Predicting the mechanisms and crack growth rates of pipelines undergoing stress corrosion cracking at high pH

Corrosion Science 51 (2009) 2657–2674

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Predicting the mechanisms and crack growth rates of pipelines undergoing stress corrosion cracking at high pH q F.M. Song * Mechanical and Materials Engineering, Southwest Research Institute, 6220 Culebra Road, San Antonio, TX 78238, USA

a r t i c l e

i n f o

Article history: Received 3 April 2009 Accepted 30 June 2009 Available online 8 July 2009 Keywords: A. Steel pipeline B. Modeling studies C. Stress corrosion cracking C. Life prediction C. Crack growth rate

a b s t r a c t A fundamentally based mathematical model was developed with the goal to predict, as a first step, the crack growth rate (CGR) of high pH stress corrosion cracking (SCC) of buried steel pipelines. Two methods were used to predict CGRs and for both methods the model has included the film rupture and repassivation mechanism. The two methods are distinguished by the expression used to determine the active anodic current density at the crack tip. In the first method, this current density is expressed by the anodic polarization curve with a large peak current density and the prediction tends to yield a larger CGR and a lower pH at the crack tip. By contrast, when the Butler–Volmer equation is used to express the crack tip anodic current density, with a predicted low CGR the chemistry at the tip does not appear to have any significant change due to the high buffer of the solution. The predicted mechanism responsible for the steady-state crack growth is shown to be the balance between the increasing stress intensity factor as the crack grows, which tends to increase the crack tip strain rate and thus the CGR, and the change of the crack tip condition, which, for large CGRs, is the significant shift in the more negative direction of the crack tip potential, and for low CGRs, the increase of ferrous ion concentration, and either tends to decrease CGR. Limitations currently existing in the model and proposal for further development of the model are discussed. Ó 2009 Elsevier Ltd. All rights reserved.

1. Introduction As many existing gas and liquid pipelines approach the end of their design life, pipeline operating companies are looking for ways of extending the lives of these pipelines. Detection and prediction of SCC damage is becoming increasingly more important to ensure the pipeline safety. Detection of stress corrosion cracks can be accomplished with pressure testing, in-line inspection (ILI), or SCC direct assessment (DA). These detection methods have been described in industrial standard documents, ASME B31.8 [1], API 1160 [2], NACE RP 0204 (SCC DA) [3], NACE RP 0102 (ILI) [4], and in the United States Code of Federal Regulations, Part 192, Subpart O [5]. SCC DA [3] involves excavating portions of the pipeline and inspecting the surface with techniques such as magnetic-particle. Since all three options are very expensive, it is important to be able to predict where SCC is most likely and where it is highly unlikely.

q Revised from Paper: ‘‘Exploring to predict pipeline high pH SCC crack growth rates”, IPC2008–64671, a best paper winner, published and presented at the 7th International Pipeline Conference held on September 29–October 3, 2008 in Calgary, Alberta, Canada. * Tel.: +1 210 522 3988; fax: +1 210 522 6965. E-mail addresses: [email protected], [email protected]

0010-938X/$ - see front matter Ó 2009 Elsevier Ltd. All rights reserved. doi:10.1016/j.corsci.2009.06.051

This prioritization of SCC-susceptible locations is especially important for SCC DA, because only a small fraction of the pipe will be visually examined, so that the examination must be done where the probability of SCC is highest. CGR determines the remaining life of a pipeline section containing SCC and, thus, a confident prediction of relative CGRs along a pipeline is very useful in prioritizing the SCC locations along that pipeline. CGR is also a significant parameter for pipeline integrity management of SCC because it must be used when the re-assessment interval for the next excavation for SCC DA or when the re-inspection interval for pressure testing or ILI is determined. A reliable estimate of the CGR can allow the industry to reduce maintenance costs by reducing the conservatism in estimating the CGR, and in the meantime, maintaining the safe operation of the pipelines. The current methods of estimating CGR in the pipeline industry are based on empirical approaches, relying on experiences, laboratory tests, field observations, empirical CGR models [6–8], or by repeated inspections. Because of difficulties in inspecting small cracks and aligning the inspection locations confidently, CGR estimation by the current approaches is fraught with significant uncertainties. This program was proposed to develop an alternative approach to estimating CGR through the use of a physics-based model.

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Many mechanisms have been proposed for SCC in general, including film rupture and slip dissolution (FRSD) [9–12], adsorption-induced cleavage [13], atomic surface mobility [14,15], film-induced cleavage [16], and localized surface plasticity [17]. However, only the FRSD mechanism [9,10] has gained the widest acceptance. Existing mechanistic models are often developed based on FRSD mechanism and such models have been used mainly for prediction of SCC in nuclear systems [18,19]. Although such models have been used in the pipeline industry [20,21], the use is very limited and the confidence of the prediction needs improvement. The need for a mechanistic model that is applicable to the pipeline industry and improves the prediction of pipeline CGR is desired. SCC in pipelines occurs under two broad conditions – alkaline pH (pH in the range of 8–10) and near-neutral or low pH (pH in 5–7.5). Both types of SCC have been observed to occur under disbonded coatings on the external pipe surface, as schematically shown in Fig. 1. High pH SCC has been associated with carbonate and bicarbonate species, which can be produced by a reaction between carbon dioxide in soil and hydroxyl ions produced by cathodic protection (CP) used to protect the pipeline steel surface from corrosion. The CP currents are partially shielded from the steel by the coating and, thus, are often insufficient to prevent SCC. Near-neutral pH SCC occurs in the presence of bicarbonate and carbonic acid at locations where the coating may totally shield the pipe from CP. For high pH SCC, it has been accepted that the crack walls are passive or inert compared to the crack tip where periodic film rupture and slip dissolution occur and the SCC growth follows the FRSD mechanism [22–29]. By contrast, for near-neutral pH SCC, the crack walls are found to be somewhat active, while diffusion of atomic hydrogen into the metal ahead of the crack tip is considered to be partially responsible for promoting crack growth [30– 33]. This hydrogen mechanism is, at present, not fully understood. Regardless of high pH SCC or near-neutral pH SCC, the crack growth involves the transport of the solution species between the bulk soil and the crack tip (see Fig. 1, for demonstration purpose). Starting from the bulk soil, the solution species first pass across the holiday and then transport further into the coating-disb-

CP

Disbonded coating -

-

H2 O+e ->H+OH +

-

-

H +e ->H -

deposits

-

(O2 +H2 O+4e ->4OH ) 2+

2-

HCO3 /CO3 bulk solution with or without O2 X X FeCO 3or Fe 3O4

onded region. They may also directly transport through the disbonded coating. These species, and the newly produced species if any, must go through the crack before they finally reach the crack tip. During each step of the transport, the solution chemistry may be modified by the homogeneous chemical reactions in solution and the electrochemical reactions at the steel surfaces. At the crack tip, the crack growth is also influenced by the stress–strain conditions in the material around the crack tip and by the steel properties, including microstructure and chemical and metallurgical factors. A critical parameter is the crack tip strain rate, which involves the complex interactions of stress, strain, steel properties, and the solution chemistry at the crack tip. A number of expressions have been proposed under different loading conditions [20,34–44] and most are for corrosion fatigue. To be described later, an expression that was developed for constant load condition has been chosen and used for this model. For the sole purpose of predicting CGRs, well-defined chemistries in the coating-disbonded region are assumed in order to simplify the computations and modeling. In other words, the chemistry and the potential at the mouth of the crack are assumed to be unaffected by the processes in the crack. This process is signified as Steps (B.1) and (B.2) in Fig. 2. Success in this endeavor will demonstrate proof of the principle that SCC can be characterized by the disbonded chemistry, which itself can be related to bulk soil chemistry through Step (A.1). The capability developed, here, for predicting CGR would be of interest to pipeline companies as a means of prioritizing the severity of SCC based on measured bulk soil chemistry. Models have been developed in the past to predict conditions such as the chemistry and potential in the coating-disbonded regions; some results have been published elsewhere [45–48]. For the purpose of practical application, the predicted SCC CGRs can be characterized for a given chemistry in the form of a relationship between CGR and stress intensity factor, signified as K. The parameter K is a function of applied stress (related to the pressure in a pipeline) and crack size. A typical relationship of this kind is shown schematically in Fig. 3, which has included the three stages in a cracking process. Stage I occurs near the threshold for SCC, characterized by a critical applied K value, KISCC. The growth rate of a defect prior to reaching KISCC is very small compared to it afterward when plastic deformation at the crack tip starts to enhance the crack growth. In Stage II, the so-called steady-state region, the crack velocity is relatively insensitive to the applied K. The onset of Stage III is characterized by an enhancement in the Stage II crack velocity, and signals the onset of the rapid crack growth, leading to crack instability and failure. This work focuses on

-

-

Bulk Soil Chemistry

-

X X

(Fe->Fe +2e )

-

(A.1)

e -

e

-

-

H2 O+e ->H+OH +

Disbonded Region: cathodic protection, transport and chemical and electrochemical reactions

-

e

e 2+

-

Fe H+

-

H +e ->H 2+

2+

-

-

e

e H H

Grain boundaries Dislocations Inlcusions

-

Fe->Fe +2e

-

(Fe->Fe +2e )

log(i)

-

-

(H2 O+e ->H+OH) +

-

H +e ->H

log(i)

In Crack: transport and chemical and electrochemical reactions (A.3)

Impurities

log(t)

Known Cracking Chemistry in Disbonded Region

(A.2)

log(t)

Fig. 1. High/near-neutral pH SCC in pipelines (For near-neutral pH SCC, the coating holiday may not be required. Also, repassivation is less pronounced due to higher dissolution rate and hydrogen effect).

(B.1)

(B.2)

Crack tip: crack tip chemistry and CGR influenced by stress and material properties ahead of the tip Fig. 2. Schematic of SCC process (A) and the process to be modeled (B and boxed).

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F.M. Song / Corrosion Science 51 (2009) 2657–2674

The model crack geometry of this work is shown in Fig. 4(a). The methods of solving the differential equations are similar to those published elsewhere [47–48] while the application of the methods in the latter papers [47–48] was for predicting chemistry and corrosion in a coating-disbonded region; model geometry shown in Fig. 4(b). The similarity of the boundary conditions used for solving for chemistry and potential within a crevice relative to in a crack is given in Table 1. When used for predicting CGRs, equations other than those used for predicting the chemistry and corrosion in a coating-disbonded region are required, since CGR also involves film rupture and repassivation kinetics at the crack tip relating to crack tip strain rate. A moving boundary condition is also needed to solve for CGR as the crack depth increases with time. It has been found that the film repassivation kinetics follows [9–10]:

log(CGR)

Stage III, failure

Stage II, steady-state

KISCC Stage I, unsteady-state

K Fig. 3. Schematic relationship between SCC CGR and K.

predicting Stages I and II cracking behaviors. These two stages are located in the Parkins’ bathtub-shaped schematic diagram [25] for pipeline SCC, as the transition between its Stages 3 and 4. The Parkins’ diagram has been broadly used in the pipeline industry to explain the overall mechanisms of field pipeline SCC. Details on this diagram can be found elsewhere [25]. The objective of this work was to develop a model that links a known coating-disbonded chemistry to the crack tip chemistry (that is affected by the crack tip stress and strain and material parameters), from which the CGR is predicted. It is noted that the current model, in its current status, is not mature enough to predict real CGRs in the field as there are so many variables and variability in the field. However, this tool may become useful to predict the relative CGRs along a pipeline with similar conditions of pipe steel property, coating, and soil conditions and the result can be used to rank severity of crack growth at relevant locations along the pipeline. To fulfill the above purpose, simplification of the model is still necessary, by which the key controlling variables or (dimensionless) groups of variables may be identified for various rate-controlling regimes. For practical application, only the values of these controlling variables or variable groups are necessary to be determined in order to predict a CGR or provide guidances for a given regime. Such methodology has been applied successfully in previous works [49–53].

it ¼ i0

 n t t0

ð3Þ

where it is repassivation current transient when time t, is greater than an initial small time t0. n is repassivation kinetic exponent, which needs to be measured from experimental tests to be described in Section 2.2.

Crack mouth y

z crack

a Corroding metal

(a) 2. The mathematical model

y

2.1. Basic equations of the model

Crevice mouth The time-dependent partial differential equations used in this program are similar to those in a previous project [45] for predicting chemistry and corrosion in a coating-disbonded region presented elsewhere [46–48]. The general transport equation for any neutral or charged species in solution can be expressed as:

X dcj Rkj ¼ 0  r  Nj þ dt

ð1Þ

where N j ¼ ðDj rcj Þ þ zj uj cj ru is flux and zj, Dj, uj and cj are valence, diffusivity, mobility, and concentration of the jth species, respectively. u is electrostatic potential of the solution. Rkj is kth irreversible reaction rate of the jth species. The above transport equations and the equation of electroneutrality below:

X

zj c j ¼ 0

ð2Þ

and the corresponding boundary conditions can be used to solve for the crack chemistry and CGR.

z

Coating Crevice

a

Corroding metal

(b) Fig. 4. Simulated (a) crack geometries for modeling in this work and (b) coatingdisbonded crevice.

Table 1 Equivalent boundaries for predicting corrosion in a disbonded region (Fig. 4(b)) and for predicting SCC CGR (Fig. 4(a)). Crack (Fig. 4(a)) Crack mouth Crack wall Centerline of the crack Crack tip

Coating-disbonded region (Fig. 4(b)) , , , ,

Crevice mouth Steel surface Disbonded coating opposite to steel Coating opposite to the crevice mouth

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F.M. Song / Corrosion Science 51 (2009) 2657–2674

When t is smaller than t0, the bare metal surface undergoes active dissolution at an anodic current density of i0, which follows the Butler–Volmer equation below:

i0 ¼

EEeqref bFe

0 iref

10

c 2þ  Fe 10 cref Fe2þ

EþEeqref bFec

Vt ¼

! ð4Þ

0

where iref and Eeqref are, respectively, exchange current density and equilibrium potential, at a reference condition for a given concen; b is Tafel slope (subscript: ‘‘Fe” refers tration of ferrous ion cref Fe2þ to the anodic portion of the iron vs. ferrous ion redox reaction and ‘‘Fec” the cathodic portion); E is a corrosion potential at the crack tip measured with respect to a reference electrode. Compared to active dissolution without any passive film at the tip, with a passive film the anodic dissolution occurs only when the film is broken. Since any bare surface at the crack tip results from film rupture, the equivalent bare surface area (A) can be expressed by [18]:

A ¼ A0

 n t t0

ð5Þ

where A0 is the total crack tip surface area. For one cycle of film rupture/repassivation, the average bare surface area can be estimated to be:

¼ A

Z 0

tf

A0

 n t n t f dt=t f ¼ A0 t0 ð1  nÞt n 0

ð6Þ

where tf is time of one cycle of film rupture and repassivation, equal to the ratio of film rupture strain divided by crack tip strain rate. The time-averaged total dissolution current is: EEeqref bFe

 0 10 I ¼ Ai ref



eqref cFe2þ EþE 10 bFec ref cFe2þ

! ð7Þ

Thus, with consideration of film rupture and repassivation, the average anodic current density at the crack tip becomes:

i ¼

!!  n eqref EEeqref cFe2þ EþE tf 1 0 bFe bFec iref 10  ref 10 1  n t0 cFe2þ

grows. The CGR is described as a moving boundary condition at the crack tip expressed by:



ð8Þ

It is clear from Eq. (8) that the time-averaged dissolution rate is a function of not only repassivation kinetics, but also potential E and solution chemistry (cFe2þ ), both at the crack tip. This indicates that although critical, repassivation kinetics alone cannot fully determine the CGR. Since repassivation kinetics is important, relating to the mechanical effect on CGR, experimental work was performed to measure the film rupture and repassivation kinetics in a standard solution, consisting of 1 N–1 N carbonate–bicarbonate ions [26]. Although this solution does not contain ferrous ions as would be the case for solution at the tip of a growing crack, it has been frequently used in laboratory tests to understand the susceptibility of pipeline steels to high pH SCC. The chemistry at the crack tip varies with time and thus, an exact measurement of the repassivation kinetics at the crack tip corresponding to each change of the chemistry can be impossible. It is an assumption of the model that the repassivation kinetic exponent ‘‘n” is a constant. The steel used for the repassivation kinetic measurement is X52, a steel in which high pH SCC has been found in the field. The repassivation kinetics of X52 in this standard solution has not been reported. The experimental test procedure and results are to be described briefly in Section 2.2 and Appendix A. For the crack geometry shown in Fig. 4(a), the chemistry at the crack mouth may be assumed to be the same as initially the general chemistry in the coating-disbonded region. The crack walls are assumed to be passive and the crack tip experiences periodic film breakdown and repassivation. When the film ruptures, the exposed metal experiences anodic dissolution and, thus, the crack

dh dt

ð9Þ

where Vt is crack velocity, and h is the depth of the crack. Since it is known that high pH SCC follows the anodic dissolution-controlled mechanism, the CGR is proportional to anodic current density following the relation derived from Faraday’s law: [25]

V t ¼ ai

ð10Þ

where a is a coefficient relating to atomic weight of steel (M), valency of solvated species (z), Faraday’s constant (F), and the density of the metal (q) by: a ¼ zFMq. Eq. (9) or (10), as a boundary condition, needs to be incorporated with other equations, including Eqs. (1) and (2) and the boundary conditions (Table 1), to solve for crack chemistry and CGR. Note that the crack walls are passive and, thus, the anodic electrochemical reaction rate there can be assumed to be constant (passive current density). Depending on the types of passive film present on crack walls, such as Fe3O4 (which is electrically conductive and, thus, cathodic reactions can occur on it) or FeCO3 (which is non-conductive and, thus, anodic and cathodic reactions do not occur on it), the cathodic reactions may or may not be present on the crack walls. Here, a small constant passive current density is used to resemble a film dominated by Fe3O4. For the system to be modeled here, it is assumed that a crack starts with a small defect, such as a narrow pit, whose width and depth are 20 lm and 0.2 mm, respectively. The dimension of the defect in the direction perpendicular to the crack geometry of Fig. 4(a) is assumed to be very large relative to its width so that only its unit dimension is of concern. The crack grows along the grain boundaries in a pipe wall and the wall thickness is assumed to be 1 cm. With high pH, dissolved CO2 can be considered in equilibrium with H2CO3. If the primary species in solution are selected as (1) Na+, (2) Cl, (3) Fe2+, and (4) H+, the concentrations of the secondary species: (5) CO2, (6) OH, (7) CO3 2 , (8) HCO3  , (9) H2CO3, (10) FeOH+, and (11) FeHCO3 þ can be calculated from the equilibrium relations of the secondary species with the primary species, similar to what was done earlier [48]. The choice of primary species does not affect the prediction results. After the reaction rates were cancelled out for the secondary species as was described elsewhere [48], the following differential equations based on Eqs. (1) and (2) can be obtained:

@c1 þ r  N1 ¼ 0 @t

ð11Þ

@c2 þ r  N2 ¼ 0 @t

ð12Þ

@ðc3  c5  c7  c8  c9 þ c10 Þ þ r  ðN 3  N 5  N7  N8 @t e p  N 9 þ N10 Þ ¼ r Fe =a @ðc4 þ 2c5  c6 þ c8 þ 2c9  c10 þ c11 Þ þ r  ðN 4 þ 2N 5  N6 @t reH þ reH2 O þ r eH2 CO3 þ N 8 þ 2N9  N10 þ N11 Þ ¼  a

r

11 X j¼1

zj N j ¼

  e p 2r Fe  r eH þ r eH2 O þ r eH2 CO3 a

ð13Þ

ð14Þ

ð15Þ

where cj and Nj (j = 1–11) are, respectively, concentration and flux of the jth species in solution. re is the electrochemical reaction rate at a crack wall with half a crack width being ‘‘a”, with r eFep being the

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F.M. Song / Corrosion Science 51 (2009) 2657–2674

passive dissolution rate of steel, reH reH2 O , reH2 CO3 follow their Tafel equations given earlier [48]. Eq. (15) is equivalent to the equation of electroneutrality. The initial condition and the mouth boundary condition can be assumed to be the same which is dictated by concentration. At the crack tip, flux boundary conditions, as well as a moving boundary condition (Eq. (10)), apply, see Table 1. At the crack tip, the boundary conditions are:

N1 ¼ 0

ð16Þ

N2 ¼ 0

ð17Þ

N3  N5  N7  N8  N9 þ N10 ¼ r eFe

ð18Þ

N4 þ 2N5  N6 þ N8 þ 2N9  N10 þ N11   ¼  r eH þ r eH2 O þ r eH2 CO3

ð19Þ

11 X

  zj Nj ¼ 2reFe  reH þ reH2 O þ reH2 CO3

exponent of a pipe carbon steel X65 varies with potential, measured in a different carbonate and bicarbonate solution with the use of scratching electrode method [56]. The potential range used in this literature [56] was wider (0.327 VCSE to 0.423 VCSE), and a wider range of the exponent (0.5–1.2) was also reported. In Table 2, the 2nd and 3rd pulls refer to the same tensile specimen being subsequently pulled twice after the first pull. During each pulling, the specimen was put under potentiostatic control so that the repassivation current generated during the pulling was recorded. A subsequent pulling was conducted after the preceding pull was completed, load released, and new passive film grown and becoming stable again. Thus, the subsequent pulling can result in strain hardening. When a material is strain hardened, more dislocations tend to be released to rupture the passive film. This results in fast dissolution and thus a higher repassivation kinetic exponent. Based on the measured results in Table 2, a repassivation kinetic exponent is taken at: n = 0.6. The critical strain for the film was estimated, roughly, in the range of 0.2–0.3%. ef ¼ 0:003 is used for this work as it was also reported by others [20].

ð20Þ 2.3. Modified crack tip anodic current density

j¼1

r eFe

Here, is the active iron dissolution rate including the film rupture and repassivation effect and can be converted from Eq. (8) using Faraday’s law below:

r eFe ¼

i nFe F

Based on data obtained for passive alloys such as stainless steels [9,10], with the film on steel being thicker and possibly more brittle, t0 = 0.0063 s is assumed for the current model. Substitution of n = 0.6 into Eq. (8) yields:

ð21Þ

i ¼ 3:9e_ 0:6 i0a 10 bFe0 þ cFe2þ i0c 10 bFec0 ct c0Fe2þ EE

where nFe = 2 and F is Faraday’s constant.

EþE

! ð22Þ

2.2. Measurement of repassivation kinetics

where the conditions at the crack month are used as the reference conditions of the model. E0 is the crack mouth potential.

The experimental technique and procedure used for measuring the repassivation kinetics of X52 pipe steel for high pH SCC is the same as for the measurement of a passive alloy, which was described in detail elsewhere [54–55]. The solution used for the test consists of 1 N–1 N carbonate–bicarbonate. The tests were conducted for two temperatures, a number of strain rates, two potentials, and different loading conditions. A description of the test and an analysis of the test results are presented in Appendix A. Table 2 is a summary of the measured repassivation kinetic exponents calculated from the film repassivation kinetics measurement. Unless stated otherwise, all potentials in this table or elsewhere in this paper are referenced to saturated Cu/CuSO4 electrode or CSE. It appears that the repassivation kinetic exponent increases, significantly, with strain hardening of the material, while the variation with applied strain rate, temperature and potential appears to be not significant within the range of conditions tested. It was reported elsewhere that the repassivation kinetic

i0a ¼ i0 10 0 ref

E Eeqref bFe

0

E0 þEeqref bFec

0 c 2þ and i0c ¼ iref cFe 10 ref

are, respectively, the

Fe2þ

anodic and cathodic portions of the Butler–Volmer equation at the initial conditions or at the crack mouth. The unit of e_ ct is s1. Note that in deriving Eq. (22), the following relation was used: e tf ¼ e_f . For crack tip strain rate, the equation below is used in this work, which was derived for crack growth at a constant load [20,40,41]:

e_ ct ¼ CK 4

ð23Þ

where K is stress intensity factor and C is the proportionality constant that accounts for the extent of reversed slip per unit of crack advance, which varies with material and its properties. Although creep is well known to contribute to the growth of high pH SCC [20,25], at this time the validity of Eq. (23) for the conditions of interest in this paper still requires confirmation through further study. By criticizing the empiricism embedded in the existing

Table 2 Repassivation kinetics parameters at different strain hardening stages and strain rates in 1 N Na2CO3 + 1 N NaHCO3 solution at temperatures 25 °C and 75 °C at 0.510 VCSE and 0.690 VCSE. Test condition

Strain rate (s1)

1st pull

2nd pull

3rd pull

c*

n

c

n

c

n

0.510 VCSE, 75 °C

102 103

5.92 E05 3.24 E06

0.69 0.62

8.47 E05 8.74 E06

0.66 0.73

Specimen failed 2.00 E05

0.85

0.510 VCSE, 75 °C

102

1.39 E04

0.34

1.86 E04

0.69

3.93 E04

0.91

0.690 VCSE, 75 °C

102 103

2.58 E04 2.10 E05

0.58 0.57

6.85 E04 8.22 E05

0.83 0.85

4.97 E03 3.10 E+01

0.98 1.00

0.510 VCSE, 75 °C

Strain rate (s1) Various

1.71 E04

0.52

1.61 E04

0.78

2.22 E+01

0.510 VCSE, 25 °C

Strain rate (s1) Various

*

0.05

0.01

0.05 2.69 E04

0.001

0.01 0.72

c is a coefficient used in determining ‘‘n” and its meaning is explained in Appendix A.

1.28 E04

1.00 0.005

0.66

9.97 E05

0.81

F.M. Song / Corrosion Science 51 (2009) 2657–2674

pffiffiffiffiffiffi K ¼ rk W

ð24Þ

pffiffiffiffiffiffiffiffiffiffiffiffi pc 2tanð Þ where r is stress, k ¼ ð0:752 þ 2:02c þ 0:37ð1  sinðp2cÞÞ3 Þ cosðpcÞ2 2 and c ¼ h=W. W is pipe wall thickness. For a crack with a very small initial depth of h0, the effect of stress on crack growth can be treated to be negligible. Once K1SCC is reached, with the effect of stress expressed by Eq. (24) the crack tip current density in Eq. (22) may be modified as:



K K ISCC

2:4

i0a 10 bFe0 þ cFe2þ i0c 10 bFec0 c0Fe2þ EE

EþE

Cracking range

100

3.1. Model assumptions and two methods used for prediction In this work, model predictions are made based on some reasonable assumptions. The crack walls are assumed to be passive, and the passive anodic current density is invariant with time for a given temperature. The cathodic reactions: water reduction and hydrogen ion reduction occur both on the crack walls and at the crack tip, following their respective Tafel equations. At the crack tip, the film experiences periodic breakdown and repassivation, with the cycle determined by the film rupture strain divided by the crack tip strain rate. This has been incorporated into the model as shown in Eqs. (6) and (22). In expressing the anodic current density on a bare surface at the crack tip (or dissolution current density immediately following film rupture), two methods are used. In the first method, the anodic polarization curve of steel, measured in a solution representing that within the crack, was used. The current density on the anodic polarization curve represents the net current density, or the sum of iron dissolution, water reduction, and hydrogen ion reduction current densities, although away from the equilibrium potential the net current density can be well approximated by the iron dissolution current density. The advantage of using this method of treating anodic current density, in comparison with the second method or using the Butler–Volmer equation to express the anodic current density directly is that the net current density during the transition from passivity to active dissolution is reflected in this curve, which no equation exists to express this current density explicitly. Even though the Bulter–Volmer equation or the less accurate Tafel equation can be useful for treating the active region, finding the most appropriate values for the exchange current density and Tafel slope can be a challenge. The limitation of using the polarization curve is that such a curve measured in one condition, such as at the beginning of a cracking process (when the in-crack chemistry is similar to the bulk or in-disbondment chemistry), may not represent the change of the condition as the crack grows, when solution chemistry can go toward being more acidic. In addition, the polarization curve varies with scan rates as shown in Fig. 5, and thus, such a curve at an arbitrary scan rate may not represent the actual dissolution rate of the bare steel surface at the crack tip.

100

1

10

20

1

0.01 -0.9

ð25Þ

3. Model assumptions and validation

10

C7 Steel 1N Na2CO3+1NNaHCO3 75 oC

!

following a combination of Eqs. (22–24). In Eq. (25), iISCC is iron dissolution current density at KISCC and may be approximated by the iron dissolution current density at the initial condition.

1000 mv/min

0.1

-0.8

-0.7

-0.6

-0.5

-0.4

Potential (V vs. SCE) Fig. 5. Polarization curves of steel in 1 N–1 N Na2CO3–NaHCO3 solution at 75 °C at different scan rates [60]. Here, the potential is referenced to saturated calomel electrode or SCE.

The advantage of using the Butler–Volmer equation or Tafel equation is that it reflects the anodic dissolution mechanisms of steel in the active region and, thus, the equation can accommodate the change of conditions, as long as the dissolution mechanism does not change. This method has traditionally been used in SCC modeling [18,19]. In both model validation and model predictions to be described below, it is assumed that for both methods, the passive current density is chosen as 0.01 times of the initial current density at the active crack tip determined from the Butler–Volmer equation when the over-voltage is 20 mV. This way of choosing passive current density is to reflect that the steel dissolution is much faster at the tip than on the crack walls, and the passive current density is small. Some authors [18] assumed a zero anodic current density or chose a current density in the passive region of the polarization curve [19]. For high pH SCC, there is a challenge to determine the passive current density, because it is not as unique as that of a passive alloy and often appears too large to reflect reality (see Figs. 5 and 6). In the field, the corrosion in the coating disbonded region has been shown to be minimal and the crack width is often small,

-0.6 Potential (V vs. Cu/CuSO 4 )

i ¼ iISCC

1000

2

expressions of crack tip strain rate such as Eq. (23) [44,57,58], alternative and claimed to be more fundamental expressions of crack tip strain rate [44] or CGR modeling [58] are provided. Such approaches deserve to be explored in the future for the conditions of interest in this paper. The expression of K for a crack within a single edge notched tension panel [59] is:

Current density (A/m )

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-0.725 V

-0.7

-0.8

-0.75 V

-0.775 V

Measured cracking range -0.810 V

-0.9

-1 -10

0

10

20

30

40

50

60

2

Current density (A/m ) Fig. 6. Potentiodynamic polarization curve for X52 steel in 1 N–1 N carbonate– bicarbonate solution at 75 °C (scan rate at 10 mV/min).

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both indicating a very small metal loss on the crack walls due to the passivity of the metal surface. In this model, although the passive current density at the crack walls is assumed to be constant with time, the anodic current density at the crack tip is not, which, to be shown in Sections 3.2 and 4, increases significantly with time immediately after KISCC. The passive current density at the crack walls varies in the model with temperature since the parameters in the Butler–Volmer equation, such as exchange current density and Tafel slope vary with temperature. The hoop stress causing SCC has been reported, in the field, to be within 47–56% of the specified minimum yield stress (SMYS) [26]. Thus, for steel X52 50% SMYS is 179.3 MPa, which was used in the model predictions, unless stated otherwise. In the high pH solution, the cathodic reduction of carbonic acid is insignificant and neglected. High concentrations of carbonate and bicarbonate were used whose total charge was approximately balanced by Na+ and the solution was assumed to be saturated by ferrous carbonate due to its very low solubility. At the crack tip, the effect of crack tip strain rate on CGR has been shown in Eqs. (22) and (25). The model was first validated with one set of experimental data and then used to predict CGR and in-crack chemistry and potential.

3.2. Qualitative validation of the model Pipe steel related CGRs measured in the laboratory or estimated from repeated ILI runs in the field, or based on crack surface fractography of failed pipe pieces, can vary within a broad range. The CGR data generated from accelerated lab tests was often reported to be in one or more orders of magnitude higher than field CGRs. This presents challenges for the model validation. Field CGR data depends on field conditions near a crack, which vary with time and are usually unknown. Extrapolation of the CGRs from repeated ILI runs is empirical and requires a number of runs before the trend of CGRs vs. time can be developed, and such data are often not publically available if at all. Besides, the error associated with the ILI tool resolution can be large and, thus, only large cracks can be detected and measured. By contrast, lab CGRs measured under well-controlled conditions provide advantages over field data for validation of SCC models. Unfortunately, abundant CGR data for high pH SCC are not available or such data were reported without providing all essential test conditions necessary for validation of the model. For the above reason, the intention for validation of the model, here, is not meant to be a direct quantitative comparison of the model results with laboratory or field data. This is because with such a large span of CGR data, the agreement of the model results with one specific set of data does not offer a convincing validation of the model. Although it is always possible to replicate one set of data generated under one specific condition by adjusting the model variables, such a way of validating the model is not useful, because this artificial adjustment of the model variables could lead to erroneous prediction for other conditions. As was already made clear in the objective (Section 1), the goal of developing this model is not attempting to accurately predict the real CGRs in the field, but is, instead, meant to assist in future endeavors in predicting the relative CGRs along a pipeline segment, for ranking the severity of different SCC susceptible locations. The validation of the model, here, is, thus, rather qualitative. A comparison of the model results with experimental data is to be made in terms of the trend and magnitude of CGR. Only Stages I and II in Fig. 3 are to be predicted from the model. It is expected that by predicting the cracking phenomena, the cracking mechanism behind the phenomena can be better understood. For this prediction, the anodic polarization curve of X52 steel in the 1 N– 1 N carbonate–bicarbonate standard solution at 75 °C was used.

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Potentiodynamic polarization was conducted in 1 N–1 N carbonate–bicarbonate solution at 75 °C, a portion of which is shown in Fig. 6, which was used for the modeling. Several potentials were labeled in the figure for later use. The figure shows that as the potential drops slightly along the polarization curve from 0.725 VCSE to 0.775 VCSE, the decrease of current density is significant. By contrast, around the peak, for the potential drop from 0.775 VCSE to 0.810 VCSE, the change of current is negligible. The solution used for the model simulation has a pH of 9.8, 1 mol/L bicarbonate concentration, Na+ concentration of 2 mol/L, and Cl concentration of 0.01 mol/L. The concentrations of the other species can be calculated from the above concentrations with the use of electroneutrality and equilibrium relations. The simulation was performed with a potential of 0.725 VCSE at the crack mouth, temperature of 75 °C and KISCC of 21.5 MPa m0.5. These conditions are consistent with those used to generate the experimental CGR data in the literature [26,60] shown in Fig. 7, to which the model results are to be compared qualitatively. Much information is unknown about the test conditions in Fig. 7, including initial crack depth, other dimensions of the crack and the stress applied. Fig. 8 is a semi-log plot showing the predicted CGR vs. K for the above condition. Three major sections of the curve are described. Prior to KISCC (incubation stage of a crack), the defect undergoes dissolution with a slight increase of the dissolution rate where the stress effect is negligible. Although the first two stages of the crack growth (unsteady (Stage I) and steady (Stage II) state crack growths) are clearly demonstrated in Fig. 8 similar to Fig. 7, the extremely sharp increase of CGR near KISCC shown in the experimental test data in Fig. 7 was not revealed. This discrepancy between the model predictions and the experimental results may be explained as below. Unlike pre-crack compact test specimen used for the experimental test, which can allow direct transport of solution species to the crack tip from the sides of pre-crack specimens, the model, here, is one-dimensional and, thus, does not have the above

Fig. 7. SCC CGR at different initial stress intensity factors for C-Mn steel in 1 N–1 N carbonate–bicarbonate solution at 75 °C and 0725 VCSE.[21].

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Fig. 8. Predicted SCC CGR for X52 steel in 1 N–1 N carbonate–bicarbonate solution at 75 °C and 0.725 VCSE.

Fig. 10. Predicted crack tip potential vs. K, and crack depth/wall-thickness vs. K, for X52 steel in 1 N–1 N carbonate–bicarbonate solution at 75 °C and 0.725 VCSE.

artificial transport effect. Such transport does not exist with field stress corrosion cracks. Besides, the modeled crack starts from a small defect with the calculated K (approximately 5 MPa m0.5 at the beginning) much smaller than KISCC of 21.5 MPa m0.5. It was found in the experimental tests conducted in other test conditions that the shape of CGR vs. K near KISCC depends on the value of initial K, which can be varied by the initial crack size or applied stress (Fig. 9) [61]. Starting with a small K, the variation of CGR vs. K is more gradual and the magnitude of the increase of CGR is not as significant as shown in Fig. 7. Although the X65 steel used for the experimental test [26,60] in Fig. 7 is different from the X52 steel for this modeling, this difference should have a negligible effect since the physical, chemical, and mechanical properties of the two steels are very similar. The predicted CGR at steady state is shown to fluctuate with K, which may result from the counterbalance between the increase of CGR by increasing K as the crack grows, and the decrease of CGR along the polarization curve after the peak is passed moving toward more negative potentials. The result of Fig. 8 can be explained further by a comparison of the crack tip potential variation vs. K in Fig. 10 and the polarization curve in Fig. 6. Prior to KISCC, the stress effect is not present or neg-

ligible and, thus, the slight increase of the defect growth rate fully results from the little decrease of potential following the polarization curve from 0.725 VCSE to the potential at KISCC or 0.727 VCSE. After KISCC, as K continues to increase but prior to reaching 36 MPa m0.5, the potential has a gradual decrease to 0.750 VCSE, while the increase of CGR is significant. This increase of CGR results from the combination of two factors. The main factor is the significant increase of the anodic current density with a slight shift of potential in the more negative direction on the polarization curve. The second factor is the increase of the crack growth by the increase of K, which increases the crack tip strain rate. With a slight increase of K from 36 to 37 MPa m0.5, the potential has a relatively large decrease from 0.75 VCSE to 0.81 VCSE (Fig. 10). Since this potential change is located in the region on the polarization curve (Fig. 6) that passes across the peak of the curve, the large change of the potential does not result in a significant increase of the CGR. The increase of CGR, rather, results mainly from the mechanical effect, because even the slight increase of K can lead to a significant increase of crack tip strain rate due to the power law relation between them, as shown in Eq. (23). As the crack continues to grow, the potential continues shifting in the more negative direction and the CGR decreases if following just the polarization curve. In the meantime, the increase of K continues to increase CGR mechanically by increasing the crack tip strain rate and, thus, the balance of the two factors resulted in a steady-state condition. The fluctuation of the curves indicates the complex balance between the above two factors. In Fig. 8, the third stage of stress corrosion cracking, due to mechanical fracture, was not included. This fracture mechanism has not been incorporated in the model. No intention is attempted in this work to manipulate the model parameters in order for the predicted CGRs to have a best fit to the experimental CGR data. The goal here is to use one set of input variables to predict the cracking phenomena observed in the lab tests or field, by which further prediction permits better understanding of the cracking mechanisms. 4. Computational results

Fig. 9. Dependence of static load CGR on initially applied stress intensity factor Ki [54].

The model predictions were conducted under two broad conditions: (1) the use of net current density on the potentiodynamic polarization curve, a method used for the model validation, and (2) anodic current density determined from the Butler–Volmer equation, similar to what others have done [18,19]. The results

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obtained from these two different methods are presented separately below. 4.1. The use of polarization curve to express the net current density at the crack tip In this section, the solution chemistry used for the model prediction is the standard 1 N–1 N carbonate–bicarbonate solution, same as used for model validation earlier. Experimental evidence exists to suggest that the pH is similar between within a crevice (simulating a crack) and the bulk [62] and between the inside of a crack of a slow strain rate specimen and outside [63]. These limited while only available test results lead some to believe that the solution pH and potential within the crack are uniform since the solution is a concentrated buffer. Since the crack growth was shown to follow an anodic dissolution-controlled mechanism, some have attempted to estimate CGR by using a polarization curve, where the peak current density represents a conservative estimate of the CGR. In the field, the CGR rarely grows at that high a velocity, which, if true, a pipe would fail within days once the steady-state condition is reached. Also, if the CGR is so high, why would the tip solution not become acidic? If the assumption is valid that the tip dissolution follows the polarization curve, and the solution chemistry does not vary, the polarization curve could be used to estimate the net current density and then determine the transient CGR and chemistry throughout the crack by solving, simultaneously, the transport equations, together with the chemical and electrochemical kinetic equations, and the equations that reflect the stress effect such as the crack tip strain rate expressed by stress intensity factor K (Eqs. (23) and (24)). The transport and equilibrium equations of the model are the same as published elsewhere [47,48]. From the model calculations, it is necessary to understand whether the premise for constant potential and chemistry within the crack can hold. The predicted crack tip potential, as well as CGR, vs. time is shown in Fig. 11. The incubation time, or time before reaching KISCC, is significant. During this time, the CGR is low. The predicted crack depth increases from the original 0.2 to 2.25 mm, or 22.5% of the wall thickness, as shown in the upper portion of Fig. 12. Moving into the stage of unsteady-state crack growth (Stage I), Fig. 11 shows a gradual shift of the potential in the more negative direction while the CGR increases significantly. As described in Section 3.2, this increase of CGR results from the decrease of potential following the polarization curve shown in Fig. 6 (prior to

Fig. 11. Predicted crack tip potential and CGR vs. time, for X52 steel in 1 N–1 N carbonate–bicarbonate solution at 75 °C and 0.725 VCSE.

Fig. 12. Predicted crack depth and stress intensity factor varying with time.

K = 36 MPa m0.5) and from the increase of stress intensity factor as crack grows (K between 36 and 37 MPa m0.5). The total crack depth increased to 36.6% of wall thickness at K = 37 MPa m0.5 as shown in the upper portion of Fig. 12. For the steady-state crack growth (Stage II), Fig. 11 shows a high CGR, which appears constant with time. In the meantime, the potential has a significant drop to more negative values. K = 37 MPa m0.5 corresponds to the tip potential of 0.810 VCSE, and this drop of potential is located below 0.810 VCSE on the polarization curve in Fig. 6. The fact that the significant drop of potential should result in a significant decrease of CGR following the polarization curve in Fig. 6, but this trend was not shown, clearly demonstrates the counterbalance caused by the significant increase of stress intensity factor, which increases CGR, as shown in the lower portion of Fig. 12. Since the steady-state CGR is high, it takes only 70 h to increase the crack depth from 36.6% to 52.9% wall thickness, in comparison with the 925 h of incubation time to increase the crack depth by 20.5% of wall thickness. It is quite clear from Fig. 11 that the variation of crack tip potential can be significant when the CGR is large, as shown on the polarization curve near the peak. The overall decrease of potential during the cracking process modeled is approximately 200 mV. This significant variation of potential is accompanied by a large change of the solution pH and resistivity, and concentrations of the soluble species at the crack tip. Fig. 13 shows the variation, with time, of the crack tip pH and solution resistivity. The crack tip pH shows a significant drop, more than 4 U. Consistent with the magnitude of CGR in Fig. 11, the pH variation is small before KISCC is reached. The pH drop predominantly occurs during the unsteady-state crack growth before reaching K = 37 MPa m0.5, where the CGR has the greatest increase. It should be noted that even though the mechanism of the crack growth during this stage is controlled by the two factors: mechanical and electrochemical, the pH variation depends on both proportionally, because both factors change CGRs by the dissolution mechanism. The pH drop results from iron dissolution followed by hydrolysis of dissolved ferrous ions. In the steady-state stage of crack growth, the pH fluctuates, and on average, has a slight increase with time. This increase of pH may result from the higher cathodic reduction rates (by hydrogen ion and water) on the crack walls that generate more alkali, which transport to the crack tip to neutralize the solution acidity there. At the crack tip, the surface is still anodically polarized because of the net current density reflected on the polarization curve, which is positive or anodic (Fig. 6).

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Fig. 13. The variation of crack tip pH and solution resistivity.

Corresponding to the pH variation at the tip, the overall solution resistivity at the crack tip experiences a slight increase during the incubation time, while the increase becomes significant during the stage of unsteady-state crack growth. In these two stages, the alkaline solution is neutralized as reflected by the decreasing pH and, therefore, the concentrated alkaline solution is diluted. As the anodically-generated ferrous ions at the crack tip repel the sodium ions away from the tip, chloride ions are attracted to reduce the pH (Fig. 14). At the same time, the saturation limit of ferrous carbonate reduces carbonate and bicarbonate ion concentrations by precipitation. In the steady-state stage, the production of ferrous ions is relatively stable at the tip, and the concentration of sodium ions slightly increases to balance the increasing charges of carbonate and bicarbonate ions. The acidic solution is becoming more concentrated and the resistivity decreases again (Fig. 13). Since the experimental tests have shown that CGR varies with applied potential, it is interesting to investigate by modeling how applied potentials, at the mouth, affect CGR. The crack growth behavior at the mouth potential of 0.81 VCSE was investigated in comparison with the results presented above for the mouth potential of 0.725 VCSE. Fig. 15 shows a comparison of CGRs and crack tip potentials vs. K for the two potentials at the mouth, respectively, at 0.725 VCSE

Fig. 14. The variation vs. time, of the concentrations of some major solution species at the crack tip.

Fig. 15. The variation, vs. K, of CGR and crack tip potential at two mouth potentials, with all other model parameters used the same as for Figs. 8 and 10.

and 0.81 VCSE. The steady-state feature shown at 0.725 VCSE is not demonstrated for the potential of 0.81 VCSE, which is shown in Fig. 6 located outside the cracking range determined experimentally. For the mouth potential of 0.81 VCSE, a potential that is located below the peak current density on the polarization curve in Fig. 6, the CGR has a more significant increase immediately after KISCC (KISCC is labeled by the vertical dashed line in Fig. 15). This increase in CGR is due to mechanical stress as is clear from earlier discussion. Followed by a sharp drop the CGR decreases to nearly zero as K just passes 40 MPa m0.5. This result indicates that the crack will arrest at this potential. This sharp decrease of CGR results from the more significant decrease of the net current density (approximately equal to CGR) vs. potential shown on the polarization curve in Fig. 6. This decrease of CGR by the decrease of the net current density is more significant than the increase of CGR due to the increase of K as the crack grows. The correspondingly significant shift of the tip potential in the more negative direction is also seen in the lower portion of this figure. Fig. 15 also shows that the peak CGR is higher at the mouth potential of 0.725 VCSE than at 0.81 VCSE, while the starting CGR is higher in the reverse direction due to their respective location on the polarization curve in Fig. 6. The variation of CGR and crack tip potential is accompanied by the change of solution pH and resistivity at the crack tip as is shown in Fig. 16. With anodic polarization at the mouth for both potentials of 0.725 VCSE and 0.81 VCSE, the tip pH first decreases with increasing K. This decrease of pH becomes more evident when the KISCC is passed. Then, the increase of CGR is more significant. During this time, the solution becomes more dilute and less conductive (shown as an increase of solution resistivity). For the potential of 0.81 VCSE, it was found that ferrous ions replace sodium ions, and carbonate ions are replaced by bicarbonate ions and chloride. When the peak CGR is passed, the pH increases while the solution resistivity decreases. The decreasing CGR generates less acidic species, and the transport of alkaline species from the mouth neutralizes the solution acidity at the crack tip and, thus, the tip pH starts to increase. As the solution pH varies from acidic to alkaline, the bicarbonate and chloride ions are replaced back by carbonate ions, and the ferrous ions by sodium ions. To understand how the above variables, and the total crack depth, vary with time, Fig. 17 is presented. At the crack mouth potential of 0.81 VCSE, the crack depth stops to increase at about 40% wall thickness, when the crack growth ceased. This is different

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ship depends on the scan rate as shown in Fig. 5 [65], and it is unknown what scan rate is most representative. Since the CGR is large, the strong buffer solution could be broken down. It is unknown if the crack tip condition is constant with time if the CGR is low. It is interesting to examine the cracking conditions at low CGRs, since low CGRs appear to be more consistent with field observations. In view of the problems associated with the use of the polarization curve for CGR prediction, the Tafel or Bulter–Volmer equation is used next, although it is recognized that problems also exist associated with this approach as also mentioned earlier (Section 3.1). Since the 1 N–1 N carbonate–bicarbonate solution has not been found in the field, the solution chemistry used for further simulation is replaced with a more dilute, but still concentrated, solution, close to observed field chemistry in coating disbonded regions near stress corrosion cracks [24].

Fig. 16. The variation, vs. K, of crack tip pH and solution resistivity at two mouth potentials, with all other model parameters used the same as for Figs. 8 and 10.

4.2. The use of Butler–Volmer equation to express the anodic current density at the crack tip The initial solution, or the solution at the crack mouth, used for this simulation has the following chemistry characteristics, unless stated otherwise: a pH of 9, Na+ concentration 1 mol/liter, Cl concentration 0.01 mol/L, and Fe2+ concentration 2.73  1010 mol/L (55 °C). The concentrations inside the crack are unknown and have to be determined from the modeling. In this solution, the charge of Na+ is balanced, mainly, by carbonate and bicarbonate species whose concentrations at 55 °C are 0.061 mol/L and 0.868 mol/L, respectively. It is assumed that K1SCC is 12.5 MPa m0.5 and the crack mouth potential is 0.75 VCSE unless otherwise stated.

Fig. 17. The variation, vs. time, of crack depth and K at two mouth potentials, with all other model parameters used the same as for Figs. 8 and 10.

from the situation at the month potential of 0.725 VCSE, where the crack continues to grow, eventually leading to a failure. The above predicted potential and chemistry or pH, which vary significantly both at the crack tip and within the crack, contradicts the speculation of some that both potential and chemistry are relatively constant within the crack due to the buffering nature of the concentrated carbonate and bicarbonate solution. The predicted results are rather consistent with the traditional understanding and bulk experimental evidence for SCC of passive alloys, that crack growth leads to a low pH at the tip, which then causes the tip to be less passivated and, thus, results in continuous growth of the crack [64]. It is recognized that using the net current density from the polarization curve for predicting CGR has some drawbacks, some of which have been mentioned earlier (Section 3.1). Briefly, as the chemistry at the tip varies with time, the polarization curve can become invalid to predict CGR with the new chemistry. Similarly, the measured repassivation kinetic exponent used for CGR prediction may also be questionable, particularly when the crack tip chemistry becomes acidic. Then, the FRSD mechanism may not be valid. Besides, the use of the polarization curve itself has its uncertainties because the current density vs. potential relation-

4.2.1. CGR prediction under a given condition The results presented below are obtained for a constant temperature of 328.15 K, and a constant external hoop stress of 179.3 MPa. The initial and boundary conditions have been given earlier, similar to as is shown in Table 1. Fig. 18 shows CGR vs. time and crack depth vs. time. For the first 8 years, the defect experiences slow growth. During this time, the CGR is shown to be relatively constant, while on a finer scale this rate can be seen to decrease slightly. This decrease in CGR appears to follow pit decay law due to increased ohmic potential drop (IR) by the increase of defect depth. At about 8 years, KISCC is reached

Fig. 18. Crack growth rate vs. time for a temperature of 328.15 K, bulk pH 9, external stress of 179.3 MPa and KISCC = 12.5 MPa m0.5. The initial crack depth was 0.2 mm.

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and the defect transforms into a stress corrosion crack. Then, the increasing crack tip strain rate starts to promote the crack growth. The CGR is shown to increase, significantly, with time and the increase is progressively more significant until approaching a peak rate, and then the CGR becomes relatively steady. Fig. 18 also shows the crack depth, which increases with time, and the slope of this curve is the CGR. The variation of CGR vs. K is shown in Fig. 19 as a semi-log plot. The CGR is seen to increase immediately after KISCC. Then, the increase slows down until a steady value vs. K is reached. The corresponding accumulated crack depth vs. K is also shown in this figure. The significant increase of CGR vs. K results from mechanically enhanced anodic dissolution at the crack tip after KISCC is reached. As the crack grows, K increases, which further promotes the growth of the crack until the increase of crack growth is counter-balanced by the environmental factors, to be elaborated on later. In Fig. 20, the pH at the crack tip is shown to decrease with time. This decrease is however negligibly small overall, slightly over half a unit drop for the entire duration of crack growth used in this calculation. This decrease of the crack tip pH results from hydrolysis at the crack tip by the increased concentrations of ferrous ion

Fig. 19. Crack growth rate vs. K for a temperature of 328.15 K, bulk pH 9, external stress of 179.3 MPa and KISCC = 12.5 MPa m0.5. The initial crack depth was 0.2 mm.

Fig. 20. Crack tip pH vs. time for a temperature of 328.15 K, bulk pH 9, external stress of 179.3 MPa and KISCC = 12.5 MPa m0.5. The initial crack depth was 0.2 mm.

and chloride or the decreased concentrations of sodium ion and carbonate and bicarbonate ions. The decrease of pH is rather small because the solution is concentrated and also buffered by the carbonate and bicarbonate species. The presence of this buffered solution makes the solution chemistry in the entire crack relatively stable with time, as shown in Fig. 21, where the in-crack pH has little variation with space and time. With a concentrated solution in the crack, the IR drop within the crack is very small. Thus, the potential within the crack shows a negligible change in Fig. 22. To explain how the crack growth is counter-balanced at steady state by the mechanical and environmental factors, a comparison of the crack tip current densities in the presence and absence of stress vs. time is shown in Fig. 23. Note that the current density, in the absence of stress, represents corrosion velocity under the artificial assumption that the crack depth were maintained the same as in the presence of stress, for a given time. This current density, without stress, is a portion of the current density in the presence of stress, and the latter represents CGR, as expressed in Eq. (25). Although the corrosion velocity decays with time due to increased IR as the crack grows, the significant increase of the

Fig. 21. In-crack pH vs. crack depth at different times, for a temperature of 328.15 K, bulk pH 9, external stress of 179.3 MPa and KISCC = 12.5 MPa m0.5. The initial crack depth was 0.2 mm.

Fig. 22. In-crack potential vs. crack depth at different times, for a temperature of 328.15 K, bulk pH 9, external stress of 179.3 MPa and KISCC = 12.5 MPa m0.5. The initial crack depth was 0.2 mm.

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crack pH and potential do not vary significantly with crack growth. However, as the crack experiences rapid growth after K just passes over KISCC, there is a significantly increased dissolution of iron into solution, which accumulates at the tip. This increase of ferrous ion concentration increases the tip-to-mouth concentration ratio (the mouth concentration c0Fe2þ is assumed to be constant in the model) in the cathodic portion of the Butler–Volmer equation (Eq. (4)). Thus, the corrosion velocity is significantly decreased, as clearly demonstrated in Fig. 25, where the anodic portion, and the cathodic portion without this concentration, are both relatively constant with time. That the increase of corrosion velocity follows the same pattern to that of the concentration ratio indicates that the latter is responsible for the former.

CGR follows a similar pattern as the increase of Coeff1 (upper porK Þ2:4 , which retion of the figure), a coefficient in Eq. (25), or ðK ISCC flects the effect of crack tip strain rate expressed by K. This similar trend of the two curves indicates that the increase of CGR is due to the increase of K as the crack grows. The increase of CGR due to the mechanical component, and the decrease of the corrosion current density, not including the effect of stress or due to change of the environmental chemistry at the crack tip, are two components of the CGR expression in Eq. (25), and are dynamically counter-balanced due to the sharp decrease of iFe (no stress factor) at the steady state as shown in Fig. 23. The above two conditions are further discussed in Fig. 24 relative to K. On a semi-log scale, the decay of corrosion velocity vs. K, in the absence of stress, is clearly demonstrated again. This significant decrease of the corrosion velocity as K passes 50 MPa m0.5 is the reason that the CGR could reach a relatively constant value even when K or Coeff1 continues to increases with crack growth. The decrease of corrosion velocity is shown to result from the increase of the concentration ratio of iron concentration since the

4.2.2. Effect of temperature on CGR Using the same concentrations of Na+ and Cl and the same pH as those at 55 °C, the Fe2+ concentration at other temperatures can be calculated from charge balance when the solution is saturated with ferrous carbonate. It is determined to be 7.39  1010 mol/L for 25 °C or 1.79  1010 mol/L for 75 °C. This variation of Fe2+ concentration with temperature results from the variation of the equilibrium constants with temperature. The results presented below are obtained for a constant external hoop stress of 179.3 MPa at 328.15 K and two other temperatures with 20 K interval from 328.15 K. The initial chemistry and stress conditions are the same as used earlier for 328.15 K. As the temperature varies, the values of the kinetic and equilibrium parameters vary, including exchange current density and the Tafel slope and, thus, the initial CGRs at different temperatures are different, as shown in Fig. 26, where CGR vs. time and crack depth vs. time are shown. Since the initial CGR increases with increasing temperature, at a higher temperature it takes less time before KISCC is reached. In Fig. 26, KISCC is located at the point where the CGR starts to increase sharply. At a higher temperature, the increase of CGR becomes steeper. The effect of temperature on CGR is significant because the time to fail a pipe is much shorter at a higher temperature. For a crack to grow to 60% of wall thickness, the model predicts 10 years at 348.15 K, in comparison with 40 years at 298.15 K. Fig. 27 shows CGR vs. K and crack depth vs. K. It is necessary to note that crack depth vs. K is invariant with temperature, and

Fig. 24. CGR vs. K in the presence and absence of stress, for a temperature of 328.15 K, bulk pH 9, external stress of 179.3 MPa and KISCC = 12.5 MPa m0.5. The initial crack depth was 0.2 mm.

Fig. 25. Crack tip current densities (anodic portion, cathodic portion, overall and cathodic portion without, including the concentration ratio of ferrous ion) and relative ferrous ion concentration vs. time, for a temperature of 328.15 K, bulk pH 9, external stress of 179.3 MPa and KISCC = 12.5 MPa m0.5. The initial crack depth was 0.2 mm.

Fig. 23. Crack tip anodic current densities in the presence and absence of stress and the stress-influencing factor to CGR vs. time, for a temperature of 328.15 K, bulk pH 9, external stress of 179.3 MPa and KISCC = 12.5 MPa m0.5. The initial crack depth was 0.2 mm.

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Fig. 26. Crack growth rate and crack depth vs. time at three different temperatures, including 328.15 K. For each temperature, bulk pH 9, external stress of 179.3 MPa and KISCC = 12.5 MPa m0.5. The initial crack depth was 0.2 mm.

Fig. 27. Crack growth rate and crack depth vs. time at three different temperatures, including 328.15 K. For each temperature, bulk pH 9, external stress of 179.3 MPa and KISCC = 12.5 MPa m0.5. The initial crack depth was 0.2 mm.

also invariant with stress (to be shown later), concentration and pH, because CGR is related to K by Eq. (24), which is independent of the above parameters. The pattern of the variation of CGR with K appears to be similar for the three temperatures. Before KISCC is reached, the CGR is approximately constant and small. After KISCC is reached, a significant increase of CGR is shown due to the increase of K and the corresponding increase of the crack tip strain rate. Then, the CGR reaches a relatively steady-state condition, counter-balanced by the decreasing anodic current density resulting from the increase of ferrous ion concentration.

Fig. 28. Crack growth rate and crack depth vs. time at three different external stresses, including 0.5 SMYS. For each stress, bulk pH 9, external potential of 0.75 VCSE, temperature of 328.15 K and KISCC = 12.5 MPa m0.5. The initial crack depth was 0.2 mm.

Fig. 29. Crack growth rate and crack depth vs. K at three different external stresses, including 0.5 SMYS. For each stress, bulk pH 9, external potential of 0.75 VCSE, temperature of 328.15 K and KISCC = 12.5 MPa m0.5. The initial crack depth was 0.2 mm.

Fig. 29 shows the variation of CGR vs. K and crack depth vs. K. That CGR vs. K does not appear to be affected by stress, because, regardless the stress level, as long as K is the same, the crack tip strain rate is the same and the CGR does not vary. This implies that it is the crack tip strain rate, instead of stress itself, which is more relevant to CGR. The significant increase of CGR after KISCC is reached results from the significant increase of K/KISCC as indicated by Eq. (25). 5. Conclusions

4.2.3. Effect of external stress on CGR In this calculation, all conditions at 328.15 K were kept the same except stress. Fig. 28 shows CGR vs. time and crack depth vs. time at four stress levels with respect to SMYS in the range of 0.47–0.56 SMYS, with an increment of 0.03 SMYS. The CGR increases with time more significantly at a higher external stress. This is because, at a higher stress, a smaller crack depth or a shorter time is required to reach KISCC. In Fig. 28, KISCC is located at the time when CGR just picks up to increase significantly.

A mathematical model was developed as a first step to predict pipeline CGRs mechanistically, potentially useful for field applications in the prioritization of SCC susceptible locations along a pipe segment. The model incorporated the effect on CGR of the mechanical stress, steel properties, the increase of crack depth with time as a moving boundary condition at the crack tip, in addition to mass transport and the kinetic relations of chemical and electrochemical reactions involved in the cracking process.

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Lacking knowledge of the film repassivation kinetic exponent as a key model parameter for predicting CGRs, experimental tests were conducted to generate this information. An exponent of n = 0.6 was used for the modeling based on the test results. Due to the significant variability of laboratory and field crack growth rate data (the former is often orders of magnitude higher than the latter), a qualitative validation of the model was performed. The model was demonstrated to be capable of predicting both the unsteady-state (Stage I) and steady-state (Stage II) crack growth. The model was used to predict CGRs by two methods of expressing the anodic current density: (1) the potentiodynamic polarization curve, and (2) the Butler–Volmer equation. The merits and shortcomings of both methods were analyzed before their use for CGR prediction. The first method predicts large CGRs and the predicted in-crack chemistry and potential were found to vary significantly with time and space. The second method has the flexibility to predict low CGRs. At low CGRs, the predicted in-crack chemistry and potential have no significant variation with time and space. The predicted mechanism responsible for the steady-state crack growth was shown to be the balance between increasing K, which tends to increase CGR as the crack grows, and the change of crack tip condition, which, for large CGRs, is the significant shift in the more negative direction of the crack tip potential, and for low CGRs, the increase of ferrous ion concentration, and either tends to decrease CGR. Acknowledgments This work was sponsored by Southwest Research Institute as an internal program. The author acknowledges the various contributions to this work by Drs. Steve Hudak, Graham Chell, Narasi Sridhar (now in DNV, OH, USA), Todd Bredbenner and Leo Caseres, and by Ms. Lori Salas, all SwRI employees. Drs. Krishnan S. Raja and Yugo Ahida of the University of Nevada Reno conducted the experimental tests. Dr. Raymond Fessler of BIZTEK Consulting Inc. served as a consultant to this program. Appendix A. Repassivation kinetics measurements A.1. Experimental technique The goal of this test was to measure the repassivation kinetics of X52 carbon steel in a standard solution containing 1 N–1 N Na2CO3–NaHCO3 and charge balanced by Na+. The effect of electrochemical potential, environmental temperature, and strain hardening of the material was explored. The experimental technique used to measure the repassivation kinetics was detailed elsewhere [54], and briefly described below. The test was conducted by using a slow strain rate test machine. Potentiostatically controlled in a test solution, a tensile specimen was cyclically loaded, unload and reloaded at a different strain rate for each cycle. The tensile specimens used for the repassivation kinetics measurement have the geometry and dimensions shown in Fig. A-1. The specimen gage section is 0.64 inch (1 in. = 2.54 cm) in length and 0.16 in. in diameter.

Fig. A-1. Cylindrical tensile specimen with dimensions in inch.

with alcohol followed by distilled water before being introduced into the test cell. The electrochemical test cell is detailed elsewhere [54]. Briefly, it consists of a specially ordered flask with a holed adapter on the bottom, another on the top, and neck adapters for counter electrode (coiled Pt wire), reference electrode, heating rod, gas purge, and thermocouple. Two rubber O-rings were used to provide a leak proof seal between the holed adapters and test specimen. The standard 1 N–1 N Na2CO3–NaHCO3 solution was prepared by using analytical grade chemicals and distilled water. Deaeration was conducted during the test by continuously purging an industrial grade nitrogen gas into the solution. The test solution was kept at 25, 55, and 75 °C by using a quart heating rod and temperature controller. The temperature deviation was controlled within ±1 °C. Before starting a tensile test, the specimen was initially immersed in the test solution under an open circuit condition. Corrosion potential was measured during the deaeration with nitrogen gas. After the corrosion potential decreased to a stable value between 0.915 VCSE and 0.955 VCSE, the specimen was shifted to a test potential (0.510 VCSE or 0.690 VCSE) and kept for about 1 h to allow the passive current to be stabilized. Subsequently, a tensile test was carried out under potentiostatic control at 0.510 VCSE or 0.690 VCSE. The above two potentials were chosen based on polarization test results to be shown. The specimens were pulled at different extension rates, such as 0.016256, 0.08128, 0.16256, and 0.8128 mm/s, corresponding to initial strain rates of 0.001/s, 0.005/s, 0.01/s, and 0.05/s, respectively. The strain rates and strain were calculated based on the initial gage length. The loading procedure has been schematically shown in Fig. A-2. During the first loading cycle, the specimen was strained to a specified value with a given extension rate and the activated transient

A.2. Experimental procedure In preparation of the test specimens, all other portions, except the gauge section with a surface area of 2.08 cm2, was masked with a thin epoxy coating. The gage section of the specimens was polished up to 1200 grit emery paper and then thoroughly washed

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Fig. A-2. Schematic illustration of the loading cycles in the tensile test.

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current was recorded. After reaching the specified strain, the pulling was stopped abruptly and the strain was maintained constant, allowing the current transient to decay and monitored for about 300 s. Then the specimen was completely unloaded, and the passivation was continued until it reached a steady passive current again. For the second pulling cycle, the unloaded specimen was strained to 15% (corresponding to still the initial gage length of 16.26 mm) with the same extension rate. The current transient was recorded during pulling, as well as after stopping pulling, while still maintaining the strain. This procedure was repeated for another cycle. For several tests, the specimen was strained to 10% of the original gauge length in the serialized three cycles with different extension rates. The effect of strain rate on the anodic dissolution behavior of X52 was examined. A.3. Experimental results A.3.1. Steel nominal composition and microstructure The nominal chemical composition of the steel used for making the specimens is in element wt.% as 0.16C, 1.32Mn, 0.31Si, 0.006S, 0.017P, 0.01Ni, 0.01Cr, 0.02Nb, and balance Fe. With a piece of the steel wet polished using 240–1200 grit emery papers followed by light etching with Nital (2% HNO3), the microstructure of X52 was examined under an optical microscope. Shown in Fig. A-3, the microstructure consisted of lighter ferrite and darker pearlite. The low magnification picture shows that the pearlite elongated and distributed along the rolling direction of the material. The high magnification picture shows that the

Fig. A-3. Microstructure of X52 showed ferrite and pearlite (a) 40; (b) 160.

1.5 25 °C

55 °C

1.0 75 °C

Potential (VCSE)

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0.5 1N-1N Na2CO3-NaHCO3 solution pH 9.5 25 °C N2 deaerated Exposed aera: 0.71 cm2

0.0 -0.5 -1.0 -1.5 0.000001 0.00001

0.0001

0.001

0.01

0.1

1

Curren density (A/cm2) Fig. A-4. Anodic polarization curves of X52 in 1 N–1 N Na2CO3–NaHCO3 solution at 25, 55, and 75 °C.

grain size of the ferrite was between 10 and 50 lm with curved grain boundaries. A.3.2. Polarization results Anodic polarizations were conducted, potentiodynamically, for three temperatures: 25, 50, and 75 °C. At a rate of 10 mV/min, as recommended in ASTM G61-86 (Reapproved 1998), the potential scan was started at 0.500 V below the corrosion potential and

Fig. A-5. Microstructure of X52 after electrochemical etching, for 5 min at (a) 0.510 VCSE; (b) 0.690 VCSE.

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(a)

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(b)

600 Extension rate 0.05 s

-1

-1

-1

0.01 s

0.001 s

500

Stress (MPa)

400

300

200

100

0 0

0.1

0.2 Strain (%)

0.3

0.4

(c)

(d)

Fig. A-6. Stress and repassivation current during tensile test of X52 steel with multiple straining cycles and multiple strain rates in 1 N–1 N Na2CO3–NaHCO3 solution at 75 °C and at a potential of 0.510 VCSE. (a) Stress, strain, and strain rate of each loading cycle, (b) first cycle to 10% strain at a strain rate of 0.05/s, (c) second cycle reloading to 15% strain at a strain rate of 0.01/s, (d) third cycle reloading to 15% strain at a strain rate of 0.001/s. c2 in this figure is the same as c in Eq. (A-1) or (A-2).

was terminated when the anodic current reached 0.1 A/cm2. All potentials were corrected with respect to saturated Cu/CuSO4 electrode at 25 °C. Fig. A-4 shows the polarization curves of X52 in standard 1 N– 1 N Na2CO3–NaHCO3 solution at the above three temperatures. It exhibits an active peak region next to the corrosion potential and a secondary peak region for all of the three temperatures. The current densities at the active peak were 0.63, 3.8, and 7.3 mA/cm2 when the temperatures were 25 °C, 50 °C, and 75 °C, respectively. Passive current density at 25 °C was obviously lower than those at 50 and 75 °C. Potentiostatically controlled at 0.510 VCSE and 0.690 VCSE, repassivation kinetic measurements were conducted. The current densities at 0.510 VCSE and 0.690 VCSE, were 0.039 and 0.527 mA/cm2 at 25 °C, 0.189 and 0.505 mA/cm2 at 55 °C, and 0.306 and 0.22 mA/cm2 at 75 °C, respectively. It was found [25,26] that the SCC sensitive potential range is likely located between the active peak and the secondary peak, where a big difference in current could be expected if polarization curves at high and low potential scan rates were generated. The potentials of 0.510 VCSE and 0.690 VCSE are within this potential range. A.3.3. Microstructure after etching in 1 N–1 N Na2CO3–NaHCO3 solution In order to confirm the effect of potential on anodic dissolution at grain boundaries, polished specimens were electrochemically

polarized for 5 min in the standard test solution of 1 N–1 N Na2CO3–NaHCO3 at 75 °C. Fig. A-5(a) and (b) show the microstructure of X52 after electrochemical etching for the potentials of 0.510 VCSE and 0.690 VCSE, respectively. In (a), darker pearlite and slightly etched ferrite grain boundaries can be seen after etching at 0.510 VCSE. By contrast, after etching at 0.690 VCSE pearlite colonies and ferrite grain boundaries were observed more clearly in (b). The difference in precipitate at the grain boundaries was due to the current difference as shown on the polarization curves. This difference may cause the variation of SCC susceptibility under a certain loading condition. A.3.4. Current transient during multiple cyclic tensile tests The current transient vs. time after onset of current increase during electrode straining was derived to follow: [55]

  pffiffiffiffiffiffiffiffiffiffiffiffiffi = 1 þ e_ t It ¼ c t1n  t 1n 0

ðA:1Þ

where t0 is time same as defined for Eq. (3), t is elapsing time starting from the increase of the transient current during straining of the tensile specimen (see experimental data in Fig. A-6), n is repassivation kinetic exponent, e_ is initial strain rate, and c is a coefficient related to n, current density of a bare metal surface in the given solution, the original specimen gauge length, and crosssection area. The decay of the transient current immediately following the abrupt stop of the specimen straining follows:

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 .pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi It ¼ c t 1n  ðt  tc Þ1n 1 þ e_ tc

F.M. Song / Corrosion Science 51 (2009) 2657–2674

ðA:2Þ

where tc is time at which the straining of the specimen stops abruptly. In the above equations, all are known except ‘‘n” and ‘‘c”, which are to be obtained from the data optimization. Fig. A-6 shows one example set of the test results obtained for 0.510 VCSE at 75 °C, with the strain rates of 5  102/s, 1  102/s, and 1  103/s applied in the three loading cycles in sequence. The applied load is shown in (a), suggesting that the material was hardened after the first loading cycle. The acquired current transients for the three loading cycles are shown in (b–d) as black dots. The model results by best fit to Eqs. (A.1) and (A.2) are shown as a gray line. The acquired ‘‘n” is also shown in the figure. Table 2 summarized the repassivation kinetic exponents derived from multi-strain rate testing of a single specimen at different potentials and temperatures. Each specimen was strained at three different strain rates, such as 0.05, 0.01, and 0.005 s1. Initially, the specimen was strained to 10% at 0.05 s1 and, subsequently, strained to 30% at 0.01 s1 and to 45% at 0.005 s1. References [1] ASME, Gas Transmission and Distribution Piping Systems, ASME Code for Pressure Piping, B31, ASME B31.8-1999 (Revision of ASME B31.8-1995). [2] API, Managing System Integrity for Hazardous Liquid Pipelines, API Standard 1160, Draft, April 2001. [3] NACE, Stress Corrosion Cracking (SCC) Direct Assessment Methodology, NACE Standard RP 0204-2004. [4] NACE, In-Line Inspection of Pipelines, NACE Standard RP 0102-2002. [5] U.S. Code of Federal Regulations (CFR), Title 49: Transportation, Part 192: Transportation of Natural and Other Gas by Pipeline Minimum Federal Safety Standards, Subpart O: Gas Transmission Pipeline Integrity Management. Available from: . [6] R.R. Fessler, Research Needs on SCC of Pipelines, Presentation at DOT/RSPA, Houston, TX, December 2, 2003. [7] B.N. Leis, R.N. Parkins, Modeling stress-corrosion cracking of high-pressure gas pipelines, in: Eighth Symposium on Line Pipe Research, AGA Catalog No. L5 1680, 1993, pp. 19.1–19.21. [8] T.M. Ahmed, S.B. Lambert, R. Sutherby, A. Plumtree, Corrosion 53 (7) (1997) 581–590. [9] F.P. Ford, P.L. Andresen, Development and use of a predictive model of crack propagation in 304/316L, A533B/A508 and Inconel 600/182 alloys in 288 °C water, in: G.J. Theus, J.R. Weeks (Eds.), The Third International Symposium on Environmental Degradation of Materials in Nuclear Power Systems – Water Reactors, Traverse City, Michigan, 1987, p. 789. [10] P.L. Andresen, F.P. Ford, Material Science and Engineering A103 (1988) 167. [11] F.A. Champion, in: Symposium on Internal Stresses in Metals and Alloys, Inst. Metals, London, 1990, p. 468. [12] H.L. Logan, Journal of Research of the National Bureau of Standards 48 (1952) 99. [13] H.H. Uhlig, in: T.N. Rhodin (Ed.), Physical Metallurgy of Stress Corrosion Fracture, Interscience, 1959, p. 1. [14] J.R. Galvele, Corrosion Science 27 (1987) 1. [15] G.S. Duffo, J.R. Galvele, in: R.P. Gangloff, M.B. Ives (Eds.), Environment-Induced Cracking of Metals, NACE, Houston, 1990, p. 261. [16] K. Sieradzki, R.C. Newman, Philosophical Magazine A 51 (1) (1985) 95. [17] D.A. Jones, Principles and Prevention of Corrosion, second ed., Prentice-Hall Inc., Upper Saddle River, NJ, 1996, p. 279. [18] D.D. Macdonald, M. Urquidi-Macdonald, Corrosion Science 32 (1) (1991) 51. [19] A. Turnbull, Corrosion 57 (2000) 175. [20] R.N. Parkins, J.A. Beavers, Corrosion 59 (3) (2003) 258–272. [21] J. Been, F. King, L. Fenyvesi, R. Sutherby, A modeling approach to high pH environmental assisted cracking, in: International Pipeline Conference, Paper IPC 04-0361, 2004. [22] R.N. Parkins, Corrosion Science 20 (1980) 147–166. [23] R.N. Parkins, Realistic stress corrosion crack velocities for life prediction estimates, in: R.N. Parkins (Ed.), Life Prediction of Corrodible Structures, vol. 1, NACE International, 1994, p. 97.

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