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Wall Thickness Distributions for Steels in Corrosive Environments and Determination
of Suitable Statistical Analysis Methods
Mark STONE, Sonomatic, Aberdeen, United Kingdom
Abstract: Corrosion mapping, in which wall thicknesses over large areas are measured by 0 degree compression probe ultrasonics at closely spaced points, is being used increasingly in oil and gas applications. The data gathered typically includes very large numbers of individual thickness readings and, as such, provides a sound basis for identifying wall thickness distributions. This paper presents cumulative thickness distributions obtained in a wide range of field applications in which corrosion of carbon steel is present including data from pressure equipment with CO2 corrosion, O2 corrosion, under deposit corrosion, naphthenic acid corrosion and corrosion under insulation. The results show that there are many situations in which the wall thickness distributions display strongly ordered behaviour. In many cases it is observed that the wall loss can be represented by an exponential distribution. Examples of wall loss distributions other than exponential are also provided. It is shown that the distributions established can be a useful basis for estimates for the uninspected areas when less than 100% coverage has been achieved. A summary covering applications of such analyses to integrity management practice is provided. This highlights the benefits of the use of underlying distributions where available. Consideration has also been given to the implications of the nature of typical underlying distributions for extreme value analysis (EVA) of data sets where only the minima are reported. Such analyses can, subject to proviso’s regarding data accuracy and random spatial distribution of damage, be expected to provide reasonably representative results when there is an exponential distribution of wall loss. The results can, however, be expected to be very conservative when there is a sharp drop off to the tail. It is shown that the results of extreme value analyses applied to situations where the wall loss distribution is other than exponential are likely to be less representative. When the wall loss distribution drops more rapidly than exponential, e.g. normal, log-normal, Weibull, EVA estimates using Type 1 (Gumbel) distributions are, in principle, conservative. Without knowledge of the underlying distribution it is, however, not possible to assess the degree of potential conservatism. This can be a restriction to the applicability of EVA, particularly for reliability type analyses. When the tail of the distribution does not drop as rapidly as exponential it is possible that the results from the EVA applied become unconservative. While the same would apply to analyses using the underlying thickness distribution, knowledge of the tail behaviour allows one to consider the implications in making decisions.
Introduction There is a growing interest in application of statistical analysis of inspection data as a tool to assist with integrity management of business and safety critical equipment. This interest is driven, in part, by increasing use of Non-Intrusive Inspection (NII) as an alternative/supplement to Internal Visual Inspection (IVI) of pressure vessels. Statistical analysis forms an essential element to demonstrating that NII carried out meets the assurance requirements in many situations and its use as such is incorporated in a widely used industry recommended practice for NII [1]. Statistical analysis is also finding increasing application in integrity management of pipework systems. In this application,
4th European-American Workshop on Reliability of NDE - We.2.A.5
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analysis is used to derive information, from the inspection data, that can be used as a key input to ongoing decision making. This adds significant value to data collected, going far beyond the traditional approach where the focus was on the inspection simply being used as an indicator of acceptable condition (albeit with some rudimentary trending in certain applications). A major reason for interest in statistical approaches for NII and pipework is that in both applications a sampling type approach is applied, i.e. coverage is typically less than 100% (and significantly less so for pipework in many cases). In these situations one has to make best possible use of the data collected (over a limited area) in estimating the condition of the uninspected area. Methods for estimation outside of a region of inspection have been established for some time. In practice, these methods mostly involve extreme value analysis (EVA). Gumbel’s text [2], while not mentioning corrosion specifically, paved the way for a range of engineering applications of EVA and over time methods were developed for assessment of corrosion ([3] provides a useful summary of some of the earlier work). Application of EVA is now quite widespread and it has proven to be a useful tool in practice in many cases with its use accepted by equipment operators and regulators. Most applications of EVA for corrosion involve measurement of extremes (minimum thickness or maximum wall loss) within grids of defined area. This is simpler, when dealing with conventional inspection tools, than determination of the complete thickness distribution. The Type 1 (Gumbel) extreme value distribution that is most commonly applied also has the advantage of providing, in principle, a conservative asymptotic estimate for the extremes associated with a wide range of underlying distributions. These include normal, lognormal and exponential distributions of wall loss and a general condition for application is that the tail of the underlying distribution should decrease at least as rapidly as an exponential distribution. As mentioned, the extreme value distribution is asymptotic and provides a bounding estimate. In the case of a true exponential distribution of wall loss the extreme value distribution will in principle, i.e. barring a range of real world uncertainties, allow “exact” prediction of wall thicknesses in the un-inspected area. For underlying distributions in which the tail decreases more rapidly than an exponential distribution the Type 1 extreme value distribution derived will be conservatively bounding. The nature of this bound and the extent of potential conservatism is not always considered in detail since assessment is not possible without knowledge of the underlying distribution. In cases where the tail does not decrease more rapidly than an exponential distribution the estimates derived may be unconservative. It is not generally possible to establish if this is the case, however, without some knowledge of the underlying distribution. The above considerations indicate that knowledge of the underlying distribution can assist in making for more robust extreme value analysis. The underlying distribution is obviously also of value in itself in providing a basis for direct estimation of the conditions in uninspected areas1. Furthermore, the underlying distributions can give some insight into the nature of corrosion, e.g. is it general or pitting corrosion. There is however very little
1 The value of the underlying thickness distribution is acknowledged in a number of publications including [4] and [5].
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published data covering typical underlying thickness distributions for steels in corrosive environments. A summary of published thickness distributions is covered in [6] and the data considered therein is derived from one or more of the following.
• Point thickness measurements at discrete locations (small overall coverage) • Measurements of maximum pit depths • Measurement at a single location on corrosion coupon samples
Hence, while some of the results provide an insight to the characteristics of corrosion and may be suitable in certain types of reliability analysis they do not yield direct information on the underlying thickness distributions. The most comprehensive available information found is that in [5]. This does include underlying distributions, in the form of probability density functions, but these relate to laboratory samples with very early stage corrosion (depths of less than 200 microns). This paper provides thickness distributions derived from a range of field collected data. The objective is to provide some insight into the nature of typical wall thickness distributions for carbon steels in service in corrosive environments (representative of a range of oil and gas applications) and to highlight the implications for statistical analysis.
Wall thickness distributions Ultrasonic corrosion mapping is widely used for inspection of vessels and pipework where local and general corrosion may be active. A range of ultrasonic corrosion mapping systems are available but the key feature is rapid collection of 0 degree compression probe thickness measurements with the location of each measurement point being recorded. The data is typically presented in the form of colourgraphic corrosion maps in which the pixel colour at each location corresponds to a wall thickness. The colour scale is graduated to allow rapid visual identification of areas of wall loss. An example of a corrosion map over an area of pitting is shown in Figure 1.
Depth(mm)0.00 1.00 2.00 3.00 4.00 5.00 6.00 7.00 8.00 9.00 10.0 11.0 12.0 13.0 14.0 15.0 16.0 17.0 18.0 19.0 20.0 21.0
Figure 1: Colourgraphic corrosion map over subsea line with localised degradation
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As an inspection company delivering advanced ultrasonic techniques, Sonomatic Ltd collects large volumes of corrosion mapping data from equipment in service. In-house methods for distribution analysis have been developed and, for the past several years, such analysis has been routinely carried out on data obtained in the field. A significant database of corrosion distributions has been built up and some typical examples are provided in this paper. Distributions can be visualised in a number of ways but the preferred method is to plot a cumulative distribution of wall thickness based on the individual thickness measurements. This has a number of advantages over plotting the probability density function (or thickness histogram), i.e.
• Local variations are naturally smoothed, making it easier to visualise overall characteristics.
• The cumulative distribution allows straightforward extrapolation using the area associated with each measurement point as a basis.
The cumulative distributions (P) are derived in a straightforward way using the Weibull definition of plot position [2], i.e.
( )1i
iP xN
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where ix is the ith largest thickness measurement and the data set consists of N measurements. The distribution so derived relates to measurement points and, subject to a number of conditions affecting accuracy, tends towards the true underlying distribution as the spacing between points is reduced. A range of factors play a role in determining the extent to which the measured distribution is an accurate representation of the true underlying distribution. The main factors are the ultrasonic beam spread, the sensitivity settings for measurement and the variation of thickness within the area (as determined by scan increment) covered by the measurement. In practice corrosion mapping scan increments should be set according to the nature of degradation, i.e. situations with small isolated pits demand a smaller scan increment. Hence in cases where the inspection is set up appropriately, the distributions based on the corrosion mapping data can be expected to be reasonably representative but there may be situations, e.g. large scan increment, rapidly changing wall thickness, where the distribution determined by corrosion mapping may deviate significantly from the true underlying distribution. Further work is needed to quantify the effects of inspection parameters and corroded surface morphology on the accuracy of distributions obtained. The area associated with each measurement point is determined by the inspection scan increment hence the distribution is related to area inspected, i.e. ( )iP x represents a proportion of total area inspected. Its important to note that we effectively assume the thickness variation within the area covered by each measurement is negligible. This means the distribution obtained is not infinitely scalable with reducing area.
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The corrosion mapping results for the lower part of a separator vessel are used to illustrate a number of key features of thickness distributions. The corrosion map presented in Figure 2 shows the thickness variations over the water wetted region of a three phase separator vessel in service offshore. The region shown in Figure 2 has suffered CO2 corrosion in service and there is wall loss down to approximately 9 mm (from an average thickness of undamaged material of approximately 17 mm). The corrosion map shows pitting corrosion distributed throughout the region, with some areas affected worse than others.
Figure 2: Corrosion map showing degradation at the bottom of a separator vessel
The distribution of wall thickness, as determined from the corrosion mapping results in Figure 2 is presented in Figure 3. This shows two distinct distributions are present with a clear change in behaviour at approximately 16 mm. If one isolates the material with a thickness of greater than 16 mm this is found to follow a near normal distribution, i.e. associated with natural, as-manufactured, variations in the thickness of the plate material. The departure from this evident at 16 mm and the change to a shallower slope indicates that wall loss from the as manufactured condition is present. This affects approximately 10% of the area (it begins at Proportion of area = 10-1).
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Figure 3: Wall thickness distribution for region of separator with corrosion The distribution for thicknesses less than 16 mm is observed to be very close to linear on the logarithmic scale used. This implies that the wall loss follows an exponential distribution and suggests that the underlying process in this case is, despite the apparently irregular distribution of pitting in Figure 2, strongly ordered at a fundamental level. The regular (exponential) nature of the tail indicates that behaviour can be described by parameters that would be expected to allow a useful statistical description The regular nature of the distribution shown in Figure 3 might be might be regarded as surprising at first sight and, indeed, questions might be raised as to whether this is simply a unique case. Review of significant numbers of corrosion data sets collected by Sonomatic Ltd reveals, however, that this is not something unique and the orderly nature of such distributions is present in most cases where the wall thickness data is taken from zones of similar corrosion conditions. To illustrate this further, distributions for the lower parts of a number of different separator vessels are presented in the figures that follow. The results shown in Figure 4 indicate that approximately 10% of area is affected by localised corrosion with a near exponential tail for wall loss. The results presented in Figures 5 and 6 also show near exponential wall loss distributions but are for cases of very isolated pitting, with approximately 0.4% (Figure 5) and 0.15% (Figure 6) of area affected by pitting.
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Figure 4: Wall thickness distribution for separator with CO2 corrosion
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Figure 5: Wall thickness distribution for separator with very isolated CO2 corrosion
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Figure 6: Wall thickness distribution for separator with very isolated CO2 corrosion
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The corrosion map shown in Figure 7 is from a vessel that is exposed internally to water and has experienced corrosion due to a high oxygen content. The corrosion mapping carried out over the vessel shell revealed areas of pitting as observed in Figure 7 which shows part of the region affected. This indicates quite extensive corrosion with localised areas of deeper pitting. The cumulative thickness plot is shown in Figure 8. The transition from normal variation to corrosion is less clear than in CO2 cases and this may be indicative of some general corrosion. The results indicate that somewhere between 20% and 25% of the area is subject to clearly discernable corrosion. The distribution is fairly regular for wall thicknesses in the 8.5 mm to 12.5 mm range. Some change in the characteristics of the corrosion is observed for the deepest pits, i.e. wall thickness below 8.5 mm, where the proportion of area affected is higher than would be indicated by the trend in the more regular 8.5 mm to 12.5 mm range. This type of behaviour, when present, can have significant implications for the reliability of results obtained by statistical analysis.
Figure 7: Corrosion mapping regions showing characteristics of corrosion
Figure 8: Cumulative wall thickness distribution for vessel with O2 corrosion Figure 9 shows the wall thickness distribution for heater tubes in a refinery application. The conditions are such that the steel is susceptible to naphthenic acid corrosion which tends to
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be manifest as localised pitting. Corrosion mapping was carried out over a number of the tubes during a shut down. This identified that corrosion was present but that areas of corrosion were very isolated, occurring apparently at random on the tubes with adjacent areas sometimes separated by several metres. The resulting cumulative thickness distribution is as shown in Figure 9. This reveals a clear transition from the normal distribution (associated with the as manufactured condition of the seamless tube) to localised corrosion. The transition is evident at a thickness of approximately 5.5 mm and approximately 0.5% of the area is affected by localised corrosion. The tail behaviour for the corrosion is regular and suggests a near exponential distribution of wall loss. This is another example of strong underlying structure being evident in the data in a situation that would, at first sight, be taken as corrosion occurring by a process that would not be amenable to parametric description.
Figure 9: Cumulative thickness distribution for tubes with naphthenic acid corrosion The corrosion maps shown in Figure 10 are from sections of an oil export line which is subject to internal corrosion. The maps shown are for sections typical of the condition of this line and indicate that there is general corrosion with areas of more localised pitting (the pitting is believed to be associated with formation of deposits). The thickness distribution for all the mapping data for straight sections of the oil line is presented in Figure 11. This shows that there is no clear transition to the corrosion tail, i.e. a large proportion of the surface area is affected by general corrosion. The tail indicates regular behaviour with a near exponential distribution of wall loss until a region of more rapid drop off below 28 mm. This type of drop off is encountered fairly frequently in practice. Another example showing a steep drop off to the tail is shown in Figure 12 (this is also for data collected on an oil line). Note that such a drop off would suggest that, in cases where it is present, extrapolation of a fitted exponential distribution for wall loss would provide conservative estimates for minimum thicknesses in regions not inspected.
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Figure 10: Corrosion over typical areas of corrosion in oil export line
Figure 11: Thickness distribution for oil export line with corrosion
The distribution in Figure 12 shows a gradual transition to the tail indicating some general corrosion in addition to pitting. Further oil line data is shown in Figure 13 which indicates a much sharper transition to pitting. This data also shows a regular, near exponential, tail to the distribution with evidence of strong drop off towards the end of the tail.
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Figure 12: Distribution for oil line showing drop off at end of exponential tail
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Figure 13: Distribution for oil line with pitting corrosion
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The data shown in the preceding examples is for cases of internal corrosion. It is interesting to note that external corrosion (due to exposure to water) has also been observed to display regular statistical behaviour. A corrosion map from an offshore vessel suffering corrosion under insulation (CUI) is shown in Figure 14. Unusually, this data was captured from the inside of the vessel (during a shut down) since this was determined as the option that would give the most accurate information on the wall thicknesses. The corresponding wall thickness distribution is shown in Figure 15 and, again, regular tail behaviour is observed with a near exponential distribution of wall loss evident up until a drop off below 6 mm.
Figure 14: Corrosion map over section of vessel suffering CUI
Figure 15: Wall thickness distribution for vessel with CUI Figure 16 shows the wall thickness distribution for a plate section on a buried vessel suffering external corrosion (the inspection here was also carried out from the vessel interior).
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DataNormal fit to tail
Figure 16: Wall thickness distribution for buried vessel with external corrosion
The data in Figure 16 shows a sharp transition to the corrosion tail, with approximately 4% of area affected by pitting. Unlike most of the preceding examples, however, the tail here is clearly not representative of an exponential distribution of wall loss. Behaviour is nevertheless visibly regular and in this case a normal distribution of wall loss offers a good approximation to the tail of the distribution as indicated by the green line in Figure 16. The normal distribution would be expected to provide a reasonable basis for extrapolation in this case. Our experience is that the corrosion tail displays the characteristics of an exponential distribution in most cases (although there is frequently a drop off from exponential at the lower end of the tail as indicated in several of the examples2). Nevertheless, wall loss distributions other than exponential are not uncommon. Another example where the tail is clearly not exponential is shown in Figure 17 (this is data from an area of early stage degradation on a condenser vessel in a refinery application). The figure indicates that the distribution obtained is represented reasonably well by either a normal or Weibull distribution of wall loss, although the latter clearly offers a better fit to the lower end of the tail.
2 Further work, from a corrosion and inspection perspective, is needed to properly understand the reasons for the tail drop off’s observed.
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Figure 17: Wall thickness distribution for area of corrosion in a condenser vessel
Application of the underlying distribution to estimates for uninspected areas A key requirement of sampling inspection, as applied to both vessel NII and pipework, is to allow reasonable estimates of the condition in the uninspected area. The ability to make such estimates depends mainly on two requirements being met:
(i) The distribution obtained properly represents the true underlying distribution for the complete area under consideration.
(ii) The nature of the tail of the distribution is such that it can be accurately described
parametrically. The extent to which both of these requirements are satisfied should be tested in some way. The applicable tests are straightforward for the second requirement since they revolve around established statistical methods for distribution fitting and assessing the applicability of distributions selected. Methods for assessing if the first condition is adequately met are less clear and further work is needed to establish accepted approaches. Simple empirical tests aimed at determining the extent to which the characteristics of the distribution are affected by removing sub-sets of data (typically associated with one or more corrosion mapping scan areas) are useful but work is needed to develop standardised methods and to quantify acceptance criteria. At this stage a number of general guidelines can be provided, i.e.
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• Caution should be exercised when it is observed that removing a relatively small quantity of data in any area significantly affects the nature of the distribution.
• Caution should be exercised when the distribution is not amenable to parametric description by one of the statistical distributions typically associated with natural phenomena, e.g. exponential, normal, lognormal, Weibull etc. Care should be taken to ensure that any distribution applied in such circumstances is at least conservative, e.g. fitting an exponential distribution in cases where there is strong tail drop off is considered acceptable. Evidence of the lower end of the tail dropping off slower than the bulk of the tail should be of concern however (see Figure 8 for an example).
• In general the larger the area covered by the data set, the more representative the distribution is likely to be. Caution should be exercised in extrapolating the results from corrosion mapping over a small area.
In cases where there is reasonable confidence that the two primary requirements are met, application of the distribution for estimation to uninspected areas is relatively straightforward. The thickness value determined from the distribution curve at a cumulative probability (proportion of area) equal to 1/(N+1) can be taken as directly indicating the expected minimum thickness for an area corresponding to N measurement points. Using Figure 16 as an example, the expected minimum thicknesses for an area corresponding to 100, 10 000 and 1 000 000 points would be approximately 21.5 mm, 19.3 mm and 18 mm respectively. It should be noted that the expected minimum in this case refers to the mean, e.g. if one takes a large number of different zones, whose area in each instance corresponds to 100 measurement points, the average of the minimums for these areas would be 21.5 mm. In a certain proportion of cases the minimum would be larger than the expected value and in the remainder of cases it would be less than the expected value. There is a probability distribution associated with the minimum thickness for a given area and this can be derived from the underlying distribution using extreme value theory. The cumulative probability that the minimum thickness, for an arbitrarily selected region corresponding to n measurement points, exceeds t is given by ( , ) ( , , )G t n F xβ α β= (2) where Fβ is the value of the (cumulative) beta distribution function evaluated at
( ) x P t= with 1 and =nα β= . Note that for t less than the minimum measured in the area inspected, ( )P t has to be estimated from the parametric description of the distribution tail. For the purpose of illustration, Figure 18 shows the probability distributions for minimum thickness for the situation covered in Figure 16 (the underlying distribution is shown for reference). The red line indicates the probability distribution for the minimum thickness associated with an area 10 times larger than the inspected area and the light blue line indicates the probability distribution for the minimum thickness associated with an area 10 times smaller than the inspected area. In a case where the uninspected area was 10 times the inspected area, the probability that the minimum thickness for the uninspected area is less than 18 mm would be estimated as 0.6 (60%). In the case that the uninspected area was one tenth of the inspected area the probability that the minimum thickness for the uninspected area is less than 18 mm would be estimated as 0.01 (1%). It should be noted that the
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distributions shown do not attempt to include confidence bounds. Further work is required to establish a consistent definition of confidence bounds for this type of analysis.
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Figure 18: Probability distributions for min thickness associated with different areas
The information in the distributions obtained from corrosion mapping can be used in the above manner to derive estimates that are directly applicable to integrity management decision making. Specifically, the results can be used as follows.
• Determination of acceptability in the uninspected area, estimation of margins remaining on the design allowances.
• Assessment of whether coverage is appropriate (lesser coverage may be justified where there is low level degradation while higher levels of coverage may be needed where there is extensive and or severe degradation).
• In cases of potential severe degradation the results can be used as input to a fitness for service assessment that uses a statistical definition of defect geometries.
• The results from successive inspections can be used to determine very accurate corrosion rate profiles. The estimates so obtained are significantly more reliable than those relying on inspection location based comparison or on theoretical prediction of a single corrosion rate value for a given situation. Accurate corrosion rate distributions facilitate more appropriate and reliable planning decisions.
• Input to risk ranking for Risk Based Inspection planning.
Implications for Extreme Value Analysis The discussion herein is not intended as a comprehensive assessment of application of EVA to situations with typical in-service corrosion. The objective is simply to highlight a number
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of points relevant to practical application of EVA and to this end the discussion is restricted to application of the Type 1 (Gumbel) extreme value distribution. EVA is typically preferred in practice where there is no or limited information on the underlying distribution available from inspection data. This situation is often encountered in routine inspection of pipework where the data is collected by manual wall thickness scanning with the minimums being recorded. The minimums allow a definition of the distribution of extremes within the range of the data collected. The Type 1 distribution is considered to provide a conservatively bounding estimate to the true extreme value distribution in cases where the underlying distribution (of wall loss) drops off at least as rapidly as an exponential distribution. In simple terms, for reference to the data presented in the preceding sections, this means a continuously decreasing gradient of the logarithm of the cumulative distribution. This condition is obviously violated if the analysis includes data above the tail transition region. Consequently validity of the EVA is dependent on the minima used being restricted to the distribution tail3. Unfortunately it is difficult to assess the extent to which this condition is met in practice when only the minimums are reported from the inspection grids. As manufactured thicknesses, for pipework in particular, can vary significantly hence it is not always obvious (to the user of the data at least) that the measurement for a particular grid is actually associated with corrosion and can therefore be included in the extreme value data set. The use of wall loss (rather than thickness) values can assist in this respect but determination of wall loss is not always straightforward. The nature of the distribution of measured minimum values can, in certain circumstances, be used to indicate the transition but this is not generally the case. It is therefore recommended that values close to the average thickness for uncorroded material should be investigated before inclusion in the extreme value data set. This is particularly important when dealing with early stage degradation. It is possible that, with early stage degradation, an EVA based on measured minimum thicknesses may not yield reliable or conservatively bounding results. In these circumstances alternatives such as compliance testing [8] may be more appropriate. It should also be noted that if significant areas do not include data from the corrosion tail, this would indicate that the degradation is not randomly distributed (by location) this has an impact on applicability of the EVA. In cases where the underlying distribution of wall loss is truly exponential, a Type 1 extreme value distribution, based on measured tail data, will in principle correspond to the true extreme value distribution. Estimates provided by EVA using Type 1 distributions can therefore be expected to be reasonably accurate in such cases4. Given that such exponential (or approximately exponential) distributions are frequently encountered in practice, this would suggest EVA using Type 1 distributions is often appropriate. A rapid drop off to distributions that have generally exponential behaviour is, however, often observed at the lower end of the tail. The EVA estimates (Type 1) in the presence of such a tail have reduced accuracy but remain conservative. In these circumstances other extreme value distributions, e.g. Weibull or Generalised Extreme Value (GEV) may be more appropriate. 3 It is noted that this condition is catered for in Peaks over Threshold methods, see for example [7], provided the threshold is defined appropriately. 4 There are a number of factors that affect accuracy of the estimates in practice, e.g. measurement error, limitations of the distribution fitting approach (it is never possible to determine exactly the cumulative probability value associated with each extreme value in the sequence) and spatial distribution of damage not being fully random. This paper does not aim to consider these important factors in detail.
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Selection of the most appropriate extreme value distribution is beyond the scope of this paper, however, it should be recognised that it is not straightforward when dealing with limited sets of minima. When the tail drops very rapidly but the extreme value data set includes measurements with thicknesses across much of the range of the tail, the estimates may be very conservative. This problem is not necessarily unique to EVA but is also encountered in analysis using the underlying distribution where conservative estimates (based on an exponential fit to the bulk of the tail data) have to be made if there is insufficient confidence to attempt a parametric description of the tail drop off. A key difference remains, however, in that there is significantly more insight into behaviour when dealing with the underlying distribution. With EVA it is, for example, possible that the analyst may not even be aware of the fact that there is a clear drop off to the tail and has no way of assessing if the estimates are likely to be conservative and the extent to which they may be conservative. When the tail of the underlying distribution is not exponential but still decreases more rapidly than exponential, the results of the EVA using a Type 1 distribution yield, in principle, conservatively bounding estimates5. The extent to which the estimates are conservative is determined by the range of thicknesses included in the extreme value data set which is, in turn, determined by defect density. The slope of the underlying distribution, as established for the mean of the minimum thicknesses included in the data set, essentially determines the slope of the tail of the extreme value distribution (which behaves much like an exponential distribution). In cases where the defect density is low the mean of the minimum thicknesses will tend to be higher, making for a flatter tail to the extreme value distribution and, consequently, more conservative estimates for larger areas. When the defect density is higher, the mean of the minimum thicknesses will tend to be lower and the slope for the tail of the extreme value distribution will be steeper6. This results in a lesser level of conservatism. The above points are illustrated by using the data from Figure 16 to estimate extreme value distributions for different grid sizes. Two different grid sizes were selected, i.e. one consisting of 3000 measurements per grid and the other with 500 measurements per grid. With random assignment of individual thicknesses to grids, the minimums for the larger grid size are typically smaller and hence have a lower average value. For the data set considered here the average of the minimums of the larger grids is 19.5 mm while for the smaller grids it is 20.2 mm. The slope of the tail for the extreme value fit distribution determined from the larger grids matches that of the underlying distribution for a thickness of 19.9 mm. This is illustrated in Figure 19 in which the red line shows the slope of the tail for the distribution of minima for the larger grids (EV1). Figure 19 also shows the slope (green line) for the distribution fitted to the minima for the smaller grids (EV2). It is evident that there is a reasonably significant difference in slope and this can be expected to affect estimates, particularly when larger areas are considered.
5 In practice a number of factors, including those mentioned in the preceding footnote, may mean that the estimates become unconservative. 6 It should be noted that references to the slope of the tail distribution here consider the gradient of the logarithm of the cumulative distribution, i.e. an exponential tail has constant slope by this definition.
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Figure 19: Slopes of tails of extreme value distributions The extent to which the estimates may be affected is illustrated in Figure 20 which shows the estimated probabilities for minimum thickness over an area equal to the inspected area from which the thickness data was obtained. The minimum thickness estimate for 50% probability is not significantly different for the two cases, i.e. 18.3 mm for EV1 and 17.7 mm for EV2. The differences become larger at lower probability levels, however, and the differences in slopes cause increasing divergence. Such differences can become very significant in reliability type analyses where consideration is usually given to the probability that the minimum thickness is less than a certain value. The blue line in Figure 20 shows the probabilities estimated using the underlying distribution which is normal in this case and considered to provide a good description of the tail behaviour. It is apparent that the estimates from both EV1 and EV2 become very conservative for lower wall thicknesses. In the above example, measurements were assigned to grids on a completely random basis, i.e. the distributions per grid would then be consistent with the conditions for applicability of statistical analysis. In practice there is not always completely random spatial distribution of corrosion and this can affect the applicability of EVA. An EVA has also been applied to the data used in the preceding example but with the minimums from grids defined by physical location, i.e. aligned with practice where the minimums would be reported from grids of a given size. The data set included 42 grids each consisting of approximately 2700
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measurement points.
14 15 16 17 18 19 2010
-6
10-5
10-4
10-3
10-2
10-1
100
Minimum thickness (mm)
Pro
babi
lity
EV1EV2UnderlyingEV3
Figure 20: Estimated probabilities for minimum thickness
The resulting distribution (EV3) is also shown in Figure 20. It is evident that in the departure from random spatial distribution of the corrosion in this instance has a significant influence on the probabilities estimated. This emphasises the need to check the extent to which the spatial distribution departs from random. Unfortunately such checks are not well established for situations where the data sets consist only of the minimums. The determination of correlation distance can be used to check for randomness [9] but the method has limitations for application to typical data sets of minima given their relatively small size. Hence, in order to ensure robust analysis when only the minima are reported, further work is needed to develop measures to identify when the spatial distribution of the real corrosion may strongly affect the results.
Summary and Conclusion This paper provides cumulative wall thickness distributions, as measured by corrosion mapping, for a range of in-service applications. The results show that corrosion often displays orderly statistical behaviour, thereby allowing for parametric description that facilitates estimates for uninspected areas. It is apparent that typical in-service corrosion mechanisms often give rise to an exponential distribution of wall loss. The ends of the tails to these distributions are sometimes observed to fall away rapidly and in these circumstances estimates made using the exponential component of the distribution may be very conservative. Further work is needed to understand the drop off frequently observed at the end of the tail so that parametric descriptions including this drop off can be applied with confidence.
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Wall loss distributions other than exponential are also encountered in practice. In cases where a parametric description is possible, these distributions can also be used for estimates for uninspected areas. The paper discusses methods for application of the underlying distributions to estimation for uninspected areas and provides a summary of such how such analyses can be applied to integrity management practice. This highlights the benefits of the use of underlying distributions where available. Consideration has also been given to the implications of the nature of typical underlying distributions for extreme value analyses of data sets where only the minima are reported. Such analyses can, subject to proviso’s regarding data accuracy and random spatial distribution of damage, be expected to provide reasonably representative results when there is an exponential distribution of wall loss. The results, using a Type 1 extreme value distribution, can be expected to be conservative when there is a sharp drop off to the tail however. It is shown that the results of extreme value analyses applied to situations where the wall loss distribution is other than exponential are likely to be less representative. When the wall loss distribution drops more rapidly than exponential, e.g. normal, log-normal, Weibull, EVA estimates using Type 1 distributions are, in principle, conservative. Without knowledge of the underlying distribution it is, however, not possible to assess the degree of potential conservatism. This can be a restriction to the applicability of EVA, particularly for reliability type analyses. It is also shown, by use of an example, that the spatial distribution of wall loss can affect the results of an EVA. It is recommended that means by which sensitivity to spatial distribution can be assessed should be developed for cases where only the minima are reported. When the tail of the distribution does not drop as rapidly as exponential it is possible that the results from the EVA applied become unconservative. The same would apply to analyses using the underlying thickness distribution but knowledge of the behaviour of the tail distribution allows one to consider the implications in making decisions.
Acknowledgements This work has been made possible through the collection of corrosion mapping data by numerous colleagues in Sonomatic. The quality of the data collected, often under challenging circumstances, is a testament to their efforts and professionalism. The author also acknowledges the input of colleagues in the Integrity Support team at Sonomatic. They have provided invaluable assistance through identification of data sets, analysis and discussions regarding application of statistical methods to field collected data.
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