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Application of SRA for Pipeline Design Operation & Maintenance
Andrew Francis
Advantica Technologies
ASRANeT, 2nd Annual Colloquium, 9th July 2001
INTRODUCTION
Uses of SRA Wall thickness determination Uprating Life extension Risk Based Inspection
Generally Determination of required level of failure mitigation
Present Study
Failure Mode
• Fatigue Crack Growth
Uncertainty
• Construction Defect Depth
Mitigation Measures
• Construction process
• Weld Inspection
• Hydrostatic Test
Mitigation effects using Bayes Theorem
Conditional probability
p( X | Y ) is the probability of event X occurring given prior knowledge that event Y has already occurred
p (XY) is the probability that both X and Y will occur before any prior knowledge has been obtained
p ( Y ) is the probability that Y will occur before any prior knowledge is obtained
)(
)()|(
Yp
YXpYXp
Construction Process
For a given weld constructed to an appropriate standard, the probability of having a defect of depth a is
p(D) is the probability that the weld contains a defect and p(a) is the probability that the defect has depth a
)()()( apDpaDp
Construction Process
The Likelihood of a Defect of Given Depth Being Present
0.0E+00
2.0E-03
4.0E-03
6.0E-03
8.0E-03
1.0E-02
0 2 4 6
a (mm)
p(D
a
)
Pre-Service Weld Inspection
Objective: To detect any defects, which are unacceptable according to the appropriate criteria
Issue: Inspection techniques often only 70% - 80% reliable, sometimes lower
Pre-Service Weld Inspection
We want to know probability, p(D a | I)
Event I: Weld was inspected and no defect was found
Using Bayes Theorem
)(
)()()|()|(
Ip
daapDpaDIpdaIaDp
Pre-Service Weld Inspection
Probability, p ( I | D a) , that the weld was inspected and nothing was found given that the weld contains a defect of depth a is given by
PoD(a) is the probability of detection of a defect of depth a
)(1)|( aPoDaDIp
Pre-Service Weld Inspection
Probability of Detection
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
0 2 4 6 8 10 12a (mm)
Po
D(a
)
Pre-Service Weld Inspection
Probability of inspecting weld and finding nothing is given by
)(1)](1)[()()(0
DpdaaPoDapDpIp
Pre-Service Weld Inspection
Probability that the weld contains a defect of depth a, given that the weld was inspected and nothing was found is given by
)(1)()](1[)(
)()](1)[()|(
0
DpdaapaPoDDp
apaPoDDpIaDp
Pre-Service Weld Inspection
The likelihood of a defect of given depth being present following inspection
0.00E+00
2.00E-03
4.00E-03
6.00E-03
8.00E-03
1.00E-02
0 2 4 6
a (mm)
p(D
a
|I)
Pre-Service Hydrostatic Test
Objective: To give assurance integrity of the pipeline prior to gassing up
Pre-Service Hydrostatic Test
We want to know the probability, P(D a | H)
Event H: Survival of the Hydrostatic Test
Using Bayes Theorem
)(
)()()|()|(
Hp
daapDpaDHpdaHaDp
Pre-Service Hydrostatic Test
Now the probability p(H | D a) , that the weld survived the hydrostatic test given that the weld contains a defect of depth a is given by
H is the Heaviside step function defined by
aH is the depth of defect that would just fail under the hydrostatic test pressure
)()|( aaHaDHp H
0,0)( xxH
0,1)( xxH
Pre-Service Hydrostatic Test
The value of aH depends on geometrical and material parameters and can be evaluated using fracture mechanics procedures such as BS7910
Since geometrical and material parameters are subject to uncertainty, aH is also subject to uncertainty. This is not considered here for simplicity
Pre-Service Hydrostatic Test
Probability of surviving the hydrostatic test is given by
)(1)()()(0
DpdaapDpHpHa
Pre-Service Hydrostatic Test
Probability that the weld contains a defect of depth given that the hydrostatic test was survived is given by
)(1)()(
)()()()|(
0
DpdaapDp
aaHapDpHaDp
HaH
Pre-Service Hydrostatic Test
Combining with effects of inspection leads to
)(1)](1)[()(
)()](1)[()()|(
0
0
0
DpdaaPoDapDp
aaHaPoDapDpIHaDp
HaH
Pre-Service Hydrostatic Test
The Effect of a Hydrotest
0.0E+00
2.0E-03
4.0E-03
6.0E-03
8.0E-03
1.0E-02
0 2 4 6
a (mm)
p(D
a
|IH
)
Fatigue Crack Growth
Fatigue crack growth rate is dependent on the instantaneous defect depth, a, giving
Function f depends on stress intensity factor which is dependent on depth and the magnitude of the cyclic loading
nKaafdt
da )(
Fatigue Crack Growth
Underlying assumption: the following continuity equation is satisfied:
This equation states that all defects which lie within the interval [a(t), a(t) +da(t)] at time t will lie within the interval [a(t+dt), a(t+dt) + da(t+dt)] at time t+dt
)(]),([)(]),([ dttdadttdttaptdattap
Fatigue Crack Growth
P(a,t) P(a,t)
a(t) a+da(t)a(t+dt)
a(t+dt)+da(t+dt)
a(t+dt)
Growth
a(t)
Fatigue Crack Growth
Distribution at time t based on distribution at time of commissioning following inspection and hydrostatic test
0,)1()1(1),( 1
1111 nnn
nn KtnapKtantap
Fatigue Crack Growth
Effect of Fatigue Crack Growth
0.00E+00
2.00E-03
4.00E-03
6.00E-03
8.00E-03
1.00E-02
0.00 1.00 2.00 3.00 4.00 5.00 6.00 7.00
a
p(a
)
Probability of Failure
Probability of failure in time interval[0, T]
aop is the defect size that would fail at the specified operating conditions
H
op
a
a
f daTapTp ),()(
Probability of Failure
Probability of Failure @105% SMYS
1.00E-09
1.00E-08
1.00E-07
1.00E-06
1.00E-05
1.00E-04
1.00E-03
1.00E-02
0 10 20 30 40 50
Time
Pf
Nothing
Just Hydrotest
Just Inspection (Rad)
Just Inspection (TOF)
Both
Probability of Failure
Probability of Failure @100% SMYS
1.00E-09
1.00E-08
1.00E-07
1.00E-06
1.00E-05
1.00E-04
1.00E-03
1.00E-02
0 10 20 30 40 50
Time
Pf
Nothing
Just Hydro
Just Insp - Rad
Just Insp - ToF
Both
Probability of Failure
Probability of Failure @90% SMYS
1.00E-05
1.00E-04
1.00E-03
1.00E-02
0 10 20 30 40 50
Time
Pf
Nothing
Just Hydrotest
Just Inspection (Rad)
Just Inspection (TOF)
Both
Conclusions
SRA & Bayes theorem used to systematically quantify effect of hydro test on construction defects taking account of inspection
An acceptable fatigue life may be achieved without any pre-service inspection if a hydrostatic test pressure of 105%SMYS is used.
Conclusions
If test pressure of 90%SMYS then pre-service inspection using a technique such as radiography or better is likely to be required.
If sophisticated TOF is used then may be possible to achieve an acceptable fatigue life without the need for a hydrostatic test.
Results are preliminary and further validation work required
