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Markov chain modelling of pitting corrosion in underground pipelines

F. Caleyo a,*, J.C. Velázquez a, A. Valor b, J.M. Hallen a

a Departamento de Ingenierı́a Metalúrgica, ESIQIE, IPN, UPALM Edif. 7, Zacatenco, México D. F. 07738, Mexicob Facultad de Física, Universidad de La Habana, San Lázaro y L, Vedado, 10400 La Habana, Cuba

a r t i c l e i n f o

Article history:Received 14 April 2009Accepted 7 June 2009Available online 14 June 2009

Keywords:A: SteelB: Modelling studiesC: Pitting corrosion

a b s t r a c t

A continuous-time, non-homogenous linear growth (pure birth) Markov process has been used to modelexternal pitting corrosion in underground pipelines. The closed form solution of Kolmogorov’s forwardequations for this type of Markov process is used to describe the transition probability function in a dis-crete pit depth space. The identification of the transition probability function can be achieved by corre-lating the stochastic pit depth mean with the deterministic mean obtained experimentally. Monte-Carlosimulations previously reported have been used to predict the time evolution of the mean value of the pitdepth distribution for different soil textural classes. The simulated distributions have been used to createan empirical Markov chain-based stochastic model for predicting the evolution of pitting corrosion depthand rate distributions from the observed properties of the soil. The proposed model has also been appliedto pitting corrosion data from pipeline repeated in-line inspections and laboratory immersionexperiments.

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1. Introduction

The stochastic nature of pitting corrosion has been recognisedsince the mid-1930s [1,2]. One of the assumptions that can bemade about this damage is that it retains no memory of the past,so only the current state of the damage influences its future devel-opment. This important characteristic allows pitting corrosion tobe categorised as a Markov process [3]. The discretisation of thepit depth space in a finite or countable set of non-overlappingstates makes pitting corrosion a good candidate for Markov chainmodelling [3].

Provan and Rodriguez [4] used a non-homogenous Markov pro-cess to model pit depth growth for the first time. The strengths andlimitations of the Provan and Rodriguez’s approach are discussedin [5]. Later, Morrison and Worthingham [6] used a continuoustime birth process with linear intensity k for determining the reli-ability of high pressure corroding pipelines. These authors dividedthe space of the load-resistance ratio into discrete states andsolved numerically the Kolmogorov equations to find the intensi-ties of transition between damage states. They showed that theestimated probability distribution function of the load-resistanceratio closely reproduced the distribution obtained from field mea-surements. Hong [7] refined the Morrison and Worthingham’smodel by using the analytical solution of the Kolmogorov equa-tions for the same homogeneous continuous type of Markov pro-

cess. He investigated the effect of corrosion defect size on theload-resistance ratio and obtained the probability transition matrixof the process to estimate the probability of failure of the pipeline.

In recent years, important advances have been made in model-ling pitting corrosion through Markov chains [5,8–10]. Bolzoniet al. [8] used a continuous-time, three-state Markov process tomodel the first stages of localised corrosion considering three pos-sible states of the metal surface: passivity, metastability and local-ised corrosion. Valor et al. [5] proposed a new stochastic model inwhich pit initiation is modelled as a Weibull process, while pitgrowth is modelled using a non-homogenous, linear growth Mar-kov process. The theory of extremes was used to combine both pro-cesses and to adequately reproduce the experimental observationsof pitting corrosion for a range of materials and environments. La-ter, Zhang et al. [9] used this stochastic model to investigate thepitting corrosion susceptibility of pure Mg and Mg alloys.

Particularly pertinent to the focus of the present study is themodel developed by Timashev et al. [10] to assess the conditionalprobability of pipeline failure and to optimise the maintenance ofoperating pipelines. This model is based on the use of a continu-ous-time, discrete state pure birth homogenous Markov process forstochastically describing the growth of corrosion-caused metal loss.The initial conditions are determined by the (measured or assumed)corrosion depth distribution at t = 0. The probability pi(t) that thedefect depth is in the i-th state at a given moment in time t is deter-mined by the (measured or assumed) depth distribution at t. Usingstandard methods, the intensity or rate (ki) of the unknown, time-independent transition probabilities are calculated by iterativelysolving the system of Kolmogorov’s forward equations:

0010-938X/$ - see front matter � 2009 Elsevier Ltd. All rights reserved.doi:10.1016/j.corsci.2009.06.014

* Corresponding author. Tel.: +52 55 57296000x54205; fax: +52 5557296000x55270.

E-mail address: fcaleyo@gmail.com (F. Caleyo).

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dp1dt ¼ �k1p1ðtÞ

dpidt ¼ ki�1pi�1ðtÞ � kipiðtÞ

8<: ð1Þ

The referenced authors illustrated that by estimating ki, the cor-rosion depth and rate distributions can be estimated for any timepoint. From that, the conditional probability of pipeline failurecan be estimated, and repair and maintenance actions can bescheduled.

The Markov chain approach to corrosion growth proposed byTimashev and co-workers has many merits and strengths, butsome of its limitations are worthy of consideration. First, the rela-tive complexity of the iterative method used to determine the tran-sition intensities ki requires that the pipe wall thickness be dividedinto a relatively small number of Markov states. Second, the use oftime-independent transition rates ki (time homogeneity condition)implies that their estimated values actually represent the averageof the time-dependent intensities ki(t) over the selected period oftime. The time homogeneity condition entails that the staying timeof the corrosion penetration front in a given damage state is expo-nentially distributed. It also implies that the corrosion growth rateis treated implicitly as a constant, while the corrosion depth istreated as a linear function of the exposure time. Due to the non-linearity of the pit growth process, these assumptions hold trueonly if the estimations are made for long exposure times over rel-atively short time spans. Finally, the solution to the Kolmogorovequations applies only for the particular conditions of a pipelineundergoing repeated in-line inspections.

One important aspect that remains unsolved in the area of reli-ability-based pipeline integrity management is the accurate esti-mation of future pit depth and growth rate distributions from a(single) measured or assumed pit depth distribution. Obviously,such estimation can be carried out only if oversimplifications aremade (see, for example, Morrison and Desjardins’ discussion inRef. [11]) or if additional information is available besides the pitdepth distribution. For example, Alamilla and Sosa [12] recentlydeveloped a stochastic model to estimate the corrosion damagevelocity in operating pipelines. In their model, the results of thein-line inspection are used not only to provide the corrosion depthdistribution at the time of the inspection but also to make infer-ences about the intensity of the homogenous Poisson process usedto simulate the generation of corrosion defects.

In the case of external pitting corrosion in underground pipe-lines, it seems reasonable to postulate that the available predictivemodels for pit growth as a function of the soil characteristics[13,14] could provide the additional information necessary to pre-dict, with a reasonable accuracy, the time evolution of the pittingdepth and rate distributions.

In this work, a non-homogenous linear growth pure birth Mar-kov process, with discrete states in continuous time, is used tomodel external pitting corrosion in underground pipelines. A mod-el for pit growth recently developed by the authors [14] has beenused to perform Monte Carlo simulations aimed at predicting thedistribution of maximum pit depths as a function of the pipelineage and the physicochemical characteristics of the soil. The bino-mial closed form solution of the forward Kolmogorov equationsis adopted to express the probability of transition between pitdepth (Markov) states in a given interval of time. In so doing, theMarkov-derived stochastic mean of the pit depth distribution issupposed to be equal to the deterministic mean of the distributionobtained through Monte-Carlo simulations. This supposition hasbeen made for different exposure times and different soil classesdefined according to soil physicochemical characteristics that areeasy to measure in the field. A multivariate regression analysishas been used to obtain a predictive model for pit growth [14],which was employed to specify the transition probability as a func-

tion of soil properties. Thus, with a knowledge of the pit depth dis-tribution at a given point in time and the measured values of thesoil characteristics, the proposed Markov framework can be usedto predict the time evolution of the pitting depth and rate distribu-tions. Real-life case studies, involving simulated and experimentalpit depth distributions, are presented to illustrate the proposedMarkov chain modelling framework.

2. Model development

2.1. Markov chain representation of pitting corrosion damage

The reader is referred to [3] and [15] for a formalism of the the-ory of Markov processes. In this section, only the definitions rele-vant to the focus of the present study are presented. As a matterof general definition, it is assumed that the pipe wall thicknesshas been divided in N discrete states and that the corrosion dam-age (pit) depth, at any point in time t, can be represented by a dis-crete random variable D(t) with P{D(t) = i} = pi(t), i = 1, 2 . . .N.Furthermore, it is assumed that the probability that the damageat the i-th state advances one state during a very short intervalof time dt can be written as ki(t)d t + o(dt). For a continuous-time,non-homogenous linear growth Markov process with intensitieski(t) = ik(t), the probability that the process presently in state i willbe in state j (j P i) at some later time obeys the following system ofKolmogorov’s forward equations (note the differences with respectto Eq. (1)):

dpi;jðtÞdt

¼kj�1ðtÞpi;j�1ðtÞ � kjðtÞpi;jðtÞ; j P iþ 1�kiðtÞpi;iðtÞ

(ð2Þ

For a Markov process defined by the system of equations (2),the conditional probability of transition from the m-th state tothe n-th state (n P m) in the interval (t0,t), that is,pm,n(t0,t) = P{D(t) = njD(t0) = m}, can be obtained in closed form(see page 304 in [15]):

pm;nðt0; tÞ ¼n� 1n�m

� �e�fqðtÞ�qðt0Þgmð1� e�fqðtÞ�qðt0ÞgÞn�m ð3Þ

where:

qðtÞ ¼Z t

0kðt0Þdt0 ð4Þ

In words, Eq. (3) means that the pitting corrosion damage in-crease in an interval of length t � t0 follows a negative binomialdistribution NegBin(r,p) with parameters r = m andp ¼ ps ¼ e�fqðtÞ�qðt0Þg. From pm,n(t0,t), it is possible to estimate theprobability distribution function f(t) of the damage rate t associ-ated with the damage process over the interval of lengthDt = t � t0, when the damage depth is at the m-th state:

f ðt; m; t0; tÞ ¼ pmðt0Þpm;mþtDtðt0; tÞDt ð5Þ

Now, from f(t;m,t0,t), it is straightforward to derive the pittingrate probability distribution associated with the entire pit popula-tion using:

f ðt; t0; tÞ ¼XN

m¼1

f ðt; m; t0; tÞ ð6Þ

Critical to the goal of this study is the fact that if the initial dam-age state at t = ti is ni, so that D(ti) = ni, then the time-dependent,stochastic mean M(t) = E[D(t)] of the linear growth Markov processcan be expressed as [3]:

MðtÞ ¼ nieqðt�tiÞ ð7Þ

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Under certain simple assumptions1, the stochastic mean of the pro-cess can be assumed to be equal to the deterministic mean D

��ðtÞ of a

damage process for which the increment in the time interval Dt is:

D D��ðtÞ ¼ k

��ðtÞD

��ðtÞDt ð8Þ

where k��

can be interpreted as the deterministic intensity of thedamage process.

If D��ðtÞ represents the deterministic mean pit depth at time t,

and if the power law model is considered to be an accurate deter-ministic representation of the pit growth process, then one canwrite [14]:

D��ðtÞ ¼ jðt � tsdÞm ð9Þ

where j and m are the pitting proportionality and exponent param-eters, respectively; and tsd is the starting time of the pitting corro-sion damage. In systems where passivity breakdown and/orinclusion dissolution are the prevalent mechanisms for pit initia-tion, tsd would represent the initiation time of stable pit growth.In the case of underground pipelines, this parameter would repre-sent the total elapsed time from pipeline commissioning to coatingdamage plus the time period in which the cathodic protection iseffective in preventing or attenuating external pitting corrosionafter coating damage.

It is easy to show that the pitting rate, obtained by taking thetime derivative of D

��ðtÞ, obeys the functional form of Eq. (8) with

k��¼ m=s and s = t � tsd:

d D��ðtÞ

dt¼ m

sD��ðsÞ ð10Þ

Valor et al. [5] showed that, when Markov processes are used tomodel pitting corrosion, a direct physical meaning can be given tothe functions k(t) and q(t), being related to the pit growth rate andpit depth, respectively. In the present work, k(t) and q(t) are alsorelated to the pitting rate and depth, respectively, though in aslightly different manner than in [5]. It is assumed that the exper-imentally observed deterministic mean D

��ðtÞ of the pitting corro-

sion depth is equal to the Markov-derived stochastic mean pitdepth M(t):

MðtÞ ¼ D��ðtÞ ð11Þ

Let dt1 represent the sojourn time of the corrosion damage in thefirst state of the chain. In this case ni = 1 between tsd and tsd + dt1.If dt1 is significantly less than the simulation time span, it is easyto show from Eqs. (7), (9) and (11) that the value of the functionq(t) can be approximated as:

qðtÞ ¼ lnðjðt � tsdÞmÞ ð12Þ

Furthermore, from Eq. (4), it follows that:

kðtÞ ¼ mt � tsd

ð13Þ

This means that the intensity of the Markov process k(t) is in-versely proportional to the exposure time, just like the determinis-tic intensity of the damage process in Eq. (10).

Now, it is relatively easy to show that the probability parameterps ¼ e�fqðtÞ�qðt0Þg in Eq. (3) can be expressed as:

ps ¼t0 � tsd

t � tsd

� �m

; t P t0 P tsd ð14Þ

Let E[n �m]/Dt be the average damage rate in the time interval(t0, t0 + Dt) for a pit with depth in the m-th state at t0. In the Appen-dix to this paper it is shown that the instantaneous rate of damagepredicted by the stochastic model for this case is:

tðm; t0Þ ¼E½n�m�

Dt!

Dt!0m

mt0

ð15Þ

Thus, the stochastically-predicted instantaneous damage rateagrees with the rate predicted in Eq. (10) by the deterministicmodel. The agreement of the stochastic and deterministic rates ofdamage supports the adequacy of the proposed in this work-Mar-kov chain approach for pitting corrosion modelling.

Let us assume that the probability distribution of the corrosiondepth at t0 is known, that is P{D(t0)) = m} = pm(t0). For example, thiscan be obtained if the corrosion damage in the pipeline is moni-tored using in-line inspection. In this case, t0 would be the timeof the inspection and the value of the probabilities pm would beestimated from the ratio of the number of corrosion pits withdepths in the m-th state to the total number of pits observed. Ifthe transition probability function pm,n(t0, t) is known, then thepit depth distribution at any future moment in time can be esti-mated using [3,15]:

pnðtÞ ¼Xn

m¼1

pmðt0Þpm;nðt0; tÞ ð16Þ

Based on the foregoing discussion, it is postulated that the pit-ting corrosion damage evolution in underground pipelines can beundertaken as follows. The measured or assumed pit depth proba-bility distribution at t0 is used as the initial corrosion damage dis-tribution pm(t0). The transition probability function pm,n(t0, t),which is completely identified if the function q(t) is known, wouldbe obtainable if a predictive model were available for relating theparameters tsd and m in Eq. (12) to the physicochemical conditionsof the soil [14]. From the results of the in-line inspection and theknowledge of the local soil characteristics pertinent to such a pre-dictive model, it would be possible to estimate the pitting corro-sion damage evolution. The next section presents the predictionmodel developed to identify pm,n(t0, t) from in-field measured soilcharacteristics.

2.2. Predictive model for the intensity of transition probabilities

The prediction of the function q(t) from the soil characteristicsrelies upon the assumptions expressed by Eqs. (11) and (12). Theestimation of D

��ðtÞ is carried out using a predictive model based

on Eq. (9), which was developed by the authors for predicting max-imum pitting damage in underground pipelines [14]. The experi-mental details and corrosion data used to produce this model canbe found elsewhere [14,16]. The soil classes considered in [14]are also used in this work. If D

��ðtÞ represents the maximum pit

depth to be predicted and if the soil and pipe characteristics givenin Table 1 are treated as independent variables, then the pittingparameters in Eq. (9) can be estimated as follows [14]:

j ¼ j0 þ jphphþ jcccc þ jrprpþ jrereþ jbcbc þ jscsc ð17Þm ¼ m0 þ mppppþ mwcwc þ mbdbdþ mctct ð18Þ

The reader is referred to the original paper [14] for details of thederivation of Eqs. (17) and (18). There, an in-depth discussion onthe relation between the model parameters and the environmentalvariables can be found. It is worth noting that, in Eq. (18), the var-iable ct is treated as a discreet ordinal variable, whose value is as-signed according to a scoring model described in detail in Ref. [14]and summarized in a footnote to Table 1.

Once the soil class has been identified and the values of thepredictor variables have been measured, Eq. (18) can be used to

1 A sufficient condition for the equality of the stochastic and deterministic means isthat for any positive integer q, the structure of the process starting from qmindividuals (states) is identical to that of the sum of q separated systems each startingfrom m (see page 159 in [3]).

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determine m and thereby identify q(t). The identification of q(t) al-lows the transition probability function pm,n(t0, t) to be determinedfor the soil and pipe characteristics in case.

It will be shown later that the proposed model can be used topredict the progression of other pitting processes. For this to bedone, the values of the pitting exponent and starting time mustbe known for the process. These can be obtained, for example, fromthe analysis of repeated in-line inspections of the pipeline, fromthe study of corrosion coupons or from the analysis of laboratorytests.

3. Results and discussion

3.1. Modelling pitting corrosion in soils

In a previous work [16], the authors used a Monte-Carlo simu-lation framework based on Eqs. (9), (17) and (18) to predict thetime evolution of the probability distribution of maximum pitdepth caused by external corrosion in underground pipelines.The observed distributions of the independent variables consid-ered in Table 1 were used as inputs in Monte-Carlo simulationsof the pitting process for each soil category. The reader is directedto reference [16] for details of the used distributions and theMonte-Carlo simulation framework. Random values drawn fromthese distributions were used to evaluate Eqs. (9), (17) and (18)5000 times for different exposure times. The regression coefficientsused in the simulations are those given in Table 1.

For each soil category and exposure time considered, the simu-lated pit depth distribution was fitted to the generalized extremevalue (GEV) distribution [17]:

GðzÞ ¼ exp � 1þ fz� l

r

� �h i�1=f� �

ð19Þ

where f, r and l are the shape, scale and location parameters,respectively.

Fig. 1 shows the time evolution of the shape (Fig. 1a), scale(Fig. 1b) and location (Fig. 1c) parameters of the fitted GEV func-tions. In order to estimate the pitting proportionality and exponentfactors associated with the typical (average) values of the predictorvariables in each soil category, the time evolution of the mean of

the simulated maximum pit depth distributions was fitted to Eq.(9) using the corresponding value of tsd given in Table 1. The modelparameters obtained by this method are shown in the last tworows of Table 1. They are assumed to be unbiased estimates of jand m for typical conditions for each soil category [14,16]. The mod-el output for the typical values of the predictor variables in a givensoil category can be interpreted as the typical or average responseof the model in that soil.

Fig. 2 shows the time evolution of q(t) (Fig. 2a) and ps (Fig. 2b)predicted from the exponent associated with the pitting damageprocess for each soil category (mt in Table 1). Given that q(t) is com-pletely determined by the extent of the damage (Eqs. (9) and (12)),its value is unique for each soil category, being larger for higher soilcorrosivities. In addition, the probability ps is unique for each soiltype. Its value at a given moment in time t P t0 increases withincreasing t0 and decreases when the soil corrosivity and thelength of the interval (t0, t) increases.

The results shown in Fig. 2 determine the stochastic behaviourof the pitting corrosion rate. Based on the properties of the Neg-Bin(m,ps) distribution, it follows that a reduction in the value ofps with soil corrosivity entails larger mean and variance values ofthe pitting rate in highly corrosive soils. The increase in ps witht0 indicates that the mean and variance of the pitting rate decreaseas the lifetime of the pitting damage increases. Finally, the form ofq(t) in Fig. 2a suggests that, for pits with equal lifetimes, the deeperthe pit, the smaller the value of ps and, therefore, the larger themean and variance of the pitting rate.

The properties of the stochastically predicted pitting rate areillustrated in Fig. 3 for the case in which the pitting damage occursin clay soils. To produce this figure, it was assumed that pm(t0) = 1for every m considered. Eqs. (3), (5) and (14) were used to computef(t; m, t0, t) using the values of mt and tsd given in Table 1 for claysoils. The pipe wall thickness was divided in 0.1-mm-thick statesso that the pitting damage is represented through Markov chainswith states ranging from m = 1 to m = N = 100. Unless otherwisespecified, this scheme of discretisation of the pipe wall thicknessis used hereafter to represent the pitting damage penetration.

The results shown in Fig. 3 fully agree with the behaviour pre-dicted in the foregoing paragraphs for the damage rate as well aswith the general body of knowledge about pitting corrosion rate.The mean and variance of the pitting rate distribution decrease

Table 1Variables and coefficients of the predictive model for pitting corrosion [14].

Variable or parameter Symbol (units) Coefficient name Coefficient or parameter value by soil class

Clay Clay loam Sandy clay loam All

Maximum pit depth dm(mm) – – – – –Pipeline age t(years) – – – – –Redox potentiala rp(mV) jrp �9.0 � 10�5 �1.1 � 10�4 �1.8 � 10�4 �1.8 � 10�4

pH ph jph �5.9 � 10�2 �1.2 � 10�1 �6.4 � 10�2 �6.5 � 10�2

Pipe/soil potentialb pp(V) mpp 4.9 � 10�1 4.6 � 10�1 5.1 � 10�1 5.2 � 10�1

Resistivity re(X–m) jre �2.2 � 10�4 �3.0 � 10�4 �2.1 � 10�4 �2.6 � 10�4

Water content wc(%) mwc 3.7 � 10�3 1.7 � 10�2 4.5 � 10�4 4.6 � 10�4

Bulk density bd(g/ml) mbd �1.0 � 10�1 �9.9 � 10�2 �1.6 � 10�1 �9.9 � 10�2

Chloride content cc(ppm) jcc 8.4 � 10�4 1.8 � 10�3 8.6 � 10�4 8.7 � 10�4

Bicarbonate content bc(ppm) jbc �1.3 � 10�3 �4.9 � 10�4 �6.8 � 10�4 �6.4 � 10�4

Sulphate content sc(ppm) jsc �5.3 � 10�5 �2.1 � 10�4 �1.1 � 10�4 �1.2 � 10�4

Coating typec ct mct 4. 7 � 10�1 57 � 10�1 4.3 � 10�1 4.3 � 10�1

Pitting starting time tsd(years) – 3.0 3.1 2.6 2.9Constant prop. term j0(mm/yrm_0) – 5.5 � 10�1 9.8 � 10�1 6.0 � 10�1 6.1 � 10�1

Constant exponent term m0 – 8.8 � 10�1 2.8 � 10�1 9.6 � 10�1 8.9 � 10�1

Typical prop. factor jt(mm/yrm_t) – 0.178 0.163 0.144 0.164Typical exponent factor mt – 0.829 0.793 0.734 0.780

a Relative to the standard hydrogen electrode.b Relative to a Cu/CuSO4(sat.) reference electrode.c The pipeline coating conditions described by the ct variable and the assigned scores (in parenthesis) are: non-coated (1.0), asphalt-enamel-coated (0.9), wrap-tape-coated

(0.8), coal-tar-coated (0.7) and fusion-bonded-epoxy-coated (0.3) [14].

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with increased pit lifetime. In Fig. 3, the distributions for the 15–20year period (first two distributions from the left) have lower meanand variance values than those predicted for the 5–10 year period(the two distributions on the right of Fig. 3). For pits with similarlifetimes, the pitting rate distribution mean and variance increasewith increasing pit depth; the deeper the pit (larger m) the larger

the pitting rate mean and variance. Fig. 3 also reveals one of theadvantages of the proposed Markov chain approach, that is, thata unique probability distribution of pitting rate can be correlatedto the damage according to its depth and lifetime.

Fig. 1. Time evolution of (a) the shape, (b) the scale and (c) the location parametersof the GEV distribution fitted to the Monte-Carlo-simulated pit depth distributionfor each soil category.

Fig. 2. Time evolution of (a) q(t) and (b) ps predicted from the exponent mt (Table 1)associated with the pitting damage process for each soil category.

Fig. 3. Dependence of the pitting rate distribution f ðm; m; t0; t) on pit depth andlifetime.

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The 10-year pit depth distribution predicted for each soil cate-gory using the Monte-Carlo framework was used as the initialdamage distribution pm(t0 = 10) from which the damage distribu-tion pn(t) was predicted for different times using the proposedMarkov chain model. Fig. 4 shows the GEV distribution fitted topm(10) for each soil category.

Fig. 5 shows the results of the Markov chain modelling of pittingcorrosion in the least corrosive (sandy clay loam, Fig. 5b) and mostcorrosive (clay, Fig. 5a) soils. The pitting exponent and startingtime used to carry out the predictions for each soil category werethose given in Table 1. The Monte-Carlo-simulated pit depth distri-butions are represented by grey shaded histograms, while the con-tinuous lines represent the Markov chain-predicted distributionspn(t).

The goodness-of-fit of the results shown in Fig. 5, and of thosenot shown here for the other soil classes, was assessed using theKolmogorov–Smirnov test. For the sake of simplicity, the one-sam-ple test was conducted based on the null hypothesis that eachMonte-Carlo-estimated distribution (empirical sample) comesfrom the corresponding Markov chain-modelled distribution(underlying reference distribution). In all the analyzed cases (a to-tal of 20, five exposure times for each one of the four soil classes), itwas not possible to reject the proposed null hypothesis at 5% sig-nificance level. This could be interpreted as evidence that there isno reason to reject the assertion that the Monte-Carlo-estimatedand Markov chain-modelled distributions are close enough to con-sider the proposed Markov model accurate.

The proposed Markov chain model is capable of reproducing thetime evolution of the pit depth distribution accurately. The qualityof the fitting in the clay loam and in all soil categories was verysimilar to that observed in Fig. 5. This can be corroborated by theresults shown in Fig. 6, which shows the time evolution of themean (Fig. 6a) and standard deviation (Fig. 6b) of the Monte-Car-lo-simulated and Markov-modelled pit depth distributions forthe soils of interest.

In addition to the qualitative (Fig. 5) and quantitative (Fig. 6)accuracy of the Markov chain predictions, the most striking featureof these results is that the model is able to reproduce the changesobserved in the shape of the pitting depth distribution withincreasing exposure time for all soil categories. Fig. 1a shows that,depending on the exposure time, any of the three maximal ex-treme value distributions, i.e., Weibull, Fréchet or Gumbel, arisesas the best fit to the pitting depth data in the investigated soils.

At the same time, Fig. 4 reveals that the starting distributions areunique for each soil class. The variation in the shape of theMonte-Carlo-simulated pit depth distributions is adequatelyreproduced by the proposed Markov chain model for all soil types,from the beginning of the damage process and over the entiremodelling time span considered, as can be seen from Figs. 5 and 6.

Fig. 7 shows the 30-year pitting rate distribution for the All soilcategory estimated using Eqs. (5) and (6) with t0 = 30 and t = 50years. It is worth noting that the application of Eq. (5) involves acareful selection of the estimation interval Dt. On one hand, atoo-short (<5 years) estimation interval might result in a largeuncertainty in the estimation of the pitting rate distribution. Onthe other hand, too-long estimation intervals would lead to exces-sively averaged estimates of the pitting rate distribution over theselected time span. A 20-year estimation interval was selected inorder to predict the long-term distribution of the pitting rate inthis category and to compare it with the long-term rate recently re-ported by the authors in [16]. The adequacy of the Markov chainapproach for estimating the pitting corrosion rate by means ofEqs. (5) and (6) is supported by the good agreement between thetwo predictions shown in Fig. 7.

3.2. Modelling pitting corrosion from repeated in-line inspections

Another application example of the proposed Markov chainmodel is shown in Fig. 8. An 82-km-long operating pipeline, usedto transport sweet gas since its commissioning in 1981, was in-spected in 2002 and 2007 using magnetic flux leakage in-lineinspection (ILI). The pipeline in case is coal-tar coated with a wallthickness of 9.52 mm (0.37500). The distributions shown in Fig. 8awere obtained by calibrating the ILI tools using a methodology de-scribed elsewhere [18]. The use of calibrated pit depth distribu-tions allows minimizing the negative impact on the results of theanalysis of the measurement errors associated with the inspectiontools. The cross-hatched histogram in Fig. 8a shows the depth dis-tribution of NT02 = 3577 pits2 located and measured by ILI at thebeginning of 2002. The grey shaded histogram shows the depth dis-tribution of NT07 = 3851 pits located and measured by ILI in mid-2007.

In order to apply the proposed model, the empirical depth dis-tribution observed in 2002 was used as the initial distribution sothat t0 = 21 years, Dt = 5.5 years and pm(t0 = 21) = Nm/NT02, withNm being the number of pits with measured depth in the m-thstate. The soil characteristics along the pipeline were assumed tobe those of the All category. Therefore, the values of tsd and m weretaken as 2.9 years and 0.780, respectively (Table 1). The applicationof Eqs. (3), (14) and (16) produced the Markov chain-predicted pit-ting depth distribution in Fig. 8a, shown with the thick line.

The good agreement between the empirical pit depth distribu-tion observed in 2007 and the Markov chain-modelled distributionalso points to the accuracy of the proposed model. The validation ofits capability to correctly predict the evolution of the pit depth dis-tribution in this pipeline points to the fact that the proposed modelcan be used to predict its reliability over a given time span. Forexample, the corrosion rate distribution f(t;t0, t) associated withthe entire population of pit defects in the 2002–2012 period canbe predicted using Eqs. (5) and (6), and then, it can be used to esti-mate the evolution of the pitting damage starting from any year inthis period. Fig. 8b shows the pitting rate distribution predicted forthe entire population of pits in the pipeline using this approachwith: t0 = 21 years, t = 31 years, m = 0.780 and tsd = 2.9 years. Theobtained pitting rate distribution is very close to a GEV distribution

Fig. 4. GEV distribution fitted to the 10-year Monte-Carlo-simulated pit depthdistributions, used as the initial pit depth distribution for the Markov chainmodelling for each soil category. The Markov chain states are 0.1-mm-thick.

2 In both ILI reports, pits were identified as corrosion-caused metal losses with adiameter equal to or less than two times the pipe wall thickness.

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(Eq. (19)) with the shape parameter f being negative, although it isvery close to zero. Therefore, the Weibull and the Gumbel subfam-

ilies of the GEV distribution seem to be appropriate for describingthe pitting rate in the pipeline over the selected estimation period.

Fig. 5. Results of the Markov chain modelling of pitting corrosion compared to the Monte-Carlo-simulated distribution in (a) clay and (b) sandy clay loam soils. The Markovchain states are 0.1-mm-thick.

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3.3. Modelling pitting corrosion in immersion test coupons

With the purpose of illustrating the versatility of the proposedmodel, the final application example involves the modelling oflaboratory-induced pitting corrosion in coupons of 1 � 1 cm2

from API-5L X52 pipeline steel immersed for different periodsof time in a solution with characteristics close to those found insoils. The experimental details and the results of these pittingexperiments were reported recently by Rivas et al. [19]. The greyshaded histograms shown in Fig. 9a represent the pit depth distri-butions of the stable pits induced in the investigated coupons forthe selected immersion times (1, 3, 7, 15, 21 and 30 days). Theobserved time evolution of the mean pit depth is shown inFig. 9b. It was fitted to a power law function with a pitting expo-nent m = 0.4175, while the pitting starting time was assumed tobe negligible (tsd = 0) with respect to the duration of the immer-sion tests.

The coupon thickness was divided in non-overlapping 1-lm-thick discrete states to represent the pitting damage through Mar-kov chains with states ranging from m = 1 to m = N = 150. Theempirical pit depth distribution for a 1-day exposure was used aspm(t0). The total number of pits in the 1-day coupons wasNT1 = 1019, and the non-parametric mean and standard deviationof this distribution were found to be 17.6 and 5.5 lm, respectively

Fig. 6. Comparison of the time evolutions of (a) the mean and (b) standarddeviation of the Markov chain-modelled and Monte-Carlo-simulated pit depthdistributions for all soils. The Markov chain states are 0.1-mm-thick.

Fig. 7. Comparison of the Monte-Carlo-simulated [16] and Markov chain-predictedlong-term pitting corrosion distributions in the All soil category. The Markov chainstates are 0.1-mm-thick.

Fig. 8. Results of the Markov chain modelling of pitting damage evolution fromrepeated in-line inspections. (a) Measured and modelled pit depth distributions. (b)Predicted pitting corrosion rate for the 2002–2012 period. The Markov chain statesare 0.1-mm-thick.

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(see Table 3 in [19]). From pm(t0 = 1) = Nm/NT1, where Nm is thenumber of pits with depth in the m-th state, the damage distribu-tion pn(t) was predicted for the immersion times at which theempirical depth distribution had been obtained.

The pit depth probability distributions predicted for the immer-sion times greater than 1 day using the proposed Markov chainmodel are shown in continuous thick lines in Fig. 9a. Qualitatively,the agreement between the observed and Markov chain-predictedpit depth distributions is reasonably good, even when some of theexperimental depth distributions (t P 15 days) are not strictly uni-modal. The bimodal character of these distributions was explainedby Rivas et al. [19] on the basis of the physicochemical dependenceamong the growing pits in the test coupons. However, this factdoes not seem to impair the suitability of the proposed Markovchain framework for modelling the results of these immersiontests.

The relatively good quality of the Markov chain estimations isfurther corroborated in Fig. 9b and c, where the time evolutionsof the mean and standard deviation of the Markov chain-predictedpit depth distributions are compared with the empirical values. Itcan be seen that, for all immersion times, the predicted meanand standard deviation are within the 95% prediction bands ofthe power law model fitted to the experimental data. It is impor-tant to underline that the largest disagreement is observedbetween the experimental and modelled mean values for the15- and 21-day pit depth distributions (Fig. 9b). This disagreementis most likely related to the fact that the empirical distributions forthese exposure times are associated with a noticeably bimodalcharacter due to the pit interdependence [19]. Nonetheless, the re-sults shown in Fig. 9 are interpreted by the authors as a confirma-tion of the accuracy and adaptability of the proposed Markov chainmodelling framework.

Fig. 9. Empirical and Markov chain-modelled distributions for immersion pitting corrosion tests in pipeline steel. (a) Pit depth distributions. (b) Time evolution of the means.(c) Time evolution of the standard deviations. The Markov chain states are 1-lm-thick.

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4. Concluding remarks

A new Markov chain model for pitting corrosion has been devel-oped and validated using both synthetic and experimental pittingcorrosion data. The use of a continuous-time, non-homogenous lin-ear growth (pure birth) Markov process for modelling pitting cor-rosion is particularly attractive due to the existence of a closedform solution (Eq. (3)) to the system of Kolmogorov’s forwardequations that describe this type of Markov process. The use of thissolution avoids a reduction of the number of states for the sake ofmathematical simplicity or the requirement of assumptions aboutthe staying time of the pitting damage in a given state.

Critical to the development of the proposed model is theassumption that the Markov chain-derived stochastic mean ofthe pitting damage equals the deterministic, empirical mean ofthe process. Such an assumption allows the transition probabilityfunction to be identified only from the pitting starting time andthe pitting exponent (Eq. (14)). Furthermore, at least for the situa-tions in which the power law model accurately describes the pit-ting process, this assumption requires that the functional form ofthe stochastic and deterministic instantaneous pitting rates alsoagree. This supports the idea that the intensities of the transitionsin the Markov process are closely related to the pitting damagerate.

The fact that the transition probability function depends onlyon the pitting starting time tsd and the pitting exponent m can beexplained by considering this function to be directly related tothe corrosion rate (see Eqs. (5), (14) and (15)). The pitting propor-tionality coefficient j in the power law model for pit depth appearsimplicitly in the model when the initial distribution pm(t0) is mea-sured or assumed. One of the advantages of the Markov chain ap-proach over deterministic [14] and other stochastic models [12] forpitting corrosion is that the Markov chain model is able to capturethe dependence of the pitting rate on the pit depth and lifetime.Therefore, it allows for an estimation of not only the probabilitydistribution of the pitting rate associated with the entire pit popu-lation but also such a distribution for a subpopulation within spe-cific lifetime and depth ranges.

A knowledge of the dependence of the pitting starting time andexponent on the environmental characteristics determining thepitting damage allows for a generalisation of the model for a rangeof pitting environments. This was illustrated in the present studyby modelling external pitting corrosion in underground pipelinesin a range of soils. Importantly, this approach could be extendedto investigate pitting corrosion in laboratory conditions. For exam-ple, the effects of chloride concentration and pH on pitting corro-sion evolution in low carbon steels and aluminium are currentlybeing investigated by the authors using the proposed Markov chainmodel.

Two questions that were not answered in this study are of par-amount significance for future work: How can the validity of theassumption about the equality of the stochastic and deterministicpit depth means be assessed? What kind of pit populations can bedescribed by means of the proposed Markov chain model?

In the theory of stochastic processes, the answer to the first ofthese questions is known for situations in which counting pro-cesses are considered (see page 189 in [3]). In the case of pittingcorrosion, the prediction and interpretation of the equality of thestochastic and deterministic pit depth means are not straightfor-ward. Still, the equality of the means seems to hold true for allthe cases studied in this work. The second question seems to be re-lated to the existence and type of the limiting distribution of thestate occupation probabilities after a sufficiently long period oftime [3]. Both aspects are of a fundamental nature, and their inter-

pretation, when Markov chains are used for modelling pitting cor-rosion, is the aim of ongoing research by the authors.

Acknowledgement

This work was performed during a stay of A. Valor at the Na-tional Polytechnic Institute (ESIQIE-IPN), Mexico, under researchproject No. 428817805. The comments provided by the reviewersare greatly appreciated.

Appendix A

The stochastic expression for the instantaneous corrosion ratecan be obtained based on the properties of the negative binomialdistribution [20] and the expression for the probability ps.

Let us assume that the pit depth is in the m-th state at t0. In thiscase, the average value of the damage rate over an interval Dt,starting at t0, can be approximated by:

�tðm; t0;DtÞ ¼ E½n�m�Dt

ðA-1Þ

According to Eq. (3), pm,n(t0, t0 + Dt) has a negative binomial dis-tribution NegBin(r,p) with parameters r = m and p = ps. Therefore,its expected value over Dt is given by [20]:

E½n�m� ¼ mð1� psÞ=ps ðA-2Þ

For the conditions considered in this deduction, the expressionfor ps is:

ps ¼ e�fqðt0þDtÞ�qðt0Þg ðA-3Þ

If expressions (A-2) and (A-3) are substituted into (A-1), theaverage damage rate over Dt becomes:

�tðm; t0;DtÞ ¼ mðefqðt0þDtÞ�qðt0Þg � 1Þ

DtðA-4Þ

Based on the expression for q(t) (Eq. (12)) and using the factthat, for sufficiently small values of x, the exponential function ex

can be approximated by 1 + x, it can be shown that:

�tðm; t0;DtÞ !Dt!0

mmDt

ln 1þ Dtt0

� �ðA-5Þ

This expression can be further simplified by taking into accountthat, for sufficiently small values of x, the function ln(1 + x) can beapproximated by x. Therefore:

tðm; t0Þ !Dt!0

mmt0

ðA-6Þ

This expression represents the stochastically-derived instanta-neous corrosion rate for a pitting damage, whose depth is in them-th state at the moment in time t0.

References

[1] T. Shibata, Statistical and stochastic approaches to localized corrosion,Corrosion 52 (1996) 813–830.

[2] P.M. Aziz, Application of the statistical theory of extreme values to the analysisof maximum pit depth data for aluminium, Corrosion 13 (1956) 495–506.

[3] D.R. Cox, H.D. Miller, The Theory of Stochastic Processes, first ed., Chapman &Hall, CRC, Boca Raton Florida, 1965.

[4] J.W. Provan, E.S. Rodrı́guez III, Part I: Development of Markov description ofpitting corrosion, Corrosion 45 (1989) 178–192.

[5] A. Valor, F. Caleyo, L. Alfonso, D. Rivas, J.M. Hallen, Stochastic modelling ofpitting corrosion: a new model for initiation and growth of multiple corrosionpits, Corros. Sci. 49 (2007) 559–579.

[6] T.B. Morrison, R.G. Worthingham, Reliability of high pressure line pipe underexternal corrosion, in: Proc. 11th ASME Int. Conf. Offshore Mechanics and ArticEng., vol. V, Part B, Book No. H0746B, Calgary, Canada, June 7–12, 1992, pp.401–408.

2206 F. Caleyo et al. / Corrosion Science 51 (2009) 2197–2207

Author's personal copy

[7] H.P. Hong, Inspection and maintenance planning of pipeline under externalcorrosion considering generation of new defects, Structural Safety 21 (1999)203–222.

[8] F. Bolzoni, P. Fassina, G. Fumagalli, L. Lazzari, E. Mazzola, Application ofprobabilistic models to localized corrosion study, Metallurgia Italiana 98(2006) 9–15.

[9] T. Zhang, L. Xiaolan, S. Yawei, M. Guozhe, W. Fuhui, Electrochemical noiseanalysis on the pit corrosion susceptibility of Mg–10Gd–2Y–0.5Zr, AZ91D alloyand pure magnesium using stochastic model, Corros. Sci. 50 (2008) 3500–3507.

[10] S.A. Timashev, M.G. Malyukova, L.V. Poluian, A.V, Bushiskaya, Markovdescription of corrosion defects growth and its application to reliabilitybased inspection and maintenance of pipelines, in: Proc. 7th ASME Int. PipelineConf. IPC2008, Calgary, Canada, Paper IPC2008-64546, September 26–29,2008.

[11] T.B. Morrison, G. Desjardin, Determination of corrosion rates from single in-line inspection of a pipeline, NACE One day seminar, Houston, TX, December 1,1998.

[12] J.L. Alamilla, E. Sosa, Stochastic modelling of corrosion damage propagation inactive sites from field inspection data, Corros. Sci. 50 (2008) 1811–1819.

[13] Y. Katano, K. Miyata, H. Shimizu, T. Isogai, Predictive model for pit growth onunderground pipes, Corrosion 59 (2003) 155–161.

[14] J.C. Velazquez, F. Caleyo, A. Valor, J.M. Hallen, Predictive model for pittingcorrosion in buried il and gas pipelines, Corrosion 65 (2009) 332–342.

[15] E. Parzen, Stochastic Processes, Classics in Applied Mathematics, Society forIndustrial and Applied Mathematics (SIAM), PA, 1999.

[16] F. Caleyo, J.C. Velazquez, A. Valor, J.M. Hallen, Probability distribution of pittingcorrosion depth and rate in underground pipelines: a Monte Carlo study,Corros. Sci. (2009), doi:10.1016/j.corsci.2009.05.019.

[17] S. Coles, An Introduction to Statistical Modelling of Extreme Values, SpringerSeries in Statistics, Springer-Verlag, London, 2001.

[18] F. Caleyo, L. Alfonso, J.H. Espina, J.M. Hallen, Criteria for performanceassessment and calibration of in-line inspections of oil and gas pipelines,Meas. Sci. Technol. 18 (2007) 1787–1799.

[19] D. Rivas, F. Caleyo, A. Valor, J.M. Hallen, Extreme value analysis applied topitting corrosion experiments in low carbon steel: comparison of blockmaxima and peak over threshold approaches, Corros. Sci. 50 (2008) 3193–3204.

[20] E. Parzen, Modern Probability Theory and its Applications, John Wiley andSons, Inc., Wiley Classics Library Edition, New York, 1992.

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