ENGINEERING STRUCTURAL INTEGRITY ASSESSMENT

A CONSIDERATION OF DATA REQUIREMENTS FOR STRUCTURAL RELIABILITY BASED ASSESSMENTS OF ONSHORE PIPELINES

A. Francis*, R.J. Espiner*, A.M. Edwards* & R.J. Hay*

The use of structural reliability based techniques to assess the fitness for purpose of onshore pipelines is increasing. The technique involves the evaluation of failure probability based on a probabilistic analysis of the limit state functions that describe the conditions that will cause failure for each relevant failure mode. The fitness for purpose assessment is based on a consideration of both the calculated failure probability and the consequences of a failure. The basic technique is now well established. However, no firm guidance currently exists on the type, quality and quantity of data that are required. This is generally recognised as a potential weakness of the technique. This paper describes an analytical consideration of some of the key issues associated with data and provides some guidance on basic data requirements.

INTRODUCTION

The use of structural reliability analysis (SRA) for the purpose of design and assessment of both onshore and offshore pipelines has significantly increased over recent years. The technique has been used to justify safe operation of transmission gas pipelines in the UK at a design factor of 0.78 (Francis et. al., [1,2], Senior & Jones [3]) and to justify the safe operation of the Britannia gas export pipeline at a design factor of 0.81 (McKinnon et. al. [4]). The basic philosophy of SRA is to specifically identify the potential causes of failure, establish the mechanism which leads to the failure (failure mode), and to evaluate the risks associated these failures. Justification for safe operation is made by demonstrating that the risks associated with harmful events are negligible or ‘As Low As Reasonably Practicable’ (ALARP). There are essentially six basic components of SRA and these are briefly described below. One of the most currently debated of these is the interpretation of the results; i.e. what can be inferred about the safety of the pipeline from computed failure probability. In order to acquire an appreciation of this issue one needs to understand the origins of the computed value and these lie firmly within the uncertainty in the available information. This paper goes on to discuss the sources of uncertainty and * BG Technology, Gas Research & Technology Centre, Ashby Road, Loughborough, Leics. LE11 3GR, United Kingdom

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describes analysis procedures based on Bayesian Statistics, which can be used to combine information in order to reduce uncertainty and hence reduce the value of the computed failure probability. STRUCTURAL RELIABILITY ANALYSIS Structural Reliability Analysis comprises six elements. These are •

Establishment of Limit States

•

Identification of Failure Modes

•

Formulation of Limit State Functions

•

Uncertainty Analysis

•

Evaluation of Failure Probability

•

Assessment of Results

A brief description of each of these elements and the role they play in the overall analysis is given below. More detailed descriptions are given in Francis et al [5]. Establishment of Limit States A limit state is defined as the state of a structure when it no longer satisfies a particular design requirement. The limit states thus determine the conditions that are to be avoided. A leak is a limit state. Identification of Failure Modes A failure mode is the mechanism that causes the pipeline to reach a limit state. For instance, corrosion growth is a failure mode that can cause a pipeline to leak. Formulation of Limit State Functions A limit state function is a mathematical relationship between the parameters characterising a particular failure mode that exists when the pipeline has reached a limit state. It is generally expressed in the form G( x1 , x 2 ,...x n ) = 0 where x1, x2,…xn denote the n parameters which characterise the failure mode under consideration. Some of the parameters may be dependent on time. In this case the limit state function determines the relationship that exists between the parameters at the current time (t = 0, say) that will result in a failure at a later time, t.

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Uncertainty Analysis In any practical engineering circumstance, each of the input values to a limit state function is subject to uncertainty. Uncertainties are accounted for in a structural reliability analysis by describing variables in statistical terms. For each limit state function, the variability in the sensitive parameters must be quantified by data analysis and ultimately the construction of probability density functions. This is achieved by performing appropriate statistical analyses of the data available from sources including construction records, test certificates and inspection records. The outputs of these calculations are mathematical functions describing the likelihood of occurrence of specified values of particular parameters. Parameters typically belong to one of four groups: pipeline geometry, material properties, defect dimensions and loads. Evaluation of Failure Probability The probability density functions for each sensitive parameter are used in conjunction with the limit state functions to determine the probability of failure. For a given limit state and failure mode, the failure probability is the sum of the likelihoods of occurrence of all combinations of the relevant parameters which cannot coexist in an un-failed state. Integral calculus is used for this purpose since the parameters are generally described by a continuum, rather than by discrete values. Assessment of Results The final stage of the SRA is to make a decision based on a consideration of the computed failure probability. Justification for safe operation is made by demonstrating that the risks associated with harmful events are negligible or â€˜As Low As Reasonably Practicableâ€™ (ALARP). There are currently several approaches available for this purpose, each having particular merits. These include comparison of the computed failure probabilities for the case under consideration with those calculated in association with previous operation and those implicit in design. However, there are currently no widely accepted methods although significant progress towards this end has recently been made within the pipeline industry. UNCERTAINTY ANALYSIS Onshore pipeline designs are characterised by nominal values of diameter, wall thickness, material yield strength and operating pressure. These nominal values are characteristic values of the statistical distributions that describe the uncertainty in the actual values. For instance pipe mill sampling procedures are used to ensure that the

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specified minimum yield stress is typically representative of the lower five percentile of the actual distribution. This ‘quality control’ of strength is one of many factors that contribute to the overall structural integrity of the pipeline. Other factors include the effects of the hydrostatic tests, limits on operating pressure, pipe mill weld inspections and on-line inspections. Each of these activities provides the engineer with information and hence reduces uncertainty. In this section a simple example is given that describes in detail how information provided by one such activity can reduce the requirement to obtain data relevant to others. The example describes the effect of the pipe mill hydrostatic test on the uncertainty in yield strength and wall thickness. Wall Thickness and Yield Strength The distributions of yield strength and wall thickness values can be inferred from information given on pipe mill test certificates. There is no reason to assume any dependency between yield strength and wall thickness and the uncertainty can thus be described by two independent probability density functions, p(σy) and p(w), respectively. Sampling procedures in the pipe mill are followed to ensure that the specified minimum values of these quantities are typically less than the lower fifth percentiles of the actual distributions. There is thus a low probability that ‘substandard’ pipes will not be detected during the sampling process. However, in addition to the sampling process each pipe section is subjected to hydrostatic test at the mill which induces a pipe wall hoop stress of 90% SMYS. Although hydrostatic testing is applied to each and every pipe, in contrast to the sampled direct measurements, the mill hydrotest provides very little extra information about yield strength and wall thickness under normal circumstances. This is because the vast majority of the yield strength distribution inferred from sampling will have a yield strength value in excess of 90% SMYS. The primary benefit of the pipe mill hydrostatic test is thus not to identify ‘rogue properties’ but rather to confirm the structural integrity of the seam weld. The consideration of seam weld defects is beyond the scope of this paper. The pipes are further hydrostatically tested in the field during commissioning to provide a final check on the overall structural integrity. This test typically induces a pipe wall hoop stress of 105% SMYS. Although, the primary reason for this test is to identify any defects that may have been introduced during construction and burial process, it also provides assurance that there are no adverse combinations of wall thickness and yield stress that are unable to resist the internal pressure. In other words, survival of the test eliminates all combinations of wall thickness, w, and yield strength, σy, that satisfy the inequality

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pH D 2w

σy ≤

The effect of the test is to reduce the uncertainty in the values of yield strength and wall thickness. However, since the yield strength required to resist the internal pressure is dependent on wall thickness the uncertainty can no longer be described by two independent pdfs; a joint pdf is required for this purpose. To this end, assuming that the pipe diameter and the test pressure are precisely known quantities, Bayes’ theorem can be invoked to update the joint pdf of the yield strength and the wall thickness based on the survival of the hydrostatic test. Denoting the event that the wall thickness lies in the infinitesimal range [w, w + dw] and the yield strength lies in the infinitesimal range [σy, σy + dσy] by Z and the event of surviving the hydrostatic test by S, Bayes’ theorem states p( Z | S ) =

p(Z ∩ S ) p( S )

Prior to the test, the probability of Z occurring is given by the prior joint pdf of w and σy. Since before the test is conducted w and σy are generally considered to be independent quantities, p(Z) may be expressed p( Z ) = p( w ) p( σ y ) dw dσ y

The probability p ( Z ∩ S ) can thus be expressed p ( Z ∩ S ) = p ( w) p (σ y ) H (σ y −

pH D )dwdσ y 2w

where H denotes the Heaviside step function. The probability of surviving the hydrostatic test is equal to the probability that only combinations of w and σy that do not satisfy the above inequality can exist, namely ∞

∞

0

pH D 2w

p( S ) = ∫ p( w ) ∫ p( σ y )dσ y dw

The probability of event Z occurring given that event S has occurred, which is the posterior joint probability density function of w and σy denoted by p(w, σy | S) is thus given by

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p( w ) p( σ y )H ( σ y −

p( w,σ y | S ) =

∞

∞

0

pH D 2w

pH D ) 2w

∫ p( w ) ∫ p( σ y ) dσ y dw

The information provided by the survival of the test has improved our knowledge of the structural integrity, which is dependent on both w and σy. The joint pdf given by the above equation, together with loadings and defect dimensions, determines the probability of failure of the pipeline during its operating life. An indication of the general effect of the survival of the hydrostatic test on the individual distributions of w and σy can be obtained by consideration of the marginal distributions, p(w | S) and p(σy | S) respectively, given by ∞

p( w ) ∫ p( σ y ) dσ y p( w | S ) =

∞

pH D 2w ∞

∫ p( w ) ∫ p( σ y ) dσ y dw

0

pH D 2w

and ∞

p( σ y ) ∫ p( w ) dw p( σ y | S ) =

pH D 2σ y

∞

∞

0

pH D 2w

∫ p( w ) ∫ p( σ y ) dσ y dw

Application The information inferred from sampling and hydrostatic testing is illustrated here. In particular, it is shown that high rates of sampling provide little extra knowledge in terms of the yield strength and wall thickness distributions, but that the commissioning hydrotest has the potential to substantially improve the characterisation of these distributions. The example discussed involves the production of X60 grade pipes (SMYS of 414 MPa) with an actual wall thickness characterised by a Normal distribution with a mean of 12.8 mm and a standard deviation of 0.304 mm, and a yield strength characterised by a lognormal distribution with a mean of 444 MPa and a standard

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deviation of 12.8 MPa. A direct measurement sample of a certain size is taken from this population and each pipe is then field hydrostatically tested at 105% SMYS. The knowledge inferred from this process is shown graphically in Figures 1 - 6. In each case the solid line represents the actual production distribution, the dot-dashed line represents the average estimate of the production distribution inferred from the sampling process (assuming sampling does not affect the underlying distribution), and the dotted line represents the estimate of the distribution after field hydrostatic testing. It can be seen that, whilst there are significant differences in inferred estimates between cases with a sample size of 5 and 20, there is little difference between cases with sample sizes of 20 and 100. This is because the average sample standard deviation rapidly tends to the actual population standard deviation as the sample size increases. However, most apparent from Figures 1 - 6 is the significant effect of the field hydrostatic test on the estimated wall thickness and yield strength distributions. The above analysis shows that the field hydrostatic test is significantly more useful than a large number of pipe mill certificates. Furthermore, in general a conservative estimate of the yield strength pdf prior to the hydrostatic test can be constructed generically based on the assumption that the mean value is 10% higher than the SMYS and that the SMYS is the lower fifth percentile of the distribution. This distribution can then be updated using the above procedure. The underlying assumption could be confirmed using a suitable hypothesis test based on information from about six mill certificates. However, even in situations that did not satisfy the test it has been shown above that only twenty or so certificates are required in order to construct the prior distribution. A similar generic approach to the prior pdf of wall thickness can be taken, in which the mean value is equal to the nominal wall thickness and the specified minimum wall thickness is the lower fifth percentile of the distribution. The distributions determined by the above analysis can be used in probabilistic studies of the more significant failure modes such as those associated with mechanical damage and corrosion. Similar approaches can be used to improve the confidence in the pdfs that describe the uncertainty in other parameters such as defect dimensions but this is beyond the scope of this paper. CONCLUSIONS Prior probability density functions (pdfs) that describe the uncertainty in yield strength and wall thickness can be constructed using data available from pipe mill certificates. In general no more than about 20 certificates are required.

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Conservative prior pdfs can be constructed generically using widely accepted assumptions for mean and minimum values. The validity of these prior generic distributions can be confirmed by suitable hypothesis tests using information from about six mill certificates. The application of Bayes’ theorem to determine the updated posterior distributions given the survival of the hydrostatic test can be used to significantly improve the pdfs of yield strength and wall thickness. The information provided by the survival of the hydrostatic test provides a major contribution to the confidence in yield strength and wall thickness and is significantly more useful than a large number of pipe mill certificates. REFERENCES (1) Francis, A., Espiner, R.J., Edwards, A.M., Cosham, A. & Lamb,M., Uprating an In-service Pipeline Using Reliability-based Limit State Methods, 2nd International Conference on Risk Based & Limit State Design & Operation of Pipelines, Aberdeen, UK, May 1997 (2) Francis, A., Espiner, R.J., Edwards, A.M., & Senior, G., The Use of Reliabilitybased Limit State Methods in Uprating High Pressure Pipelines, International Pipeline Conference, Calgary, Canada, June 1998 (3) Senior, G. & Jones, W.P., Justifying the Uprating of a Transmission Pipeline to a Stress Level of Over 72% SMYS – An Operator’s Experience, 18th International Conference on Offshore Mechanics & Arctic Engineering, St. Johns, Newfoundland, Canada, July 1999 (4) McKinnon, C., Bell, G., Patel, J. & Hawkins, P., A Case Study of the Britannia Export Pipeline Reliability Based Limit State Design, Conference on Risk & Reliability and Limit States in Pipeline Design & Operations, Aberdeen, UK, May 1996 (5) Francis, A., Espiner, R.J. & Edwards, A.M., Guidelines for the Use of Structural Reliability and Risk-based Techniques to Justify the Operation of Onshore Pipelines at Design Factors Greater Than 0.72, World Gas Conference, Nice, France, June 2000

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Figure 1 Probability Density Functions for Wall Thickness 2 1.8

Actual Population Population inferred from 5 Samples Population inferred after Hydrotest

Probability Density (/mm)

1.6 1.4 1.2 1 0.8 0.6 0.4 0.2 0 11.5

12

12.5

13

13.5

14

14.5

Wall Thickness (mm)

Figure 2 Probability Density Functions for Yield Strength 0.05 0.045

Actual Population Population inferred from 5 Samples Population inferred after Hydrotest

Probability Density (/MPa)

0.04 0.035 0.03 0.025 0.02 0.015 0.01 0.005 0 380

400

420

440

460

Yield Strength (MPa)

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Figure 3 Probability Density Functions for Wall Thickness 2 1.8

Actual Population Population inferred from 20 Samples Population inferred after Hydrotest

Probability Density (/mm)

1.6 1.4 1.2 1 0.8 0.6 0.4 0.2 0 11.5

12

12.5

13

13.5

14

14.5

Wall Thickness (mm)

Figure 4 Probability Density Functions for Yield Strength 0.05 0.045

Actual Population Population inferred from 20 Samples Population inferred after Hydrotest

Probability Density (/MPa)

0.04 0.035 0.03 0.025 0.02 0.015 0.01 0.005 0 380

400

420

440

460

Yield Strength (MPa)

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Figure 5 Probability Density Functions for Wall Thickness 2 1.8

Actual Population Population inferred from 100 Samples Population inferred after Hydrotest

Probability Density (/mm)

1.6 1.4 1.2 1 0.8 0.6 0.4 0.2 0 11.5

12

12.5

13

13.5

14

14.5

Wall Thickness (mm)

Figure 6 Probability Density Functions for Yield Strength 0.05 0.045

Actual Population Population inferred from 100 Samples Population inferred after Hydrotest

Probability Density (/MPa)

0.04 0.035 0.03 0.025 0.02 0.015 0.01 0.005 0 380

400

420

440

460

Yield Strength (MPa)

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Churchill

Published on Mar 25, 2011

A. Francis*, R.J. Espiner*, A.M. Edwards* & R.J. Hay* E NGINEERING S TRUCTURAL I NTEGRITY A SSESSMENT © EMAS 2000 1