Markov chain modelling for time evolution of internal pitting corrosion distribution of oil and gas pipelines
Introduction
Pitting process can be metastable in nature — a situation in which a pitting process starts and stops after a while or immediately [1], [2] or it can be a stable pitting that nucleates and grows indefinitely. Stable pits generally show stochastic behaviour [1], [3] and are the focus of many researches. Pitting corrosion is initiated due to:
- i.
Electrochemical reactions of the carbon steel surfaces with the environment resulting in the formation of surface layers;
- ii.
discontinuity of the carbon steel material as a result of inclusions; and
- iii.
removal of an already formed surface layer due to erosion [4].
Forecasting of pitting corrosion rate has been done by modelling, extrapolation or using expert judgement [5]. Modelling technique can follow either probabilistic, deterministic or both approaches and has widespread application as exemplified by numerous publications [1], [6], [7], [8], [9]. Yusof et al. [6] studied pitting corrosion of offshore pipelines with Markov chain model and discovered that the prediction was not conservative due to the assumption that the model is linear. The data for the analysis was from repeated in-line inspection (ILI) of internal corroded offshore pipelines.
The authors assumed time of initiation of internal pitting corrosion as 2.9 years (after Velazquez et al. [10]) which is time of initiation of underground pipeline external pitting corrosion. This assumption may invalidate the result of these authors since the environmental condition of the soil is definitely different from that inside the pipeline. Although the future predicted pit depth distribution in this work was based on the exponential parameter (Vp) of power law being 1, the authors proposed Eq. (1) for predicting the value of Vp for future pit depth distribution if the initial pit depth (Pd1) and time (t1) and future pit depth (Pd2) and time (t2) are known with the pitting initiation time (tint).
The work of Valor et al. [1] focused on pitting corrosion of underground pipelines and corrosion coupons. The authors used discrete pit depths in non-homogenous, continuous time Markov chain modelling to determine the transition probability function by correlating the stochastic mean pit depth with the empirical deterministic pit depth. They used Weibull process for simulation of the pitting induction time. Other researchers such as Bolanos-Rodriguez et al. [11], Valor et al. [12] and Rodriguez III et al. [13] also applied non-homogenous, continuous time pure birth Markov chain modelling to estimate the pit depth distribution of pipelines by using a closed form of Kolmogorov forward equation for computation of the transition probability function whilst assuming that the pit depth follows a stochastic process. Similarly, Camacho et al. [14] applied Fokker–Planck equation for transition probability function estimate of pitting corrosion of underground pipelines based on a continuous time, non-homogenous pit depth evolution and Hong [15] worked on pit initiation and growth processes by modelling pit initiation as a homogenous Poisson process whilst estimating the pit growth with time as a non-homogenous, continuous time Markov process.
Pipeline failures resulting from pitting have been attributed to pin-hole type pit [8] hence, the need for extreme value modelling of maximum pit depths of corroded pipelines to predict the distribution in the future. Valor et al. [8] applied a stochastic modelling approach to estimate the extreme value distribution of corroded low carbon steel using API-5L X52 pipeline corrosion coupons experimental data. Melchers [16] showed that extreme value analysis can be carried out with limited data if it is combined with Bayesian approach and demonstrated this feat with carbon steel coupons exposed to marine environment. Similarly, Melchers [17] used a bi-modal probability density function to represent the maximum pit depth distribution of mild steel exposed to marine environment and concluded that maximum pit depth distribution is better represented with Fretchet distribution for a long-time exposure of the material than Gumbel distribution that is traditionally used for the extreme value distribution plotting [1], [2], [3], [5], [18], [19] however, Sheikh et al. [9] showed that the initial pitting corrosion followed a normal distribution and lognormal distribution for long-time exposure of carbon steel material to a corrosive environment.
Sulphate Reducing Bacteria (SRB) are the most active contributor to pitting in long-time exposure of carbon steel materials to marine environment [2] because their metabolic activities results in sulphate ion reduction to hydrogen and sulphide. The sulphide ion attacks the steel electrochemically causing more pitting corrosion due to an increase in anodic/cathodic reactions necessitated by sulphate reduction. Other researchers also found out experimentally that sulphur reducing bacteria starved of organic energy sources cause severe pitting corrosion of carbon steel materials [20]. Although cathodic protection and other forms of coating have the ability of protecting marine infrastructures like pipelines from external pitting corrosion, ageing infrastructures exposed to marine environment have serious problem of pitting corrosion which can predominantly cause assets failures. Rivas et al. [3] used block maxima and peak over threshold approach for extreme value analysis of laboratory simulated field data of buried carbon steel pipeline and concluded that the peak over threshold approach was more robust in estimating the maximum pit depth of the samples. In their own work, Valor et al. [21] described pit initiation and propagation as a stochastic process of non-homogenous Poisson process and non-homogenous continuous time Markov process respectively. They used extreme value statistics for modelling maximum pit depth growth for data obtained from literature. Although the work produced better results than those obtained from available literature (see ref. [21]), however, the assumption that the entire pits tested nucleates instantaneously may not always be the case practically.
Corrosion can result in unscheduled downtime especially for pitting corrosion, crevice corrosion, stress corrosion cracking and fatigue corrosion since they occur without outward signs on the facilities [22]. Hence, corrosion modelling is used for integrity management via prediction of expected time of pipeline failure so that mitigation actions that could include inspection and repairs will be initiated [7], [8], [23], [24]. To establish the time dependent reliability of corroded high pressure offshore pipelines, Zhang and Zhou [25] determined the expected future internal corrosion wastage distribution due to internal pressure using Poisson square wave process. The authors established the time of pipeline failure with respect to small leak, large leak and rupture by using in-line inspection data after modelling stochastic pit depth growth with homogenous gamma distribution according to Eq. (2):
where fG(Pd(t)| α(t - t0), β) is the probability density function of the pit depth at time t, α(t - t0) is the time dependent shape parameter, Γ(.) is the gamma function, I(t) is an indicator function with values given in Eq. (3).
Bazán and Beck [26] also used Poisson square wave process to model external pitting corrosion of underground pipelines and concluded that power model gave a more conservative estimate of the future corrosion wastage than random linear model after comparing the results with field inspection data. Similarly, Valor et al. [27] used historic data to determine the reliability of corroded non-piggable upstream pipelines exposed to external corrosion by statistically analysing the acquired data, determining the corrosion distribution at a future time and correlating the results with the designed pipeline failure pressure. The aim of these researchers is to establish a mitigation programme aimed at enhancing the lifecycle of the pipelines [27]. The work of Rodriguez III et al. [28] was also aimed at mitigation and control of pitting corrosion by applying Markov chain modelling to determine the future pit depth whilst predicting the remaining useful life of the pipeline at future times based on the pit depth distribution.
Other pitting corrosion related researches that are noteworthy includes the work of Valor et al. [23] that used Monte Carlo reliability framework to model different corrosion distributions that included linear growth model, time dependent and time independent models, Markov model and single value distribution model. They utilized both synthetic and field data in evaluating these models whilst considering defect sizes, age and depth of corrosion with time. They concluded that Markov chain predictive model was best for describing the corrosion distributions [23]. Caleyo et al. [19] also used Monte Carlo simulation to model pit depth growth of underground pipelines in different soil conditions and fitted the three maximal extreme value distributions — Weibull, Fretchet and Gumbel to the resulting best fit models of the studied soils, however, Fretchet distribution was best for describing the model over a long-time exposure as was already stated in this work. Again, another work on experimental determination of internal pitting rate in pipelines concluded that increases in pitting rate occurs due to increased chloride concentration, temperature, subcutaneous substances (such as sand) and flow rate whereas decrease in pitting rate was observed with increase in bicarbonate, CO2 and H2S partial pressures and operating pressure [4]; however, the results in this research were validated with limited field data. Pitting corrosion rate has also been modelled by researchers using damage function analysis by considering pit nucleation, growth rate and re-passivation of carbon steel in chloride solution [22].The pitting rate for underground pipelines was also predicted with lognormal linear model in consideration of environmental variables [29] and the time of initiation of pitting has also been predicted for different soil categories using Monte Carlo simulation of field observed soil conditions [10].
The above reviewed literatures show that limited work has been done on Markov modelling of internal pitting corrosion of oil and gas pipelines and the few works are either flawed due to limited field validation data or are not based on pure birth non-homogenous Markov modelling, hence, the need for a holistic field data analysis of pitting rate distribution using a continuous time non-homogenous linear growth pure birth Markov model. Since effective corrosion modelling requires a combination of electrochemical activities relating to water and oxides transport within the metal surface, macro-environment (such as temperature, pH, humidity, salinity, porosity) and external environment (such as rainfall, seasonal rainfall and temperature fluctuations) [30], it is possible to model internal pitting corrosion of oil and gas pipelines by considering the operating conditions of the pipelines and the pit depths at different ages. The present work is aimed at determining the future distribution of pit depths of internally corroded oil and gas pipelines by using non-homogenous, continuous time linear growth pure birth Markov process. A multivariate regression analysis of field data was used in a Monte Carlo simulation framework to estimate the time of initiation of the pitting for different categories of pitting rates based on NACE classification. The work used initial knowledge of pit depth distribution to determine the transition probability function of pit depth growth in future time based on the closed form of negative binomial distribution solution of Kolmogorov's forward equation.
Section snippets
Finite Markov chain modelling of internal pitting corrosion of pipelines
A Markov process has no memory because future events are independent of past ones but dependent on the present event [31] hence, if the pit depth of oil and gas pipeline at time t is represented by (Pd), then the probability at such a time can be written as Eq. (3):
where N represents the number of states the pipeline wall is divided, Pi(t) is the probability that the pit depth is at ith state at time t and can be determined by measuring the pit depth distribution at such a
Prediction of the model parameters
To predict the transition probabilities for the deterministic pit depth requires the estimation of the deterministic pitting rate and the time of initiation of the pitting in Eq. (23). To estimate these parameters, pit depths and operating parameters (pH, temperature, flow rate and CO2 partial pressure) measurements of oil transmission pipelines obtained from a producing company in Nigerian Niger Delta region were used for numerical analysis and determination of the multivariate coefficients
Results and discussion
The minimum and maximum observed pit depths of each of the 60 pipelines were used as boundary conditions for a Monte Carlo simulation experiment aimed at predicting the variation of the pit depth growth with time of exposure of the pipelines for different categories of pitting rate shown in Table 1. This simulation executed in Matlab version R2014a was used to obtain results for 1 year to 40 years exposure of the studied pipelines. The cumulative simulated pit depths for each of the simulated
Conclusions
To estimate the future pit depth distribution of oil and gas pipelines, a non-homogenous, continuous time pure birth Markov process was used. The work focused on internal pitting corrosion of oil and gas pipelines by considering the effects of some operating parameters — temperature, CO2 partial pressure, pH and flow rate on the pit depth growth at different pitting categories stipulated by NACE. The pipeline wall thickness was divided into a number of states and the pit depths categorized into
References (45)
Extreme value statistics and long-term marine pitting corrosion of steel
Probab. Eng. Mech.
(2008)- et al.
Extreme value analysis applied to pitting corrosion experiments in low carbon steel: comparison of block maxima and peak over threshold approaches
Corros. Sci.
(2008) - et al.
Inverse Gaussian process-based corrosion growth model for energy pipelines considering the sizing error in inspection data
Corros. Sci.
(2013) - et al.
Stochastic approach to pitting-corrosion-extreme modelling in low-carbon steel
Corros. Sci.
(2010) - et al.
Chain modelling of pitting corrosion in underground pipelines
Corros. Sci.
(2009) Inspection and maintenance planning of pipeline under external corrosion considering generation of new defects
Struct. Saf.
(1999)Representation of uncertainty in maximum depth of marine corrosion pits
Struct. Saf.
(2005)- et al.
Probability distribution of pitting corrosion depth and rate in underground pipelines: a Monte Carlo study
Corros. Sci.
(2009) - et al.
Stochastic modeling of pitting corrosion: a new model for initiation and growth of multiple corrosion pits
Corros. Sci.
(2007) - et al.
Reliability assessment of buried pipelines based on different corrosion rate models
Corros. Sci.
(2013)
System reliability of corroding pipelines considering stochastic process-based models for defect growth and internal pressure
Int. J. Press. Vessel. Pip.
Stochastic process corrosion growth models for pipeline reliability
Corros. Sci.
The science of pipe corrosion: a review of the literature on the corrosion of ferrous metals in soils
Corros. Sci.
Markov decision process
Eur. J. Oper. Res.
Reliability estimation of pressurised pipelines subject to localised corrosion defects
Int. J. Press. Vessel. Pip.
Predicting corrosion remaining life of underground pipelines with a mechanically-based probabilistic model
J. Pet. Sci. Eng.
A study on the reliability assessment methodology for pipelines with active corrosion defects
Int. J. Press. Vessel. Pip.
Probabilistic estimation of remaining life of a pipeline in the presence of active corrosion defects
Int. J. Press. Vessel. Pip.
Amine type inhibitor effect on corrosion–erosion wear in oil gas pipes
Wear
Advanced method for the development of an empirical model to predict time-dependent corrosion wastage
Corros. Sci.
Markov chain models for the stochastic modeling of pitting corrosion
Math. Probl. Eng.
Model to predict internal pitting corrosion of oil and gas pipelines
Corrosion
Cited by (53)
A risk-based maintenance decision model for subsea pipeline considering pitting corrosion growth
2024, Process Safety and Environmental ProtectionDevelopment of HGAPSO-SVR corrosion prediction approach for offshore oil and gas pipelines
2023, Journal of Loss Prevention in the Process IndustriesAdvances in corrosion growth modeling for oil and gas pipelines: A review
2023, Process Safety and Environmental ProtectionCitation Excerpt :For relatively short exposure periods, the Gumbel and Weibull distributions are the most likely to be suitable for corrosion depth and rate (Bhandari et al., 2017). In the literature on pitting corrosion, the two-parameter Gumbel (Marsh and Taylor, 1988; Strutt et al., 1985) and Weibull (Khan et al., 2021; Ossai et al., 2015) distributions are commonly used. This is because the mathematical complexity of the three-parameter distribution is high, and in most cases, two-parameter distributions can well fit the corrosion data.
Evaluation of the green inhibitor effect on the corrosion of pipeline steel in NS4 medium
2022, Procedia Structural IntegrityEarly warning method for overseas natural gas pipeline accidents based on FDOOBN under severe environmental conditions
2022, Process Safety and Environmental Protection