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Assessing Probability ofFailure for Pressure

Equipment: Part 1

The Equity Engineering Group, CanadaSherwood Park, Alberta

Boyd McKay, P.Eng.

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Overview

Introduction

Statistics Background

Pressure Equipment Risk Calculation Process

Damage Models and Damage Rates

Probability Of Failure Calculation Pressure Equipment Reliability Model

Examples Summary

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Introduction

Importance and value of RBI concepts in thesafe operation of pressure equipment

Fundamental physical and mathematicalprinciples are valid irrespective of how theProbability Of Failure (POF) is calculated

Most RBI methodologies are generally used forscreening purposes

The general risk process as well as the API RBIProbability Of Failure calculation will also bereviewed along with several applications

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What are the location, shape and scaleparameters ?

How to test for goodness of fit? How are the distributions ranked for fit? Is the sample representative of the population?

4 questions requireanswers for validanalysis.

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The calculation of POF can be generally divided

into 3 classes:(1) Non-statistical- Same weight on data points

(2) Statistical can censor data

(3) Hybrid:

- Combination of statistical and non statistical- Expert judgement

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Risk Calculation Process General steps in the process:

Establish time period

Determine damage mechanisms

Determine final risk scenario

Calculate damage rate usingspecified damage model

Select damage rate model(s)

Calculate POF over specified time interval

Calculate CONS over specif ied time interval

Calculate RISK over specified time interval

Compare RISK torisk matrix threshold

Perform mitigationactivitiesDo nothing

RISKacceptable

Calculate RISK overspecified time interval

Yes No

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Risk Calculation Process

Time Period Under Consideration

A time period that is too long (e.g. 15 years) may

overestimate POF (e.g. no clear calculation of whento perform mitigating activities. A short time period(e.g. 1 day) will tend to produce a low POF.

The time period under consideration is important forseveral reasons:(i) The POF is influenced by time.

(ii) The consequence is influenced by time (butusually less so than the POF time dependence).(iii) The stream composition may change as a

function of time

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Damage Mechanisms

Stream components and conditions are such that areaction or susceptibility can exist

damage

mechanism exists

Assumptions made regarding valve position,isolation, operating pressure, temperature andvelocity parameters

Stream CompositionComponents may either be sampled frequently anddifficult to test or sampled infrequently and easy totest.

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Risk Calculation ProcessDetermining The Final Risk Scenario (FRS)

1. Localized internal corrosion that may requirerepair.

2. A corrosion pinhole resulting in a leak to atm. 3. A corrosion pinhole resulting in leak to atmosphere

with exposure and business loss.

FRS POF COF Risk1 P 1 PMO COF 1 P 1* COF 1 2 P 2

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Determining The Final Risk Scenario

(a) Decreased confidence in pressure equipmentintegrity program within the organization.

(b) Decrease in funds for inspecting equipment thatmay really need to be inspected and repaired(e.g. unacceptable risk not being addressed).

(c) Increased effort to conduct multiple equipmentreviews, with same data, and arriving at usuallydifferent answers .

The PMO can be used for all three FRS. This

conservative approach can lead to the following:

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Determining The Final Risk Scenario

An important point to note is that Fitness-For-Service (FFS) can be applied to FRS-1. A FFSanalysis could be run to determine limiting flawsizes prior to inspection (internal or external).

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Damage Models & Distribution of Damage Rates

Damage mechanisms for different industries arepublished in API 571, WRC 488, 489 and 490.

Deliverable from damage models is usually thelowest wall thickness in the system of interest.

A set of damage rates over the time interval can begenerated.

Determine what statistical distribution fits the data.

Calculate the probability of a damage rate occurringthat would result in a specified thickness (e.g. t min )being present in the system.

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Conversely, for a given data set of wallthicknesses over the time interval of interest, theuser can determine what statistical distribution (ifany) will fit the data.

Using the distribution parameters, the user cancalculate the probability of a thickness occurring

that would less than a specified thickness (e.g.t min ) being present in the system.


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The data set may come from one or morepopulations that have different distributions.

Re-sample from data set based on populationcharacteristics.

Characteristics such as no flow region, bottom/topof pipe, extrados/intrados of elbow, etc., are allvalid ways of producing another sample data setthat can be tested against known distributions.

It is not uncommon to calculate that the data doesnot come from any known distributions.


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Distribution can change type over time.

Re-sample from data set based on populationcharacteristics as well as time (morecomplicated).

Changes in damage drivers/operation can aid inre-sampling.


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Calculation of Cumulative Probability Of Failure

Integrate continuous random variable to get POF:( ) ( )

t

F x f t dt

=

For a maximum/threshold POF of P cr require:

( ) cr F x P

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(a) Can theoretically be used for any continuousdistribution and censored data.

(b) Approximate confidence bounds can be calculated.(c) Less suitable for less than 5 (approximately) data

points.


Maximum Likelihood Estimate (MLE), linearunbiased estimators, method of moments, etc.,procedures used to calculate parameters

The MLE method has the following significant

characteristics:

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Parameters must be tested using a test statisticsuch as Anderson-Darling, Kolmogorov-Smirnov(KS), etc.

Alternatively, a correlation coefficient (CC) can beused.

Graphically viewing the probability plots is also

used to validate the correlation coefficient and tosee if there may be different failure modes presentin the data.


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Correlation Coefficient and Scale

Parameter plot


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Limit State Function

Creates regions of acceptable and unacceptableresults

( ) Resistance( ) - Load( )G t t t =

There are 3 cases to consider:

( ) 0

( ) 0( ) 0

G t

G t G t

Unacceptable (loaded beyond capacity)

Acceptable

Acceptable

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is the variable of interest in calculating the POF.

-3 -2 -1 0 321

Load L

-3

-2

-1

0

1

2

3

R = L +

G*i i iCOV =2

1i

V

G vi i

dGS S dv=

=

G

GS

=

NORM(- )POF =

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The preceding discussion is the core of the API

RBI POF methodology. The POF calculation shown here is the seed for

the POF to be used in a thinning risk calculation.

There are other considerations such as acomparison to a Generic Failure Frequency (GFF)for equipment type, etc. These considerationscan be modeled numerically and applied to give afinal POF result.

The same techniques can be applied to othervessel failures modes such as cracking.

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Pressure Vessel Example

Vessel Data

Vessel: Atmospheric Overhead AccumulatorMaterial: SA 285 GR C Corrosion Allowance: 3/16Design Pressure: 50 psig Prior Inspection Data: None

Diameter: 6-6 COV (pressure): 20% Age: 6 years COV (CR): 30%

Thickness: 3/8 COV ( UTS ): 10%

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Analysis Equations

( ) 000

0.25, , , , ln

0.277

n

G UTS UTS G

e R R R t d n RSF

n n R t d

= + +

( )applied applied L P P=

*Gd time CR=0.25

ln0.277 *

n

oUTS applied

o init

e RG RSF Pn n R t CR time

= + +

( )0.250.277 *

n

UTS o init

dG e time RSF

dCR n n R t CR time

= +

( )0.25 ln0.277 *

no

UTS o init

dG e R RSF d n n R t CR time

= +

1applied

dGdP

=

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Analysis Equations2 22

applied UTS G P CRapplied UTS

dG dG dGS S S S

dP dCR d

= + +

NORM(- )=NORMG

GPOF

S

=

ResultsVessel: Atmospheric Overhead Accumulator

Material: SA 285 GR C Corrosion Allowance: 3/16Design Pressure: 50 psig Prior Inspection Data: None

Diameter: 6-6 COV (pressure): 20%

Age: 6 years COV (CR): 30%Thickness: 3/8 COV ( UTS ): 10%Damage State Corrosion Rate POF

Corrosion Rate 1: 0.030/year 1.85388E-07Corrosion Rate 2: 0.035/year 1.05330E-05Corrosion Rate 3: 0.040/year 2.16402E-04

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Pressure Relief Valve Example

PRV Data

Failure Mode: Failure to Open on Demand (FOD)

Service Interval(Years)

FODPass Fail

Intermediate Hydrocarbon 36 2 1

Intermediate Hydrocarbon 48 2 1Intermediate Hydrocarbon 54 4 2Intermediate Hydrocarbon 60 2 1Intermediate Hydrocarbon 66 3 3

Intermediate Hydrocarbon 72 3 4Intermediate Hydrocarbon 78 1 5Intermediate Hydrocarbon 84 0 4Intermediate Hydrocarbon 90 - 1

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Analysis Equations

Use Maximum Likelihood Estimate of parametersDistrib ution Parameters Type Random

VariableDomain

Probability Density Function Variable Maximum Likelihood Estimates For Parameters

Normal location= shape

=

x < < + 21exp

2( )

2

x

f x

=

1

1 N

j j

x x N

=

= =

( )2

2 2

1

1 N

x j j

S x x N

=

= =

Standard x < < +( )21exp

2( )2

Z f x

=

x Z

=

Weibull(2P)

0= shape = scale

=

( )1

( ) exp x x

f x

=

( ){ }{ }

1

1

11

1 1

1

1 log

N

j j

N N

j j j j j j

x N

x x x x N

=

= =

=

=

Standard ( )( ) ( )1( ) exp f x Z Z

=

x Z

=

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Results

The PRV POF calculation is the seed for the POF

to be used in an API RBI risk calculation for aPRV.

There are other considerations such as a

comparison to a GFF for PRVs, PRV dischargelocation, chattering. These considerations can bemodeled numerically and applied to give a finalPOF result.

Same techniques can be applied to other PRVfailures modes such as leaking, fail partiallyopen, etc.

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Summary

The API RBI POF general POF calculation methodhas been demonstrated for thinning damagemechanisms and a PRV.

The methodology is in the public domain and isdefensible from a technical perspective.

The API RBI methodology can be of significantbenefit in obtaining more information frominspection data.

The methodology generates reproducible anddefensible inputs to a risk analysis for pressureequipment.

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P.O. Box 3648

Sherwood Park, Alberta CanadaPhone: 780-449-4180 Fax: 866-775-1528Cell: 780-722-3246

2008 boyd mckay

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