Markov approach to early diagnostics, reliability assessment, residual life and optimal maintenance of pipeline systems

Structural Safety 56 (2015) 68–79

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Markov approach to early diagnostics, reliability assessment, residual life and optimal maintenance of pipeline systems S.A. Timashev a,b, A.V. Bushinskaya a,b,⇑ a b

Science and Engineering Center ‘‘Reliability and Safety of Large Systems and Machines’’, Ural Branch, Russian Academy of Sciences, Russia Ural Federal University, Yekaterinburg, Russia

a r t i c l e

i n f o

Article history: Received 3 September 2014 Received in revised form 3 April 2015 Accepted 14 May 2015

Keywords: Pipeline systems Markov processes Reliability Residual life Entropy Optimal repair time

a b s t r a c t In this paper the pipeline degradation – simultaneous growth of many corrosion defects and reduction of pipe residual strength (burst pressure) is described by Markov processes of pure birth and pure death type, respectively. This allows considering collective (joint) behavior of the set of actively growing defects in the pipeline as a distributed system, and to eliminate restrictions of the classical approach. On the basis of constructed Markov models following methods are proposed: (1) a method for assessing the probability of failure (POF)/reliability of a single defective pipeline cross-section and of a pipeline as a distributed system; (2) a practical assessment of the gamma-percent residual life of pipeline systems (PS); (3) an adequate economic model for assessing the optimum time for performing the next inline inspection (ILI) or PS maintenance/repair, which minimizes maintenance expenditures; (4) method of estimating the information entropy generated by degradation of the defective pipeline cross-section. This permits establishing relations between different physical and probabilistic states of the PS and opens new possibilities for its early diagnostics and optimizing its maintenance. Ó 2015 Elsevier Ltd. All rights reserved.

1. Introduction The main cause of the degradation of pipeline systems is destruction of pipe walls due to corrosion, fatigue damage accumulation, effects of shock loads, etc. The distinctive feature of degradation of such systems is, as a rule, the presence of multiple actively growing defects, each of which is a potential threat to its integrity. Violation of pipeline integrity usually leads to enormous losses, which can reach several million dollars per accident. There are two types of pipeline integrity loss: leak and rupture. In general, a distributed system is a system for which the location of its elements (or groups of its elements) plays an important role from the standpoint of its functioning and, therefore, its analysis. In classical structural reliability theory the pipeline systems is modeled as a chain of series-connected elements (defects). In this case the probability of failure (POF) of a PS is equal to the product of the POFs of all the elements in the structure. The reliability of such a system is lower than the reliability of its elements and with increase of the number of elements (defects) the system reliability ⇑ Corresponding author. E-mail addresses: [email protected] (S.A. Timashev), [email protected] (A.V. Bushinskaya). http://dx.doi.org/10.1016/j.strusafe.2015.05.006 0167-4730/Ó 2015 Elsevier Ltd. All rights reserved.

rapidly decreases. If the number of elements in the system is large, it is practically impossible to create a system with required (high) reliability. The main cause of this is that in the chain model all defects are involved in the POF calculation and essentially influence its value. But, in distributed pipeline systems not all the defects present are capable of creating an input into its POF. To account for this circumstance it was suggested to take into consideration only «significant»defects which can actually affect the system reliability. At the same time there are no recommendations on to how to select the «significant»defects. Practically, to select from the entire set of defects, those which possess this quality, it is necessary to perform fairly complex calculations. From the above it is clear that in order to consistently describe and analyse pipeline reliability a mathematical model is needed which is able to simultaneously account for each defect and at the same time, for the collective behavior of the whole system of defects present in the pipeline. One of the most suitable models for this are Markov processes.Among them, simplest are Markov chains (discrete states, discrete time). Markov chains are used in [1] for describing cumulative damage in the form of fatigue cracks and wear in structures and its elements, using the so called B-models. In [2,3] the theory of Markov models was applied to assess the state of high pressure pipelines. In [2] the growth of corrosion pittings is considered as

S.A. Timashev, A.V. Bushinskaya / Structural Safety 56 (2015) 68–79

69

Nomenclature DEMC MSOP

differential equation and Monte-Carlo (method) maximum safe operating pressure

a Markov chain. In [3] a Markov chain in the form of the Yule model was chosen for consideration because it is the simplest model, which operates with only one transition intensity. However, Markov processes (with a discrete number of states and continuous time) are more universal and adequately describe the true state of thin-wall pipeline systems. Markov processes are described by systems of differential equations and do not depend on the nature of objects and their physical properties. In this sense they are universal and are widely and successfully used in various fields of science and technology: nuclear physics, biology, astronomy, queueing theory, reliability theory, etc. [4–9,1]. Unlike Markov chains, they permit assessment of the probability of finding the system in each of the states and the intensity of transition from one state to another at any time. Examination of literature shows [10–13] that there are no studies on the construction of such Markov models as pure birth (death) Markov process (MP) which describe the degradation of the bearing capacity of a distributed system with a finite set of discrete defects. In order to use these processes, the transition probabilities must not depend on the past, and the sojourn time for a process to be in any particular state should be exponentially distributed. Multiple empirical studies show [10–13] that both conditions take place in most types of technical systems, including pipelines. Assessment of reliability of such systems usually is based on an exponential distribution of pipeline defective cross sections uptime, and does not depend on the previous time of safe operation. 2. Formal description of the pure birth (death) Markov process In reliability analysis of technical systems, their operation is generally regarded as a random process of transition from one state to another, caused by the degradation and failure of its components (elements). This process under certain conditions can be quite strictly described by a Markov process. Consider a system S, which can be in one of the following states S0 ; S1 ; S2 ; . . ., which form a set that is finite or countable and the time t is continuous, i.e., the transition of the system from one state to another is occurring at random unknown beforehand moments of time t. Denote as Sðt Þ the state of system S at the moment of time t. The probability Pi ðtÞ of the system being in the i-th state at time t is called ‘‘probability of an event’’, which consists in that at the moment of time t the system S is in the Si state:

Pi ðt Þ ¼ P½Sðt Þ ¼ Si : A random process that evolves in the system S with discrete states S0 ; S1 ; S2 ; . . ., is called a Markov process, if for any arbitrary moment of time t1 the probability of each of the system states in the future (at t > t 1 ) depends only on the current state the system is in (at t ¼ t1 ), and does not depend on when and how did the system enter this state, i.e., does not depend on system’s behavior in the past (at t < t1 ). The system S is changing only by transition from one state to the closest, adjacent state (from Sn to Snþ1 or to Sn1 ). If at some moment of time t the system S is in the state Sn , then the probability that during an incremental time Dt, which immediately follows t, a transition En ! Enþ1 ðEn1 Þ will take place, is approximately

SDE SF

system of differential equations safety factor

    equal to kn ln Dt, where the quantity kn ln  0 [1=time] and does not depend on how the system S arrived at the current state. This means that the process in consideration is a Markov process [4]. The probability that during the small time span of Dt more than one transition will occur is of higher order of magnitude smaller   than Dt. The quantity kn ln for the pure birth (death) Markov process is called «transition intensity»of system S from one state to another. 3. Pure birth Markov model of corrosion defects growth This section presents a new effective model which describes simultaneous growth of a set of independent corrosion defects (see Fig. 1). Divide the pipe wall thickness into M non-overlapping intervals with numbers i ¼ 1; . . . ; M. The defect depth at moment of time t is the random value dðt Þ, which takes values from the interval ð0; wt, where wt is the pipe wall thickness. The process of the depth growth of a set of defects is considered as a pure birth Markov process. The defect depth with time can only monotonically grow, i.e., at random moments of time can transit from the i-th state only to the ði þ 1Þ-th state. The system of differential equations (SDE) describing this process, which is characterized by a discrete number of states and continuous time, has the form

( dP

1 ðt Þ ¼ k1 P 1 ðtÞ; dt dPi ðt Þ ¼ ki1 Pi1 ðt Þ dt

 ki P i ðtÞ; i ¼ 2; . . . ; M;

ð1Þ

where P i ðtÞ is the probability that the defect depth is in the i-th state at time t; ki is the intensity of transition of the defect depth from the i-th state to the ði þ 1Þ-th state. Carry out a consistent solution of the system of differential equations (1) with initial conditions corresponding to the distribution of states (intervals) of defects depths at the initial moment of time t ¼ 0:

Pi ðT 0 Þ ¼ pi ; i ¼ 1; 2; . . . ; k; Pi ðT 0 Þ ¼ 0; i > k: Here

pi ¼

ni ð0Þ ; i ¼ 1; 2; . . . ; k; N  ð0Þ

ð2Þ

pi is the frequency of occurrence of defect depth in the i-th interval at the initial time t ¼ 0; ni ð0Þ and N  ð0Þ is the number of defects,

Fig. 1. Pipeline segment with multiple corrosion type defects.

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S.A. Timashev, A.V. Bushinskaya / Structural Safety 56 (2015) 68–79

whose depth is in the i-th interval and the total number of defects detected in the pipeline at the initial time t ¼ 0 respectively. The general solution of system (1) with the initial conditions(2) can be written as:

Pi ðt Þ ¼

i X





lij exp kj t ði ¼ 1; . . . ; MÞ;

ð3Þ

j¼1

where

8 l ¼ p ; > > > 11 81 > > > li1; j kkii1 ; i – j; > > kj > > > > > > > > > i1 < X > > < p  li q ; i ¼ j; i 6 k; i > l ¼ > ij > q¼1 > > > > > > i1 > > > > > X > > >  li q ; i ¼ j; i > k: > > : :

The unknown intensities ki (i ¼ 2; 3; . . . ; M) are calculated numerically by sequentially solving equations i X





li;j exp kj ðt1 Þ ;

j¼1 i ðt 1 Þ where pi ¼ nNðt ; i ¼ 2; . . . ; M; ni ðt1 Þ is the number of defects, whose 1Þ

depths are in the i-th interval; Nðt 1 Þ is the total number of defects detected in the pipe segment at the initial time t ¼ t 1 . 4. Adequacy check of the Markov model for corrosion defect growth Model of defects parameters (length, depth and width) growth can be constructed not only on the basis of using ILI and verification results, but also by using a combination of the Monte-Carlo simulation method and the differential equations (DE) method, which adequately describe the defect parameter growth and has the form [6]:

dxðtÞ n ¼ kx ðtÞ; dt

ð5Þ

where t is time, xðtÞ is the defect parameter at time t; k is the proportionality coefficient; n > 0 is the defect parameter growth rate. DE (5) is solved for the initial condition

xð0Þ ¼ x0 :

ð6Þ

For n ¼ 1 the solution of DE (5) with the initial condition (6) has the form

xðtÞ ¼ x0 exp ½kt:

ð7Þ

For n – 1 1  1n xðtÞ ¼ kð1  nÞt þ x1n : 0

Interval #

1 2 3 4 5 6 7 8 9

MPPD

DEMC

P i ð18Þ

P i ð19Þ

P i ð20Þ

P i ð18Þ

P i ð19Þ

P i ð20Þ

0.1369 0.0552 0.0346 0.0176 0.0189 0.0162 0.0136 0.0208 0.6860

0.1226 0.0495 0.0311 0.0158 0.0170 0.0146 0.0122 0.0188 0.7182

0.1098 0.0444 0.0279 0.0142 0.0153 0.0132 0.0110 0.0169 0.7472

0.1445 0.0565 0.0395 0.0230 0.0145 0.0135 0.0155 0.0245 0.6685

0.1365 0.0555 0.0355 0.0225 0.0160 0.0115 0.0115 0.0215 0.6895

0.1290 0.0560 0.0285 0.0250 0.0175 0.0095 0.0110 0.0165 0.7070

ð4Þ

q¼1

Pi ðt 1 Þ ¼ pi ¼

Table 1 Probabilities of finding defects depths (at intervals 1–9 at t ¼ 18; 19; 20 years) using MPPD and DEMC methods.

ð8Þ

To verify the adequacy of Markov process of pure birth (MPPB) compare the results of two independent methods – MPPB and DE with Monte-Carlo (DEMC). This comparison for moments of time t ¼ 18; 19; 20 years is given in Table 1, according to which both methods give probability values P i ðt Þ, which are in very good agreement. The observed discrepancy for the 9-th, failure state is in the range of 2–4.6%; hence, the MPPB method provides for a more conservative estimate. This comparison shows that the DEMC method is an effective tool for verifying adequacy of the MPPB method.

Example 4.1. Apply the empirical Markov model to the growth of defects depths found in a pipeline with following parameters: length L ¼ 10 km, pipe wall thickness wt ¼ 11:2 mm, diameter D ¼ 1200 mm, design life t p ¼ 20 years. Consider the MPPB method applied to simulated defects depths from the start of pipeline operation. The defects depths xi ðt 0 Þ, i ¼ 1; 2; . . . ; N, at the time t ¼ 0 (start of pipeline operation) are simulated as realizations of a random variable with a lognormal distribution by adjusting its parameters m and r so that all N ¼ 2000 realizations are placed in the range ð0; 0:1wt: X ¼ exp ðr  g þ mÞ, where g 2 N ð0; 1Þ. Thus, it is assumed that at initial time t ¼ 0 all existing defects depths are less than 10% of the pipe wall thickness, which is quite plausible. Divide the interval [0; 0.8wt] into eight equal intervals (states), which width h ¼ 1:1 mm (interval size; for example, can be equal to the ILI tool sensitivity threshold (the smallest size of defect which still can be detected by the tool). For a magnetic flux leakage (MFL) tool this threshold is 5–10% wt, for an ultrasound (UT) tool it is 1–2 mm). In our case the last, ninth interval is [0.8wt; wt]. Find the initial distribution of defects depths over constructed states. Since the initial time t0 ¼ 0 the defects depths are less than 10% wt, the initial conditions (2) take the form

P1 ðt 0 Þ ¼ p1 ¼ 1; Pðt 0 Þ ¼ 0; i > 1 To determine the size of defects depths at time t ¼ t 1 use the DEMC combination of methods as described by formulas (5)–(8). MPPB is applicable when the defects depths are spread among all the states, without exclusion. To describe the defect depth growth use the solution (7) of DE (5) for n ¼ 1. From a certain moment of time the defects depths will be distributed over all intervals. The growth rate of each defect is given through the coefficient k from Eq. (7). Modeling of coefficients k is conducted for a uniform distribution with parameters a ¼ 0 and b ¼ kmax . The maximum value of coefficient kmax is chosen such that for t1 ¼ 5 years (after the start of pipeline operation) at least one defect passes into the last interval (at least one maintenance/repair has occurred), i.e., the set of defects is distributed over all states. In this example, kmax was found to be 0.68 (1/year). Now, for any time t > t 1 , using formula (7),for each of the defects its new depth can be assessed and the new PDF, spread over the pipe wall thickness, can be obtained. Figs. 2–4 show the distributions of the obtained defects depth values over the pipe wall thickness at the initial time t0 ¼ 0, as well as at t ¼ 5 and 17 years. The intensities ki ðt1 Þ ¼ ki ; i ¼ 1; 2; . . . ; M of transition from the i-th to the ði þ 1Þ-th state were obtained using formulas

k1 ¼ 

  ln P1 ðt 1 Þ=p1 ; t1

ð9Þ

while other intensities ki ; i ¼ 2; ::; M were determined sequentially by solving the equation

S.A. Timashev, A.V. Bushinskaya / Structural Safety 56 (2015) 68–79

Fig. 2. The initial distribution of defects depths over the pipe wall thickness at t0 ¼ 0 (initial state).

71

Fig. 5. Transition intensities ki of defects depths over pipe wall thickness at t ¼ 5 years.

Fig. 6. Transition intensities ki of defects depths over pipe wall thickness at t ¼ 17 years. Fig. 3. The distribution of defects depths over the pipe wall thickness at t ¼ 5 years (DEMC method).

by substituting in them the found values of t ¼ 5 years (Fig. 7).

lij ; ki for the time

5. Markov model for the pipeline residual strength degradation process Apply now the empirical Markov model to describe the degradation (reduction of the residual strength) of the pipeline, which is assessed by the international design codes: B31G [14], B31Gmod [15], Shell92 [16], DNV [17], PCORRC (Battelle) [18]. 5.1. Residual strength degradation model for a pipeline cross section with a growing defect

Fig. 4. The distribution of defects depths over the pipe wall thickness at t ¼ 17 years (DEMC method).

Pi ðt 1 Þ ¼ pi ¼

i X





lij exp kj t1 ;

ð10Þ

j¼1

where parameters lij were determined by formula (4). The transition intensities for t ¼ 5; 17 years are shown in Figs. 5, 6. The predictive probabilities of finding defects depths in the i-th state for times t > 5 years were calculated by formula

Pi ðt Þ ¼

i X





lij exp kj t :

j¼1

ð11Þ

Consider a pipeline cross section with a defect. The burst (failure) pressure (BP) of a performing pipeline defective cross section at some fixed time t is a random variable (RV) Pf ðtÞ P Pop , where Pop is the pipeline operating pressure. The defect failure pressure is assessed using one of the above five internationally recognized pipeline design codes. Divide the possible range of change of the burst pressure of a   into M  1 pipeline defective cross section P op ; P f ð0Þ non-overlapping equal intervals Ii ði ¼ M  1; . . . ; 1Þ. Here P f ð0Þ is the defect failure pressure at initial time t ¼ 0. The last interval (conditional failure state) IM , which includes the lowest values of failure pressure, is taken as ð0; Pop . The failure pressure of the defective cross section can only monotonically decrease over time, i.e., transit at random moments of time from the i-th state only to the ði þ 1Þ-th state, where state is one of the intervals Ii ði ¼ 1; . . . ; MÞ. The system of differential equations (SDE) that describes this process has the form

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S.A. Timashev, A.V. Bushinskaya / Structural Safety 56 (2015) 68–79

Fig. 7. Predictive probabilities of finding defects depths in the i-th state (MPPB method) for the moments of time t ¼ 6; 7 and 8 years.

8 dP ðtÞ 1 > ¼ l1 ðtÞP1 ðt Þ; > < dt dP i ðt Þ ¼ li1 ðtÞPi1 ðtÞ  li ðtÞPi ðtÞ; ði ¼ 2; ::; M  1Þ; dt > > : dPM ðtÞ ¼ lM1 ðt ÞPM1 ðtÞ; dt

ð12Þ

where Pi ðtÞ is the probability that the BP of defective cross section is in the i-th state at time t; li ðt Þ is the intensity of transition from the i-th state to the ði þ 1Þ-th state. System (12) describes the non-homogeneous pure death Markov process (NPDMP) which is characterized by discrete number of states and continuous time. For the pure birth/pure death Markov processes the flow of events, which transit the system from one state to another, is a non-stationary Poisson process (it is the only flow of events without after-effects, which provides the Markov property) [4]. Hence, according to the definition of a non-stationary Poisson flow of events, the expected number of events, which change the value of burst pressure within a time interval ð0; t, is calculated by formula:

qðtÞ ¼

Z

t

lðsÞds:

ð13Þ

0

The physical meaning of the intensity (density) lðtÞ of the non-stationary Poisson flow of events (transitions from one state to another) is the average number of events per unit time for an

elementary (infinitesimal) time interval ½0; t. The quantity qðtÞ is the average number of states through which the random variable Pf ðt Þ passes within a small time interval [0; t]. The intensity lðtÞ can be expressed by any non-negative function and has the dimension [1/time] [4]. Consequently, the quantity lðt Þ may be associated with the rate of change of RV Pf ðtÞ as follows:

lðtÞ ¼ 

P0f ðt Þ ; DI

where DI is the interval length, P0f ðt Þ is the derivative of the function Pf ðt Þ with respect to time at time t. The minus sign in this formula is due to the fact that the derivative of monotonously decreasing function has negative values in the whole domain of its definition. Now the system (12) can be rewritten as

8 dP ðtÞ 1 > ¼ lðt ÞP1 ðt Þ; > < dt dPi ðt Þ ¼ lðt ÞPi1 ðt Þ  lðtÞPi ðt Þ; ði ¼ 2; ::; M  1Þ; dt > > : dPM ðtÞ ¼ lðt ÞPM1 ðtÞ: dt

ð14Þ

It is obvious that at the initial moment of time t ¼ 0 the RV Pf ð0Þ 2 I1 . Hence, the initial conditions for the SDE (14) will be:

P1 ð0Þ ¼ 1; Pi ð0Þ ¼ 0; ði ¼ 2; ::; M Þ: The general solution of SDE (14) will be as follows:

ð15Þ

S.A. Timashev, A.V. Bushinskaya / Structural Safety 56 (2015) 68–79

8 i1 ðt Þ > P ðtÞ ¼ qði1Þ!  exp fqðt Þg; i ¼ 1; . . . ; M  1; > < i " # M1 X qi1 ðtÞ > >  exp fqðt Þg ; : PM ðt Þ ¼ 1  exp fqðtÞg þ ði1Þ!

ð16Þ

i¼2

where P i ðtÞ is the probability that the BP of defective cross section is in the i-th state at the moment of time t, and the unique value of qðtÞ is calculated by formula

qðtÞ ¼

Z

t

lðsÞds ¼ 

0

Z 0

t

P 0f ðsÞ Pf ðt Þ  Pf ð0Þ : ds ¼ DI DI

ð17Þ

5.2. Model of residual strength degradation of a pipeline with multiple defects as a distributed system The Markov model for a PS with a set of defects regarded as distributed system is constructed using the same reasoning as for a single defective cross section of a pipeline. Assume that at the initial moment of time t ¼ 0, using some inspection tools N defects were found, their geometric parameters sized, and for each defect an estimate of burst pressure was obtained. Now calculate the frequency of occurrence of the failure pressure in each of the i ¼ 1; ::; M intervals (states) at time t ¼ 0:

Pi ð0Þ ¼ pi ¼

ni ði ¼ 1; 2; . . . ; M Þ: N

ð18Þ

where ni is the number of defects, which burst pressure at the moment of time t ¼ 0 is in the i-th interval, N is the overall number of defects. Expressions (18) are the initial conditions for the SDE (14). Solving consistently the SDE (14) with initial conditions (18) by the variation-of-the-constant method, the general solution (14) for a distributed PS with a set of defects takes the form (compare with Eqn. (16))

8 k h i X > ki > > > P ðt Þ ¼ pi qðkiðÞ!tÞ exp fqðtÞg; k ¼ 1; . . . ; M  1; > < k i¼1 " !# M1 k h i > X X > ki >   q ðt Þ > ð t Þ ¼ 1  p þ p exp fqðt Þg; P > 1 i ðkiÞ! : M k¼2

i¼1

where Pi ðtÞ is the probability that the value of burst pressure Pf ðtÞ of the PS with a set of defects is in the i-th state at time t. For a PS with a set of defects the value of its qðtÞ at a given moment of time t is a problem which has to be solved. In this paper the generalized value of qðtÞ which characterizes the PS as a whole is defined as the sample quantile of order a of the set of values qi ðtÞ ði ¼ 1; . . . ; NÞ, where N is the number of defects discovered in the PS. Here qi ðtÞ is the value of qðtÞ of the i-th defect, which is calculated by formula (17). In other words, qðtÞ ¼ qðkÞ ðtÞ, where qðkÞ ðtÞ is the k-th order statistic of the ordered series qð1Þ ðtÞ 6 qð2Þ ðtÞ 6; . . . ; 6 qðNÞ ðtÞ of the sample values of q1 ðtÞ; q2 ðtÞ; . . . ; qN ðtÞ at time t and k ¼ ½aN þ 1, where ½. . . is the integer part of the value in brackets. The order a is chosen out of safety reasons. It should be noted that the quantile approach has an important place in the arsenal of probabilistic tools for analysis of statistical data. By controlling (assigning) the quantile it is possible to assess POF of the whole PS with needed reliability (confidence level). 5.3. Assessment of the gamma-percentile pipeline residual strength Assessment of gamma-percentile of pipeline residual strength is based on Markov pure death process, constructed in Sections 5.1 and 5.2.

73

Assume that the Markov process of residual strength degradation of a pipeline system or its defective cross section is successfully constructed. Denote as T i the time the burst pressure P f ðtÞ is in the subset of states Si ¼ ½I1 ; . . . ; Ii ; 1 6 i < M. According to [4], the cumulative distribution function of RV T i is equal to

e i ðt Þ: F T i ðt Þ ¼ PðT i < t Þ ¼ P e i ðtÞ is the probability that at time t the burst pressure (BP) The P will transit from subset Si to the subset of states Siþ1 ¼ fIiþ1 ; . . . ; IM g, where M is the number of the subset states. In other e i ðtÞ is the probability that BP is found in the subset Siþ1 . words, P The subset Siþ1 is considered to be absorbing, i.e., that burst pressure, once entering this subset, cannot leave it. By this a Markov process is constructed, which has only two states, represented by subsets Si and Siþ1 . Now find, sequentially, the cumulative distribution function of time the burst pressure is in each of the subsets of states. Let time T 1 be the time when the BP is in the first state I1 . Then the second state of this process is the subset S2 ¼ fI2 ; . . . ; IM g. Since the subset S2 is absorbing, from formula (16) it follows that the probability of finding the BP in the first state I1 is equal to exp fqðtÞg. Then the cumulative distribution function of time T 1 takes the form

e 1 ðt Þ ¼ 1  exp fqðt Þg: F T1 ¼ P In this case, for a distributed PS with defects it is necessary that, at the initial time, the failure pressure of all defects is in the first state, as only the time of being in this state is considered. Now find the cumulative distribution function of time T 2 of the BP being in the first and second states, that is, the time after which the BP will transit to the third state. In this case we have the following two subsets of states: S2 ¼ fI1 ; I2 g and S3 ¼ fI3 ; . . . ; IM g. Since in this format subset S3 is absorbing, from formula (16) it follows that the probability of finding the BP in the first and second states is equal to exp fqðtÞg þ qðtÞ exp fqðtÞg. Then the cumulative distribution function of time T 2 takes the form

e 2 ðt Þ ¼ 1  ½1 þ qðt Þ exp fqðt Þg: F T2 ¼ P For a distributed PS with a set of defects

  e 2 ðt Þ ¼ 1  p þ p qðtÞ þ p exp fqðtÞg; F T2 ¼ P 1 1 2 on the condition that the failure pressure for all defects at the initial time t ¼ 0 is found only in the first and second states. Reasoning similarly, the distribution function of time T i the BP being in subset Si ¼ ½I1 ; . . . ; Ii ; 2 6 i 6 M  2, that is, of the time after which the BP will transit to the ði þ 1Þ-th state, has the form

" # i X qj1 ðtÞ e i ðt Þ ¼ 1  exp fqðt Þg þ F Ti ¼ P  exp fqðt Þg : ðj  1Þ! j¼2 For a distributed PS with a set of defects

" !# i k  kj X X ðt Þ   q e F T i ¼ P i ðt Þ ¼ 1  p1 þ pj exp fqðtÞg; ðk  jÞ! j¼1 k¼2 provided the failure pressure of all defects at initial time t ¼ 0 is found only in the first i-th states. Then the distribution function of time T M1 the burst pressure is in the first (M  1)-th states or, in other words, the time after which the burst pressure will transit to the last (conditional failure, or limit) state, is given by formula: for a single defect (cross-section)

74

S.A. Timashev, A.V. Bushinskaya / Structural Safety 56 (2015) 68–79

" F T M1

e M1 ðt Þ ¼ 1  exp fqðtÞg þ ¼P

M 1 X

qj1 ðtÞ

j¼2

ðj  1Þ!

#  exp fqðt Þg :

The pipeline strength related safety factor K for each defective cross section is found, using GPRL and the corresponding value of burst pressure, using following expression:

ð19Þ for a distributed PS with a set of defects

" e M1 ðt Þ ¼ 1  p þ F T M1 ¼ P 1

M1 X

k  X

k¼2

i¼1

pi

qki ðtÞ ðk  iÞ!

K¼ !#

exp fqðt Þg; ð20Þ

provided the failure pressure of all defects at initial time t ¼ 0 is located only in the first ðM  1Þ-th states. Knowing the cumulative distribution function of time T i of failure pressure being in the first i-th states, the time tic can be estimated, at which the failure pressure transits into the ði þ 1Þ-th state with probability c. This is achieved by numerically solving the following equation relative to time t ic :

  t ic : F T i tic  c ¼ 0:

ð21Þ

For time T M1 function (21) takes the form

  t c : F T M1 tc  c ¼ 0:

ð22Þ

Thus, formula (22) gives an estimate of the gamma-percent residual life (GPRL) from time of pipeline diagnostics (t ¼ 0) to time of failure – the moment of time t c , at which the burst pressure   Pf tc 6 P op with probability c, where P op is the pipeline operating pressure. The formulas (19), (20), (22) allows evaluating the GPRL of a single defective cross section as well as a distributed PS with a set of defects, that is, the time during which the defect (PS with defects) does not reach the ‘‘rupture’’ type of failure (limit) state with probability c, given as a percentage:

  c t c : F T M1 tc ¼ 1  : 100

ð23Þ

It should be noted that in formulas (21)–(23) gamma is a value prescribed by codes [19,20]. Hence, the corresponding quantile will  c  be 1  100 . 5.4. Adequacy test of Markov model for the residual strength degradation Check adequacy of the Markov model of residual strength degradation by comparing safety factors (SF) of defects, which correspond to the GPRL calculated by the Markov model, with SF prescribed by the Pipeline Design Code [21]. Estimate the GPRL of a real pipeline, 325 mm in diameter, with a 9 mm thick wall, made out of pipe steel with specified minimum yield strength 245 MPa, ultimate tensile strength 410 MPa and operating under a 6.4 MPa pressure. The last ILI revealed 3372 defects of the «metal loss»type in the pipeline and assessed their true depths. Using the method of conditional maximum rate of defects growth [22,23] it was found that the growth rate of the defects depth and length is 0.20 mm/year and 2.34 mm/year, respectively. Predicted future defects depths and lengths were calculated. Comparative analysis was performed using all the five pipeline design codes: B31G, B31Gmod, DNV, Battelle and Shell92. According to Guidance documents [19,20], assessment of GPRL for potentially dangerous objects, supervised by the Federal Environmental, Industrial and Nuclear Supervision Service, should be performed with c  90%. Consequently, a minimum value of 90% is used below as the value of c.

  Pf tc ; Pop

ð24Þ

  where t c is the defect GPRL obtained using formula (23); Pf t c is the burst pressure at time t c obtained using one of internationally recognized pipeline design codes; Pop is the pipeline operating pressure. Thus, estimates of each defect SF are obtained, in terms of burst pressure, which corresponds to GPRL found at c ¼ 90%. This permits estimating the minimal SF as obtained by the developed Markov model. The calculation results, averaged over the number of defects N ¼ 3372, are listed in Table 2 and in Figs. 8, 9 for two pipeline design codes (B31Gmod and DNV). According to Table 2 the defects GPRLs at c ¼ 90% correspond to average SF K value range from 1.5 to 2.0. In all cases the standard deviation of these SF is less than 0.03 (1.5–2.0% of their mean value). According to the Pipeline Design Code [21], the pipeline design SF K d is in the range from 1.64 to 3.42. Fig. 10 shows the estimated SF values of 3372 defects under consideration which correspond to the above GPRL, assessed via the developed Markov model. Vertical lines in Fig. 10 denote the span of SF values; dots denote the SF mean values. Hence, the SFs of pipeline defective cross sections obtained with GPRL at gamma c ¼ 90% are within the design values as specified by the Pipeline Design Codes (except for B31G) [21].

Table 2 Statistical characteristics of safety factors of defects which correspond to gammapercent residual life (GPRL) at c ¼ 90%. Pipeline design code

Sample mean, years

Sample standard deviation, years

Minimum and maximum values

B31G B31Gmod Battelle DNV Shell-92

1.51 1.67 1.98 2.03 1.84

0.1 0.02 0.02 0.02 0.03

1.39; 1.48; 1.75; 1.78; 1.57;

1.53 1.69 2.01 2.05 1.87

Fig. 8. Histogram of SF of defects, which correspond to GPRL at c ¼ 90% (B31Gmod code).

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 when using MPPB the probability of finding defects depths in the last, conditional failure state, which is equal to ðk  wt; wt, is the PS conditional probability of failure according to the ‘‘leak’’ criterion.  when using MPPD the probability of finding the burst pressure of defective cross section (PS with defects) in the last, conditional failure, state, which is equal to ðPop ; Pa , is the conditional probability of failure of the specific defect (PS with defects), according to the ‘‘rupture’’ criterion.

Fig. 9. Histogram of SF of defects, which correspond to GPRL at c ¼ 90% (DNV code).

Implementation of the method is shown in Figs. 11 and 12, which visualize the movement of the probability mass in time for growing defects using MPPB method. It can be seen that the probability mass is moving from left to right over time, accumulating in the last, conditional failure, state. The probability of finding defects depths (see Fig. 11) and burst pressure (see Fig. 12) in the last state is the POF of the whole PS with defects by the ‘‘leak’’ and ‘‘rupture’’ criterion respectively. This POF is one of the two components needed for risk assessment of operating PS, because risk is POF multiplied by the consequences of failure. The constructed Markov models of pure birth (death) allow assessing the probability of finding the whole set of depths and burst pressures of defects in any of the possible states, the probability of their transition from one state to another, and allow determining the conditional pipeline POF and, on this basis, assessing its integrity and operational risk. 7. Optimization of the timing for performing maintenance/ repair work on a pipeline system The above Markov models allow determining optimal timing of the next inline inspection (ILI) or maintenance/repair [24–26]. The essence of the method proposed below is in minimizing the total cost function for keeping pipeline in safe operation. It has the following form

Sðsor Þ ¼ C 0 P M ðsor Þ þ

Fig. 10. SF of defects, which correspond to GPRL at c ¼ 90%.

6. Method of assessing pipeline system POF using Markov processes Consider the problem of evaluating reliability of a pipeline with multiple actively growing defects of arbitrary size, using Markov models of pure birth (Section 4) and pure death (section 5). The generalized models (MPPB and MPPD) of PS degradation processes of the corrosion growth type and burst pressure reduction type are constructed for a set of defects, which are discovered, identified and sized by ILI tools in a specific pipeline section. Fitting and calibrating of the model is performed using actual ILI data. The method is implemented by analyzing the behavior in time of the whole set of defects which were found in a certain section of the pipeline. In this sense, the method transforms the set of quantities which describe the defects growth and residual strength reduction, into a random process. Evaluation of the pipeline POF with defects can be made using the above MPPB and MPPD. The probability of finding the depth of defects and their burst pressure in the last state is, actually, the probability of failure of the whole pipeline as a system:

sd C ; sor þ sr ILI

ð25Þ

where sor is the optimized time period after which the inspection and repair should be conducted; sd is the design life (or remaining life) of the pipeline; sr is the average time needed to perform the maintenance/repair/rehabilitation (these numbers vary for different companies); C ILI is the cost of in-line inspection or direct assessment of the pipeline; C 0 is the total cost of pipeline failure; P M ðsor Þ is the conditional probability that the defects depth (burst pressure) are in the last (failure) state at moment of time sor , determined by using the pure birth (death) Markov process. The parameter C 0 includes costs of maintenance (repair, renewal), downtime, elimination/mitigation of environmental damage, all types of fines, etc. The relationship sd =ðsor þ sr Þ is the frequency of maintenance/repair. Expression sd C ILI =ðsor þ sr Þ is the amount of expenditures required to perform maintenance/repair, C 0 P M ðsor Þ is the conditional probable loss due to pipeline failure for the cases when the depth or burst pressure of defects are in the last (failure) state. The optimal period between the inspections/repairs can be found as the minimum of the total cost function (25) with respect to sor :

sor : Sðsor Þ ! min : s or

Function Sðsor Þ is continuous and differentiable at any sor : dðsor Þ < wt, where dðsor Þ is the defect depth at time sor , wt is the pipe wall thickness. Hence

S0sor ¼ 

sd ðsor þ sr Þ2

C ILI þ P 0M ðsor ÞC 0 ¼ 0

ð26Þ

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S.A. Timashev, A.V. Bushinskaya / Structural Safety 56 (2015) 68–79

Fig. 11. Probability mass movement (MPPB method) for times t ¼ 6; 7; 8 years.

Fig. 12. Probability mass movement (MPPB method) for times t ¼ 1; 2; ::; 5 years.

where P 0M ðsor Þ is the derivative of function P M ðsor Þ with respect to sor . Thus, optimal timing of diagnostics/control and maintenance/repair sor is determined by numerical solution of Eq. (26). Maintenance/inspection of the pipeline makes sense when the cost of services is less than cost of failure. The optimal time for the next pipeline diagnostics, depending on the ratio C ILI =C 0 is shown in Fig. 13. It can be seen that when this ratio tends to zero the optimal time for conducting the next inspection rapidly decreases. Therefore, it is very important to correctly quantify all the components of risk of pipeline failure. The above model allows optimizing the time of the next diagnostics and maintenance/repair, taking into account the Markov description of defects parameters growth and degradation of pipeline residual strength. Example 7.1. Calculate the optimal time for conducting repair of defective pipeline cross sections for the case described in Section 5.1. Let the cost of the consequences of pipeline failure be C 0 ¼ 106 virtual monetary units (m.u.), cost of ILI C ILI ¼ 250  103 m.u. The pipeline was in operation for sd ¼ 26 years, the average duration

Fig. 13. Optimal time for the next inspection depending on ratio C ILI =C 0 .

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of ILI and repair sr ¼ 0:1 years. Numerical solution of the Eq. (26) with regard to sor gives the optimal time for repair of the defect #1 – 16.5 years and for defect #2 – 9.7 years. 8. Pipeline system entropy as a tool for early diagnostics Preliminary considerations. If the state of a physical system is known in advance it obviously would not make sense to monitor its condition. The data accumulated via monitoring is meaningful only if the current state of the system is not known in advance. Having this in mind, consider a physical system X with some stochastic properties (in our case, a pipeline) which state is known with a degree of uncertainty, which may randomly be in one state or another, as an object, from which data is being gathered. Consider a stochastic physical system X, which can be in a finite set of states: x1 ; x2 ,. . ., xn with probabilities P 1 ; P 2 ,. . ., P n , where P i is the probability that the system X is in state xi . In this case, obviP ously ni¼1 Pi ¼ 1. The entropy HX of system X is the sum of products of probabilities of various system states and the logarithm of these probabilities [2]: n X HX ¼  Pi loga Pi :

ð27Þ

i¼1

Entropy HX has a number of important properties: (1) It is equal to zero when the system can be found only in one state (its probability being equal to unity), while all other states are impossible (their probabilities equal to zero). In this case the information about the system is completely predictable; (2) Its entropy has a maximum, if the system can be found in several states, when the states are equiprobable (all probabilities are equal, and the uncertainty is maximal); (3) Finally, entropy is additive – when several systems are combined into one, the entropy of the composite system is equal to the sum of entropies of the elements which form the whole system. According to [2], the logarithm in (27) may be taken at any base a > 1. A change of base is equivalent to simple multiplication of entropy by a constant, and the choice of the base is equivalent to choosing a particular unit of entropy. For simplicity, in further calculations the natural logarithm is used.

HP ðt Þ ¼

n n X M X X   Hdi ðt Þ ¼  Pij ðt Þ ln Pij ðtÞ ; i¼1

ð29Þ

i¼1 j¼1

where n is the number of defects; Pij ðtÞ is the probability calculated using Eqs. (16) that the RV BP of i-th defect is in the j-th state at time t. According to the properties of entropy, it is equal to zero, when any one probability is equal to unity, and all other probabilities are zero. It occurs at the initial time t ¼ 0 (see initial conditions 15) and at the moment of time when P M ðtÞ ¼ 1 (100% probability of finding the failure pressure in the last, conditional failure, state). Entropy takes its maximal value when all probabilities P i ðtÞ are the same. After reaching its maximal value entropy begins to monotonically decrease to zero, which happens at the moment of time when P M ðtÞ ¼ 1. Hence, entropy can serve as a diagnostic tool suitable for early diagnostics and as a precursor to pipe failure. 8.2. Entropy analysis of a pipeline and its defective cross-sections Construct a graph of entropy change depending on possible values of burst pressure. Consider the previously described 325 mm diameter pipeline with 9 mm wall thickness, made out of steel which specified minimum yield strength is 245 MPa, ultimate tensile strength is 410 MPa and the operating pressure is 6.4 MPa. Use all five design codes for calculating residual strength of the PS. Obviously, the failure pressure of any defect will be somewhere between the operating pressure Pop and the failure pressure of an ideal pipe (without defects) P0f , determined by formula

P0f ¼

2wt  rf : D

Fig. 14 shows the entropy of the considered pipeline as a function of the failure pressure. Failure pressure decreases over time due to defect growth down to the operating pressure P 0f (failure state). Hence, in Fig. 14 entropy changes from right to left, from P0f to Pop . The vertical lines on the left denote the values of P op and the maximum safe operating pressure Ps , calculated using the safety factor K:

K¼

P f ðt Þ Pop

)

P s ðtÞ ¼ K  P op :

In this case K ¼ 1:4; Ps ¼ 8:96 MPa. 8.1. Pipeline system entropy generated by degradation of the residual strength of corrosion type defects In Section 5 it was shown that the burst pressure of a pipeline cross section with defect can only monotonically decrease and move, at random times, from an i-th state only to the ði þ 1Þ-th with probability Pi ðtÞ. Entropy as a function of time, generated by the degradation process taking place in a cross section of the pipeline with a growing corrosion defect which may fail by the «rupture»type of failure is calculated using the formula M X   Hd ðtÞ ¼  Pj ðt Þ ln Pj ðtÞ ;

ð28Þ

j¼1

where Pj ðt Þ, calculated by (16), is the probability that the RV BP of defective cross section in the j-th state at the moment of time t. Assuming that the defects are independent, their failure pressures are also independent. In the considered format all residual strengths of defective pipe cross sections are RVs, which evolve in time. Therefore, entropy of the pipeline as a system with n defective cross sections, can be calculated as

Fig. 14. Pipeline entropy as function of possible values of defects burst pressure.

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Fig. 14 shows that the maximum value of entropy is invariant, being independent of the design code used, but is reached at different values of failure pressure. The same conclusion can be made for the size and growth rate of defects, because the values of BP from P0f to Pop cover the entire set of possible values of failure pressure of pipeline defects – from the moment of the defects formation to the failure condition. Maximum entropy for the considered pipeline is 2.12 and is reached at 10.09 MPa for the B31G code; 11.16 MPa for the B31Gmod code; 12.47 MPa for the Shell92 code, 13.44 MPa for the Battelle code and 13.72 MPa for the DNV code. Moreover, the entropy value at the time of failure (BP ¼ Pop ) is equal to 1.45, and is also invariant in respect to used design codes. Now consider one of the defects of the pipeline in consideration which depth is 10% wt and length is 100 mm. Assume the growth rate of the defect depth ad is equal to 0.3 mm/year, and the growth rate of defect length al is 3 mm/year. Estimate the time needed for the defect to reach its conditional critical and limit states:  times t dp ; t df for the defect depth to reach respectively 80% (critical state) and 100% of the pipe wall thickness (limit state – leak type failure);  time t s for the defect failure pressure to reach the value of maximum safe operating pressure (MSOP) (critical state – rupture type failure);  time tf for the defect failure pressure to reach the value of operating pressure (limit state – rupture type failure).

Fig. 16. Entropy, critical and limit state moments of time for the defective cross section as a function of time (according to the DNV code).

Table 3 Comparison of time of occurrence (years) of critical limit states of the defect and the time of reaching maximum entropy. Pipeline design code

t dp

tdf

ts

tf

Time of reaching maximum entropy

26.99

19.79 18.79 20.77 20.14 17.19

26.55 22.97 22.69 22.26 20.16

16.22 14.56 16.19 15.34 12.49

Construct for this defect its entropy as a function of time and mark on the entropy curve the times of occurrence of the above critical and limit states. The obtained results are shown in Figs. 15, 16 and in Table 3. Vertical dashed lines correspond to the conditional rupture type failure, and the solid lines - to the leak type failure of the defect. According to Figs. 15, 16 and Table 3, the times of occurrence of the deterministic critical and limit states of the defect are located soon after the time entropy reaches its maximum value. This opens the possibility of using entropy analysis (the entropy index – maximum entropy point) for early diagnostics of the pipeline critical condition. Calculate entropy of the whole pipeline, considering it as a system of six sections with defects, which depths are correspondingly: 10%, 20%, 30%, . . ., 60% wt, using formula (29). Assume that the

growth rate of defects depths ad is equal to 0.5 mm/year and the growth rate of their lengths al ¼ 5 mm=year. Results of calculus are shown in Fig. 17, according to which maximum entropy of the whole pipeline system is not invariant with respect to used design codes. This can be explained by that system entropy is defined as the sum of partial entropies of individual pipeline cross sections, each of which generates entropy at its own rate.

Fig. 15. Entropy, critical and limit state moments of time for the defective cross section as a function of time (according to the B31Gmod code).

Fig. 17. Entropy of the whole pipeline system (using all the five pipeline design codes).

B31G B31Gmod Battelle DNV Shell92

21.00

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9. Conclusion The developed and described Markov models of pipeline degradation allow estimating:  the probability of failure/reliability of pipeline systems;  the gamma-percent residual life of pipeline systems;  the optimum time for performing the next ILI or pipeline systems maintenance/repair;  the information entropy generated by degradation of the defective pipeline cross-section. The developed Markov models are capable of describing the joint behavior of the set of growing defects in the pipeline as a distributed system, eliminates the drawback of the classical approach of the structural reliability theory when applied to a pipeline system. A methodology has been developed for estimating the information entropy generated by the Markov process of degradation of corrosion type defects and establishing relationships between physical and probabilistic conditions of structural systems as shown on examples of pipelines. According to the results of conducted analysis, the value of the defect entropy maximumis invariant with respect to the used design codes, defect depth and its corrosion rate, all of which confirms its fundamental nature. This entropy measure could be useful for early diagnostics of pipeline systems condition. In general, the described methodology provides consistent and transparent mathematical logistics for solving main problems associated with reliability and risk based pipeline design, operation and predictive maintenance. The specifics of this approach is in the form of a string of interconnected problems when the output of the first problem serves as the input to the second problem, and so on, until the solution of the last problem does not yield the needed result. Acknowledgments Authors would like to express their sincere gratitude to the Reviewers for their insightful comments which provided authors with a deeper understanding of their own research and facilitated substantial improvement of the text of the manuscript. References [1] Bogdanoff J, Kozin F. Probabilistic models of cumulative damage. N.Y.: John Wiley & Sons; 1985. [2] Ventsel E. Theory of probability. Moscow: Nauka; 1969 [in Russian]. [3] Hong H. Inspection and maintenance planning of pipeline under external corrosion considering generation of new defects. Structural Safety 1999:203–22.

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