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Influence of grade on the reliability of corroding pipelines
ARTICLE IN PRESS
Reliability Engineering and System Safety 93 (2008) 447–455 www.elsevier.com/locate/ress
Influence of grade on the reliability of corroding pipelines M.A. Maesa,, M. Danna, M.M. Salamab a
Department of Civil Engineering, University of Calgary, Calgary, Alta, Canada T2N 1N4 b ConocoPhillips, IE&PM Facilities Engineering, Houston, TX, USA Accepted 11 December 2006 Available online 12 January 2007
Abstract This paper focuses on a comparative analysis of the reliability associated with the evolution of corrosion between normal and highstrength pipe material. The use of high strength steel grades such as X100 and X120 for high pressure gas pipeline in the arctic is currently being considered. To achieve this objective, a time-dependent reliability analysis using variable Y/T ratios in a multiaxial finite strain analysis of thin-walled pipeline is performed. This analysis allows for the consideration of longitudinal grooves and the presence of companion axial tension and bending loads. Limit states models are developed based on suitable strain hardening models for the ultimate behavior of corroded medium and high strength pipeline material. In an application, the evolution of corrosion is modeled in pipelines of different grades that have been subjected to an internal corrosion inspection after a specified time which allows for a Bayesian updating of long-term corrosion estimates and, hence, the derivation of annual probabilities of failure as a function of time. The effect of grade and Y/T is clearly demonstrated. r 2007 Elsevier Ltd. All rights reserved. Keywords: Pipeline corrosion modeling; Y/T ratio; Pipeline reliability; High-strength pipelines
1. Introduction The use of high strength steel grades such as X100 and X120 for high pressure gas pipeline in the arctic is currently being considered [1]. The economic advantage is clear: by increasing strength, the wall thickness and the material cost are reduced. However, there are some problems with this objective: first, most of the popular design codes specify that the allowable stress is either solely or partially dependent on the yield strength. This partly ignores the effect of the strain hardening or the lack thereof. It is well known that high strength steels are marked by a much larger yield to ultimate tensile strength ratio Y/T, which means that much less strain hardening is available in situations involving extreme pressures. Second, high strength pipelines will tend to operate at higher pressures (compared with medium grade pipeline segments having equivalent wall thickness) and higher
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E-mail address: [email protected] (M.A. Maes). 0951-8320/$ - see front matter r 2007 Elsevier Ltd. All rights reserved. doi:10.1016/j.ress.2006.12.009
pressures mean that the material is more susceptible to internal corrosion due to carbon dioxide. Third, an increase of the depth of a corrosion groove along the wall of a pipeline causes a corresponding loss in burst resistance that is quite different depending on grade. High strength steels do not possess the same ability to engage plasticity in zones away from the groove, hence plasticity is locally limited to the groove and the burst pressure is lower than for normal grade steel. There exists therefore a clear need to consider the distinctive properties of high strength steels and to cover a range of yield to ultimate ratios that can extend to 0.95. This need to take into account the effects of a high Y/T ratio and the resulting reduced strain hardening capacity has been pointed out by several authors [2–4]. This paper focuses on a comparative analysis of the reliability associated with the evolution of corrosion in both normal and high-strength pipe material (4X80). This is achieved by considering a variable Y/T ratio in the use of the multiaxial generalization of the flexible Ludwik Law [2] together with a finite strain analysis based on Stewart [3] and Klever [2]. This also allows the consideration of
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longitudinal corrosion grooves of variable depth width and length and the presence of companion axial tension and bending loads. Limit states (LS) models are developed and structural reliability analysis (SRA) is used to explore differences in the risk of failure between various pipeline steel grades. SRA has been used extensively in assessing pipeline design risks [5–7]; however, in this study the emphasis is on the use of suitable strain hardening models for the ultimate behavior of medium and high strength pipeline material. As an example, the evolution of corrosion is modeled in pipelines of different grades; an internal corrosion inspection after a specified time allows for a Bayesian updating of long-term corrosion estimates and, hence, the derivation of annual probabilities of failure as a function of time. The effect of grade and Y/T is clearly demonstrated. 2. Constitutive model A suitable and flexible uniaxial stress–strain constitutive equation that models the strain hardening of pipeline steels accurately is the Ludwik Law [2]. It allows investigation of normal pipeline steel grades as well as high strength steel such as X100 and X120. The model requires the use of true stress and true strain rather than engineering stress and strain: ¼ ln ð1 þ eng Þ,
(1)
s ¼ ð1 þ eng Þ seng .
(2)
The Ludwik Law is based on two parameters sUTS and n. The uniaxial constitutive stress–strain relationship can easily be fitted to data (see, for example [8]) in a true stress vs. true-strain graph where the relationship is a strict power law: e n s ¼ sUTS n , (3) n where s is the (modeled) true stress, e is the true strain, sUTS is the ultimate tensile strength, n is the strain hardening index, and e is the base of the natural logarithm. It has been experimentally verified for a large range of structural steels [3] and it has been applied successfully for normal grade pipeline steels. The strain hardening index n can in fact be interpreted as the true (logarithmic) strain at the maximum load in a simple tensile test [2]:
describe the strain hardening behavior instead of the strain hardening index n for describing the strain hardening behavior. SMYS and SMTS represent the 5%-quantile of the yield stress sy and the ultimate tensile strength sUTS. The expression Y/T is used as an abbreviation for this ratio of SMYS to SMTS: Y =T ¼
SMYS . SMTS
(5)
To relate the ratio Y/T in Eq. (5) to the strain hardening index n, the Ludwik Law (3) and the yield point with (sy, ey) can be used. e n y , (6) Y =T ¼ n where ey is the true yield strain, n is the strain hardening index in the Ludwik Law and e is the base of the natural logarithm. ey is a function of the engineering yield strain ey,eng by virtue of (6) but since y;eng 1, (6) is also valid if the true yield strain ey is replaced by the engineering yield strain ey,eng. In the pipeline industry the yield strain is commonly given by ey,eng ¼ 0.5%. It is not possible to write (6) in a closed-form inverse relationship, but instead we can easily develop [8] an excellent linear approximation: n ¼ 0:462 0:445ðY =TÞ.
(7)
In order to deal with pressure-based limit states, the Prandtl–Reuss equations for incremental plasticity [9] are used. These can most conveniently be expressed using the ‘‘effective stress’’ and the ‘‘octahedral strain’’. The effective stress is a factor square root of 3 times the square root of the distortion energy, which is most easily defined as the second invariant of the deviatoric stress tensor. In the current case, the thin-wall assumption causes the radial stress srr to be negligible with respect to the hoop stress syy and the longitudinal stress sLL. Hence, the effective stress is, for every state of (bi-axial) stress in a thin-walled cylinder, equal to qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi s¯ ¼ s2LL sLL syy þ s2yy . (8)
(4)
Similarly, based on the effective plastic strain increment in the Prandtl–Reuss theory, the expression for the octahedral strain is equal to rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 2 ð þ 2yy þ 2rr Þ. ¯ ¼ (9) 3 LL
allowing easy calibration of n and flexible modeling of materials having different yield/ultimate ratios. Typical values for normal strength grades such as X60 are n ¼ 0:13 . . . 0:15 with experimental values ranging from 0.09 to 0.19. They are smaller for high grade steels (0.04–0.06) due to the smaller yield plateau and the smaller strain hardening capacity. In many limit state functions the ratio of the values of the specified minimum yield strength, SMYS, and the specified minimum tensile strength, SMTS, are used to
It can be checked that both of the above expressions reduce to the simple uniaxial stress–strain relationship in case the loading is fully uniaxial. Since the subsequent results are based on finite strains, the definition of true strains relate the strains to the logarithms of the ratios of the current diameter D and the current wall thickness t to the original parameters D0 and t0 as follows: D yy ¼ ln , (10) D0
n ¼ C ,
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t rr ¼ ln . t0
(11)
The use of effective stress and octahedral strain in the incremental theory of plasticity allows any state of stress and strain to be described using the generalized stress– strain relationship between s¯ and ¯. This isotropic strain–hardening relationship is mathematically identical to the uniaxial relationship (3) but now involves both the effective stress and the octahedral strain: e n ¯n . s¯ ¼ sUTS (12) n 3. Burst limit state without corrosion defects As shown in [3], the Ludwik Law and all of the above assumptions can be applied to a thin-walled cylinder with outer diameter D0 and wall thickness t0 subject to extreme internal pressure, leading to the exact burst pressure in algorithmic form. With the slenderness ratio D0/t0 be defined as d¼
D0 . t0
(13)
A Taylor expansion with respect to d leads to a convenient first-order approximation of the exact algorithmic solution in the following closed-form that is accurate to at least order d2 [3]: n 2 2 n pb ¼ ð1þnÞ=2 sY , (14) d1 eY 3 where n is the Ludwik Law strain hardening coefficient and sY is the yield stress. The model scatter error C1 is an additive error which represents the fact that stress data for given strain levels in the Ludwik Law results are inherently statistically uncertain. The procedure for doing this is discussed in [8] and results in the following expression for the burst pressure including C1: n 2 2 n 2C 1 2 pb ¼ ð1þnÞ=2 sY þ pffiffiffi . (15) d1 eY 3 en d 1 3 The above result is valid for a von-Mises yield surface, and it is a factor 1.15 larger than the result for a Tresca yield surface. Experimental evidence shows that the average of the von Mises and Tresca predictions provides an unbiased prediction of the mean burst capacity [3]. If furthermore the model uncertainty C2 associated with the burst pressure prediction is included, then the following stochastic burst pressure expression including all uncertainties is obtained: 2 pb ¼ 0:935 C 2 d1 n 2 n 2C 1 sY þ pffiffiffi ð1þnÞ=2 , ð16Þ eY 3 en 3
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which is a function of five basic random variables: the yield strength sY, the strain hardening index n, the wall thickness t0, and the LS model uncertainties C1 and C2 which represents the scatter in the stress–strain data as well as the overall burst model uncertainty. The random variable C2 is, as a result of the introduction of the factor 0.935 representing the averaging of the von Mises and Tresca predictions, namely the average of 1 and (1/1.15), unbiased and has therefore zero mean. 4. Limit state for burst combined with tension The limit state for the combined loading condition marked by extreme internal pressure (leading to burst) together with a companion effective axial tension F (which by itself is not large enough to cause collapse or instability) is now considered. Use of the above assumptions [10] yields the following expression for the burst pressure of an undamaged thin tube pb,F: sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 pb;F F , (17) ¼ 1 ð1 nÞ FC pb;F ¼0 where the critical tensile load FC is the ultimate effective tensile load on the cylinder: n n F C ¼ pt20 ðd 1Þ sY . (18) eY It can be noticed that while the interaction equation (17) is dependent on the steel grade and the strain hardening potential, the reduction of the burst pressure for increasing F/FC is quite small because tension mainly affects longitudinal stresses and burst is chiefly caused by plasticity in the high hoop direction. 5. Limit state for burst combined with bending The limit state for the combined loading condition marked by extreme internal pressure (leading to burst) together with a companion bending moment M (which by itself is not large enough to cause collapse or instability) is considered. Use of the above assumptions [10] yields the following expression for the burst pressure of an undamaged thin tube pb,M: sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2ffi pb;M 5 M , (19) ¼ 1 ð1 n13n Þ 1 þ d MC pb;M¼0 where the pure bending collapse moment MC can be derived as [10] 0:82ð1:05 0:0015dÞ n n 2 n
MC ¼ D0 t0 sY . (20) eY exp 2 ð1 nÞ Note that this expression takes into account large strain ovalization and thinning, as well as buckling on the compression side which occurs in more slender sections. The latter aspect is calibrated against the SUPERB [11] findings for bending of pressured pipes.
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6. Burst and combined burst limit states for grooved pipelines In the present section we consider the effect of an infinitely long longitudinal groove. This case represents a ‘‘limit case’’ for internal grooves caused by corrosion. In a subsequent section the case of finite grooves and pits is considered. The assumption of the infinitely long groove allows a plastic plane strain analysis to be performed as shown in [3]. Consider a groove with uniform depth d and width w, such that y¼
d , t0
(21)
j¼
w , pðD0 t0 Þ
(22)
then the analysis can be performed for one of two cases: 1. In the case of a shallow groove (yoy*), the entire crosssection is able to mobilize its strain hardening capacity so that the resulting burst pressure pN of the grooved section is based on plastic strains developing in the entire hoop direction. This is shown by part I of the broken line in Fig. 1 which represents the relative decrease in burst pressure as a function of groove depth. Part II in this figure represents the fact that the burst pressure cannot be less than the burst pressure corresponding to a cylinder having a constant thickness equal to (t0d). 2. In the case of deeper grooves (y4y*), plasticity only occurs locally in or near the groove while the remainder of the cross-section remains elastic. Clearly, this condition causes a steeper decline of burst pressure with defect depth as shown in part III in Fig. 1. The critical consideration is that the threshold value of defect depth y* ¼ d*/t0 at which this transition occurs is itself a function of grade and Y/T, as shown by [3] pffiffiffi n 3eY ð1 þ jÞ j exp . (23) y ¼1 2n 1þj
This is not unexpected as high strength steel will have less ability to re-distribute its plastic stresses and will therefore transit from the first into the second regime more quickly than a medium strength steel grade. Typically, for Y/T values around 0.7 the transition occurs around y* ¼ 0.32 while for Y/T ¼ 0.9 it occurs around y* ¼ 0.1. In the subsequent reliability analysis it is more effective to replace the discontinuous parts I and II in Fig. 1 by the solid line shown for small defect depths, with hardly any loss of accuracy. As a result, the burst pressure pN for the pipeline with infinite groove, is equal to 1 f ð1 y Þ p1 ¼ 1 y pb for ypy , (24) y p1 ¼ f ð1 yÞpb
for yXy ,
(25)
where f is the circumferential corrosion factor given by n 2 (26) f ¼ 1þj and where pb is the undamaged pipe burst expression given by Eq. (16). It can be observed that f is equal to 1 for a fully corroded section (j tends to 100%) in which case, the burst pressure (25) is equal to the burst pressure of a section having a thickness equal to (t0d) rather than t0. On the other hand, when the groove width is very small and corrosion is very localized (j tends to 0), the burst pressure is greater than the one corresponding to a wall thickness of (t0d). The maximum value for f is equal to 2n and is therefore also clearly a function of Y/T albeit a rather weak function. The increase is the smallest for high strength material and the largest for the more ductile lower grades. For instance, it is equal to 4% for X100 and 9.4% for X60. The above groove defect burst expressions have been verified experimentally in [3]. Nine different tests on various defect geometries reveal a bias of 1 (meaning that Eq. (25) is accurate in the mean) and a standard deviation of about 3%. Hence, in the context of an uncertainty analysis we can follow the same procedure as for the full burst LS expressions by including two uncertainty random variables C1 and C2 similar to (16). By extension of the expressions (17) and (19) for companion tension and bending effects, respectively, the limiting burst pressures pN for grooved pipelines subject to extreme internal pressure and companion effective axial tension F and companion bending moment M, are equal to sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2ffi p1;F F (27) ¼ 1 ð1 nÞ FC p1;F ¼0 and p1;M ¼ p1;M¼0
Fig. 1. Assumed burst capacity for infinitely long groove.
sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2ffi 5 M . 1 ð1 n13n Þ 1 þ d MC
(28)
In Eqs. (27) and (28) we subsequently assume that the infinitely long groove only affects the pressure and not
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the critical values of the tension force and the bending moment.
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Eq. (29) to determine the burst resistance associated with a finite parabolic groove of depth d, length L, and width w.
7. Limit states involving finite corrosion defects When the longitudinal groove of depth d has finite length L and width w, then the true burst pressure pL will take on a value that is greater than the burst pressure pN corresponding to the infinitely long groove, while being less than the burst pressure pb for the undamaged pipe section. The most popular approach implemented in both ANSI/ASME B31.4 and B31.8 is based on the so-called ‘‘effective’’ area method, but it is well known to underestimate the true burst pressure [12] and also fails to account directly for the effect of defect width w. A more attractive approach is based on the so-called weighted depth difference method WDDM [13]. The ultimate burst pressure pL is linearly interpolated between the two above bounds using a defect-geometry factor g which varies between 0% and 100%: pL ¼ p1 ð1 gÞ þ pb g.
(29)
The expression for g is purely geometric and independent of crack width w. The original expression for g in [13] can be extended to the case of a defect with variable depth d(x) with a groove-aligned coordinate x running from 0 to L, as shown in [14]. If the crack depth along the groove has an autocorrelation function r(x) then it can be shown [14] that the defect-geometry factor g in Eq. (29) is equal to R þ1 1 ðdðx þ xÞÞ=ðdðx ÞÞ rðxÞ dx gðx Þ ¼ 1 . (30) R þ1 1 rðxÞ dx Often, however, it is worthwhile exploring assumed geometrical defect shapes rather than complex stochastic depth profiles, in order to use them as a basis for convenient user-friendly design or service checks for (1) simple design check equations; or (2) a simple burst rating of in situ corrosion defect measurements [13]. Assuming a parabolic depth variation along the groove of finite length L, with the deepest point dmax ¼ d occurring in the middle of the groove [8], and assuming an exponential correlation pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi function with correlation length D0 ðt0 dÞ [13]: ! jxj (31) rðxÞ ¼ exp pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi D0 ðt0 d Þ then the corrosion groove geometry factor g in Eq. (30) is equal to "pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi # dð1 yÞ 2dð1 yÞ 8dð1 yÞ þ g¼ 4 l l2 l2 ! l exp pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi . ð32Þ 2 dð1 yÞ The results are expressed in terms of the above defect geometry parameters y in Eq. (21), d in Eq. (13), l ¼ L/t0 and w in Eq. (22). This expression can now be used in
8. Application: corrosion inspection and long-term reliability prediction To illustrate the use of the above grade and Y/Tdependent expressions, consider a gas pipeline that has been in service for a period of time [t0 ¼ 0, tinsp ¼ 5 years]. At time t ¼ tinsp it is inspected using a conventional pig. We consider three possible outcomes of this inspection: 1. no corrosion is detected at time t ¼ tinsp ¼ 5 years: dobsodmin,obs, where dmin,obs is the smallest observable crack depth taken to be a random variable with mean 0.06t0 (see Table 1), 2. ‘‘moderate corrosion’’ is detected at time t ¼ tinsp ¼ 5 years: dobs ¼ 0.20t, and 3. ‘‘severe corrosion’’ is detected at time t ¼ tinsp ¼ 5 years: dobs ¼ 0.35t. It is assumed that the average operating pressure of the pipeline section where the largest corrosion is detected decreases linearly from poper;t¼0 to 75% of its original value after 20 years: poper ðtÞ poper ðtÞ t ¼1 ¼ 80 poper;t¼0 poper ðt ¼ 0Þ ðt is expressed in yearsÞ.
ð33Þ
The pipe slenderness ratio d and the wall thickness t0 are taken to be constant irrespective of steel grade (and the Y/T ratio), as shown in Table 1. The original wall thickness contains an allowance for corrosion. The pipeline is assumed to be designed using the three steel grades shown in Table 1. The corresponding Y/T ratios are also given. To ensure fair comparison among the different grades, we assume that the (undamaged) pipeline at time t0 ¼ 0 is initially designed for an annual burst limit state probability of 1 105 irrespective of grade. Because of this approach, the operating pressure poper;t¼0 increases as a function of grade. The operating temperature T, the mean value of the influence factor a of the inhibitor used to reduce corrosive action, the mean mol fraction of CO2 in the gas, and the corrosion groove width factor j are given in Table 1. The groove length is considered to be a random variable with a mean of 20t0 and cov ¼ 10% as indicated in Table 1. The annual corrosion rate n (considered as the prior information with respect to corrosion evolution) is estimated based on the empirical de Waard and Williams formula [11] involving CO2 content nCO2 in mol fraction, time t in years, temperature T in K, and the mean actual operating pressure poper(t) in MPa. This has been widely used for predicting the corrosivity of gas and condensate
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Table 1 Parameters and random variables used in the application Parameters
Value
Pipe slenderness ratio d Original wall thickness t0 Specified yield/ultimate ratio Y/T Operating temperature T Influence factor of inhibitor a Mol fraction of CO2 in the gas nCO2 Corrosion groove width factor j
30 22 mm 0.7, 0.8, 0.9 298 K 0.5 0.02 0.17
Random variables Strain hardening index Yield strength for Y/T ¼ 0.7 Yield strength for Y/T ¼ 0.8 Yield strength for Y/T ¼ 0.9 Wall thickness Max. annual pressure ratio Groove length Model error burst pressure Model error burst pressure Model error corrosion rate Observed corrosion depth Minimum observable corrosion depth Companion axial tension Companion bending moment
n sY sY sY t0/tnom p/poper(t) L C1 C2 Cd dobs dobs,min F/Fc M/Mc
Pdf type
Mean
Cov (in %) or Stand. dev.
Normal Lognormal Lognormal Lognormal Normal Gumbel Normal Normal Lognormal Lognormal Normal Normal Normal Normal
0.462–0.445Y/T 350 MPa 450 MPa 760 MPa 1.02 1.07 20t0 0 1.0 1.0 Actual depth d 0.06t0 0 or 0.3 0 or 0.3
10 6.6 6.6 6.6 3 2 10 10.8–37(0.462–0.445Y/T) 5 20 0.05t0 20 15 15
flow lines. In combination with (33), this expression becomes in mm/year: t 0:67 nðtÞ ¼ 0:935ðnCO2 poper;t¼0 Þ0:67 10ð5:8ð1710=TÞÞ 1 . 80 (34) The maximum depth d of the corrosion groove at time t, including the effect of the corrosion inhibitor a, is thus equal to the integral of (35): Z t dðtÞ ¼ aC d nðzÞ dz, (35) 0
which involves a model uncertainty random variable Cd having mean 1.0 and cov ¼ 20%. The resulting expression for corrosion groove depth is thus as follows: dðtÞ ¼ 44:8C d aðnCO2 poper;t¼0 Þ0:67 10ð5:8ð1710=TÞÞ t 1:67 1 1 . 80
ð36Þ
This random variable can now be updated based on one of the above three inspection outcomes at time t ¼ tinsp ¼ 5 years. This Bayesian updating procedure takes the following form: f ðd jd obs ; t ¼ 5 yearsÞ ¼ cl ðd obs ; t ¼ 5 yearsjd Þf ðdÞ,
(37)
where the left-hand side of the equation represents the posterior pdf given the inspection outcome, f(d) is the prior pdf of the random variable (36) and c is a normalizing constant. The likelihood expression l in Eq. (37) reflects the pig’s probability of detection (POD) potential to observe
dobs given a groove of actual depth d. This is taken as a normal pdf with a mean equal to the actual corrosion depth and a standard deviation of 5% of the original wall thickness. The additional random variables used in the reliability analysis are also given in Table 1. The limit states used in the SRA are all of type gðx; tÞ ¼ pL ðx; tÞ pðx; tÞ,
(38)
where t is time, x is the vector of random variables (based on Table 1), pL(x) is the ultimate burst resistance given by the selected burst expression (one of the above burst capacities) and where p(x) is the actual extreme pressure differential to which the pipeline is subjected during one year. The latter variable is Gumbel distributed with a mean based on the current operating pressure, as shown in Table 1. The tension and moment companion loads are normal random variables with the means depending on the scenario considered (see Table 1). The annual failure probability PF(t) at time t, is then given by PF ðtÞ ¼ Prðgðx; ðtÞÞo0Þ ¼ Pr pL ðx; ðtÞ pðx; tÞÞo0 . (39) It should be emphasized that all of these failure probabilities are conditional on the outcome of the 5-year inspection. Fig. 2 shows the estimated evolution of the annual failure probability as a function of time without inspection. It shows that despite the gradual pressure decrease, the pipeline experiences increasing failure probabilities as time goes on. It is important to notice the effect of Y/T: for high
ARTICLE IN PRESS M.A. Maes et al. / Reliability Engineering and System Safety 93 (2008) 447–455 PF, without inspection, F/FC=0, M/MC=0
PF, obs=0.20, F/FC=0, M/MC=0 1.E-01 Probability of failure PF
Probability of failure PF
1.E-01
1.E-02
1.E-03
1.E-04
1.E-05
453
0
5
10
1.E-03
1.E-04
1.E-05
15
Fig. 2. Annual failure probability for different Y/T ratios, no inspection.
1.E-02
0
5
10
15
Fig. 4. Annual failure probability for different Y/T ratios, 5-year inspection reveals moderate corrosion damage.
PF, obs=0.06, F/FC=0, M/MC=0 PF, obs=0.35, F/FC=0, M/MC=0
1.E-01
Probability of failure PF
1.E-02
1.E-03
1.E-04
1.E-05
0
5
10
15
Fig. 3. Annual failure probability for different Y/T ratios, 5-year inspection reveals no corrosion damage.
strength steel (large Y/T) the limited strain hardening potential causes the uncorroded part of the section to remain elastic under extreme pressure. This clauses the slope of risk vs. time to be steeper than that for milder steels. Fig. 3 shows that if the 5-year pigging inspection reveals no detectable corrosion, then this outcome significantly reduces the need for further inspection in the next 10 years. Only when Y/T is large, a noticeable increase in failure risk can be observed during the next 10 years. Figs. 4 and 5 represent the cases of moderate and severe corrosion damaged observed after the first 5 years of service. Figs. 6 and 7 illustrate the effect of companion effective tension and companion bending, respectively. It appears that the same Y/T dependency can be observed, with the effect being slightly more pronounced for bending than for tension. Fig. 8 represents the annual failure probability after 10 years, given that a 5-year inspection shows moderate corrosion damage. Here the effect of the Y/T ratio of the pipeline material is quite clear: for Y/T ratios exceeding 0.8, the structural safety against burst of the corroded pipe both with and without companion axial actions decreases rapidly with increasing grade.
1.E-02
1.E-03
1.E-04
1.E-05
0
5
10
15
Fig. 5. Annual failure probability for different Y/T ratios, 5-year inspection reveals severe corrosion damage. PF, obs=0.20, F/FC=0 and F/FC=0.30, M/MC=0 1.E-01 Probability of failure PF
Probability of failure PF
1.E-01
1.E-02
1.E-03
1.E-04
1.E-05 0
5
10
15
Fig. 6. Annual failure probability for different Y/T ratios, 5-year inspection reveals moderate corrosion damage: effect of companion effective axial tension.
9. Conclusions This paper focuses on a comparative analysis of the reliability associated with the evolution of corrosion in both normal and high-strength pipe material (4X80). This
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PF , obs=0.20, F/FC=0, M/MC=0 and M/MC=0.30
Probability of failure PF
1.E-01
M/Mc = 0 M/Mc = 0.3
1.E-02
Y/ T = 0.7 Y/ T = 0.8 Y/ T = 0.9
1.E-03 1.E-04 1.E-05
0
5
10
15
T [ years]
Fig. 7. Annual failure probability for different Y/T ratios, 5-year inspection reveals moderate corrosion damage: effect of companion bending moment.
Probability of failure PF
1.E-01
1.E-02
1.E-03
Acknowledgements
1.E-04
1.E-05
derivation of annual probabilities of failure as a function of time. The effect of grade is clearly demonstrated. Despite the economic advantage of increased pipeline strength (mainly in material cost in long pipeline sections), this paper confirms that the long-term action of corrosion in high strength pipelines ought to be properly managed. Since much less strain hardening is available in situations involving extreme pressures, and since high strength pipelines will tend to operate at higher pressures (compared with medium grade pipeline segments having equivalent wall thickness) the susceptibility to internal corrosion increases. But in addition, it appears from the present analysis that the increase of risk per unit time is larger for material with high Y/T ratios. This is because high strength steels do not possess the same ability to engage plasticity in zones away from the groove, hence plasticity is locally limited to the corrosion groove with the result that both the burst pressure and the associated reliability decrease. This has important ramifications for the re-qualification of existing pipelines, for optimal inspection planning, for the required detection ability of inspection tools, and, last but not least, for durability design of pipelines involving high strength steel grades.
0.7
0.75
0.8
0.85
0.9
Y/T
Fig. 8. Annual failure probability after 10 years, given a 5-year inspection showing moderate corrosion damage: effect of the Y/T ratio of the pipeline material.
is achieved by considering a variable Y/T ratio in the use of the multiaxial generalization of the flexible Ludwik Law together with a finite strain plastic analysis based on Stewart [3] and Klever [2]. This also allows the consideration of longitudinal grooves and the presence of companion axial tension and bending loads. It is important to consider the distinctive properties of high strength steels and to cover a range of yield to ultimate ratios that can extend to 0.95 since it has been pointed out by several authors that the reduced strain hardening capacity can potentially give rise to a decrease in structural safety. Limit states models are developed for various extreme pressure situations and for various assumptions about grooves and pits in the interior wall of pipelines. These are used as the basis of a time-dependent structural reliability analysis to illustrate how corrosion and Y/T properties interact. In an application, the evolution of corrosion is modeled in pipelines of different grades; an internal corrosion inspection after five years allows for a Bayesian updating of long-term corrosion estimates and, hence, the
The first author is grateful for funding from the NSERC discovery grant program (Canada) and from ConocoPhillips. The second author acknowledges the support of Prof. Dr. Graubner’s research group at the Technical University of Darmstadt. References [1] Corbett KT, Bowen RR, Petersen CW. High-strength steel pipeline economics. J Offshore Polar Eng 2004;14(1):75–80. [2] Klever FJ. Burst strength of corroded pipe: flow stress revisited. Offshore technology conference (OTC), Houston, 1992. [3] Stewart G, Klever FJ, Ritchie D. An analytical model to predict the burst capacity of pipelines. In: Proceedings of the offshore mechanics & arctic engineering (OMAE1994), vol. V, Houston, 1994. p. 177–87. [4] Zhu X, Leis BN. Influence of yield-to-tensile strength ratio on failure assessment of corroded pipelines. J Pressure Vessel Technol 2005;127: 436–42. [5] Nessim MA, Zhou J. Guidelines for reliability based design and assessment of onshore natural gas pipelines. GRI report no. GRI-04/ 0229, 2005. [6] Leira B. Reliability aspects of Riser and Subsea pipeline design. In: Proceedings of the offshore mechanics & arctic engineering (OMAE1996). 1996. [7] CSA Z662-03. Oil and gas pipeline systems. Mississauga, Ontario: Canadian Standards Association; 2003. [8] Maes MA, Salama MM, Dann M. Reliability of burst limit states for damaged and corroded high strength pipelines. In: Proceeding of the offshore mechanics & arctic engineering (OMAE2006), Hamburg: 2006. [9] Mendelson A. Plasticity: theory and application. New York: MacMillan; 1968. [10] Maes MA. Effect of grade and Y/T on the reliability of the combined bending, tension, and burst limit states for pipelines, Part 1: limit
ARTICLE IN PRESS M.A. Maes et al. / Reliability Engineering and System Safety 93 (2008) 447–455 state model development. Report submitted to ConocoPhillips, February 2006. [11] SUPERB. Buckling and collapse limit state. JIP report, December 1996. [12] Bubenick TA, Oslon RJ, Stephens DR, Francini RB. Analyzing the pressure strength of corroded pipe. In: Proceedings of the offshore
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mechanics & arctic engineering (OMAE1992), Calgary, vol. V, 1992. p. 225–31. [13] Cronin DS, Pick RJ. Prediction of failure pressure for complex corrosion defects. Pressure vessels and piping, 2002. p. 279–87. [14] Maes MA. Burst limit state reliability for corroded pipe based on X100 data. Report submitted to ConocoPhillips, November 2005.
Reliability Engineering and System Safety 93 (2008) 447–455 www.elsevier.com/locate/ress
Influence of grade on the reliability of corroding pipelines M.A. Maesa,, M. Danna, M.M. Salamab a
Department of Civil Engineering, University of Calgary, Calgary, Alta, Canada T2N 1N4 b ConocoPhillips, IE&PM Facilities Engineering, Houston, TX, USA Accepted 11 December 2006 Available online 12 January 2007
Abstract This paper focuses on a comparative analysis of the reliability associated with the evolution of corrosion between normal and highstrength pipe material. The use of high strength steel grades such as X100 and X120 for high pressure gas pipeline in the arctic is currently being considered. To achieve this objective, a time-dependent reliability analysis using variable Y/T ratios in a multiaxial finite strain analysis of thin-walled pipeline is performed. This analysis allows for the consideration of longitudinal grooves and the presence of companion axial tension and bending loads. Limit states models are developed based on suitable strain hardening models for the ultimate behavior of corroded medium and high strength pipeline material. In an application, the evolution of corrosion is modeled in pipelines of different grades that have been subjected to an internal corrosion inspection after a specified time which allows for a Bayesian updating of long-term corrosion estimates and, hence, the derivation of annual probabilities of failure as a function of time. The effect of grade and Y/T is clearly demonstrated. r 2007 Elsevier Ltd. All rights reserved. Keywords: Pipeline corrosion modeling; Y/T ratio; Pipeline reliability; High-strength pipelines
1. Introduction The use of high strength steel grades such as X100 and X120 for high pressure gas pipeline in the arctic is currently being considered [1]. The economic advantage is clear: by increasing strength, the wall thickness and the material cost are reduced. However, there are some problems with this objective: first, most of the popular design codes specify that the allowable stress is either solely or partially dependent on the yield strength. This partly ignores the effect of the strain hardening or the lack thereof. It is well known that high strength steels are marked by a much larger yield to ultimate tensile strength ratio Y/T, which means that much less strain hardening is available in situations involving extreme pressures. Second, high strength pipelines will tend to operate at higher pressures (compared with medium grade pipeline segments having equivalent wall thickness) and higher
Corresponding author. Tel.: +1 403 220 7400; fax: +1 403 282 7026.
E-mail address: [email protected] (M.A. Maes). 0951-8320/$ - see front matter r 2007 Elsevier Ltd. All rights reserved. doi:10.1016/j.ress.2006.12.009
pressures mean that the material is more susceptible to internal corrosion due to carbon dioxide. Third, an increase of the depth of a corrosion groove along the wall of a pipeline causes a corresponding loss in burst resistance that is quite different depending on grade. High strength steels do not possess the same ability to engage plasticity in zones away from the groove, hence plasticity is locally limited to the groove and the burst pressure is lower than for normal grade steel. There exists therefore a clear need to consider the distinctive properties of high strength steels and to cover a range of yield to ultimate ratios that can extend to 0.95. This need to take into account the effects of a high Y/T ratio and the resulting reduced strain hardening capacity has been pointed out by several authors [2–4]. This paper focuses on a comparative analysis of the reliability associated with the evolution of corrosion in both normal and high-strength pipe material (4X80). This is achieved by considering a variable Y/T ratio in the use of the multiaxial generalization of the flexible Ludwik Law [2] together with a finite strain analysis based on Stewart [3] and Klever [2]. This also allows the consideration of
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longitudinal corrosion grooves of variable depth width and length and the presence of companion axial tension and bending loads. Limit states (LS) models are developed and structural reliability analysis (SRA) is used to explore differences in the risk of failure between various pipeline steel grades. SRA has been used extensively in assessing pipeline design risks [5–7]; however, in this study the emphasis is on the use of suitable strain hardening models for the ultimate behavior of medium and high strength pipeline material. As an example, the evolution of corrosion is modeled in pipelines of different grades; an internal corrosion inspection after a specified time allows for a Bayesian updating of long-term corrosion estimates and, hence, the derivation of annual probabilities of failure as a function of time. The effect of grade and Y/T is clearly demonstrated. 2. Constitutive model A suitable and flexible uniaxial stress–strain constitutive equation that models the strain hardening of pipeline steels accurately is the Ludwik Law [2]. It allows investigation of normal pipeline steel grades as well as high strength steel such as X100 and X120. The model requires the use of true stress and true strain rather than engineering stress and strain: ¼ ln ð1 þ eng Þ,
(1)
s ¼ ð1 þ eng Þ seng .
(2)
The Ludwik Law is based on two parameters sUTS and n. The uniaxial constitutive stress–strain relationship can easily be fitted to data (see, for example [8]) in a true stress vs. true-strain graph where the relationship is a strict power law: e n s ¼ sUTS n , (3) n where s is the (modeled) true stress, e is the true strain, sUTS is the ultimate tensile strength, n is the strain hardening index, and e is the base of the natural logarithm. It has been experimentally verified for a large range of structural steels [3] and it has been applied successfully for normal grade pipeline steels. The strain hardening index n can in fact be interpreted as the true (logarithmic) strain at the maximum load in a simple tensile test [2]:
describe the strain hardening behavior instead of the strain hardening index n for describing the strain hardening behavior. SMYS and SMTS represent the 5%-quantile of the yield stress sy and the ultimate tensile strength sUTS. The expression Y/T is used as an abbreviation for this ratio of SMYS to SMTS: Y =T ¼
SMYS . SMTS
(5)
To relate the ratio Y/T in Eq. (5) to the strain hardening index n, the Ludwik Law (3) and the yield point with (sy, ey) can be used. e n y , (6) Y =T ¼ n where ey is the true yield strain, n is the strain hardening index in the Ludwik Law and e is the base of the natural logarithm. ey is a function of the engineering yield strain ey,eng by virtue of (6) but since y;eng 1, (6) is also valid if the true yield strain ey is replaced by the engineering yield strain ey,eng. In the pipeline industry the yield strain is commonly given by ey,eng ¼ 0.5%. It is not possible to write (6) in a closed-form inverse relationship, but instead we can easily develop [8] an excellent linear approximation: n ¼ 0:462 0:445ðY =TÞ.
(7)
In order to deal with pressure-based limit states, the Prandtl–Reuss equations for incremental plasticity [9] are used. These can most conveniently be expressed using the ‘‘effective stress’’ and the ‘‘octahedral strain’’. The effective stress is a factor square root of 3 times the square root of the distortion energy, which is most easily defined as the second invariant of the deviatoric stress tensor. In the current case, the thin-wall assumption causes the radial stress srr to be negligible with respect to the hoop stress syy and the longitudinal stress sLL. Hence, the effective stress is, for every state of (bi-axial) stress in a thin-walled cylinder, equal to qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi s¯ ¼ s2LL sLL syy þ s2yy . (8)
(4)
Similarly, based on the effective plastic strain increment in the Prandtl–Reuss theory, the expression for the octahedral strain is equal to rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 2 ð þ 2yy þ 2rr Þ. ¯ ¼ (9) 3 LL
allowing easy calibration of n and flexible modeling of materials having different yield/ultimate ratios. Typical values for normal strength grades such as X60 are n ¼ 0:13 . . . 0:15 with experimental values ranging from 0.09 to 0.19. They are smaller for high grade steels (0.04–0.06) due to the smaller yield plateau and the smaller strain hardening capacity. In many limit state functions the ratio of the values of the specified minimum yield strength, SMYS, and the specified minimum tensile strength, SMTS, are used to
It can be checked that both of the above expressions reduce to the simple uniaxial stress–strain relationship in case the loading is fully uniaxial. Since the subsequent results are based on finite strains, the definition of true strains relate the strains to the logarithms of the ratios of the current diameter D and the current wall thickness t to the original parameters D0 and t0 as follows: D yy ¼ ln , (10) D0
n ¼ C ,
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t rr ¼ ln . t0
(11)
The use of effective stress and octahedral strain in the incremental theory of plasticity allows any state of stress and strain to be described using the generalized stress– strain relationship between s¯ and ¯. This isotropic strain–hardening relationship is mathematically identical to the uniaxial relationship (3) but now involves both the effective stress and the octahedral strain: e n ¯n . s¯ ¼ sUTS (12) n 3. Burst limit state without corrosion defects As shown in [3], the Ludwik Law and all of the above assumptions can be applied to a thin-walled cylinder with outer diameter D0 and wall thickness t0 subject to extreme internal pressure, leading to the exact burst pressure in algorithmic form. With the slenderness ratio D0/t0 be defined as d¼
D0 . t0
(13)
A Taylor expansion with respect to d leads to a convenient first-order approximation of the exact algorithmic solution in the following closed-form that is accurate to at least order d2 [3]: n 2 2 n pb ¼ ð1þnÞ=2 sY , (14) d1 eY 3 where n is the Ludwik Law strain hardening coefficient and sY is the yield stress. The model scatter error C1 is an additive error which represents the fact that stress data for given strain levels in the Ludwik Law results are inherently statistically uncertain. The procedure for doing this is discussed in [8] and results in the following expression for the burst pressure including C1: n 2 2 n 2C 1 2 pb ¼ ð1þnÞ=2 sY þ pffiffiffi . (15) d1 eY 3 en d 1 3 The above result is valid for a von-Mises yield surface, and it is a factor 1.15 larger than the result for a Tresca yield surface. Experimental evidence shows that the average of the von Mises and Tresca predictions provides an unbiased prediction of the mean burst capacity [3]. If furthermore the model uncertainty C2 associated with the burst pressure prediction is included, then the following stochastic burst pressure expression including all uncertainties is obtained: 2 pb ¼ 0:935 C 2 d1 n 2 n 2C 1 sY þ pffiffiffi ð1þnÞ=2 , ð16Þ eY 3 en 3
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which is a function of five basic random variables: the yield strength sY, the strain hardening index n, the wall thickness t0, and the LS model uncertainties C1 and C2 which represents the scatter in the stress–strain data as well as the overall burst model uncertainty. The random variable C2 is, as a result of the introduction of the factor 0.935 representing the averaging of the von Mises and Tresca predictions, namely the average of 1 and (1/1.15), unbiased and has therefore zero mean. 4. Limit state for burst combined with tension The limit state for the combined loading condition marked by extreme internal pressure (leading to burst) together with a companion effective axial tension F (which by itself is not large enough to cause collapse or instability) is now considered. Use of the above assumptions [10] yields the following expression for the burst pressure of an undamaged thin tube pb,F: sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 pb;F F , (17) ¼ 1 ð1 nÞ FC pb;F ¼0 where the critical tensile load FC is the ultimate effective tensile load on the cylinder: n n F C ¼ pt20 ðd 1Þ sY . (18) eY It can be noticed that while the interaction equation (17) is dependent on the steel grade and the strain hardening potential, the reduction of the burst pressure for increasing F/FC is quite small because tension mainly affects longitudinal stresses and burst is chiefly caused by plasticity in the high hoop direction. 5. Limit state for burst combined with bending The limit state for the combined loading condition marked by extreme internal pressure (leading to burst) together with a companion bending moment M (which by itself is not large enough to cause collapse or instability) is considered. Use of the above assumptions [10] yields the following expression for the burst pressure of an undamaged thin tube pb,M: sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2ffi pb;M 5 M , (19) ¼ 1 ð1 n13n Þ 1 þ d MC pb;M¼0 where the pure bending collapse moment MC can be derived as [10] 0:82ð1:05 0:0015dÞ n n 2 n
MC ¼ D0 t0 sY . (20) eY exp 2 ð1 nÞ Note that this expression takes into account large strain ovalization and thinning, as well as buckling on the compression side which occurs in more slender sections. The latter aspect is calibrated against the SUPERB [11] findings for bending of pressured pipes.
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6. Burst and combined burst limit states for grooved pipelines In the present section we consider the effect of an infinitely long longitudinal groove. This case represents a ‘‘limit case’’ for internal grooves caused by corrosion. In a subsequent section the case of finite grooves and pits is considered. The assumption of the infinitely long groove allows a plastic plane strain analysis to be performed as shown in [3]. Consider a groove with uniform depth d and width w, such that y¼
d , t0
(21)
j¼
w , pðD0 t0 Þ
(22)
then the analysis can be performed for one of two cases: 1. In the case of a shallow groove (yoy*), the entire crosssection is able to mobilize its strain hardening capacity so that the resulting burst pressure pN of the grooved section is based on plastic strains developing in the entire hoop direction. This is shown by part I of the broken line in Fig. 1 which represents the relative decrease in burst pressure as a function of groove depth. Part II in this figure represents the fact that the burst pressure cannot be less than the burst pressure corresponding to a cylinder having a constant thickness equal to (t0d). 2. In the case of deeper grooves (y4y*), plasticity only occurs locally in or near the groove while the remainder of the cross-section remains elastic. Clearly, this condition causes a steeper decline of burst pressure with defect depth as shown in part III in Fig. 1. The critical consideration is that the threshold value of defect depth y* ¼ d*/t0 at which this transition occurs is itself a function of grade and Y/T, as shown by [3] pffiffiffi n 3eY ð1 þ jÞ j exp . (23) y ¼1 2n 1þj
This is not unexpected as high strength steel will have less ability to re-distribute its plastic stresses and will therefore transit from the first into the second regime more quickly than a medium strength steel grade. Typically, for Y/T values around 0.7 the transition occurs around y* ¼ 0.32 while for Y/T ¼ 0.9 it occurs around y* ¼ 0.1. In the subsequent reliability analysis it is more effective to replace the discontinuous parts I and II in Fig. 1 by the solid line shown for small defect depths, with hardly any loss of accuracy. As a result, the burst pressure pN for the pipeline with infinite groove, is equal to 1 f ð1 y Þ p1 ¼ 1 y pb for ypy , (24) y p1 ¼ f ð1 yÞpb
for yXy ,
(25)
where f is the circumferential corrosion factor given by n 2 (26) f ¼ 1þj and where pb is the undamaged pipe burst expression given by Eq. (16). It can be observed that f is equal to 1 for a fully corroded section (j tends to 100%) in which case, the burst pressure (25) is equal to the burst pressure of a section having a thickness equal to (t0d) rather than t0. On the other hand, when the groove width is very small and corrosion is very localized (j tends to 0), the burst pressure is greater than the one corresponding to a wall thickness of (t0d). The maximum value for f is equal to 2n and is therefore also clearly a function of Y/T albeit a rather weak function. The increase is the smallest for high strength material and the largest for the more ductile lower grades. For instance, it is equal to 4% for X100 and 9.4% for X60. The above groove defect burst expressions have been verified experimentally in [3]. Nine different tests on various defect geometries reveal a bias of 1 (meaning that Eq. (25) is accurate in the mean) and a standard deviation of about 3%. Hence, in the context of an uncertainty analysis we can follow the same procedure as for the full burst LS expressions by including two uncertainty random variables C1 and C2 similar to (16). By extension of the expressions (17) and (19) for companion tension and bending effects, respectively, the limiting burst pressures pN for grooved pipelines subject to extreme internal pressure and companion effective axial tension F and companion bending moment M, are equal to sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2ffi p1;F F (27) ¼ 1 ð1 nÞ FC p1;F ¼0 and p1;M ¼ p1;M¼0
Fig. 1. Assumed burst capacity for infinitely long groove.
sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2ffi 5 M . 1 ð1 n13n Þ 1 þ d MC
(28)
In Eqs. (27) and (28) we subsequently assume that the infinitely long groove only affects the pressure and not
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the critical values of the tension force and the bending moment.
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Eq. (29) to determine the burst resistance associated with a finite parabolic groove of depth d, length L, and width w.
7. Limit states involving finite corrosion defects When the longitudinal groove of depth d has finite length L and width w, then the true burst pressure pL will take on a value that is greater than the burst pressure pN corresponding to the infinitely long groove, while being less than the burst pressure pb for the undamaged pipe section. The most popular approach implemented in both ANSI/ASME B31.4 and B31.8 is based on the so-called ‘‘effective’’ area method, but it is well known to underestimate the true burst pressure [12] and also fails to account directly for the effect of defect width w. A more attractive approach is based on the so-called weighted depth difference method WDDM [13]. The ultimate burst pressure pL is linearly interpolated between the two above bounds using a defect-geometry factor g which varies between 0% and 100%: pL ¼ p1 ð1 gÞ þ pb g.
(29)
The expression for g is purely geometric and independent of crack width w. The original expression for g in [13] can be extended to the case of a defect with variable depth d(x) with a groove-aligned coordinate x running from 0 to L, as shown in [14]. If the crack depth along the groove has an autocorrelation function r(x) then it can be shown [14] that the defect-geometry factor g in Eq. (29) is equal to R þ1 1 ðdðx þ xÞÞ=ðdðx ÞÞ rðxÞ dx gðx Þ ¼ 1 . (30) R þ1 1 rðxÞ dx Often, however, it is worthwhile exploring assumed geometrical defect shapes rather than complex stochastic depth profiles, in order to use them as a basis for convenient user-friendly design or service checks for (1) simple design check equations; or (2) a simple burst rating of in situ corrosion defect measurements [13]. Assuming a parabolic depth variation along the groove of finite length L, with the deepest point dmax ¼ d occurring in the middle of the groove [8], and assuming an exponential correlation pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi function with correlation length D0 ðt0 dÞ [13]: ! jxj (31) rðxÞ ¼ exp pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi D0 ðt0 d Þ then the corrosion groove geometry factor g in Eq. (30) is equal to "pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi # dð1 yÞ 2dð1 yÞ 8dð1 yÞ þ g¼ 4 l l2 l2 ! l exp pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi . ð32Þ 2 dð1 yÞ The results are expressed in terms of the above defect geometry parameters y in Eq. (21), d in Eq. (13), l ¼ L/t0 and w in Eq. (22). This expression can now be used in
8. Application: corrosion inspection and long-term reliability prediction To illustrate the use of the above grade and Y/Tdependent expressions, consider a gas pipeline that has been in service for a period of time [t0 ¼ 0, tinsp ¼ 5 years]. At time t ¼ tinsp it is inspected using a conventional pig. We consider three possible outcomes of this inspection: 1. no corrosion is detected at time t ¼ tinsp ¼ 5 years: dobsodmin,obs, where dmin,obs is the smallest observable crack depth taken to be a random variable with mean 0.06t0 (see Table 1), 2. ‘‘moderate corrosion’’ is detected at time t ¼ tinsp ¼ 5 years: dobs ¼ 0.20t, and 3. ‘‘severe corrosion’’ is detected at time t ¼ tinsp ¼ 5 years: dobs ¼ 0.35t. It is assumed that the average operating pressure of the pipeline section where the largest corrosion is detected decreases linearly from poper;t¼0 to 75% of its original value after 20 years: poper ðtÞ poper ðtÞ t ¼1 ¼ 80 poper;t¼0 poper ðt ¼ 0Þ ðt is expressed in yearsÞ.
ð33Þ
The pipe slenderness ratio d and the wall thickness t0 are taken to be constant irrespective of steel grade (and the Y/T ratio), as shown in Table 1. The original wall thickness contains an allowance for corrosion. The pipeline is assumed to be designed using the three steel grades shown in Table 1. The corresponding Y/T ratios are also given. To ensure fair comparison among the different grades, we assume that the (undamaged) pipeline at time t0 ¼ 0 is initially designed for an annual burst limit state probability of 1 105 irrespective of grade. Because of this approach, the operating pressure poper;t¼0 increases as a function of grade. The operating temperature T, the mean value of the influence factor a of the inhibitor used to reduce corrosive action, the mean mol fraction of CO2 in the gas, and the corrosion groove width factor j are given in Table 1. The groove length is considered to be a random variable with a mean of 20t0 and cov ¼ 10% as indicated in Table 1. The annual corrosion rate n (considered as the prior information with respect to corrosion evolution) is estimated based on the empirical de Waard and Williams formula [11] involving CO2 content nCO2 in mol fraction, time t in years, temperature T in K, and the mean actual operating pressure poper(t) in MPa. This has been widely used for predicting the corrosivity of gas and condensate
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Table 1 Parameters and random variables used in the application Parameters
Value
Pipe slenderness ratio d Original wall thickness t0 Specified yield/ultimate ratio Y/T Operating temperature T Influence factor of inhibitor a Mol fraction of CO2 in the gas nCO2 Corrosion groove width factor j
30 22 mm 0.7, 0.8, 0.9 298 K 0.5 0.02 0.17
Random variables Strain hardening index Yield strength for Y/T ¼ 0.7 Yield strength for Y/T ¼ 0.8 Yield strength for Y/T ¼ 0.9 Wall thickness Max. annual pressure ratio Groove length Model error burst pressure Model error burst pressure Model error corrosion rate Observed corrosion depth Minimum observable corrosion depth Companion axial tension Companion bending moment
n sY sY sY t0/tnom p/poper(t) L C1 C2 Cd dobs dobs,min F/Fc M/Mc
Pdf type
Mean
Cov (in %) or Stand. dev.
Normal Lognormal Lognormal Lognormal Normal Gumbel Normal Normal Lognormal Lognormal Normal Normal Normal Normal
0.462–0.445Y/T 350 MPa 450 MPa 760 MPa 1.02 1.07 20t0 0 1.0 1.0 Actual depth d 0.06t0 0 or 0.3 0 or 0.3
10 6.6 6.6 6.6 3 2 10 10.8–37(0.462–0.445Y/T) 5 20 0.05t0 20 15 15
flow lines. In combination with (33), this expression becomes in mm/year: t 0:67 nðtÞ ¼ 0:935ðnCO2 poper;t¼0 Þ0:67 10ð5:8ð1710=TÞÞ 1 . 80 (34) The maximum depth d of the corrosion groove at time t, including the effect of the corrosion inhibitor a, is thus equal to the integral of (35): Z t dðtÞ ¼ aC d nðzÞ dz, (35) 0
which involves a model uncertainty random variable Cd having mean 1.0 and cov ¼ 20%. The resulting expression for corrosion groove depth is thus as follows: dðtÞ ¼ 44:8C d aðnCO2 poper;t¼0 Þ0:67 10ð5:8ð1710=TÞÞ t 1:67 1 1 . 80
ð36Þ
This random variable can now be updated based on one of the above three inspection outcomes at time t ¼ tinsp ¼ 5 years. This Bayesian updating procedure takes the following form: f ðd jd obs ; t ¼ 5 yearsÞ ¼ cl ðd obs ; t ¼ 5 yearsjd Þf ðdÞ,
(37)
where the left-hand side of the equation represents the posterior pdf given the inspection outcome, f(d) is the prior pdf of the random variable (36) and c is a normalizing constant. The likelihood expression l in Eq. (37) reflects the pig’s probability of detection (POD) potential to observe
dobs given a groove of actual depth d. This is taken as a normal pdf with a mean equal to the actual corrosion depth and a standard deviation of 5% of the original wall thickness. The additional random variables used in the reliability analysis are also given in Table 1. The limit states used in the SRA are all of type gðx; tÞ ¼ pL ðx; tÞ pðx; tÞ,
(38)
where t is time, x is the vector of random variables (based on Table 1), pL(x) is the ultimate burst resistance given by the selected burst expression (one of the above burst capacities) and where p(x) is the actual extreme pressure differential to which the pipeline is subjected during one year. The latter variable is Gumbel distributed with a mean based on the current operating pressure, as shown in Table 1. The tension and moment companion loads are normal random variables with the means depending on the scenario considered (see Table 1). The annual failure probability PF(t) at time t, is then given by PF ðtÞ ¼ Prðgðx; ðtÞÞo0Þ ¼ Pr pL ðx; ðtÞ pðx; tÞÞo0 . (39) It should be emphasized that all of these failure probabilities are conditional on the outcome of the 5-year inspection. Fig. 2 shows the estimated evolution of the annual failure probability as a function of time without inspection. It shows that despite the gradual pressure decrease, the pipeline experiences increasing failure probabilities as time goes on. It is important to notice the effect of Y/T: for high
ARTICLE IN PRESS M.A. Maes et al. / Reliability Engineering and System Safety 93 (2008) 447–455 PF, without inspection, F/FC=0, M/MC=0
PF, obs=0.20, F/FC=0, M/MC=0 1.E-01 Probability of failure PF
Probability of failure PF
1.E-01
1.E-02
1.E-03
1.E-04
1.E-05
453
0
5
10
1.E-03
1.E-04
1.E-05
15
Fig. 2. Annual failure probability for different Y/T ratios, no inspection.
1.E-02
0
5
10
15
Fig. 4. Annual failure probability for different Y/T ratios, 5-year inspection reveals moderate corrosion damage.
PF, obs=0.06, F/FC=0, M/MC=0 PF, obs=0.35, F/FC=0, M/MC=0
1.E-01
Probability of failure PF
1.E-02
1.E-03
1.E-04
1.E-05
0
5
10
15
Fig. 3. Annual failure probability for different Y/T ratios, 5-year inspection reveals no corrosion damage.
strength steel (large Y/T) the limited strain hardening potential causes the uncorroded part of the section to remain elastic under extreme pressure. This clauses the slope of risk vs. time to be steeper than that for milder steels. Fig. 3 shows that if the 5-year pigging inspection reveals no detectable corrosion, then this outcome significantly reduces the need for further inspection in the next 10 years. Only when Y/T is large, a noticeable increase in failure risk can be observed during the next 10 years. Figs. 4 and 5 represent the cases of moderate and severe corrosion damaged observed after the first 5 years of service. Figs. 6 and 7 illustrate the effect of companion effective tension and companion bending, respectively. It appears that the same Y/T dependency can be observed, with the effect being slightly more pronounced for bending than for tension. Fig. 8 represents the annual failure probability after 10 years, given that a 5-year inspection shows moderate corrosion damage. Here the effect of the Y/T ratio of the pipeline material is quite clear: for Y/T ratios exceeding 0.8, the structural safety against burst of the corroded pipe both with and without companion axial actions decreases rapidly with increasing grade.
1.E-02
1.E-03
1.E-04
1.E-05
0
5
10
15
Fig. 5. Annual failure probability for different Y/T ratios, 5-year inspection reveals severe corrosion damage. PF, obs=0.20, F/FC=0 and F/FC=0.30, M/MC=0 1.E-01 Probability of failure PF
Probability of failure PF
1.E-01
1.E-02
1.E-03
1.E-04
1.E-05 0
5
10
15
Fig. 6. Annual failure probability for different Y/T ratios, 5-year inspection reveals moderate corrosion damage: effect of companion effective axial tension.
9. Conclusions This paper focuses on a comparative analysis of the reliability associated with the evolution of corrosion in both normal and high-strength pipe material (4X80). This
ARTICLE IN PRESS M.A. Maes et al. / Reliability Engineering and System Safety 93 (2008) 447–455
454
PF , obs=0.20, F/FC=0, M/MC=0 and M/MC=0.30
Probability of failure PF
1.E-01
M/Mc = 0 M/Mc = 0.3
1.E-02
Y/ T = 0.7 Y/ T = 0.8 Y/ T = 0.9
1.E-03 1.E-04 1.E-05
0
5
10
15
T [ years]
Fig. 7. Annual failure probability for different Y/T ratios, 5-year inspection reveals moderate corrosion damage: effect of companion bending moment.
Probability of failure PF
1.E-01
1.E-02
1.E-03
Acknowledgements
1.E-04
1.E-05
derivation of annual probabilities of failure as a function of time. The effect of grade is clearly demonstrated. Despite the economic advantage of increased pipeline strength (mainly in material cost in long pipeline sections), this paper confirms that the long-term action of corrosion in high strength pipelines ought to be properly managed. Since much less strain hardening is available in situations involving extreme pressures, and since high strength pipelines will tend to operate at higher pressures (compared with medium grade pipeline segments having equivalent wall thickness) the susceptibility to internal corrosion increases. But in addition, it appears from the present analysis that the increase of risk per unit time is larger for material with high Y/T ratios. This is because high strength steels do not possess the same ability to engage plasticity in zones away from the groove, hence plasticity is locally limited to the corrosion groove with the result that both the burst pressure and the associated reliability decrease. This has important ramifications for the re-qualification of existing pipelines, for optimal inspection planning, for the required detection ability of inspection tools, and, last but not least, for durability design of pipelines involving high strength steel grades.
0.7
0.75
0.8
0.85
0.9
Y/T
Fig. 8. Annual failure probability after 10 years, given a 5-year inspection showing moderate corrosion damage: effect of the Y/T ratio of the pipeline material.
is achieved by considering a variable Y/T ratio in the use of the multiaxial generalization of the flexible Ludwik Law together with a finite strain plastic analysis based on Stewart [3] and Klever [2]. This also allows the consideration of longitudinal grooves and the presence of companion axial tension and bending loads. It is important to consider the distinctive properties of high strength steels and to cover a range of yield to ultimate ratios that can extend to 0.95 since it has been pointed out by several authors that the reduced strain hardening capacity can potentially give rise to a decrease in structural safety. Limit states models are developed for various extreme pressure situations and for various assumptions about grooves and pits in the interior wall of pipelines. These are used as the basis of a time-dependent structural reliability analysis to illustrate how corrosion and Y/T properties interact. In an application, the evolution of corrosion is modeled in pipelines of different grades; an internal corrosion inspection after five years allows for a Bayesian updating of long-term corrosion estimates and, hence, the
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ARTICLE IN PRESS M.A. Maes et al. / Reliability Engineering and System Safety 93 (2008) 447–455 state model development. Report submitted to ConocoPhillips, February 2006. [11] SUPERB. Buckling and collapse limit state. JIP report, December 1996. [12] Bubenick TA, Oslon RJ, Stephens DR, Francini RB. Analyzing the pressure strength of corroded pipe. In: Proceedings of the offshore
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