finally, the proposed model is believed to be useful for dealing with a complex system that has a variety of corrosion growth behaviors, such as the oil refinery system, as well as it can be applied in other real-time applications.

In the context of Bayesian inversion for scientific and engineering modeling, Markov chain Monte Carlo (MCMC) sampling strategies have become the benchmark due to their flexibility and robustness in dealing with arbitrary posterior probability density functions (PDFs). However, these algorithms have been shown to be inefficient when sampling from high-dimensional posterior distributions or exhibit multimodality and/or strong parameter correlations. In such contexts, transitional MCMC (TMCMC) provides a more efficient alternative. Despite the recent applicability for Bayesian updating and model selection across a variety of disciplines, TMCMC may require a prohibitive number of tempering stages when the prior pdf is significantly different from the target posterior. Furthermore, the need to start with an initial set of samples from the prior distribution may present a challenge when dealing with implicit priors, e.g., based on feasible regions. Finally, TMCMC cannot be used for inverse problems with improper prior PDFs that represent a lack of prior knowledge on all or a subset of parameters. A generalization of TMCMC is proposed that alleviates such challenges and limitations toward providing a more robust and efficient tempering sampling strategy. We present convergence analysis, proving that the distance between the intermediate distributions and the target posterior distribution monotonically decreases as the algorithm proceeds. We also demonstrate the advantages of the proposed generalization through a series of test problems and an engineering application in the oil and gas industry.

Pipeline companies are facing challenges in maintaining the integrity and reliability of their pipelines. They are working towards predictive maintenance using machine learning-based approaches to predicting anomalies. Training machine learning models requires sufficient data. Data quality is therefore becoming important because inaccurate data will lead to an inaccurate or wrong decision on pipeline condition assessment and the following management. This research paper intends to address the data quality issues of pipeline inspection data such as in-line inspection (ILI) data using machine learning models. Different machine learning models developed by random forest regression, linear regression, and nearest neighbors’ methods were tested to detect outliers in the ILI data. In this paper, the ILI data collected from an oil pipeline over a period of 22 years was applied to testing and analysis. To verify the outlier detection results of machine learning models, we used statistical analysis including Z-score method to check and find if there are any gaps in the analysis. It verifies that all these methods show almost the same or very similar results for the detection of the outliers. Hence, this study presents a robust method for the field applications in the pipeline industry.

Corrosion growth rate (CGR) is one of the key elements in corrosion threat management. It is used to determine metal loss In-Line Inspection (ILI) re-assessment interval, identify high growth features that require excavation before the next ILI and predict probability of failure of an unmitigated feature. Overestimated CGR may result in unnecessary expenditures for excavations and remediation, while underestimated CGR could result in pipeline failure. Although there are many methods to derive CGR from successive ILIs, there is no particular approach widely accepted by industry. In this paper, several CGR methodologies used by Enbridge to manage corrosion threats on pipelines including CGR based on feature matching and signal matching are discussed and validated. CGR based on feature matching refers to the rate that is calculated based on ILI reported depths of two back-to-back ILIs. CGR based on signal matching refers to the rate that is determined based on ILI raw data review provided from the ILI vendor. Both feature level and joint level CGR have been used in corrosion threat management. A probabilistic CGR methodology based on Pipeline Research Council International (PRCI)(1) EC-1-2 project (Development of detailed procedures for comparing successive ILI runs to establish corrosion growth rate) that accounts for uncertainty of measurement tools is also included in this study. The ILI data and field non-destructive evaluation (NDE) results of six onshore pipelines are used for this study. Each of the six pipelines has at least three high resolution Magnetic Flux Leakage (MFL) inspections. The various CGRs are established based on the 1st and 2nd inspections. The CGR methodologies are validated based on comparing the predicted depth using CGR with the field measured depth as well as the depth reported by the 3rd ILI. Over 700 field measured feature depths are used in this study and over 60,000 ILI features are used to validate the estimated depth at the time of field measurements and at the time of the 3rd ILI. This paper demonstrates the validation results of feature level and joint level CGR based on both feature matching and signal matching. The probabilistic CGR methods are also validated using the same method. The CGR validation study enables pipeline operators to establish defect repair schedules and re-inspection intervals with increased confidence.

The current finite element (FE) assessment methods of dented pipes are based on specific dent profiles, which are generally created based on the shape of indenters. However, the actual dent profile in real case scenarios is mostly irregular in shape, depending on the cause of damage. In this paper, FE analyses of dented pipes using inline inspection (ILI) data are presented. Based on the ILI data, the dent profile is generated by applying the nodal displacements to all the pipe nodes. The validation of this nodal displacement approach is discussed in this paper. Besides, a parametric study is carried out to study the behavior of dent for different dent depth, pipe geometry, and pipe grades. The significance of residual stresses generated during the dent formation on the behavior of dented pipe during the service life is also discussed. Finally, the remaining life estimation of dented pipes according to the API 579-1 is presented using FE analysis results.

The Bayesian approach to inverse problems, in which the posterior probability distribution on an unknown field is sampled for the purposes of computing posterior expectations of quantities of interest, is starting to become computationally feasible for partial differential equation (PDE) inverse problems. Balancing the sources of error arising from finite-dimensional approximation of the unknown field, the PDE forward solution map and the sampling of the probability space under the posterior distribution are essential for the design of efficient computational Bayesian methods for PDE inverse problems. We study Bayesian inversion for a model elliptic PDE with an unknown diffusion coefficient. We provide complexity analyses of several Markov chain Monte Carlo (MCMC) methods for the efficient numerical evaluation of expectations under the Bayesian posterior distribution, given data δ. Particular attention is given to bounds on the overall work required to achieve a prescribed error level ε. Specifically, we first bound the computational complexity of ‘plain’ MCMC, based on combining MCMC sampling with linear complexity multi-level solvers for elliptic PDE. Our (new) work versus accuracy bounds show that the complexity of this approach can be quite prohibitive. Two strategies for reducing the computational complexity are then proposed and analyzed: first, a sparse, parametric and deterministic generalized polynomial chaos (gpc) ‘surrogate’ representation of the forward response map of the PDE over the entire parameter space, and, second, a novel multi-level Markov chain Monte Carlo strategy which utilizes sampling from a multi-level discretization of the posterior and the forward PDE. For both of these strategies, we derive asymptotic bounds on work versus accuracy, and hence asymptotic bounds on the computational complexity of the algorithms. In particular, we provide sufficient conditions on the regularity of the unknown coefficients of the PDE and on the approximation methods used, in order for the accelerations of MCMC resulting from these strategies to lead to complexity reductions over ‘plain’ MCMC algorithms for the Bayesian inversion of PDEs.

The Bayesian framework is commonly used to quantify uncertainty in seismic inversion. To perform Bayesian inference, Markov chain Monte Carlo (MCMC) algorithms are regarded as the gold standard technique for sampling from the posterior probability distribution. Consistent MCMC methods have trouble for complex, high-dimensional models, and most methods scale poorly to large datasets, such as those arising in seismic inversion. As an alternative, approximate MCMC methods based on unadjusted Langevin dynamics offer scalability and more rapid sampling at the cost of biased inference. However, when assessing the quality of approximate MCMC samples for characterizing the posterior distribution, most diagnostics fail to account for these biases. In this work, we introduce the kernel Stein discrepancy (KSD) as a diagnostic tool to determine the convergence of MCMC samples for Bayesian seismic inversion. We demonstrate the use of the KSD for measuring sample quality and selecting the optimal Langevin MCMC algorithm for two Gaussian Bayesian inference problems.

Bayesian model updating provides a rigorous framework to account for uncertainty induced by lack of knowledge about engineering systems in their respective mathematical models through updates of the joint probability density function (PDF), the so-called posterior PDF, of the unknown model parameters. The Markov chain Monte Carlo (MCMC) methods are currently the most popular approaches for generating samples from the posterior PDF. However, these methods often found wanting when sampling from difficult distributions (e.g., high-dimensional PDFs, PDFs with flat manifolds, multimodal PDFs, and very peaked PDFs). This paper introduces a new multi-level sampling approach for Bayesian model updating, called Sequential Gauss-Newton algorithm, which is inspired by the Transitional Markov chain Monte Carlo (TMCMC) algorithm. The Sequential Gauss-Newton algorithm improves two aspects of TMCMC to make an efficient and effective MCMC algorithm for drawing samples from difficult posterior PDFs. First, the statistical efficiency of the algorithm is enhanced by use of the systematic resampling scheme. Second, a new MCMC algorithm, called Gauss-Newton MCMC algorithm, is proposed which is essentially an M-H algorithm with a Gaussian proposal PDF tailored to the posterior PDF using the gradient and Hessian information of the negative log posterior. The effectiveness of the proposed algorithm for solving the Bayesian model updating problem is illustrated using three examples with irregularly shaped posterior PDFs.

Operators wish to understand the condition of their pipelines to manage ongoing integrity. Information on the condition of the pipeline along its entire length can be obtained using in-line inspection (ILI). However, some pipelines cannot be internally inspected due, for example, to tee connections, tight bends, low flow or to a lack of launcher and receiver facilities. The condition of these ‘unpiggable’ lines can sometimes be largely unknown. To aid the understating of the pipeline condition without ILI data, operators will often rely on alternative sources of information, such as localised external inspections, model predictions and company and individual experience. However, there may be significant uncertainty associated with these alternative data sources when using them to assess the condition of the entire pipeline. This uncertainty may be understood by applying a probabilistic approach to the assessment of pipeline integrity using structural reliability analysis (SRA) methods. An SRA approach applies probabilistic input parameters to a failure prediction model for a defined limit state function. Previous IPC papers[1,2,3] have presented guidance on probabilistic assessments to model pipeline failure. Recommended probability distributions are presented which account for uncertainties associated with line pipe properties, defect sizing and the error associated with the predicted failure model. However, there is little published guidance readily available on recommended defect characteristic distributions specific to internal corrosion features. Parameter distributions are recommended for defect sizing based on empirical data, which are mainly used for external corrosion features. In this paper, a case study is used to present a practical application of an SRA methodology for assessment of pipeline integrity with respect to internal corrosion. Discussion is presented on alternative sources of information for the assessment when ILI data is unavailable, including targeted external inspections of unpiggable lines and data sets from comparable piggable lines. Probability distributions are derived from the available inspection data for the internal corrosion feature size and corrosion rate input parameters to the SRA. Probabilistic analysis is used to account for the expected population of unknown features in the uninspected parts of the pipelines. The expected feature size, corrosion rate and feature density calculated are used in the SRA to estimate the total probability of failure due to internal corrosion over time for the entire length of the pipeline. Recommendations are provided on the application of an SRA methodology to assess pipeline failure due to internal corrosion.

The paper describes a new practical method of updating the stochastic remnant life of pipelines with defects using ILI data. The paper describes a comprehensive algorithm for assessing pipeline remnant life taking into account the stochastic results of in-line inspection (ILI). It is assumed that the pipeline segment wall has a longitudinal external crack of semi-elliptical form and is described by the J-integral. The limit state function (LSF) is described as the difference of the critical and current value of the J-integral. The latter is calculated for the current time of pipe performance and is assumed known due to monitoring of the pumping equipment. The critical crack depth is defined using the notion of fracture toughness and the J-integral approach. The algorithm contains solutions of three sequentially interconnected problems. First, the deterministic problem of fatigue crack growth (FCG) is analyzed. Then the stochastic FCG is analyzed. The probability of failure assessment algorithm is designed on the basis of the authors’ version of the adaptive important sampling (AIS) procedure. The main steps of the AIS algorithm are described in detail. The samples are generated in such a way, that at all times a majority of samples belong to the fracture region. Finally, the results of the latest ILI are fused into the algorithm, providing best possible assessment of pipeline remnant life as a random variable. The remnant life update for pipeline segment with crack-like defects using ILI data takes into account three possible outcomes: defect not discovered: defect is discovered but not measured; defect is discovered and measured. This result permits solving most important problems of pipeline maintenance: prioritization of pipeline segments for repair/rehabilitation; optimization of the time between ILI; minimization of pipe operational risk. Two real cases are described of assessing the probability of fracture/leak of a pipeline section with an external crack at different periods of its performance. The described approach currently is being generalized for the case of multiple stress corrosion SC cracks.

Multistate models, that is, models with more than two distributions, are preferred over single-state probability models in modeling the distribution of travel time. Literature review indicated that the finite multistate modeling of travel time using lognormal distribution is superior to other probability functions. In this study, we extend the finite multistate lognormal model of estimating the travel time distribution to unbounded lognormal distribution. In particular, a nonparametric Dirichlet Process Mixture Model (DPMM) with stick-breaking process representation was used. The strength of the DPMM is that it can choose the number of components dynamically as part of the algorithm during parameter estimation. To reduce computational complexity, the modeling process was limited to a maximum of six components. Then, the Markov Chain Monte Carlo (MCMC) sampling technique was employed to estimate the parameters’ posterior distribution. Speed data from nine links of a freeway corridor, aggregated on a 5-minute basis, were used to calculate the corridor travel time. The results demonstrated that this model offers significant flexibility in modeling to account for complex mixture distributions of the travel time without specifying the number of components. The DPMM modeling further revealed that freeway travel time is characterized by multistate or single-state models depending on the inclusion of onset and offset of congestion periods.

Corrosion is still one of the major threats to the integrity of onshore and offshore pipelines. Realistic corrosion growth rates are essential inputs to safe and effective pipeline integrity management decisions. For example, corrosion rates are needed to predict pipeline reliability as a function of time, to identify the need for and timing of field investigations and/or repairs and to determine optimum re-inspection intervals to name just a few applications. The consequences associated with using wrong corrosion growth rates range from the inefficient use of resources (time, people and money) on unnecessary repairs and/or inspections to unexpected pipeline releases. The identification of where corrosion is active on a pipeline and how fast it is growing is a complex process which is understood in the general sense but is highly variable. Corrosion is therefore difficult to predict due to the very localised nature of its behaviour and the many parameters that influence the corrosion reaction. Running an in-line inspection (ILI) tool in a pipeline identifies the internal and/or external corrosion located along the full length of the pipeline. The ILI inspection also determines the depth, length and width measurements for each corrosion site and for the overall feature. The use of repeat ILI data to match and compare metal loss sites in order to estimate the corrosion growth rates at individual defects along a pipeline is a well-used and established practice in the industry. The use of such corrosion rates to make predictions of the future integrity of a pipeline started in earnest approximately 5 to 10 years ago and over that time considerable experience has been gained. Now that we are starting to collect 3, 4 or even 5 or more ILI data sets for the same pipelines we are able to test and validate our earlier ILI based growth rate predictions versus what actually occurred in the pipeline over time. With the benefit of this hindsight, the methodologies employed for evaluating and applying ILI based corrosion rates can be further improved and refined to give more accurate predictions of the future pipeline condition, the response schedule and for setting the timing of re-inspections. This paper shares the experience gained and the improvements that can be made to the determination of corrosion rates and application of these rates in a pipeline integrity assessment. These topics are illustrated and investigated via the use of case studies on real ILI data sets.