Reliability Analysis Of Pipelines Containing Cracks And ......rupture of the pipeline. Based on...
Transcript of Reliability Analysis Of Pipelines Containing Cracks And ......rupture of the pipeline. Based on...
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Reliability Analysis Of Pipelines Containing Cracks And
Corrosion Defects
Main authors
Lie Zhang, Robert Adey C M BEASY Ltd
United Kingdom
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© Copyright 2008 IGRC2008
1. ABSTRACT
This paper is concerned with predicting the probability of failure (POF) of pipelines containing
cracks and corrosion defects. The current work was conducted within the framework of the
European Union NATURALHY project.
The aim of the NATURALHY project is to investigate the possibility of using the existing natural
gas transmission pipelines to deliver hydrogen or natural gas/hydrogen mixtures. Hydrogen has
been demonstrated to accelerate defect growth which may affect the safety of pipeline and
make it more expensive to operate and maintain. Failure of a structural member such as a
pipeline may occur when a crack becomes unstable or metal loss corrosion leads to leakage or
rupture of the pipeline. Based on fatigue reliability analysis, statistical methods are applied to
assess the probability of failure of pipelines containing cracks like defects and corrosion defects.
A software package has been developed based on the stochastic approach to assess the POF of
the gas pipeline due to the existence of cracks and corrosion defects. The inspection and repair
procedures are also included in the tool. With various parameters such as defect sizes, material
properties, internal pressure modelled as uncertainties, a reliability analysis based on BS 7910
level-2 failure assessment diagram is conducted through stratified Monte Carlo simulation.
Inspection and repair programmes and realistic pipeline maintenance scenarios can be
simulated. In the data preparation process, the accuracy of the probabilistic definition of the
uncertainties is crucial as the results show that the POF is very sensitive to certain variables
such as the defect sizes and the defect growth rate. The POF for a single defect with known
dimensions and the POF for a system containing multiple defects can be computed separately.
Various inspection and repair criteria are available in the Monte Carlo simulation whereby an
optimal maintenance strategy can be achieved by comparing different combinations of repair
procedures. In addition, the hazard function can be obtained based on the POF results, so that
the POF calculation tool can be used to satisfy certain target reliability requirement. Several
examples are presented in the paper comparing the POF for a natural gas pipeline with the POF
for a pipeline containing natural gas/hydrogen mixtures.
Nomenclature ()f probability distribution function c half crack length
a crack depth B pipe wall thickness
( )g x ( ) 0g x ≤ represents the failure domain Φ normal distribution function
ix stochastically independent variable PK primary stress intensity factor
SK secondary stress intensity factor thK∆ threshold stress intensity factor
ρ plastic correction factor maxrL permitted limit of rL
Yσ yield strength of material Uσ ultimate tensile strength
ICK Toughness of material refσ reference stress
n number of loading cycles *
N total number of simulations
( )f
p n cumulative POF of a single defect q total number of cracks
AN average number of cracks repaired
pfC probable cost of have a failure
|f totalP probability of failure for multiple defects
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© Copyright 2008 IGRC2008
C O N T E N T S
1. Abstract ................................................................................................. 2
2. Introduction ........................................................................................... 4
3. Theory.................................................................................................... 4
3.1. Fundamentals of structural reliability theory ...........................................................4
3.2. Failure assessment for corrosion defects and cracks.................................................5
3.3. Modelling of inspection and repair program.............................................................7
4. Monte-Carlo simulation as POF evaluation method................................. 9
4.1. Direct Monte-Carlo simulation ...............................................................................9
4.2. Stratified Monte-Carlo simulation method .............................................................10
4.2.1. Random number generation..........................................................................10 4.2.2. Stratified Monte-Carlo simulation ..................................................................11
5. Sample problems.................................................................................. 12
5.1. Probability of failure calculation for corrosion defects .............................................12
5.2. Probability of failure calculation for cracks ............................................................14
6. Conclusions .......................................................................................... 15
7. Acknowledgements .............................................................................. 15
8. References ........................................................................................... 16
9. List of Tables ........................................................................................ 17
10. List Of Figures ...................................................................................... 17
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© Copyright 2008 IGRC2008
2. INTRODUCTION
Currently major research effort is focused on the use of hydrogen as an energy vector both in
Europe and the US. An obvious pragmatic solution to transporting hydrogen, which will be
necessary to make the European hydrogen economy feasible, is to transport a mixture of
natural gas and hydrogen in the existing natural gas pipeline network. Within the European
project NATURALHY†, 39 European partners have combined their efforts to assess the effects of
the presence of hydrogen on the existing gas network. Key issues are durability of pipeline
material, integrity management, safety aspects, life-cycle and socio-economic assessment and
end-use. The work described in this paper was performed within the NATURALHY work package
on ’Integrity Management’.
Failure of a structural member such as a pipeline may occur when a crack or a corrosion defect
propagates in an unstable manner to cause leakage or rupture of the pipeline. Failure analysis
combined with a probabilistic approach has been utilized in many fields involving important
structural components such as pressure vessels and nuclear piping. Based on the failure
analysis for cracks and corrosion defects, statistical methods can be applied to assess the
reliability of pipeline containing crack-like defects [1], in other words, to provide a figure which
represents the probability that a pipeline will fail. One aim of the Naturalhy project is to assess
the durability and integrity of natural gas infrastructures for transporting and distributing
hydrogen/natural gas mixtures. In particular, the mechanical integrity of the gas piping is a
matter of great importance for both economical and safety reasons. Among all the parameters
the probability of failure is one of the most important factors for the integrity management of
pipeline infrastructures. Although evidence suggests that crack-like defects are most likely to be
affected by the presence of hydrogen due to hydrogen embrittlement the probability of failure
calculation for corrosion defects is also included in this paper for the sake of completeness.
3. THEORY
3.1. Fundamentals of structural reliability theory
According to the reliability theory if one and only one defect exists in a pipeline, the failure
probability is given by:
( ) ( ) ( ) ( )1
11 1
( ) 0 NNf X X N N
g x xP f x f x d x d x
⋅⋅⋅ ≤= ⋅⋅⋅ ⋅ ⋅⋅∫ (3.1)
where 1 Nx x⋅ ⋅ ⋅ are random variables such as defect sizes, yield strength, defect growth
parameters and applied stresses. ( )Nx N
f x denotes the probability density function of the input
variable N
x . The integration will be performed over the failure domain where 1( ) 0Ng x x⋅ ⋅ ⋅ ≤ .
1( )Ng x x⋅ ⋅ ⋅ is also called a limit state in some cases. For the case of crack-like defects, semi-
elliptical surface cracks and the through-thickness cracks will be considered. The initial surface
crack has the crack depth a and the crack length 2c as shown in Figure 1. The above
integration is rarely solved directly once the number of unknowns rises above 3. Therefore, in
order to solve practical engineering problems alternative approaches are required.
††NATURALHY is an Integrated Project, co-financed by the European Commission’s Sixth Framework Programme
(2002-2006) for research, technological development and demonstration (RTD). For more details please visit
www.naturalhy.net
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© Copyright 2008 IGRC2008
Figure 1 Geometry of surface/through thickness crack model used in the analysis
One such approach is called First Order Reliability Method (FORM). The concept behind the
approach is that instead of integrating the density functions directly a reliability index β is
computed to define the reliability of the structure thereby the probability of failure being
obtained. β is computed through the following equation [2].
( )f
P β= Φ − (3.2)
where Φ is the standard cumulative normal distribution function. However, there are some restrictions and disadvantages associated with FORM. First, the limit state equation has to be
continuous and smooth. Secondly, all variables must be continuous. Thirdly, all non-normally
distributed variables must be transformed to normal distribution. These conditions are often
infeasible in solving a practical engineering problem and some times multiple solutions/design
points exist and thus a more reliable method is needed to tackle the problem.
One promising approach is called Monte-Carlo simulation (MCS), which is essentially an
approximation of the exact integration but it is very flexible and reliable as long as there are
enough samples. The details of MCS and its enhanced version will be discussed in section 4.
3.2. Failure assessment for corrosion defects and cracks
For pipeline industry leakage normally refers to the situation where a defect penetrates through
the wall of a pipeline so the gas inside start leaking from the pipe. If a leak becomes unstable,
although not always the case, it could cause rupture, meaning a section of a pipe is torn apart
as a result of the fast propagation of the through-thickness defect. Different levels of failure
severity such as leakage or rupture etc. can be incorporated into the POF calculation. For
example the transition between leakage and rupture is modelled in the simulation. For each
failure mode a limit state function is defined.
For corrosion defects two levels of severity are considered: i.e. leakage and rupture. The limit
state function for a leakage failure takes the form:
( )g x r P= − (3.3)
where r is the estimated pressure resistance and P is the internal pressure of the pipeline.
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2
burstW S
rD
×= (3.4)
1
1burst flow
d
WS Sd
m W
−
= −
×
(3.5)
20.8
1l
mD W
= +×
(3.6)
where W is the wall thickness, D for pipe diameter, burstS for burst stress, flowS for flow
stress, d for depth of a defect, m for Folias factor and l is the total axial length of the
corrosion defect.
For cracks the basic failure criterion is based on the BS 7910 level two failure assessment
diagram (FAD). Equation (3.7) is used to describe the FAD curve, which has been shown in
Figure 2, where the region under the curve is considered safe. As with corrosion defects two
levels of severity are taken into consideration i.e. leakage and rupture, therefore a surface crack
model and a through thickness model are used in the analysis to calculate the probability of
failure for these two situations
Kr=f(Lr)
Lr
Kr
0.20
0.40
0.60
0.80
1.00
0.20 0.40 0.60 0.80 1.00 1.20 1.400
Safe region
Plastic collapse
cut-off
Brittle fracture dominated region
Figure 2 The BS7910 FAD curve used to define the limit state function
{ }2 6max
max
for (1 0.14 ) 0.3 0.7exp( 0.65 )
for 0
r r r r r
r r r
L L K L L
L L K
≤ = − + −
> = (3.7)
where rK measures the proximity to brittle fracture and rL represents the likelihood of
plastic collapse. The failure criterion includes both brittle fracture and plastic collapse. For
BS7910 level 2A FAD, they are given by:
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P Sr
IC
ref
r
Y
K KK
K
L
ρ
σ
σ
+= +
=
(3.8)
ρ is a parameter that takes plastic interaction between primary and secondary stress into
consideration. For materials that exhibit a yield discontinuity (referred to as Lüders plateau) rL
is restricted to 1.0 if no additional data is available [3]. Otherwise it is calculated through:
max2
Y ur
Y
Lσ σ
σ
+= (3.9)
Since the cumulative probability of failure over a given timeframe is required, crack propagation
due to cyclic loading must be included. The PARIS law with a threshold th
K∆ is selected to
calculate the crack length and depth with regard to the corresponding number of load cycles:
0 for <
for
th
m
th
K Kda
dn C K K K
∆ ∆=
∆ ∆ ≥ ∆ (3.10)
where C and m are fatigue growth parameters. An equivalent equation applies for the
second axis of the semi-ellipse. However it has been observed during studies that the crack
length hardly changes during fatigue analysis for a surface crack until it penetrates through the
pipe wall and becomes a through-thickness crack. The initial crack depth and length are
modelled as variables and the fatigue property parameters can also be modelled as random
variables based on certain distributions. As crack propagation could lead to fracture or leakage
of the pipeline after a certain period of time, f
P is a function of loading cycles n .
( )f
P f n= (3.11)
fP denotes the cumulative probability which monotonically increases with time or load cycles.
The inclusion of inspection and repair program can slow down this process so as to meet certain
target reliability targets.
3.3. Modelling of inspection and repair program
As part of the maintenance program of pipelines, inspections are performed using intelligent
‘pigs’ to detect defects in the pipeline. However, due to the sensitivity of the pigs not all defects
can be detected. Also it is not practical and economical to repair all the cracks detected
therefore according to Bayes' theory the process of inspection and repair at a given time
interval will change the distribution of defect depths and lengths in the pipeline. The exact
distribution will depend upon the repair strategy adopted, the frequency of inspection and the
sensitivity of the inspection tool. Not all the remaining defects will lead to failure but only those
missed by the inspection tool could cause either leakage or rupture of the pipeline. In order to
illustrate the nature of the inspection and repair program an event tree for crack-like defects
has been plotted to help understand the continuous updating process. From Figure 3, we can
clearly see that only the defects not repaired or missed by the inspection tool will contribute to
the cumulative probability of failure.
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Load cyclesn=0 n1 n2 n3
D
ND
R
NR
1st Inspection 2nd Inspection 3rd Inspection
D
ND
D
ND
R
NR
R
NR
Maintenance event tree:n,n1,n2..: load cyclesD: cracks detected; ND: cracks not detected;R: cracks repaired; NR: cracks not repaired;
Figure 3 Maintenance event tree for crack like defects
The Probability of Detection (POD) is mainly dependant on the defect depth and is usually
treated as an increasing function of defect depth and defined as an exponential function [4].
/ 1a
D aP e
λ−= − (3.12)
where a is the defect depth and λ is a parameter defining the POD curve. /D aP can be
expressed as the cumulative distribution function for the detectable depth of the inspection tool.
Hence, the detectable depth of the inspection tool follows the exponential distribution function
and both the average detectable depth and the standard deviation of the crack depth
equal1/ λ . Therefore the overall defect population can be divided into detected defects and undetected defects as shown in Figure 3. According to Bayes' theory [5] the posterior
probability density function for the un-detected defect with a depth of a is equal to:
( )0
( ) ( )
( ) ( )
NDUD
ND
P a f af a
P a f a da∞
=
∫ (3.13)
where ( )f a is the overall distribution of the defect depth and ( )NDP a can be interpreted as
the non-detection probability, which is equal to /1 D aP− . ( )UDf a can be used to calculate the
hazard function *
fP i.e. the probability of failure per year given that the failure has not yet
occurred. *
/
1
f
f
f
dP dnP
P=
−, where
fP is the cumulative probability of failure and n is the number
of load cycles.
The measurement error of the inspection tool is also taken into account during the POF
calculation, which is modelled by adding a measurement error to the actual defect depth such
that:
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m a
a a e= + (3.14)
where m
a denotes the measurement error; a for the actual defect depth and a
e the random
measurement error associated with the measurement of a . The measurement error can follow
normal, Weibull or other common types of distribution.
4. MONTE-CARLO SIMULATION AS POF EVALUATION METHOD
4.1. Direct Monte-Carlo simulation
The solution to equation (3.1) with MCS is obtained by interpreting the integral as the mean
value of a stochastic experiment after a large number of independent tests. A flowchart of the
whole simulation process is shown in Figure 4. There are two separate models used in the
analysis in order to compute leakage and rupture rate, which have been shown in Figure 4. As
the leakage always happens before the pipeline ruptures, rupture failures are considered as part
of leakage failure following the relation:
(leakage) (leakage only/no rupture) (rupture)f f f
P P P= + (4.1)
If Nf(n) failures i.e. leakages or ruptures are observed after N* repetitions, the POF is obtained
via:
*
( )( )
f
f f
N nP P n
N= = (4.2)
Equation (4.2) is only valid for pipelines which contain one defect. If there is more than one
defect in the pipeline, the cumulative probability of failure of a pipeline containing q defects
after n load cycles can be derived according to the definition:
| 1 (1 )q
f total fP P= − − (4.3)
where q is the number of cracks per km length of the pipeline. In fact, a kilometre long pipe
can be regarded as a series system [2] with q elements such that the failure of any individual
element contributes to the overall probability of failure.
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Start
Generate random
variables?
Surface crack growth for time period t
Leakage?
Nf(leakage)=Nf(leakage)+1
Yes
Yes
Yes
NoSimulation
ends
Through crack growth for time period t
No
Rupture?
Nf(rupture)=Nf(rupture)+1
No
Leakage happened in the previous year?
Nf(leakage)=Nf(leakage)-1
No
Yes
Figure 4 MC simulation including a transition from leakage to rupture failure modes
4.2. Stratified Monte-Carlo simulation method
4.2.1. Random number generation
Once a random number between 0 and 1 has been generated by the multiplicative congruent
random generator, it can be used to generate required pseudo-random variables with a given
probability distribution function. The inverse transform method is a common method to achieve
this, which is based on the observation that continuous cumulative distribution functions (cdfs)
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range uniformly over the interval (0, 1). If u is a uniform random number on (0, 1), then a
random number x from a continuous distribution with selected cdf F can be obtained using:
1( )x
x F u−= (4.4)
However, for normal distribution the inverse of the cumulative distribution function cannot be
found analytically. Hence, other methods such as the Box-Müller method and the envelop-
rejection method [6] are used in the simulation .
4.2.2. Stratified Monte-Carlo simulation
The variance of f
P is *
(1 )f f
P P
N
−. There are some variance reduction techniques which can
help to 'speed up' the simulation process. One of them is called Stratified Monte-Carlo
Simulation method (SMCS) [6, 7] and it has been employed in the current version of
PipeSafety© developed for the Naturalhy project. The basic idea is that the sampling of the
variables is carried out in the integration domain { } { }[0, ] 2 [0, ]R a B c= ∈ × ∈ ∞ which is
further partitioned into mutually disjunctive strata ( )1iD i m= K as shown in Figure 5.
Defect depth
De
fect
len
gth
Figure 5 The stratified Monte-Carlo simulation
Therefore, the original formula for calculating f
P can be written as:
( ) ( ) ( ) ( )1
11 1
( ) 0,
( ) ( )
( )
NN i
m
f X X N Ng x x D
i i
m
f i i
i i
m
f i
i i
P f x f x d x d x
Q D B D
P D
⋅⋅⋅ ≤=
=
=
= ⋅⋅⋅ ⋅ ⋅ ⋅
=
=
∑∫
∑
∑
(4.5)
For each stratum i
D , ( )f i
Q D is calculated through MCS with i
n number of samples for each
stratum and ij
q is used to count the number of failures in that stratum. Furthermore, a
normalization factor ( )i
B D is multiplied by the ( )f i
P D . ( )i
B D is calculated through:
( ) ( ) ( )i
iD
B D f a f c dcda= ∫ (4.6)
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Figure 6 Snapshot of PipeSafery software
5. SAMPLE PROBLEMS
5.1. Probability of failure calculation for corrosion defects
The fist example is used to demonstrate how the proposed framework can be used for the
integrity management of the pipeline. As the probability of failure (either leakage or rupture)
will increase as time goes by, therefore a proper inspection and repair program is required to
ensure that the POF will not rise above the target value. The following table summarises the
input data used in the analysis. In this analysis, the distribution of the corrosion defects is
assumed to be known beforehand and there is an average of 1 corrosion defect per km. In
addition it is assumed that no new corrosion defect will be considered for this pipeline during its
lifetime. If new corrosion defects are to be included, a series system needs to be constructed
therefore the total probability of failure can be evaluated. The target cumulative failure
probability value set for leakage is 1e-5 per km and 1e-7 per km for rupture.
Corrosion
defect
Internal
pressure
(Mpa)
Wall
thickness
(mm)
Pipe
diameter
(mm)
Flow
stress
(MPa)
defect
depth
(mm)
Defect
length
(mm)
defect
depth
growth rate
(mm/year)
defect
length
growth rate
(mm/year)
Distribution
type constant normal constant normal normal normal normal normal
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Mean 6.5 13 900 416 3.7 97 0.2 1.2
Standard
deviation 0 0.3 0 15 1 12 0.02 0.3
Table 1 Summary of the input data for POF evaluation of corrosion defects
The simulation results for cumulative POF evaluation without introducing inspection and repair
program are shown in Figure 7. As can be seen from the results the probability of leakage
reaches 1e-5 per km at the beginning of year 8, therefore we should conduct inspection and
repair work to prevent the probability of leakage from reaching this target value.
POF v.s year
1.00E-14
1.00E-12
1.00E-10
1.00E-08
1.00E-06
1.00E-04
1.00E-02
1.00E+00
0 5 10 15 20 25 30 35 40 45
year
pro
bab
ilit
y o
f fa
ilu
re
Leakage probability Rupture probability
Target leakage probability Target rupture probability
Figure 7 Leakage and rupture probabilities with regard to service life
Assume we have an inspection tool with the POD of 90% at 30% wall thickness and the
minimum detectable corrosion depth being 0.2mm. Also due to the limited repair budget the
detected defects with depths smaller than 1mm will not be repaired. It can be expected that the
probability of leakage and rupture will fall sharply following the inspection and repair program.
Therefore we should arrange an inspection and repair program by the end of year 7. It is worth
mentioning that the cumulative POF after inspection is conditional, assuming no actual failure
has happened before inspection. The results are shown in Figure 8.
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POF v.s year
1.00E-14
1.00E-13
1.00E-12
1.00E-11
1.00E-10
1.00E-09
1.00E-08
1.00E-07
1.00E-06
1.00E-05
1.00E-04
1.00E-03
1.00E-02
1.00E-01
1.00E+00
0 5 10 15 20 25 30 35 40 45
year
pro
bab
ilit
y o
f fa
ilu
re
Leakage probability Rupture probability
Target leakage probability Target rupture probability
Figure 8 Leakage and rupture probabilities following the inspection
It is evident that the probability of having a leakage and a rupture is now within the target
range for the first 15 years. A second inspection and repair program should therefore be
arranged by the end of year 15 in order to meet the same goal as before.
5.2. Probability of failure calculation for cracks
In this section we will investigate the impact of hydrogen on the probability of leakage and
rupture. We assume that the cracks are longitudinally oriented as the tensile stresses applied to
the crack are the maximal in this orientation. The input data is summarised in the list below
(data is based on the experiments carried out within the NATURALHY project):
No. Parameter Mean Standard
deviation
Distribution
type
1 Pipe diameter (mm) 600 0 Fixed
2 Wall thickness (mm) 10 0 Fixed
3 Initial crack depth (mm) 2.85 0.3 Log-normal
4 Initial crack length (mm) 130 70 Log-normal
5 Pressure (MPa) 6.06 0.2 Normal
6 Residual stress (MPa) Base metal- 0 0 Fixed
7 Fracture toughness
(MPa*mm1/2)
4743.4 for 100% NG
2383.9 for 100% H2
2383.9 for 50% H2/NG
0 Fixed
8 Yield strength (MPa) 358 20 Normal
9 Tensile strength (MPa) 455 25 Normal
10 Threshold toughness
(MPa*mm1/2)
338.36 for 100% NG
275.12 for 100% H2
316.23 for 50% H2/NG
10 Normal
11 Fatigue parameter c 2.32e-14 for 100% NG
2.6e-19 for 100% H2
5.2e-14 for 50% H2/NG
0 fixed
12 Fatigue parameter m 3.4 for 100% NG
5.57 for 100% H2
3.32 for 50% H2/NG
0 fixed
13 Pressure drop ratio 0.35 0 fixed
14 Service life (years) 50 0 fixed
15 No. of load cycles per
year
365 0 fixed
Table 2 Input data for crack model
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As with the previous example it is assumed that there is only one crack per km pipeline. Figure
9 shows how the cumulative POFs for pipelines carrying natural gas-hydrogen mixtures change
over time. It is clear from the results that hydrogen does have a significant impact on the life of
the pipeline and therefore risk mitigation measures must be taken to ensure that the pipelines
carrying hydrogen are checked and repaired in time.
POF v.s year
1.00E-11
1.00E-10
1.00E-09
1.00E-08
1.00E-07
1.00E-06
1.00E-05
1.00E-04
1.00E-03
1.00E-02
1.00E-01
1.00E+00
0 10 20 30 40 50 60year
pro
bab
ilit
y o
f fa
ilu
re
100% natural gas 100% hydrogen 50% natural gas/hydrogen mixtures
Figure 9 Probability of leakage for pipelines carrying natural gas-hydrogen mixtures
6. CONCLUSIONS
A methodology has been developed to assess the integrity of pipelines carrying natural gas and
hydrogen mixtures. The simulation based method has demonstrated its flexibility and reliability
in solving complex problems involving large number of variables and complex procedures. The
stratified Monte-Carlo sampling scheme has been successfully employed to accelerate the
simulation process without compromising the accuracy of results. The proposed method for POF
evaluation is of practical importance as the integrity and risk mitigation procedures can be
planned based on the information provided by PipeSafety©. The sample results demonstrate
that POFs of pipelines increase in the presence of hydrogen but the extent of such increase can
be reduced by mixing hydrogen and natural gas.
In addition to the material properties defect size distributions also have a huge influence on the
final POF results. Therefore the POF is very sensitive to the means and standard deviations
defined for the defect depth and length. In this sense, data collection and interpretation is the
most important step for a successful simulation. There are many effective ways to approximate
the distribution of variables such as the maximum likelihood method and the bootstrap
sampling method (when data is scarce). These methods should be first implemented to obtain
as accurate data as possible in advance of the use of PipeSafety©.
7. ACKNOWLEDGEMENTS
The authors would like to thank the NATURALHY project partners for providing data and
information and the European Union for kindly providing the financial support for this study.
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8. REFERENCES
1. A.Bruckner and D.Munz, Determination of Crack Size Distributions from Incomplete Data
Sets for the Calculation of Failure Probabilities. Reliability Engineering, 1985. 11: p.
191-202.
2. Thoft-Christensen, P. and M. J.Baker, Structural Reliability Theory and Its Applications.
1982: Springer-Verlag.
3. BS7910: Guide to methods for assessing the acceptability of flaws in metallic structures.
2005: British Standards. 23-60.
4. E.S.Rodrigues and J.W.Provan, Development of a general failure control system for
estimating the reliability of deteriorating structures. Corrosion, 1989. 45: p. 3.
5. R.Benjamin, J. and C.A. Cornell, Probability, Statistics, and Decision for Civil Engineers.
1970: McGraw-Hill Book Company.
6. J.S.Dagpunar, Simulation and Monte Carlo with Applications in Finance and MCMC.
2007: John Wiley & Sons, Ltd.
7. A.Bruckner-Foit, Th.Schmidt, and J.Theodoropoulos, A comparison of the PRAISE code
and the PARIS code for the evaluation of the failure probability of crack-containing
components. Nuclear Engineering and Design, 1989. 110: p. 395-411.
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9. LIST OF TABLES
Table 1 Summary of the input data for POF evaluation of corrosion defects ...........................13 Table 2 Input data for crack model...................................................................................14
10. LIST OF FIGURES
Figure 1 Geometry of surface/through thickness crack model used in the analysis................... 5 Figure 2 The BS7910 FAD curve used to define the limit state function .................................. 6 Figure 3 Maintenance event tree for crack like defects ......................................................... 8 Figure 4 MC simulation including a transition from leakage to rupture failure modes...............10 Figure 5 The stratified Monte-Carlo simulation...................................................................11 Figure 6 Snapshot of PipeSafery software .........................................................................12 Figure 7 Leakage and rupture probabilities with regard to service life ...................................13 Figure 8 Leakage and rupture probabilities following the inspection......................................14 Figure 9 Probability of leakage for pipelines carrying natural gas-hydrogen mixtures ..............15