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Extreme value modeling of localized internal corrosion in unpiggable pipelines
Journal Pre-proof Extreme value modeling of localized internal corrosion in unpiggable pipelines Carlos Melo, Markus R. Dann, Ron Hugo, Alberto Janeta PII:
S0308-0161(20)30033-8
DOI:
https://doi.org/10.1016/j.ijpvp.2020.104055
Reference:
IPVP 104055
To appear in:
International Journal of Pressure Vessels and Piping
Received Date: 8 December 2018 Revised Date:
14 January 2020
Accepted Date: 24 January 2020
Please cite this article as: Melo C, Dann MR, Hugo R, Janeta A, Extreme value modeling of localized internal corrosion in unpiggable pipelines, International Journal of Pressure Vessels and Piping, https:// doi.org/10.1016/j.ijpvp.2020.104055. This is a PDF file of an article that has undergone enhancements after acceptance, such as the addition of a cover page and metadata, and formatting for readability, but it is not yet the definitive version of record. This version will undergo additional copyediting, typesetting and review before it is published in its final form, but we are providing this version to give early visibility of the article. Please note that, during the production process, errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain. © 2020 Elsevier Ltd. All rights reserved.
Highlights • • •
Internal localized corrosion with microbiologically influenced corrosion Extreme value model to predict localized corrosion Application of structural reliability analysis for pipeline decision making
Extreme value modeling of localized internal corrosion in unpiggable pipelines Carlos Meloa , Markus R. Danna,∗, Ron Hugoa , Alberto Janetab a
Schulich School of Engineering, University of Calgary, 2500 University Dr NW, Calgary, Alberta, Canada, T2N 1N4 b Departamento de Mantenimiento, Petroamazonas EP, Av. 6 de Diciembre y Gaspar Ca˜ nero, Quito, Pichincha, Ecuador, 170504
Abstract Localized internal corrosion is one of the main failure mechanisms of energy pipelines. Microbiologically influenced corrosion (MIC) is a major contributor to internal corrosion in pipelines. Existing research uses extreme value analysis to fit extreme value distributions from data collected through either inspection or experiment to predict localized corrosion. The paper introduces an extreme value model to estimate the depth of localized internal corrosion considering MIC for unpiggable pipelines where inspection data is not available. The depth of material loss is estimated though a combination of corrosion analysis and extreme value modeling. A case study is provided to illustrate the proposed model. The application of a probabilistic approach supports risk-based inspection and maintenance planning for pipelines subject to internal corrosion. Keywords: extreme value modeling, pitting corrosion, microbiologically influenced corrosion, unpiggable pipelines, integrity, risk ∗
Corresponding Author Email address: [email protected] (Markus R. Dann)
Preprint submitted to International Journal of Pressure Vessels and Piping
January 8, 2020
1
1. Introduction
2
Pipelines are infrastructure systems that transport oil and gas from pro-
3
duction fields to processing facilities, and then from processing facilities to
4
end users. Pipelines are considered the safest and most economical method to
5
transport liquid and gaseous energy products [1–3]. However, pipelines can
6
fail due to a number of different mechanisms, with internal corrosion [4, 5]
7
being one of these mechanisms. According to data from the Pipeline Haz-
8
ardous Materials Safety Administration (PHMSA) [6] in the United States,
9
internal corrosion was the cause of approximately 10% of the 6,240 reported
10
failures of hazardous liquids pipelines between 2002 and 2010 [6]. Of these
11
internal corrosion failures, 6 in 10 involve localized pitting corrosion, and 5 of
12
these 6 identify microbiologically influenced corrosion (MIC) was identified
13
as one of the main degradation mechanisms [6].
14
Pipeline operators develop and implement pipeline integrity programs to
15
prevent these types of failures. Pipeline integrity programs utilize different
16
assessment methodologies to obtain information about a pipeline system and
17
using this information, establish maintenance plans for safe and reliable oper-
18
ation. In-line inspection (ILI) is the most detailed inspection method used for
19
integrity assessment, however, due to geometrical and/or operational restric-
20
tions, between 40 to 50% of the existing pipelines are unpiggable [7, 8] and
21
ILI tools cannot be used. Operators of unpiggable pipelines use alternative
22
integrity assessment methods, such as direct assessment [9–12], to perform
23
integrity assessment of these pipelines. However, most models used by direct
24
assessment methodologies are only able to predict general corrosion [9–12], 2
25
meaning that the effects of localized corrosion are not included yet from
26
PHMSA data it is known that 6 out of 10 internal corrosion failures involve
27
localized corrosion. This research focuses on developing a model to facilitate
28
the integrity assessment of unpiggable pipelines by estimating localized inter-
29
nal corrosion through a combination of advanced general corrosion analysis,
30
probabilistic analysis, and extreme value modeling (EVM). Probabilistic
31
modeling of corrosion has been introduced by Ahammed & Melchers in 1994
32
[13], Kale in 2004 [14],
and Lawson in 2005 [15]. To the best knowledge
33
of the authors, the proposed method is the first to combine advanced flow
34
and corrosion models with probabilistic and extreme value methods.
35
In the model, average general corrosion is estimated using advanced cor-
36
rosion analysis [16, 17]. A probabilistic analysis is implemented to include
37
the aleatory and epistemic uncertainties in the probability density func-
38
tion (PDF) of general corrosion. The respective models by Papavinasam
39
et al. [18, 19] and Sooknah et al. [19–21], are used to estimate two multi-
40
plication factors, one for localized corrosion and the other for MIC. Finally,
41
the EVM considers localized corrosion as the maximum of general corrosion
42
obtaining the PDF for localized corrosion.
43
In this paper, Section 2 provides an overview of localized corrosion, MIC,
44
and EVM. Section 3 presents the probabilistic analysis and the EVM to
45
estimate localized (pitting) corrosion as the maximum of the general corrosion
46
process in a pipeline. Section 4 applies the model to a case study where the
47
depth of pitting corrosion is estimated for a gathering pipeline in an oil
48
production field. Finally, conclusions are presented in Section 5.
3
49
2. Background
50
2.1. Localized Corrosion
51
After fabrication and during exposure to the atmosphere, a passive film is
52
created on metals as they undergo the corrosion process and this film protects
53
the metal from further corrosion. In carbon steel pipelines, however, the
54
film created is not compact and, therefore, unable to protect from corrosion
55
during operation [22]. Contact between the carbon steel pipeline and water
56
in the produced fluids creates a secondary layer that absorbs some of the
57
ions present in the solution [22]. Locations without the secondary layer that
58
are near regions with localized corrosion are prone to initiation of further
59
localized corrosion, which in turn accelerates layer deterioration [22].
60
Models to predict internal corrosion in pipelines have been under de-
61
velopment since 1975 [23]. They can be broadly classified as mechanistic
62
[16, 17, 24–27] or empirical [26, 28, 29], however, at times some prefer to
63
classify the models as either mechanistic or deterministic [30]. Most of these
64
models predict general corrosion rates, however the main contributor to cor-
65
rosion induced pipeline failures is due to localized corrosion, not general
66
corrosion. An empirical model for the estimation of localized corrosion in
67
pipelines was published by Papavinasam et al. [18]. It shows that localized
68
corrosion rates can not be obtained from general corrosion models. Despite
69
this findings, others have argued that general corrosion models can be used to
70
predict localized corrosion rates [30]. Dissenting arguments aside, the study
71
by Papavinasam et al. [18] does stress the need for extreme value analysis
72
when estimating localized corrosion.
4
73
2.2. Microbiologically influenced corrosion models
74
MIC has been identified as being responsible for many corrosion failures
75
in the oil and gas industry [8, 31–33]. MIC is the acceleration of the corrosion
76
process through the activity of microorganisms [34]. Most MIC is localized,
77
resulting in deep metal penetration in the form of pitting [34]. The envi-
78
ronment that accelerates corrosion by MIC is generated by biofilms that are
79
created when free floating planktonic microorganisms adhere to the surface
80
of the metal and become sessile [34]. An overview of some of the models used
81
to predict MIC are discussed in the following.
82
The first model to estimate the influence of MIC on internal corrosion
83
for application to the oil and gas industry was presented by Pots [35]. This
84
model was modified by Maxwell and Campbell [36] in 2006 to include the
85
influence of additional factors contributing to MIC such as system age and
86
the frequency at which cleaning tools are applied. In 2008, Sooknah et al.
87
presented a score-based model for quantifying the influence of MIC on lo-
88
calized internal corrosion rates in pipelines [20, 21]. All of these models use
89
a deterministic approach where the depth of MIC-influenced is fixed. This
90
limit their application to reliability estimates in pipelines as they are unable
91
to provide information about depth uncertainty, requirement for reliability
92
estimates.
93
2.3. Extreme value modeling
94
The objective of EVM is to make probabilistic statements about events
95
that are more extreme than any that have already occurred [37]. Figure 1
96
shows a section of a pipeline where the depth of the internal corrosion pit
5
97
i (i = 1, . . . , m) is described by the random variable Xi with a cumula-
98
tive distribution function (CDF) Fi (x ). A pipeline leaks when the depth
99
Xmax = max {X1 , . . . , Xm } of the deepest pit is equal to the wall thickness
100
wt of the pipeline. If X1 , . . . , Xm are independent and identically distributed
101
random variables with a common parent distribution CDF FX (x ), the ex-
102
treme value CDF of the m th order statistics Xmax is determined as follows
103
[37–39]: Fmax (x) = Pr{Xmax ≤ x} =
m Y
Pr{Xi ≤ x} =
i=1
m Y
FX (x) = {FX (x)}m (1)
i=1
Flow
X1
Xi
... Xm
wt
Figure 1: Pipeline subject to internal pitting corrosion where X1 , . . . , Xm are the depths of the m pits. Leak failure occurs when the maximum depth of the pits approaches the wall thickness wt of the pipeline.
104
Research by Melchers in 2008 [40] showed that under the influence of
105
MIC there is a high degree of randomness involved that makes a statis-
106
tical dependence between pits unlikely. Therefore, statistical independence
107
between pits is a reasonable assumption. In Equation 1 the CDF of the
108
deepest pit in a pipeline is Fmax (x ) = {FX (x )}m . Equation 1 can usually
109
not be applied for estimating the deepest pit because the parent distribution
6
110
FX (x ) of the corrosion damage is often unknown. Additionally, the number
111
m of corrosion pits in a pipeline is generally unknown [37].
112
3. Model
113
3.1. Overview
114
The objective of the proposed model is to estimate the probability distri-
115
bution of the depth of localized corrosion within a pipeline considering the
116
influence of MIC. The large amount of solids and liquids in upstream un-
117
piggable production and gathering pipelines increases their susceptibility to
118
internal corrosion. Pipelines have two modes of failure: leak and burst [42].
119
The depth of MIC-influenced localized corrosion is used in the model given
120
that the leak mode of failure is the most frequent for production and gath-
121
ering pipelines [43]. Figure 2 presents an overview of the proposed model.
122
A discretization in both space and time (Step 1) allows for general corrosion
123
growths to be estimated using corrosion rates from an advanced electrochem-
124
ical corrosion model (Step 2) [16, 17]. A probabilistic analysis is then applied
125
for determining the PDF of the depth of the population of corrosion features
126
(Step 3). The PDF of the depth of localized corrosion is estimated as the
127
maximum of the general corrosion using an EVM approach (Step 5) where
128
the effect of MIC is included (Step 4).
7
Sections s = 1, …, S and time periods t = 1, …, T
Step 1
s
1
Step 2 Average general Δμst
corrosion growth ωst
S
Step 4 Localized corrosion
(MIC)
time t = 1, …, T PDF
Step 3 General corrosion
time t = 1, …, T
PDF
Step 5 Localized corrosion
feature depth
feature depth
Figure 2: Overview of the 5-Steps to estimate the PDF of the depth localized corrosion. In Step 1, a discrete analysis with respect to space and time is implemented. In Step 2, general corrosion growths ∆µst are calculated. In Step 3, a probabilistic analysis includes aleatory and epistemic uncertainties in the PDF of the depth of general corrosion. In Step 4, localized corrosion rates including MIC allow to estimate a multiplication factor ωst . In Step 5, the PDF of the depth of localized corrosion is found using the PDF of general corrosion and the multiplication factor in an EVM.
129
3.2. Discretization
130
Corrosion is a continuous spatio-temporal process. A discrete analysis
131
with respect to space and time is proposed to reduce the computational de8
132
mands, while maintaining an acceptable level of accuracy for the estimation
133
of the corrosion rates and growths. In Step 1, the pipeline shown in Figure 2 is
134
discretized into S sections (s = 1, . . . , S ). Using industry standards, a maxi-
135
mum length of 15 meters per section is recommended to accurately estimate
136
the depth of general corrosion [11]. In the proposed analysis, a length of 6 me-
137
ters is used to justify a constant corrosion rate within each section. The 6-me-
138
ter discretization is based on a sensitivity analysis that was performed when
139
the applied general corrosion model [16, 17] was developed to achieve the
140
stated accuracy of the National Association of Corrosion Engineers (NACE)
141
SP0116 [11], which is one of the main references for prediction of internal
142
corrosion in pipelines.
143
The discretization of time is based on the variability of both operational
144
conditions (flow rate, temperature, and pressure) and flow composition (con-
145
tent of acids gases CO2 and H2 S, content of sulfates, bicarbonates, and chlo-
146
ride ions) to achieve approximately constant conditions within each time
147
increment. For each discretized section these variations are used to create
148
the temporal discretization ∆τt (t = 1, . . . , T ) which is pipeline-specific. A
149
constant corrosion rate is then assumed and computed for each time step and
150
section.
151
3.3. Average depth of general corrosion
152
153
The purpose of Step 2 in Figure 2 is to obtain the average depth of general corrosion µ in section s at time τT using Equation 2: µsT =
T X
∆µst
for s = 1, . . . , S
t=1
9
(2)
154
This enables the average depth of general corrosion at time τT to be esti-
155
mated. If it were desired to compute future corrosion growth rates, more
156
complex models would be required, and this is beyond the scope of the cur-
157
rent study. The general corrosion growth increments ∆µst are calculated
158
using the corrosion rates CRst from an advanced electrochemical general cor-
159
rosion model [16, 17]. In this model the fluid behavior and the mass transfer
160
of protons are the main factors influencing the corrosion process. ∆µst = CRst ∆τt
for s = 1, . . . , S and t = 1, . . . , T
(3)
161
The general corrosion growth increments in Equation 3 depend on the size
162
of the time step (∆τt = τt − τt−1 with τ0 = 0). The time step size is selected
163
based on the discretization process described in Section 3.2.
164
3.4. PDF of general corrosion
165
The purpose of Step 3 in Figure 2 is to obtain the PDF of the depth
166
of general corrosion that includes aleatory and epistemic uncertainties [44].
167
Aleatory or type 1 uncertainty is related to the natural variability of the
168
corrosion process and cannot be reduced, while epistemic or type 2 uncer-
169
tainty considers model errors and statistical uncertainties due to our lack of
170
knowledge. Epistemic or type 2 uncertainty can be reduced by increasing
171
the amount of data in the analysis [44]. Equation 4 computes XsT , a ran-
172
dom variable that represents the depth of all corrosion features at time τT ,
173
as the combination of both the aleatory uncertainty εpop,s and the epistemic
174
uncertainty εmodel,s : XsT = εpop,s µsT + εmodel,s 10
for s = 1, . . . , S
(4)
175
Section 5.2.5 of NACE SP0116 [11] specifies that the prediction of the depth
176
of corrosion by the corrosion model shall not exceed ±10% wall thickness
177
(wt) of the measured depth of corrosion at the verification sites. In despite
178
of NACE SP0116 ignoring the error in the measuring tool, in estimating
179
the depth of general corrosion, it is conservative to assume that the standard
180
deviation due to the model error is calculated using a 10%wt of error with an
181
80% confidence interval. It is considered that the model error εmodel follows
182
a normal PDF with zero mean and a standard deviation σmodel of 7.8%wt,
183
which is based on the assumed normal distribution of the model error to
184
obtain a ±10%wt error band at an 80% confidence level. This assumption
185
considers that on average the predictions of the general corrosion model are
186
accurate but not perfect due to the epistemic uncertainty.
187
Using a population-based approach for the corrosion features [45] the
188
aleatory uncertainty of the corrosion process is included in the analysis
189
through use of a coefficient of variation covpop obtained using inspection
190
information of a pipeline with similar operating conditions and thus simi-
191
lar expected corrosion damage. The covpop depends on the accuracy of the
192
inspection tool and is obtained based on minimizing the error between the
193
selected distribution and the inspection information. Therefore, in this pa-
194
per the covpop is considered as a single value for the entire pipeline that is
195
not section specific. The aleatory uncertainty due to the population effect
196
εpop,s is assumed to follow a log-normal PDF with a mean value µpop = 1 and
197
a standard deviation σpop,s = covpop µsT . The mean value µpop = 1 is used
198
considering that on average all sections have similar aleatory uncertainties.
199
Finally, to avoid negative numbers that are physically not possible the PDF
11
200
of XsT is considered to be log-normally distributed according to Equation 5:
201
p 2 2 XsT | µsT , σs ∼ log –normal µsT , σpop,s + σmodel
for s = 1, . . . , S (5)
202
where σs is the standard deviation of the depth of general corrosion and
203
is calculated using the standard deviations of the population effect and the
204
model error.
205
3.5. Effect of localized corrosion and MIC
206
The main idea behind the proposed model is that localized corrosion is
207
considered as the maximum of general corrosion. In Step 4 in Figure 2, the
208
expected value of localized corrosion E[XL,sT ] is determined by scaling the
209
expected value of general corrosion E[XsT ] with ω ¯ sT , as shown in Equation 6:
210
E[XL,sT ] = ω ¯ sT E[XsT ]
(6)
211
The extreme value model (Equation 1) accounts for the effects of the scal-
212
ing ω ¯ sT on the variance of the PDF of localized corrosion. XL,sT is the size
213
of localized corrosion features and its expected value is used as a reference
214
point to obtain the PDF of localized corrosion fmax ,sT (x ) from the PDF
215
of general corrosion fsT (x ). The variance of the size of localized corrosion
216
features can also be calculated from: Z Var[XL,sT ] ≈
∞
x2 fmax,sT (x)dx − E2 [fmax,sT (x)]
(7)
0 217
A weighted multiplication factor ω ¯ sT is used in Equation 6 instead of the
218
combined multiplication factor ωsT , which is obtained as the product of the 12
219
two multiplication factors λst and κst : ωst = λst κst
for s = 1, . . . , S and t = 1, . . . , T
(8)
220
The weighted multiplication factor ω ¯ sT used to relate the expected values of
221
general E[XsT ] and localized E[XL,sT ] corrosion is calculated only at present
222
time τT . However, the multiplication factors λst and κst are obtained at
223
each specific time period τt for t = 1, . . . , T . Therefore, in the model the
224
weighted multiplication factor ω ¯ sT is used to properly estimate the relation-
225
ship between general and localized corrosion. The weighted multiplication
226
factor considers the individual multiplication factors ωst as a function of their
227
influence ∆µst in the average depth of general corrosion at present time µsT ,
228
and it is calculated using Equation 9: ω ¯ sT =
T 1 X ∆µst ωst µsT t=1
for s = 1, . . . , S
(9)
229
The first multiplication factor λst is for localized corrosion and is calculated
230
as the ratio between localized PCRst [18, 19] and general corrosion rates CRst
231
[16, 17]. The PCRst are calculated using an empirical model that considers
232
eleven variables affecting pitting corrosion including oil, water, gas, and solid
233
compositions, temperature, pressure, partial pressures of CO2 and H2 S, and
234
concentrations of sulfate, bicarbonate and chloride ions [18, 19]. General
235
corrosion rates CRst are estimated with an advanced corrosion model that
236
includes a mechanistic flow model, a mass transfer model, a scale model, a
237
solid deposition model, and an electrochemical corrosion model [16, 17]. λst =
P CRst CRst
for s = 1, . . . , S and t = 1, . . . , T
13
(10)
238
The second multiplication factor κst considers the influence of MIC and it is
239
estimated using a qualitative MIC risk model that considers nine variables
240
including flow rates, temperature, ratio of partial pressure between acid gases
241
CO2 and H2 S, pH, Langelier saturation index, total suspended and dissolved
242
solids, Redox potential, and sulfur content [20, 21].
243
3.6. PDF of localized corrosion
244
The objective of Step 5 in Figure 2 is to obtain the PDF of localized
245
corrosion which is considered the maximum of general corrosion as detailed
246
in Section 3.5, therefore: fmax,sT (x) = m fsT (x) FsT (x)m−1
(11)
247
where fmax ,sT (x ) is the PDF of localized corrosion at time T , m is the
248
m th order statistics, and fsT (x ) and FsT (x ) are the general corrosion PDFs
249
and CDFs obtained using Equation 5. A short algorithm, which is stopped
250
when δsT ≤ 0 , was developed to calculate m ∈ N. The algorithm is provided
251
in the Appendix. δsT = ω ¯ sT E[XsT ] − E[fmax (x)]
for s = 1, . . . , S
(12)
252
where ω ¯ sT is the weighted multiplication factor from Equation 8 that includes
253
the effect of localized corrosion and MIC, E[XsT ] is the expected value of the
254
depth of general corrosion at time τT , and E[fmax ,sT ] is the expected value
255
of the maximum extreme. The expected value is used as the base point
256
for the proposed model. However, future analyses should consider other
257
base points to review their influence in the reliability estimates of pipelines.
258
The reason the expected values are used as the base point is that the link 14
259
between localized and general corrosion is calculated using average values
260
from deterministic models.
261
4. Case Study
262
A multiphase gathering pipeline, located in one of the oil fields of Petroa-
263
mazonas EP in the Amazon basin of Ecuador, is considered in this case study.
264
The pipeline has been in operation for 10 years. Acid gases, such as CO2 ,
265
and a considerable amount of water have been present in the produced fluids
266
since the beginning of operation. These conditions increase the likelihood
267
that the pipeline has been subject to internal corrosion. The objective of
268
this analysis is to estimate the size of localized corrosion using the propose
269
method. Table 1 provides the input used for the estimation of general and
270
localized corrosion rates in this gathering pipeline. The pipeline is discretized
271
into S =58 sections, each having a length of 6 meters, and the analysis is dis-
272
cretized into T =10 time points. The spatial and temporal discretization are
273
based on the criteria detailed in Section 3.2 . Table 2 shows the discretized
274
production flow rates for oil, gas, and water along with the operating tem-
275
perature for the gathering pipeline. In Table 2 there is an increase in the
276
production flow rates of water and oil between the first τ1 and last τ10 time
277
points, as normally observed in an oil production field.
15
Table 1: Pipeline geometry and pipeline operational variables for the general and localized corrosion analysis
Nominal external diameter
436.57 mm
Length Inlet pressure Fluid type Nominal wall thickness Mole fraction Pipe roughness
348 m 3.44 MPa Multiphase 10.3 mm 0.201 CO2 0.04572 mm
Table 2: Production scenarios for general and localized corrosion analysis Time point τt
years ◦
Operating temperature
1
2
3
4
5
6
7
8
9
10
C
90
92
89
91
92
93
95
95
92
92
Oil rate
m3 /day × 103
2.27
2.28
3.62
4.89
5.21
4.89
4.85
4.87
4.70
4.48
Gas rate
m3 /day × 103
38.39
42.22
43.75
60.74
117.37
174.00
80.56
52.24
59.32
60.23
3.46
2.14
1.65
4.19
4.99
7.05
7.33
7.69
9.02
9.16
Water rate
3
3
m /day × 10
278
Using the advanced general corrosion model described in Section 3.3,
279
the general corrosion rates CRst are estimated and the average growth of
280
general corrosion ∆µst for each section and time period is calculated using
281
Equation 3. Figure 3 summarizes the average depth of general corrosion
282
µs10 for each section at τ10 .
283
the pipeline that was used for the calculations of the general corrosion pro-
284
file. Applying the qualitative categorization criteria for general corrosion of
285
steel in oil production systems from NACE SP-0775-2013 [46] severe general
286
corrosion (indicated by a red triangle) is in Sections 11 and 33, high general
287
corrosion is in Sections 2 and 58 are (orange circle), and moderate general
Figure 3 also shows the elevation profile of
16
288
corrosion in in Sections 5 and 19 (yellow triangle). These 6 sections are
289
selected to demonstrate application of the proposed method.
290
The localized multiplication factors λst , and the MIC multiplication fac-
291
tors κst for each section and time period are estimated with the method
292
described in Section 3.5. These two multiplication factors, which are pro-
293
vided in Tables 4 to 7 in the Appendix, are used to estimate the combined
294
multiplication factors ωst (Equation 8) for each section and time point. Fi-
295
nally, the weighted multiplication factor ω ¯ s10 for each section at present time
296
is calculated using Equation 9. Figure 4 shows the computed weighted multi-
297
plication factor where sections with severe general corrosion (11 and 33) are
298
seen to have lower multiplication factors, while a section with high general
299
corrosion (58) has the highest multiplication factor. A possible explanation
300
for these results lies in the greater influence of proton mass transfer, which
301
is directly affected by produced fluid flow conditions, in estimating general
302
corrosion rates compared to localized corrosion rates. 1.6 Severe Corrosion High Corrosion Moderate Corrosion
ωs10
1.4
58 19
1.2
5 2
1.0
11 1
5
10
33 15
20
25
30
35
40
45
50
55
58
Section
Figure 4: Weighted multiplication factor ω ¯ s10 for each section at present time. The factor includes the localized corrosion and the MIC factors.
17
60 11
μs10 [%wt]
50
33
40 30
2
20 10 0
5 1
5
19 10
15
58
Severe Corrosion High Corrosion Moderate Corrosion
20
25
30 Section
35
40
45
50
55
58
(a) Average depth of general corrosion for all sections at present time 230
Elevation[m]
220 210 200 190 180 1
5
10
15
20
25
30 Section
35
40
45
50
55
58
(b) Elevation profile Figure 3: (a) Average depth of general corrosion for all sections at present time T = 10. Representative sections for severe, high, and moderate general corrosion are highlighted as red triangle up, orange circle, and yellow triangle down, respectively. (b) Elevation profile of pipeline
18
303
The PDFs of the depth of general corrosion for each section at present
304
time are found based on Equation 5 with µs10 , a σmodel of 7.8%wt, and a covpop
305
of 0.43. Finally the PDFs and CDFs of localized corrosion are estimated
306
using the method described in Section 3.6. Figure 5 shows the PDFs (5a
307
on left) and CDFs (5b on right) for general and localized corrosion for one
308
section. There is an increase in the variance of the PDF of localized cor-
309
rosion compared to general corrosion, which is a reasonable result based on
310
the proposed method.
311
The computational demand of the proposed model is lower than a fea-
312
ture-specific approach. The analysis of the case study took 317 seconds on
313
an Intel Xeon ES-2670 2.6 GHz processor. (a) Section 58
(b) Section 58
1.0 General Corrosion Localized Corrosion
0.0004
0.0002
0.0000
General Corrosion Localized Corrosion
0.8 CDF
PDF
0.0006
0.6 0.4 0.2
0
20
40 60 feature depth [% wt]
80
100
0.0
0
20
40 60 feature depth [% wt]
80
100
Figure 5: PDFs and CDFs of general and localized corrosion for one section of the gathering pipeline.
314
For the purpose of comparison the PDFs of general and localized corro-
315
sion are utilized in a structural reliability analysis [47] using the convolution
316
integral between the wall thickness (resistance) and the estimated depth of
317
general or localized corrosion features (load effect) to calculate the probabil-
19
318
ity of failure for general and localized corrosion: Z ∞ Fwt (x) fmax,s10 (x) dx for s = 1, . . . , S PF,L =
(13)
0 319
where PF ,L is the probability of failure for localized corrosion at T =10, and
320
Fwt (x ) represents a normal distribution with a mean of 100%wt and a stan-
321
dard deviation of 1.5%wt [48] . Equation 13 is also used to calculate the
322
probability of failure PF ,G for general corrosion by replacing fmax ,s10 (x ) with
323
fs10 (x ). Table 3 summarizes the results of these calculations and also shows
324
the ratios between the localized and general corrosion probabilities at present
325
time for the selected sections. Table 3: Summary of probability of failure for selected sections at present time
Section
General Corrosion PF ,L [ failures ] PF ,G [ failures ] section section
PF ,L PF ,G
2
High
1.38 × 10−5
6.89 × 10−6
2.00
5
Moderate
1.48 × 10−5
7.42 × 10−6
2.00
11
Severe
6.08 × 10−4
3.09 × 10−4
1.97
19
Moderate
1.49 × 10−5
7.48 × 10−6
2.00
33
Severe
5.53 × 10−4
2.81 × 10−4
1.97
58
High
9.72 × 10−6
3.24 × 10−6
3.00
326
In Table 3, the ratio of probabilities of failure for localized corrosion to
327
general corrosion is seen to range from 1.97 to 3.00, indicating that significant
328
more risk mitigation actions will be applied to the pipeline when localized
329
corrosion probabilities are used. Comparing the results with the exceedance
330
probability analysis and assuming an acceptable probability of failure of 10−3
331
], at present time all of the selected sections are operating below this [ failures section 20
332
acceptable limit. However, it is important to note that Sections 11 and 33
333
may require risk mitigation actions including inspection and maintenance in
334
the near future. This example clearly demonstrates the advantages of using
335
a reliability analysis in comparison to an exceedance probability analysis, as
336
the latter approach that may result in conservative actions.
337
5. Conclusions
338
The approach presented in this paper supports the reliability and risk-in-
339
formed assessment of unpiggable pipelines subject to internal corrosion. The
340
proposed population-based approach is computationally more efficient than
341
a feature-specific approach. Hence, its advantage is the scalability to longer
342
pipelines.
343
The model includes aleatory and epistemic uncertainties of the localized
344
corrosion process. If field data is available, Bayesian updating can be applied
345
to reduce epistemic uncertainties in the model. However, a greater benefit
346
of the model is the use of the results to select optimal sections for field
347
verification.
348
Future research will focus on prediction of the probability distributions
349
of the depth of localized corrosion at future times. The use of stochastic
350
processes will facilitate these estimations required to implement risk-based
351
inspection and maintenance planning for unpiggable pipelines.
21
352
353
Appendix. Algorithm to compute the m th order statistics The following algorithm was developed to determine iteratively the m th order statistics m ∈ N is stopped when δsT ≤ 0 in Equation 12. Algorithm 1 Algorithm to calculate the m th order statistics and the CDF and PDF for the size of localized corrosion features is stopped when δsT ≤ 0 . Input: fsT (x) and FsT (x) from Equation 5,
E[XL,sT ]
from
tion 6, and fmax (x) from Equation 11
Initialize: m = 1 repeat set m = m + 1 compute E[fmax (x)] =
R∞ 0
xfmax (x)dx
until δsT ≤ 0 (Equation 12)
Output: CDF Fmax (x) and PDF fmax (x) of localized corrosion 354
22
Equa-
355
356
357
Appendix. Multiplication factors for case study The following tables include the multiplication factors for localized corrosion and MIC used in the case study. Table 4: First multiplication factor λst for localized corrosion sections 1 to 28 Section/Year
1
2
3
4
5
6
7
8
9
1
3.4
3.6
3.4
3.8
4.0
3.8
3.4
1.0
1.0
1.0
2
3.3
3.5
3.3
3.6
3.9
3.7
1.0
1.0
1.0
1.0
3
3.3
3.5
3.3
3.6
3.9
3.8
3.6
1.0
3.2
3.1
4
3.3
3.5
3.3
3.7
3.9
3.8
3.6
1.0
3.2
3.1
5
3.3
3.5
3.3
3.6
3.9
3.8
3.6
1.0
3.2
3.1
6
3.3
3.5
3.3
3.6
3.9
3.8
3.5
1.0
3.2
3.1
7
3.3
3.5
3.3
3.6
3.8
3.8
1.0
1.0
1.0
1.0
8
3.5
3.6
3.5
3.6
3.8
3.8
1.0
1.0
1.0
1.0
9
3.5
3.7
3.5
3.6
3.8
3.8
1.0
1.0
1.0
1.0
10
3.5
3.7
3.5
3.9
4.1
3.7
1.0
1.0
1.0
1.0
11
1.0
1.0
1.0
1.0
1.0
1.0
1.0
1.0
1.0
1.0
12
1.0
1.0
1.0
1.0
1.0
1.0
1.0
1.0
1.0
1.0
13
1.0
1.0
1.0
1.0
1.0
1.0
1.0
1.0
1.0
1.0
14
1.0
1.0
1.0
1.0
1.0
1.0
1.0
1.0
1.0
1.0
15
1.0
1.0
1.0
1.0
1.0
1.0
1.0
1.0
1.0
1.0
16
1.0
1.0
1.0
1.0
1.0
1.0
1.0
1.0
1.0
1.0
17
3.7
3.9
3.7
3.9
4.0
4.0
3.8
1.0
1.0
1.0
18
3.5
3.7
3.4
3.9
4.1
4.0
3.8
1.0
1.0
1.0
19
3.5
3.7
3.5
3.9
4.1
4.0
3.8
3.2
3.5
3.4
20
3.5
3.7
3.5
3.9
4.1
4.0
3.8
3.2
3.5
3.4
21
3.5
3.7
3.4
3.9
4.1
4.0
3.8
3.2
3.4
3.4
22
3.5
3.7
3.4
3.9
4.1
4.0
3.8
3.2
3.4
3.4
23
3.5
3.7
3.4
3.9
4.0
4.0
3.8
3.2
3.4
3.4
24
3.5
3.7
3.4
3.9
4.0
4.0
3.8
3.2
3.4
3.4
25
3.5
3.7
3.4
3.8
4.0
3.9
3.8
3.2
3.4
3.4
26
3.5
3.7
3.4
3.8
4.0
3.9
3.8
3.2
3.4
3.3
27
3.5
3.7
3.4
3.8
4.0
3.9
1.0
1.0
1.0
1.0
28
3.7
3.9
3.6
3.8
4.0
3.9
1.0
1.0
1.0
1.0
23
10
Table 5: First multiplication factor λst for localized corrosion sections 29 to 58 Section/Year
1
2
3
4
5
6
7
8
9
29
3.6
3.9
3.6
3.8
4.0
3.9
1.0
1.0
1.0
1.0
30
3.7
3.9
3.6
4.1
4.0
3.9
1.0
1.0
1.0
1.0
31
3.7
3.9
3.6
4.0
4.1
3.9
1.0
1.0
1.0
1.0
32
3.7
3.9
3.6
4.1
4.3
3.9
1.0
1.0
1.0
1.0
33
1.0
1.0
1.0
1.0
1.0
1.0
1.0
1.0
1.0
1.0
34
1.0
1.0
1.0
1.0
1.0
1.0
1.0
1.0
1.0
1.0
35
1.0
1.0
1.0
1.0
1.0
1.0
1.0
1.0
1.0
1.0
36
1.0
1.0
1.0
1.0
1.0
1.0
1.0
1.0
1.0
1.0
37
1.0
1.0
1.0
1.0
1.0
1.0
1.0
1.0
1.0
1.0
38
1.0
1.0
1.0
1.0
1.0
1.0
1.0
1.0
1.0
1.0
39
1.0
1.0
1.0
1.0
1.2
1.0
1.0
1.0
1.0
1.0
40
1.0
1.0
1.0
1.0
1.2
1.0
1.0
1.0
1.0
1.0
41
1.0
1.0
1.0
1.0
1.2
1.0
1.0
1.0
1.0
1.0
42
1.0
1.0
1.0
1.2
1.2
1.0
1.0
1.0
1.0
1.0
43
1.0
1.0
1.0
1.2
1.2
1.0
1.0
1.0
1.0
1.0
44
1.0
1.0
1.0
1.2
1.0
1.0
1.0
1.0
1.0
1.0
45
1.0
1.0
1.0
1.2
1.0
1.0
1.0
1.0
1.0
1.0
46
1.0
1.0
1.0
1.2
1.0
1.0
1.0
1.0
1.0
1.0
47
4.3
4.4
4.1
4.7
4.8
4.4
4.3
1.0
1.0
1.0
48
4.2
4.4
4.1
4.6
4.8
4.4
4.6
1.0
1.0
1.0
49
4.2
4.4
4.1
4.6
4.8
4.4
4.6
1.0
1.0
1.0
50
4.2
4.4
4.1
4.6
4.8
4.4
4.6
1.0
1.0
1.0
51
1.0
1.0
1.0
1.3
1.3
1.0
1.0
1.0
1.0
1.0
52
1.0
1.0
1.0
1.3
1.3
1.0
1.0
1.0
1.0
1.0
53
4.3
4.5
4.2
4.7
4.9
4.5
4.7
1.0
1.0
1.0
54
4.2
4.4
4.1
4.7
4.9
4.5
4.7
1.0
1.0
1.0
55
4.2
4.4
4.1
4.7
4.9
4.5
4.7
1.0
1.0
1.0
56
4.2
4.4
4.1
4.7
4.9
4.5
4.7
1.0
1.0
1.0
57
4.2
4.4
4.1
4.6
4.7
4.5
4.4
1.0
1.0
1.0
58
4.2
4.4
4.1
4.6
4.7
4.4
4.3
1.0
1.0
1.0
24
10
Table 6: Second multiplication factor κst for MIC section 1 to 28 Section/Year
1
2
3
4
5
6
7
8
9
1
0.9
0.9
0.9
0.9
0.8
0.6
0.6
1.1
1.1
1.1
2
0.5
0.5
0.5
0.6
0.6
0.4
1.1
1.1
1.1
1.1
3
0.4
0.4
0.4
0.4
0.4
0.4
0.4
1.1
0.4
0.4
4
0.4
0.4
0.4
0.4
0.4
0.4
0.4
1.1
0.4
0.4
5
0.4
0.4
0.4
0.4
0.4
0.4
0.4
1.1
0.4
0.4
6
0.4
0.4
0.4
0.4
0.4
0.4
0.4
1.1
0.4
0.4
7
0.5
0.6
0.4
0.4
0.4
0.4
1.1
1.1
1.1
1.1
8
0.9
0.9
0.9
0.5
0.4
0.4
1.1
1.1
1.1
1.1
9
0.9
0.9
0.9
0.6
0.6
0.4
1.1
1.1
1.1
1.1
10
0.9
0.9
0.9
0.9
0.9
0.6
1.1
1.1
1.1
1.1
11
1.1
1.1
1.1
1.1
1.1
1.1
1.1
1.1
1.1
1.1
12
1.1
1.1
1.1
1.1
1.1
1.1
1.1
1.1
1.1
1.1
13
1.1
1.1
1.1
1.1
1.1
1.1
1.1
1.1
1.1
1.1
14
1.1
1.1
1.1
1.1
1.1
1.1
1.1
1.1
1.1
1.1
15
1.1
1.1
1.1
1.1
1.1
1.1
1.1
1.1
1.1
1.1
16
1.1
1.1
1.1
1.1
1.1
1.1
1.1
1.1
1.1
1.1
17
0.8
0.9
0.9
0.6
0.6
0.6
0.6
1.1
1.1
1.1
18
0.5
0.5
0.5
0.4
0.4
0.4
0.4
1.1
1.1
1.1
19
0.4
0.4
0.4
0.4
0.4
0.4
0.4
0.4
0.4
0.4
20
0.4
0.4
0.4
0.4
0.4
0.4
0.4
0.4
0.4
0.4
21
0.4
0.4
0.4
0.4
0.4
0.4
0.4
0.4
0.4
0.4
22
0.4
0.4
0.4
0.4
0.4
0.4
0.4
0.4
0.4
0.4
23
0.4
0.4
0.4
0.4
0.4
0.4
0.4
0.4
0.4
0.4
24
0.4
0.4
0.4
0.4
0.4
0.4
0.4
0.4
0.4
0.4
25
0.4
0.4
0.4
0.4
0.4
0.4
0.4
0.4
0.4
0.4
26
0.4
0.4
0.4
0.4
0.4
0.4
0.4
0.4
0.4
0.4
27
0.4
0.6
0.5
0.4
0.4
0.4
1.1
1.1
1.1
1.1
28
0.9
0.9
0.9
0.4
0.4
0.4
1.1
1.1
1.1
1.1
25
10
Table 7: Second multiplication factor κst for MIC sections 29 to 58 Section/Year
1
2
3
4
5
6
7
8
9
29
0.9
0.9
0.9
0.6
0.5
0.4
1.1
1.1
1.1
1.1
30
0.9
0.9
0.9
0.9
0.6
0.5
1.1
1.1
1.1
1.1
31
0.9
0.9
0.9
0.9
0.8
0.6
1.1
1.1
1.1
1.1
32
0.9
0.9
0.9
0.9
0.9
0.6
1.1
1.1
1.1
1.1
33
1.1
1.1
1.1
1.1
1.1
1.1
1.1
1.1
1.1
1.1
34
1.1
1.1
1.1
1.1
1.1
1.1
1.1
1.1
1.1
1.1
35
1.1
1.1
1.1
1.1
1.1
1.1
1.1
1.1
1.1
1.1
36
1.1
1.1
1.1
1.1
1.1
1.1
1.1
1.1
1.1
1.1
37
1.1
1.1
1.1
1.1
1.1
1.1
1.1
1.1
1.1
1.1
38
1.1
1.1
1.1
1.1
1.1
1.1
1.1
1.1
1.1
1.1
39
1.1
1.1
1.1
1.1
0.9
1.1
1.1
1.1
1.1
1.1
40
1.1
1.1
1.1
1.1
0.9
1.1
1.1
1.1
1.1
1.1
41
1.1
1.1
1.1
1.1
0.9
1.1
1.1
1.1
1.1
1.1
42
1.1
1.1
1.1
0.9
0.9
1.1
1.1
1.1
1.1
1.1
43
1.1
1.1
1.1
0.9
0.9
1.1
1.1
1.1
1.1
1.1
44
1.1
1.1
1.1
0.9
1.1
1.1
1.1
1.1
1.1
1.1
45
1.1
1.1
1.1
0.9
1.1
1.1
1.1
1.1
1.1
1.1
46
1.1
1.1
1.1
0.9
1.1
1.1
1.1
1.1
1.1
1.1
47
0.9
0.9
0.9
0.9
0.9
0.6
0.6
1.1
1.1
1.1
48
0.9
0.9
0.9
0.9
0.9
0.6
0.8
1.1
1.1
1.1
49
0.9
0.9
0.9
0.9
0.9
0.6
0.8
1.1
1.1
1.1
50
0.9
0.9
0.9
0.9
0.9
0.6
0.8
1.1
1.1
1.1
51
1.1
1.1
1.1
0.9
0.9
1.1
1.1
1.1
1.1
1.1
52
1.1
1.1
1.1
0.9
0.9
1.1
1.1
1.1
1.1
1.1
53
0.9
0.9
0.9
0.9
0.9
0.6
0.8
1.1
1.1
1.1
54
0.9
0.9
0.9
0.9
0.9
0.6
0.8
1.1
1.1
1.1
55
0.9
0.9
0.9
0.9
0.9
0.6
0.8
1.1
1.1
1.1
56
0.9
0.9
0.9
0.9
0.9
0.6
0.8
1.1
1.1
1.1
57
0.9
0.9
0.9
0.9
0.8
0.6
0.6
1.1
1.1
1.1
58
0.9
0.9
0.9
0.9
0.8
0.6
0.6
1.1
1.1
1.1
26
10
358
Acknowledgments
359
The first author gratefully acknowledge the financial support provided
360
by The Secretariat of Higher Education, Science, Technology and Innovation
361
from the National Government of the Republic of Ecuador. The authors are
362
thankful to the Maintenance Department of Petroamazonas EP for providing
363
the required data.
364
Data availability
365
The raw/processed data required to reproduce these findings cannot be
366
shared at this time as the data also forms part of an ongoing study.
367
References
368
[1] S. Papavinasam, Oil and Gas Industry Network, in: Corrosion Control
369
in the Oil and Gas Industry, Elsevier, San Diego, CA, 2014, Ch. 2, pp.
370
41–131.
371
[2] R. Goodfellow, K. Jonsoon, Pipeline Integrity Management Systems
372
(PIMS), in: R. W. Revie (Ed.), Oil and Gas Pipelines: Integrity and
373
Safety Handbook, John Wiley & Sons, Hoboken, NJ, 2015, Ch. 1, pp.
374
3–12.
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33
S0308-0161(20)30033-8
DOI:
https://doi.org/10.1016/j.ijpvp.2020.104055
Reference:
IPVP 104055
To appear in:
International Journal of Pressure Vessels and Piping
Received Date: 8 December 2018 Revised Date:
14 January 2020
Accepted Date: 24 January 2020
Please cite this article as: Melo C, Dann MR, Hugo R, Janeta A, Extreme value modeling of localized internal corrosion in unpiggable pipelines, International Journal of Pressure Vessels and Piping, https:// doi.org/10.1016/j.ijpvp.2020.104055. This is a PDF file of an article that has undergone enhancements after acceptance, such as the addition of a cover page and metadata, and formatting for readability, but it is not yet the definitive version of record. This version will undergo additional copyediting, typesetting and review before it is published in its final form, but we are providing this version to give early visibility of the article. Please note that, during the production process, errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain. © 2020 Elsevier Ltd. All rights reserved.
Highlights • • •
Internal localized corrosion with microbiologically influenced corrosion Extreme value model to predict localized corrosion Application of structural reliability analysis for pipeline decision making
Extreme value modeling of localized internal corrosion in unpiggable pipelines Carlos Meloa , Markus R. Danna,∗, Ron Hugoa , Alberto Janetab a
Schulich School of Engineering, University of Calgary, 2500 University Dr NW, Calgary, Alberta, Canada, T2N 1N4 b Departamento de Mantenimiento, Petroamazonas EP, Av. 6 de Diciembre y Gaspar Ca˜ nero, Quito, Pichincha, Ecuador, 170504
Abstract Localized internal corrosion is one of the main failure mechanisms of energy pipelines. Microbiologically influenced corrosion (MIC) is a major contributor to internal corrosion in pipelines. Existing research uses extreme value analysis to fit extreme value distributions from data collected through either inspection or experiment to predict localized corrosion. The paper introduces an extreme value model to estimate the depth of localized internal corrosion considering MIC for unpiggable pipelines where inspection data is not available. The depth of material loss is estimated though a combination of corrosion analysis and extreme value modeling. A case study is provided to illustrate the proposed model. The application of a probabilistic approach supports risk-based inspection and maintenance planning for pipelines subject to internal corrosion. Keywords: extreme value modeling, pitting corrosion, microbiologically influenced corrosion, unpiggable pipelines, integrity, risk ∗
Corresponding Author Email address: [email protected] (Markus R. Dann)
Preprint submitted to International Journal of Pressure Vessels and Piping
January 8, 2020
1
1. Introduction
2
Pipelines are infrastructure systems that transport oil and gas from pro-
3
duction fields to processing facilities, and then from processing facilities to
4
end users. Pipelines are considered the safest and most economical method to
5
transport liquid and gaseous energy products [1–3]. However, pipelines can
6
fail due to a number of different mechanisms, with internal corrosion [4, 5]
7
being one of these mechanisms. According to data from the Pipeline Haz-
8
ardous Materials Safety Administration (PHMSA) [6] in the United States,
9
internal corrosion was the cause of approximately 10% of the 6,240 reported
10
failures of hazardous liquids pipelines between 2002 and 2010 [6]. Of these
11
internal corrosion failures, 6 in 10 involve localized pitting corrosion, and 5 of
12
these 6 identify microbiologically influenced corrosion (MIC) was identified
13
as one of the main degradation mechanisms [6].
14
Pipeline operators develop and implement pipeline integrity programs to
15
prevent these types of failures. Pipeline integrity programs utilize different
16
assessment methodologies to obtain information about a pipeline system and
17
using this information, establish maintenance plans for safe and reliable oper-
18
ation. In-line inspection (ILI) is the most detailed inspection method used for
19
integrity assessment, however, due to geometrical and/or operational restric-
20
tions, between 40 to 50% of the existing pipelines are unpiggable [7, 8] and
21
ILI tools cannot be used. Operators of unpiggable pipelines use alternative
22
integrity assessment methods, such as direct assessment [9–12], to perform
23
integrity assessment of these pipelines. However, most models used by direct
24
assessment methodologies are only able to predict general corrosion [9–12], 2
25
meaning that the effects of localized corrosion are not included yet from
26
PHMSA data it is known that 6 out of 10 internal corrosion failures involve
27
localized corrosion. This research focuses on developing a model to facilitate
28
the integrity assessment of unpiggable pipelines by estimating localized inter-
29
nal corrosion through a combination of advanced general corrosion analysis,
30
probabilistic analysis, and extreme value modeling (EVM). Probabilistic
31
modeling of corrosion has been introduced by Ahammed & Melchers in 1994
32
[13], Kale in 2004 [14],
and Lawson in 2005 [15]. To the best knowledge
33
of the authors, the proposed method is the first to combine advanced flow
34
and corrosion models with probabilistic and extreme value methods.
35
In the model, average general corrosion is estimated using advanced cor-
36
rosion analysis [16, 17]. A probabilistic analysis is implemented to include
37
the aleatory and epistemic uncertainties in the probability density func-
38
tion (PDF) of general corrosion. The respective models by Papavinasam
39
et al. [18, 19] and Sooknah et al. [19–21], are used to estimate two multi-
40
plication factors, one for localized corrosion and the other for MIC. Finally,
41
the EVM considers localized corrosion as the maximum of general corrosion
42
obtaining the PDF for localized corrosion.
43
In this paper, Section 2 provides an overview of localized corrosion, MIC,
44
and EVM. Section 3 presents the probabilistic analysis and the EVM to
45
estimate localized (pitting) corrosion as the maximum of the general corrosion
46
process in a pipeline. Section 4 applies the model to a case study where the
47
depth of pitting corrosion is estimated for a gathering pipeline in an oil
48
production field. Finally, conclusions are presented in Section 5.
3
49
2. Background
50
2.1. Localized Corrosion
51
After fabrication and during exposure to the atmosphere, a passive film is
52
created on metals as they undergo the corrosion process and this film protects
53
the metal from further corrosion. In carbon steel pipelines, however, the
54
film created is not compact and, therefore, unable to protect from corrosion
55
during operation [22]. Contact between the carbon steel pipeline and water
56
in the produced fluids creates a secondary layer that absorbs some of the
57
ions present in the solution [22]. Locations without the secondary layer that
58
are near regions with localized corrosion are prone to initiation of further
59
localized corrosion, which in turn accelerates layer deterioration [22].
60
Models to predict internal corrosion in pipelines have been under de-
61
velopment since 1975 [23]. They can be broadly classified as mechanistic
62
[16, 17, 24–27] or empirical [26, 28, 29], however, at times some prefer to
63
classify the models as either mechanistic or deterministic [30]. Most of these
64
models predict general corrosion rates, however the main contributor to cor-
65
rosion induced pipeline failures is due to localized corrosion, not general
66
corrosion. An empirical model for the estimation of localized corrosion in
67
pipelines was published by Papavinasam et al. [18]. It shows that localized
68
corrosion rates can not be obtained from general corrosion models. Despite
69
this findings, others have argued that general corrosion models can be used to
70
predict localized corrosion rates [30]. Dissenting arguments aside, the study
71
by Papavinasam et al. [18] does stress the need for extreme value analysis
72
when estimating localized corrosion.
4
73
2.2. Microbiologically influenced corrosion models
74
MIC has been identified as being responsible for many corrosion failures
75
in the oil and gas industry [8, 31–33]. MIC is the acceleration of the corrosion
76
process through the activity of microorganisms [34]. Most MIC is localized,
77
resulting in deep metal penetration in the form of pitting [34]. The envi-
78
ronment that accelerates corrosion by MIC is generated by biofilms that are
79
created when free floating planktonic microorganisms adhere to the surface
80
of the metal and become sessile [34]. An overview of some of the models used
81
to predict MIC are discussed in the following.
82
The first model to estimate the influence of MIC on internal corrosion
83
for application to the oil and gas industry was presented by Pots [35]. This
84
model was modified by Maxwell and Campbell [36] in 2006 to include the
85
influence of additional factors contributing to MIC such as system age and
86
the frequency at which cleaning tools are applied. In 2008, Sooknah et al.
87
presented a score-based model for quantifying the influence of MIC on lo-
88
calized internal corrosion rates in pipelines [20, 21]. All of these models use
89
a deterministic approach where the depth of MIC-influenced is fixed. This
90
limit their application to reliability estimates in pipelines as they are unable
91
to provide information about depth uncertainty, requirement for reliability
92
estimates.
93
2.3. Extreme value modeling
94
The objective of EVM is to make probabilistic statements about events
95
that are more extreme than any that have already occurred [37]. Figure 1
96
shows a section of a pipeline where the depth of the internal corrosion pit
5
97
i (i = 1, . . . , m) is described by the random variable Xi with a cumula-
98
tive distribution function (CDF) Fi (x ). A pipeline leaks when the depth
99
Xmax = max {X1 , . . . , Xm } of the deepest pit is equal to the wall thickness
100
wt of the pipeline. If X1 , . . . , Xm are independent and identically distributed
101
random variables with a common parent distribution CDF FX (x ), the ex-
102
treme value CDF of the m th order statistics Xmax is determined as follows
103
[37–39]: Fmax (x) = Pr{Xmax ≤ x} =
m Y
Pr{Xi ≤ x} =
i=1
m Y
FX (x) = {FX (x)}m (1)
i=1
Flow
X1
Xi
... Xm
wt
Figure 1: Pipeline subject to internal pitting corrosion where X1 , . . . , Xm are the depths of the m pits. Leak failure occurs when the maximum depth of the pits approaches the wall thickness wt of the pipeline.
104
Research by Melchers in 2008 [40] showed that under the influence of
105
MIC there is a high degree of randomness involved that makes a statis-
106
tical dependence between pits unlikely. Therefore, statistical independence
107
between pits is a reasonable assumption. In Equation 1 the CDF of the
108
deepest pit in a pipeline is Fmax (x ) = {FX (x )}m . Equation 1 can usually
109
not be applied for estimating the deepest pit because the parent distribution
6
110
FX (x ) of the corrosion damage is often unknown. Additionally, the number
111
m of corrosion pits in a pipeline is generally unknown [37].
112
3. Model
113
3.1. Overview
114
The objective of the proposed model is to estimate the probability distri-
115
bution of the depth of localized corrosion within a pipeline considering the
116
influence of MIC. The large amount of solids and liquids in upstream un-
117
piggable production and gathering pipelines increases their susceptibility to
118
internal corrosion. Pipelines have two modes of failure: leak and burst [42].
119
The depth of MIC-influenced localized corrosion is used in the model given
120
that the leak mode of failure is the most frequent for production and gath-
121
ering pipelines [43]. Figure 2 presents an overview of the proposed model.
122
A discretization in both space and time (Step 1) allows for general corrosion
123
growths to be estimated using corrosion rates from an advanced electrochem-
124
ical corrosion model (Step 2) [16, 17]. A probabilistic analysis is then applied
125
for determining the PDF of the depth of the population of corrosion features
126
(Step 3). The PDF of the depth of localized corrosion is estimated as the
127
maximum of the general corrosion using an EVM approach (Step 5) where
128
the effect of MIC is included (Step 4).
7
Sections s = 1, …, S and time periods t = 1, …, T
Step 1
s
1
Step 2 Average general Δμst
corrosion growth ωst
S
Step 4 Localized corrosion
(MIC)
time t = 1, …, T PDF
Step 3 General corrosion
time t = 1, …, T
Step 5 Localized corrosion
feature depth
feature depth
Figure 2: Overview of the 5-Steps to estimate the PDF of the depth localized corrosion. In Step 1, a discrete analysis with respect to space and time is implemented. In Step 2, general corrosion growths ∆µst are calculated. In Step 3, a probabilistic analysis includes aleatory and epistemic uncertainties in the PDF of the depth of general corrosion. In Step 4, localized corrosion rates including MIC allow to estimate a multiplication factor ωst . In Step 5, the PDF of the depth of localized corrosion is found using the PDF of general corrosion and the multiplication factor in an EVM.
129
3.2. Discretization
130
Corrosion is a continuous spatio-temporal process. A discrete analysis
131
with respect to space and time is proposed to reduce the computational de8
132
mands, while maintaining an acceptable level of accuracy for the estimation
133
of the corrosion rates and growths. In Step 1, the pipeline shown in Figure 2 is
134
discretized into S sections (s = 1, . . . , S ). Using industry standards, a maxi-
135
mum length of 15 meters per section is recommended to accurately estimate
136
the depth of general corrosion [11]. In the proposed analysis, a length of 6 me-
137
ters is used to justify a constant corrosion rate within each section. The 6-me-
138
ter discretization is based on a sensitivity analysis that was performed when
139
the applied general corrosion model [16, 17] was developed to achieve the
140
stated accuracy of the National Association of Corrosion Engineers (NACE)
141
SP0116 [11], which is one of the main references for prediction of internal
142
corrosion in pipelines.
143
The discretization of time is based on the variability of both operational
144
conditions (flow rate, temperature, and pressure) and flow composition (con-
145
tent of acids gases CO2 and H2 S, content of sulfates, bicarbonates, and chlo-
146
ride ions) to achieve approximately constant conditions within each time
147
increment. For each discretized section these variations are used to create
148
the temporal discretization ∆τt (t = 1, . . . , T ) which is pipeline-specific. A
149
constant corrosion rate is then assumed and computed for each time step and
150
section.
151
3.3. Average depth of general corrosion
152
153
The purpose of Step 2 in Figure 2 is to obtain the average depth of general corrosion µ in section s at time τT using Equation 2: µsT =
T X
∆µst
for s = 1, . . . , S
t=1
9
(2)
154
This enables the average depth of general corrosion at time τT to be esti-
155
mated. If it were desired to compute future corrosion growth rates, more
156
complex models would be required, and this is beyond the scope of the cur-
157
rent study. The general corrosion growth increments ∆µst are calculated
158
using the corrosion rates CRst from an advanced electrochemical general cor-
159
rosion model [16, 17]. In this model the fluid behavior and the mass transfer
160
of protons are the main factors influencing the corrosion process. ∆µst = CRst ∆τt
for s = 1, . . . , S and t = 1, . . . , T
(3)
161
The general corrosion growth increments in Equation 3 depend on the size
162
of the time step (∆τt = τt − τt−1 with τ0 = 0). The time step size is selected
163
based on the discretization process described in Section 3.2.
164
3.4. PDF of general corrosion
165
The purpose of Step 3 in Figure 2 is to obtain the PDF of the depth
166
of general corrosion that includes aleatory and epistemic uncertainties [44].
167
Aleatory or type 1 uncertainty is related to the natural variability of the
168
corrosion process and cannot be reduced, while epistemic or type 2 uncer-
169
tainty considers model errors and statistical uncertainties due to our lack of
170
knowledge. Epistemic or type 2 uncertainty can be reduced by increasing
171
the amount of data in the analysis [44]. Equation 4 computes XsT , a ran-
172
dom variable that represents the depth of all corrosion features at time τT ,
173
as the combination of both the aleatory uncertainty εpop,s and the epistemic
174
uncertainty εmodel,s : XsT = εpop,s µsT + εmodel,s 10
for s = 1, . . . , S
(4)
175
Section 5.2.5 of NACE SP0116 [11] specifies that the prediction of the depth
176
of corrosion by the corrosion model shall not exceed ±10% wall thickness
177
(wt) of the measured depth of corrosion at the verification sites. In despite
178
of NACE SP0116 ignoring the error in the measuring tool, in estimating
179
the depth of general corrosion, it is conservative to assume that the standard
180
deviation due to the model error is calculated using a 10%wt of error with an
181
80% confidence interval. It is considered that the model error εmodel follows
182
a normal PDF with zero mean and a standard deviation σmodel of 7.8%wt,
183
which is based on the assumed normal distribution of the model error to
184
obtain a ±10%wt error band at an 80% confidence level. This assumption
185
considers that on average the predictions of the general corrosion model are
186
accurate but not perfect due to the epistemic uncertainty.
187
Using a population-based approach for the corrosion features [45] the
188
aleatory uncertainty of the corrosion process is included in the analysis
189
through use of a coefficient of variation covpop obtained using inspection
190
information of a pipeline with similar operating conditions and thus simi-
191
lar expected corrosion damage. The covpop depends on the accuracy of the
192
inspection tool and is obtained based on minimizing the error between the
193
selected distribution and the inspection information. Therefore, in this pa-
194
per the covpop is considered as a single value for the entire pipeline that is
195
not section specific. The aleatory uncertainty due to the population effect
196
εpop,s is assumed to follow a log-normal PDF with a mean value µpop = 1 and
197
a standard deviation σpop,s = covpop µsT . The mean value µpop = 1 is used
198
considering that on average all sections have similar aleatory uncertainties.
199
Finally, to avoid negative numbers that are physically not possible the PDF
11
200
of XsT is considered to be log-normally distributed according to Equation 5:
201
p 2 2 XsT | µsT , σs ∼ log –normal µsT , σpop,s + σmodel
for s = 1, . . . , S (5)
202
where σs is the standard deviation of the depth of general corrosion and
203
is calculated using the standard deviations of the population effect and the
204
model error.
205
3.5. Effect of localized corrosion and MIC
206
The main idea behind the proposed model is that localized corrosion is
207
considered as the maximum of general corrosion. In Step 4 in Figure 2, the
208
expected value of localized corrosion E[XL,sT ] is determined by scaling the
209
expected value of general corrosion E[XsT ] with ω ¯ sT , as shown in Equation 6:
210
E[XL,sT ] = ω ¯ sT E[XsT ]
(6)
211
The extreme value model (Equation 1) accounts for the effects of the scal-
212
ing ω ¯ sT on the variance of the PDF of localized corrosion. XL,sT is the size
213
of localized corrosion features and its expected value is used as a reference
214
point to obtain the PDF of localized corrosion fmax ,sT (x ) from the PDF
215
of general corrosion fsT (x ). The variance of the size of localized corrosion
216
features can also be calculated from: Z Var[XL,sT ] ≈
∞
x2 fmax,sT (x)dx − E2 [fmax,sT (x)]
(7)
0 217
A weighted multiplication factor ω ¯ sT is used in Equation 6 instead of the
218
combined multiplication factor ωsT , which is obtained as the product of the 12
219
two multiplication factors λst and κst : ωst = λst κst
for s = 1, . . . , S and t = 1, . . . , T
(8)
220
The weighted multiplication factor ω ¯ sT used to relate the expected values of
221
general E[XsT ] and localized E[XL,sT ] corrosion is calculated only at present
222
time τT . However, the multiplication factors λst and κst are obtained at
223
each specific time period τt for t = 1, . . . , T . Therefore, in the model the
224
weighted multiplication factor ω ¯ sT is used to properly estimate the relation-
225
ship between general and localized corrosion. The weighted multiplication
226
factor considers the individual multiplication factors ωst as a function of their
227
influence ∆µst in the average depth of general corrosion at present time µsT ,
228
and it is calculated using Equation 9: ω ¯ sT =
T 1 X ∆µst ωst µsT t=1
for s = 1, . . . , S
(9)
229
The first multiplication factor λst is for localized corrosion and is calculated
230
as the ratio between localized PCRst [18, 19] and general corrosion rates CRst
231
[16, 17]. The PCRst are calculated using an empirical model that considers
232
eleven variables affecting pitting corrosion including oil, water, gas, and solid
233
compositions, temperature, pressure, partial pressures of CO2 and H2 S, and
234
concentrations of sulfate, bicarbonate and chloride ions [18, 19]. General
235
corrosion rates CRst are estimated with an advanced corrosion model that
236
includes a mechanistic flow model, a mass transfer model, a scale model, a
237
solid deposition model, and an electrochemical corrosion model [16, 17]. λst =
P CRst CRst
for s = 1, . . . , S and t = 1, . . . , T
13
(10)
238
The second multiplication factor κst considers the influence of MIC and it is
239
estimated using a qualitative MIC risk model that considers nine variables
240
including flow rates, temperature, ratio of partial pressure between acid gases
241
CO2 and H2 S, pH, Langelier saturation index, total suspended and dissolved
242
solids, Redox potential, and sulfur content [20, 21].
243
3.6. PDF of localized corrosion
244
The objective of Step 5 in Figure 2 is to obtain the PDF of localized
245
corrosion which is considered the maximum of general corrosion as detailed
246
in Section 3.5, therefore: fmax,sT (x) = m fsT (x) FsT (x)m−1
(11)
247
where fmax ,sT (x ) is the PDF of localized corrosion at time T , m is the
248
m th order statistics, and fsT (x ) and FsT (x ) are the general corrosion PDFs
249
and CDFs obtained using Equation 5. A short algorithm, which is stopped
250
when δsT ≤ 0 , was developed to calculate m ∈ N. The algorithm is provided
251
in the Appendix. δsT = ω ¯ sT E[XsT ] − E[fmax (x)]
for s = 1, . . . , S
(12)
252
where ω ¯ sT is the weighted multiplication factor from Equation 8 that includes
253
the effect of localized corrosion and MIC, E[XsT ] is the expected value of the
254
depth of general corrosion at time τT , and E[fmax ,sT ] is the expected value
255
of the maximum extreme. The expected value is used as the base point
256
for the proposed model. However, future analyses should consider other
257
base points to review their influence in the reliability estimates of pipelines.
258
The reason the expected values are used as the base point is that the link 14
259
between localized and general corrosion is calculated using average values
260
from deterministic models.
261
4. Case Study
262
A multiphase gathering pipeline, located in one of the oil fields of Petroa-
263
mazonas EP in the Amazon basin of Ecuador, is considered in this case study.
264
The pipeline has been in operation for 10 years. Acid gases, such as CO2 ,
265
and a considerable amount of water have been present in the produced fluids
266
since the beginning of operation. These conditions increase the likelihood
267
that the pipeline has been subject to internal corrosion. The objective of
268
this analysis is to estimate the size of localized corrosion using the propose
269
method. Table 1 provides the input used for the estimation of general and
270
localized corrosion rates in this gathering pipeline. The pipeline is discretized
271
into S =58 sections, each having a length of 6 meters, and the analysis is dis-
272
cretized into T =10 time points. The spatial and temporal discretization are
273
based on the criteria detailed in Section 3.2 . Table 2 shows the discretized
274
production flow rates for oil, gas, and water along with the operating tem-
275
perature for the gathering pipeline. In Table 2 there is an increase in the
276
production flow rates of water and oil between the first τ1 and last τ10 time
277
points, as normally observed in an oil production field.
15
Table 1: Pipeline geometry and pipeline operational variables for the general and localized corrosion analysis
Nominal external diameter
436.57 mm
Length Inlet pressure Fluid type Nominal wall thickness Mole fraction Pipe roughness
348 m 3.44 MPa Multiphase 10.3 mm 0.201 CO2 0.04572 mm
Table 2: Production scenarios for general and localized corrosion analysis Time point τt
years ◦
Operating temperature
1
2
3
4
5
6
7
8
9
10
C
90
92
89
91
92
93
95
95
92
92
Oil rate
m3 /day × 103
2.27
2.28
3.62
4.89
5.21
4.89
4.85
4.87
4.70
4.48
Gas rate
m3 /day × 103
38.39
42.22
43.75
60.74
117.37
174.00
80.56
52.24
59.32
60.23
3.46
2.14
1.65
4.19
4.99
7.05
7.33
7.69
9.02
9.16
Water rate
3
3
m /day × 10
278
Using the advanced general corrosion model described in Section 3.3,
279
the general corrosion rates CRst are estimated and the average growth of
280
general corrosion ∆µst for each section and time period is calculated using
281
Equation 3. Figure 3 summarizes the average depth of general corrosion
282
µs10 for each section at τ10 .
283
the pipeline that was used for the calculations of the general corrosion pro-
284
file. Applying the qualitative categorization criteria for general corrosion of
285
steel in oil production systems from NACE SP-0775-2013 [46] severe general
286
corrosion (indicated by a red triangle) is in Sections 11 and 33, high general
287
corrosion is in Sections 2 and 58 are (orange circle), and moderate general
Figure 3 also shows the elevation profile of
16
288
corrosion in in Sections 5 and 19 (yellow triangle). These 6 sections are
289
selected to demonstrate application of the proposed method.
290
The localized multiplication factors λst , and the MIC multiplication fac-
291
tors κst for each section and time period are estimated with the method
292
described in Section 3.5. These two multiplication factors, which are pro-
293
vided in Tables 4 to 7 in the Appendix, are used to estimate the combined
294
multiplication factors ωst (Equation 8) for each section and time point. Fi-
295
nally, the weighted multiplication factor ω ¯ s10 for each section at present time
296
is calculated using Equation 9. Figure 4 shows the computed weighted multi-
297
plication factor where sections with severe general corrosion (11 and 33) are
298
seen to have lower multiplication factors, while a section with high general
299
corrosion (58) has the highest multiplication factor. A possible explanation
300
for these results lies in the greater influence of proton mass transfer, which
301
is directly affected by produced fluid flow conditions, in estimating general
302
corrosion rates compared to localized corrosion rates. 1.6 Severe Corrosion High Corrosion Moderate Corrosion
ωs10
1.4
58 19
1.2
5 2
1.0
11 1
5
10
33 15
20
25
30
35
40
45
50
55
58
Section
Figure 4: Weighted multiplication factor ω ¯ s10 for each section at present time. The factor includes the localized corrosion and the MIC factors.
17
60 11
μs10 [%wt]
50
33
40 30
2
20 10 0
5 1
5
19 10
15
58
Severe Corrosion High Corrosion Moderate Corrosion
20
25
30 Section
35
40
45
50
55
58
(a) Average depth of general corrosion for all sections at present time 230
Elevation[m]
220 210 200 190 180 1
5
10
15
20
25
30 Section
35
40
45
50
55
58
(b) Elevation profile Figure 3: (a) Average depth of general corrosion for all sections at present time T = 10. Representative sections for severe, high, and moderate general corrosion are highlighted as red triangle up, orange circle, and yellow triangle down, respectively. (b) Elevation profile of pipeline
18
303
The PDFs of the depth of general corrosion for each section at present
304
time are found based on Equation 5 with µs10 , a σmodel of 7.8%wt, and a covpop
305
of 0.43. Finally the PDFs and CDFs of localized corrosion are estimated
306
using the method described in Section 3.6. Figure 5 shows the PDFs (5a
307
on left) and CDFs (5b on right) for general and localized corrosion for one
308
section. There is an increase in the variance of the PDF of localized cor-
309
rosion compared to general corrosion, which is a reasonable result based on
310
the proposed method.
311
The computational demand of the proposed model is lower than a fea-
312
ture-specific approach. The analysis of the case study took 317 seconds on
313
an Intel Xeon ES-2670 2.6 GHz processor. (a) Section 58
(b) Section 58
1.0 General Corrosion Localized Corrosion
0.0004
0.0002
0.0000
General Corrosion Localized Corrosion
0.8 CDF
0.0006
0.6 0.4 0.2
0
20
40 60 feature depth [% wt]
80
100
0.0
0
20
40 60 feature depth [% wt]
80
100
Figure 5: PDFs and CDFs of general and localized corrosion for one section of the gathering pipeline.
314
For the purpose of comparison the PDFs of general and localized corro-
315
sion are utilized in a structural reliability analysis [47] using the convolution
316
integral between the wall thickness (resistance) and the estimated depth of
317
general or localized corrosion features (load effect) to calculate the probabil-
19
318
ity of failure for general and localized corrosion: Z ∞ Fwt (x) fmax,s10 (x) dx for s = 1, . . . , S PF,L =
(13)
0 319
where PF ,L is the probability of failure for localized corrosion at T =10, and
320
Fwt (x ) represents a normal distribution with a mean of 100%wt and a stan-
321
dard deviation of 1.5%wt [48] . Equation 13 is also used to calculate the
322
probability of failure PF ,G for general corrosion by replacing fmax ,s10 (x ) with
323
fs10 (x ). Table 3 summarizes the results of these calculations and also shows
324
the ratios between the localized and general corrosion probabilities at present
325
time for the selected sections. Table 3: Summary of probability of failure for selected sections at present time
Section
General Corrosion PF ,L [ failures ] PF ,G [ failures ] section section
PF ,L PF ,G
2
High
1.38 × 10−5
6.89 × 10−6
2.00
5
Moderate
1.48 × 10−5
7.42 × 10−6
2.00
11
Severe
6.08 × 10−4
3.09 × 10−4
1.97
19
Moderate
1.49 × 10−5
7.48 × 10−6
2.00
33
Severe
5.53 × 10−4
2.81 × 10−4
1.97
58
High
9.72 × 10−6
3.24 × 10−6
3.00
326
In Table 3, the ratio of probabilities of failure for localized corrosion to
327
general corrosion is seen to range from 1.97 to 3.00, indicating that significant
328
more risk mitigation actions will be applied to the pipeline when localized
329
corrosion probabilities are used. Comparing the results with the exceedance
330
probability analysis and assuming an acceptable probability of failure of 10−3
331
], at present time all of the selected sections are operating below this [ failures section 20
332
acceptable limit. However, it is important to note that Sections 11 and 33
333
may require risk mitigation actions including inspection and maintenance in
334
the near future. This example clearly demonstrates the advantages of using
335
a reliability analysis in comparison to an exceedance probability analysis, as
336
the latter approach that may result in conservative actions.
337
5. Conclusions
338
The approach presented in this paper supports the reliability and risk-in-
339
formed assessment of unpiggable pipelines subject to internal corrosion. The
340
proposed population-based approach is computationally more efficient than
341
a feature-specific approach. Hence, its advantage is the scalability to longer
342
pipelines.
343
The model includes aleatory and epistemic uncertainties of the localized
344
corrosion process. If field data is available, Bayesian updating can be applied
345
to reduce epistemic uncertainties in the model. However, a greater benefit
346
of the model is the use of the results to select optimal sections for field
347
verification.
348
Future research will focus on prediction of the probability distributions
349
of the depth of localized corrosion at future times. The use of stochastic
350
processes will facilitate these estimations required to implement risk-based
351
inspection and maintenance planning for unpiggable pipelines.
21
352
353
Appendix. Algorithm to compute the m th order statistics The following algorithm was developed to determine iteratively the m th order statistics m ∈ N is stopped when δsT ≤ 0 in Equation 12. Algorithm 1 Algorithm to calculate the m th order statistics and the CDF and PDF for the size of localized corrosion features is stopped when δsT ≤ 0 . Input: fsT (x) and FsT (x) from Equation 5,
E[XL,sT ]
from
tion 6, and fmax (x) from Equation 11
Initialize: m = 1 repeat set m = m + 1 compute E[fmax (x)] =
R∞ 0
xfmax (x)dx
until δsT ≤ 0 (Equation 12)
Output: CDF Fmax (x) and PDF fmax (x) of localized corrosion 354
22
Equa-
355
356
357
Appendix. Multiplication factors for case study The following tables include the multiplication factors for localized corrosion and MIC used in the case study. Table 4: First multiplication factor λst for localized corrosion sections 1 to 28 Section/Year
1
2
3
4
5
6
7
8
9
1
3.4
3.6
3.4
3.8
4.0
3.8
3.4
1.0
1.0
1.0
2
3.3
3.5
3.3
3.6
3.9
3.7
1.0
1.0
1.0
1.0
3
3.3
3.5
3.3
3.6
3.9
3.8
3.6
1.0
3.2
3.1
4
3.3
3.5
3.3
3.7
3.9
3.8
3.6
1.0
3.2
3.1
5
3.3
3.5
3.3
3.6
3.9
3.8
3.6
1.0
3.2
3.1
6
3.3
3.5
3.3
3.6
3.9
3.8
3.5
1.0
3.2
3.1
7
3.3
3.5
3.3
3.6
3.8
3.8
1.0
1.0
1.0
1.0
8
3.5
3.6
3.5
3.6
3.8
3.8
1.0
1.0
1.0
1.0
9
3.5
3.7
3.5
3.6
3.8
3.8
1.0
1.0
1.0
1.0
10
3.5
3.7
3.5
3.9
4.1
3.7
1.0
1.0
1.0
1.0
11
1.0
1.0
1.0
1.0
1.0
1.0
1.0
1.0
1.0
1.0
12
1.0
1.0
1.0
1.0
1.0
1.0
1.0
1.0
1.0
1.0
13
1.0
1.0
1.0
1.0
1.0
1.0
1.0
1.0
1.0
1.0
14
1.0
1.0
1.0
1.0
1.0
1.0
1.0
1.0
1.0
1.0
15
1.0
1.0
1.0
1.0
1.0
1.0
1.0
1.0
1.0
1.0
16
1.0
1.0
1.0
1.0
1.0
1.0
1.0
1.0
1.0
1.0
17
3.7
3.9
3.7
3.9
4.0
4.0
3.8
1.0
1.0
1.0
18
3.5
3.7
3.4
3.9
4.1
4.0
3.8
1.0
1.0
1.0
19
3.5
3.7
3.5
3.9
4.1
4.0
3.8
3.2
3.5
3.4
20
3.5
3.7
3.5
3.9
4.1
4.0
3.8
3.2
3.5
3.4
21
3.5
3.7
3.4
3.9
4.1
4.0
3.8
3.2
3.4
3.4
22
3.5
3.7
3.4
3.9
4.1
4.0
3.8
3.2
3.4
3.4
23
3.5
3.7
3.4
3.9
4.0
4.0
3.8
3.2
3.4
3.4
24
3.5
3.7
3.4
3.9
4.0
4.0
3.8
3.2
3.4
3.4
25
3.5
3.7
3.4
3.8
4.0
3.9
3.8
3.2
3.4
3.4
26
3.5
3.7
3.4
3.8
4.0
3.9
3.8
3.2
3.4
3.3
27
3.5
3.7
3.4
3.8
4.0
3.9
1.0
1.0
1.0
1.0
28
3.7
3.9
3.6
3.8
4.0
3.9
1.0
1.0
1.0
1.0
23
10
Table 5: First multiplication factor λst for localized corrosion sections 29 to 58 Section/Year
1
2
3
4
5
6
7
8
9
29
3.6
3.9
3.6
3.8
4.0
3.9
1.0
1.0
1.0
1.0
30
3.7
3.9
3.6
4.1
4.0
3.9
1.0
1.0
1.0
1.0
31
3.7
3.9
3.6
4.0
4.1
3.9
1.0
1.0
1.0
1.0
32
3.7
3.9
3.6
4.1
4.3
3.9
1.0
1.0
1.0
1.0
33
1.0
1.0
1.0
1.0
1.0
1.0
1.0
1.0
1.0
1.0
34
1.0
1.0
1.0
1.0
1.0
1.0
1.0
1.0
1.0
1.0
35
1.0
1.0
1.0
1.0
1.0
1.0
1.0
1.0
1.0
1.0
36
1.0
1.0
1.0
1.0
1.0
1.0
1.0
1.0
1.0
1.0
37
1.0
1.0
1.0
1.0
1.0
1.0
1.0
1.0
1.0
1.0
38
1.0
1.0
1.0
1.0
1.0
1.0
1.0
1.0
1.0
1.0
39
1.0
1.0
1.0
1.0
1.2
1.0
1.0
1.0
1.0
1.0
40
1.0
1.0
1.0
1.0
1.2
1.0
1.0
1.0
1.0
1.0
41
1.0
1.0
1.0
1.0
1.2
1.0
1.0
1.0
1.0
1.0
42
1.0
1.0
1.0
1.2
1.2
1.0
1.0
1.0
1.0
1.0
43
1.0
1.0
1.0
1.2
1.2
1.0
1.0
1.0
1.0
1.0
44
1.0
1.0
1.0
1.2
1.0
1.0
1.0
1.0
1.0
1.0
45
1.0
1.0
1.0
1.2
1.0
1.0
1.0
1.0
1.0
1.0
46
1.0
1.0
1.0
1.2
1.0
1.0
1.0
1.0
1.0
1.0
47
4.3
4.4
4.1
4.7
4.8
4.4
4.3
1.0
1.0
1.0
48
4.2
4.4
4.1
4.6
4.8
4.4
4.6
1.0
1.0
1.0
49
4.2
4.4
4.1
4.6
4.8
4.4
4.6
1.0
1.0
1.0
50
4.2
4.4
4.1
4.6
4.8
4.4
4.6
1.0
1.0
1.0
51
1.0
1.0
1.0
1.3
1.3
1.0
1.0
1.0
1.0
1.0
52
1.0
1.0
1.0
1.3
1.3
1.0
1.0
1.0
1.0
1.0
53
4.3
4.5
4.2
4.7
4.9
4.5
4.7
1.0
1.0
1.0
54
4.2
4.4
4.1
4.7
4.9
4.5
4.7
1.0
1.0
1.0
55
4.2
4.4
4.1
4.7
4.9
4.5
4.7
1.0
1.0
1.0
56
4.2
4.4
4.1
4.7
4.9
4.5
4.7
1.0
1.0
1.0
57
4.2
4.4
4.1
4.6
4.7
4.5
4.4
1.0
1.0
1.0
58
4.2
4.4
4.1
4.6
4.7
4.4
4.3
1.0
1.0
1.0
24
10
Table 6: Second multiplication factor κst for MIC section 1 to 28 Section/Year
1
2
3
4
5
6
7
8
9
1
0.9
0.9
0.9
0.9
0.8
0.6
0.6
1.1
1.1
1.1
2
0.5
0.5
0.5
0.6
0.6
0.4
1.1
1.1
1.1
1.1
3
0.4
0.4
0.4
0.4
0.4
0.4
0.4
1.1
0.4
0.4
4
0.4
0.4
0.4
0.4
0.4
0.4
0.4
1.1
0.4
0.4
5
0.4
0.4
0.4
0.4
0.4
0.4
0.4
1.1
0.4
0.4
6
0.4
0.4
0.4
0.4
0.4
0.4
0.4
1.1
0.4
0.4
7
0.5
0.6
0.4
0.4
0.4
0.4
1.1
1.1
1.1
1.1
8
0.9
0.9
0.9
0.5
0.4
0.4
1.1
1.1
1.1
1.1
9
0.9
0.9
0.9
0.6
0.6
0.4
1.1
1.1
1.1
1.1
10
0.9
0.9
0.9
0.9
0.9
0.6
1.1
1.1
1.1
1.1
11
1.1
1.1
1.1
1.1
1.1
1.1
1.1
1.1
1.1
1.1
12
1.1
1.1
1.1
1.1
1.1
1.1
1.1
1.1
1.1
1.1
13
1.1
1.1
1.1
1.1
1.1
1.1
1.1
1.1
1.1
1.1
14
1.1
1.1
1.1
1.1
1.1
1.1
1.1
1.1
1.1
1.1
15
1.1
1.1
1.1
1.1
1.1
1.1
1.1
1.1
1.1
1.1
16
1.1
1.1
1.1
1.1
1.1
1.1
1.1
1.1
1.1
1.1
17
0.8
0.9
0.9
0.6
0.6
0.6
0.6
1.1
1.1
1.1
18
0.5
0.5
0.5
0.4
0.4
0.4
0.4
1.1
1.1
1.1
19
0.4
0.4
0.4
0.4
0.4
0.4
0.4
0.4
0.4
0.4
20
0.4
0.4
0.4
0.4
0.4
0.4
0.4
0.4
0.4
0.4
21
0.4
0.4
0.4
0.4
0.4
0.4
0.4
0.4
0.4
0.4
22
0.4
0.4
0.4
0.4
0.4
0.4
0.4
0.4
0.4
0.4
23
0.4
0.4
0.4
0.4
0.4
0.4
0.4
0.4
0.4
0.4
24
0.4
0.4
0.4
0.4
0.4
0.4
0.4
0.4
0.4
0.4
25
0.4
0.4
0.4
0.4
0.4
0.4
0.4
0.4
0.4
0.4
26
0.4
0.4
0.4
0.4
0.4
0.4
0.4
0.4
0.4
0.4
27
0.4
0.6
0.5
0.4
0.4
0.4
1.1
1.1
1.1
1.1
28
0.9
0.9
0.9
0.4
0.4
0.4
1.1
1.1
1.1
1.1
25
10
Table 7: Second multiplication factor κst for MIC sections 29 to 58 Section/Year
1
2
3
4
5
6
7
8
9
29
0.9
0.9
0.9
0.6
0.5
0.4
1.1
1.1
1.1
1.1
30
0.9
0.9
0.9
0.9
0.6
0.5
1.1
1.1
1.1
1.1
31
0.9
0.9
0.9
0.9
0.8
0.6
1.1
1.1
1.1
1.1
32
0.9
0.9
0.9
0.9
0.9
0.6
1.1
1.1
1.1
1.1
33
1.1
1.1
1.1
1.1
1.1
1.1
1.1
1.1
1.1
1.1
34
1.1
1.1
1.1
1.1
1.1
1.1
1.1
1.1
1.1
1.1
35
1.1
1.1
1.1
1.1
1.1
1.1
1.1
1.1
1.1
1.1
36
1.1
1.1
1.1
1.1
1.1
1.1
1.1
1.1
1.1
1.1
37
1.1
1.1
1.1
1.1
1.1
1.1
1.1
1.1
1.1
1.1
38
1.1
1.1
1.1
1.1
1.1
1.1
1.1
1.1
1.1
1.1
39
1.1
1.1
1.1
1.1
0.9
1.1
1.1
1.1
1.1
1.1
40
1.1
1.1
1.1
1.1
0.9
1.1
1.1
1.1
1.1
1.1
41
1.1
1.1
1.1
1.1
0.9
1.1
1.1
1.1
1.1
1.1
42
1.1
1.1
1.1
0.9
0.9
1.1
1.1
1.1
1.1
1.1
43
1.1
1.1
1.1
0.9
0.9
1.1
1.1
1.1
1.1
1.1
44
1.1
1.1
1.1
0.9
1.1
1.1
1.1
1.1
1.1
1.1
45
1.1
1.1
1.1
0.9
1.1
1.1
1.1
1.1
1.1
1.1
46
1.1
1.1
1.1
0.9
1.1
1.1
1.1
1.1
1.1
1.1
47
0.9
0.9
0.9
0.9
0.9
0.6
0.6
1.1
1.1
1.1
48
0.9
0.9
0.9
0.9
0.9
0.6
0.8
1.1
1.1
1.1
49
0.9
0.9
0.9
0.9
0.9
0.6
0.8
1.1
1.1
1.1
50
0.9
0.9
0.9
0.9
0.9
0.6
0.8
1.1
1.1
1.1
51
1.1
1.1
1.1
0.9
0.9
1.1
1.1
1.1
1.1
1.1
52
1.1
1.1
1.1
0.9
0.9
1.1
1.1
1.1
1.1
1.1
53
0.9
0.9
0.9
0.9
0.9
0.6
0.8
1.1
1.1
1.1
54
0.9
0.9
0.9
0.9
0.9
0.6
0.8
1.1
1.1
1.1
55
0.9
0.9
0.9
0.9
0.9
0.6
0.8
1.1
1.1
1.1
56
0.9
0.9
0.9
0.9
0.9
0.6
0.8
1.1
1.1
1.1
57
0.9
0.9
0.9
0.9
0.8
0.6
0.6
1.1
1.1
1.1
58
0.9
0.9
0.9
0.9
0.8
0.6
0.6
1.1
1.1
1.1
26
10
358
Acknowledgments
359
The first author gratefully acknowledge the financial support provided
360
by The Secretariat of Higher Education, Science, Technology and Innovation
361
from the National Government of the Republic of Ecuador. The authors are
362
thankful to the Maintenance Department of Petroamazonas EP for providing
363
the required data.
364
Data availability
365
The raw/processed data required to reproduce these findings cannot be
366
shared at this time as the data also forms part of an ongoing study.
367
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