A statistical approach to determine Microbiologically Influenced Corrosion (MIC) Rates of underground gas pipelines.

Transcription

1 A statstcal approach to determne Mcrobologcally Influenced Corroson (MIC) Rates of underground gas ppelnes. by Lech A. Grzelak A thess submtted to the Delft Unversty of Technology n conformty wth the requrements for the degree of Master of Scence. Delft, The Netherlands Copyrght 006 by Lech A. Grzelak All rghts reserved

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3 Abstract A statstcal approach to determne Mcrobologcally Influenced Corroson (MIC) Rates of underground gas ppelnes. by Lech A. Grzelak Charperson of Graduate Commttee: Prof. dr Roger M. Cooke Department of Mathematcs Graduate Commttee: Prof. dr Roger M. Cooke Prof. Jolanta Msewcz Ir Paul M. Wesselus Dr Dorota Kurowcka Ir Danel Lewandowsk N.V. Nederlandse Gasune s the leadng Dutch gas transportaton company. Its man am to manage, mantan the gas transport system. The grd of gas ppelnes belongng to Gasune conssts of over km steel ppelnes (dameters from 4 to 48 nch) constructed n the 60s. Integrty management s based on the ablty of the ppelne operator to predct the growth of defects detected n nspecton programs. The predctons of the corroson and defect rates can be based on envronmental nput parameters. Accurate predctons allow nterventons/re-nspectons to be scheduled n order to elmnate defects whch pose a hgh potental rsk. Ths thess nvestgates three man ssues. Frstly, t shows an approprate tool for the corroson rate modelng when data from n-lne nspectons are avalable. A low number of nspectons contrbute to hgh uncertanty about the corroson rate estmaton. In many cases, a poor dataset combned wth hgh uncertanty about the measurements cause corroson estmates that are not agreeable wth realty; for example corroson s decreasng n tme. The outputs from the corroson rate model are ncorporated as nput to the second secton, where analyss s focused on nvestgatng parameters nfluencng Mcrobologcally Induced Corroson (MIC) rates. The last part of the thess presents the desgn and results of the defect rate. Keywords: Corroson, Corroson rate modelng, In-lne nspecton, MIC, factors nfluencng MIC, defect rate M.Sc. Thess Lech A. Grzelak

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5 v Acknowledgments The author wshes to express a grattude to Roger M. Cooke, Jolanta Msewcz and Dorota Kurowcka for the opportunty to spend unforgettable two years n the Rsk and Envronmental Modelng group. The author would also lke to acknowledge Thomas Mazzuch and Danel Lewandowsk for coordnaton whle dong the thess project; moreover; the author would lke to thank to the people at Gasune for a gven opportunty to work wthn ther organzaton, especally to Paul M. Wesselus, Karne Kutrowsk and Robert Kuk. Many people have nspred, guded and helped the author durng the two years at TUDelft, and the wrter would lke to thank them all for a great fun and experence. The author owes a huge debt of thankfulness to hs best frends: Poorwa, Msha and Gosa, for ther support, humor throughout not only the courses of M.Sc. program but all the way through entre tme. Last but not the least; the author would lke to express hs apprecaton to hs famly for the support and love they provded through author s entre lfe. M.Sc. Thess Lech A. Grzelak

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7 v Lst of contents Abstract... Acknowledgments...v Lst of contents...v Lst of tables...x Lst of fgures...x Introducton...xv The goals of the research...xv Outlne of the thess...xv Part I Corroson rate modelng... 1 Descrpton of the corroson data Introducton Magnetc Flux Leakage (MFL)-pg Matchng of the reported defects Number of matched defects Avalable data Reported defects Excavaton data Measurement data from matched defects...8 Descrpton of the corroson rate model Introducton Data calbraton Data calbraton procedures...11 Algorthm 1 (Calbraton data algorthm)...11 Algorthm (Measurement error dstrbuton algorthm) Data calbraton results General model for data calbraton General calbraton procedure: General corroson rate model Example Dealng wth mssng data Conclusons Implementaton (CoroGas 1.0v) Numercal results Introducton Approach M.Sc. Thess Lech A. Grzelak

8 v A statstcal approach to determne the MIC rate of underground gas ppelnes 3.3 Approach Approach Depth nfluence on the corroson rate Conclusons and recommendatons...4 Part II Parameters nfluencng mcrobologcally nduced corroson rate Potental analyss Introducton Measurements and smoothng method On-potental Grd constructon Correlaton between the corroson rate and average potentals recorded between frst and last nspecton Approach Approach Correlaton between the corroson rate and potentals standard devaton recorded between frst and last nspecton Conclusons Mcrobal Data analyss Introducton Mcrobologcally Influenced Corroson Dataset descrpton Independent varables Dependent varable Correlaton analyss Multple regresson analyss Analyss structure Mssng data Model wthout nteractons Model wth nteractons Stepwse regresson for ncluded varables Model wthout nteractons Model wth nteractons Senstvty analyss of the parameters nfluencng the corroson rate Conclusons and recommendatons...50 Part III Parameters nfluencng mcroblologcally nduced defect rate Ppelne characterstcs Defect dstrbuton Depth of cover Ppelne elevaton Sol data analyss Introducton Descrpton of avalable dataset Sol data collecton Sol type nfluence on defect rate...6 Delft Unversty of Technology

9 v Defect rate- Approach Defect rate- Approach Correlaton analyss Correlaton between sol exposures Correlaton between defect rates wrt sol types Correlaton between defect rates wrt sol types where bad coatng was assumed Conclusons and recommendatons Water table analyss Introducton Approach Approach Approach Conclusons and recommendatons Factors nfluencng defect rate Introducton Model wthout nteractons Model wth nteractons Conclusons and recommendatons Conclusons...86 Bblography...88 A: Analyss methods and nterpretaton...90 B: Tutoral fle for CoroGas M.Sc. Thess Lech A. Grzelak

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11 x Lst of tables Table 1: number of matched defects per combnaton of pgruns... 6 Table : szng accuracy of the pgs and number of reported defects... 7 Table 3: number of reference ponts per pgrun... 7 Table 4: calbraton results...11 Table 5: standard devatons of the measurement errors...1 Table 6: corroson rate and ntaton tme for approach...1 Table 7: corroson rate and ntaton tme for approach Table 8: correlaton between the corroson rate and average on-potental recorded between frst and last nspecton...34 Table 9: correlaton between the corroson rate and on-potental standard devaton recorded between frst and last nspecton...36 Table 10: groups of mcroorgansms...39 Table 11: factors and weghts for MCA score...41 Table 1 Mcrobal data descrpton...4 Table 13: corroson rate summary for 16 measurements ndcated by bo-analysts...4 Table 14 correlaton between corroson rate and the varables, cell wth red background ndcate that normalty assumpton doesn't hold, yellow rows ndcate varables for whch the null hypothess was rejected for sgnfcance level Table 15: parameters and assocated statstcs...46 Table 16: regresson model standard descrpton...46 Table 17: parameters and assocated statstcs...46 Table 18: regresson model standard descrpton...46 Table 19: parameters and assocated statstcs (model wthout nteractons)...47 Table 0: regresson model standard descrpton (model wthout nteractons)...48 Table 1: parameters and assocated statstcs (model wth nteractons)...48 Table : regresson model standard descrpton (model wth nteractons)...48 Table 3 senstvty analyss of the sgnfcant parameters...49 Table 4: general ppelnes descrpton...61 Table 5 Overall defect rate for the ppelnes...63 Table 6 defect rate for each specfc sol type...64 Table 7: descrpton of the potentally defectve and not defectve envronments...65 Table 8: comparson of potentally defectve and not defectve envronments...67 Table 9: defect rate for sectons where bad coatng was appled...69 Table 30: correlaton between sol compostons of the ppelnes...70 Table 31: correlaton between defect rates of the sol types for ppelnes...70 Table 3: correlaton between defect rates of the sol types for ppelnes, n the clusters where bad coatng was appled...70 Table 33: ground water step levels (legend)...7 Table 34: groundwater step level summary per each secton of.5 km...74 Table 35 defect rate summary- per each secton of.5 km...74 Table 36 correlaton between defect rate and water level for sectons of.5 km...75 Table 37: statstcs and confdence bounds for estmate (ppelne A1 and A)...75 Table 38: descrptve statstcs (ppelnes A1 and A)...75 Table 39: statstcs and confdence bounds for estmate (ppelnes A1, A and A3)...77 Table 40: descrptve statstcs (ppelnes A1, A and A3)...77 Table 41: defects and average groundwater level...78 Table 4 Stepwse regresson estmates (model wthout nteractons)...8 Table 43: standard model statstcs (model wthout nteractons)...8 Table 44 Stepwse regresson estmates (model wth nteractons)...83 Table 45: standard model statstcs (model wth nteractons)...84 Table 46: excavaton dataset [mm] Table 47: example of the measurements Table 48: The nomnal wall thckness M.Sc. Thess Lech A. Grzelak

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13 x Lst of fgures Fgure 0-1: deteroratng gas ppelnes... xv Fgure 0-: corroson modelng schema... xv Fgure 1-1: small and large dameter MFL-pg... 5 Fgure 1-: vsualzaton of matched defects... 5 Fgure 1-3: clock sprng nstallaton (left), gas ppelne excavaton (rght)... 8 Fgure 1-4: reported vs. measured defect depth... 8 Fgure 1-5: reported defect depth... 9 Fgure -1: error dstrbuton for pg A...1 Fgure -: example of defects clusterng...13 Fgure -3: corroson rate determnaton- general model...16 Fgure -4: corroson rate modelng- dealng wth mssng data...17 Fgure -5: calbraton data & optmzer wndow of CoroGas...17 Fgure 3-1: defect depth n tme...0 Fgure 3-: corroson rate dstrbuton...1 Fgure 3-3: dstrbuton of ntaton tme...1 Fgure 3-4: corroson rate dstrbuton... Fgure 3-5: dstrbuton of ntaton... Fgure 3-6: dstrbuton of corroson rates for shallow defect...4 Fgure 3-7: dstrbuton of corroson rates for deep defects...4 Fgure 4-1 cathodc protecton for a gas ppelne (left), voltage drops n a measurng crcut (rght)...9 Fgure 4-: Cu/CuSO 4 reference electrode...9 Fgure 4-3: Test posts dstrbuton over tme...30 Fgure 4-4: on-potentals measurements and results of appled smoothng method...30 Fgure 4-5: on-potentals before smoothng...31 Fgure 4-6: on-potentals after smoothng...31 Fgure 4-7: on-potental contour plot...3 Fgure 4-8: on-potental grd...33 Fgure 4-9: 5 dstngushed defects and estmated corroson rates...33 Fgure 4-10: concordance and dscordance...35 Fgure 5-1: mcrobologcally nfluenced corroson on gas ppelnes...39 Fgure 5-: mage of a sulphate-reducng bacteral culture wth a carbonate precptate, the bactera on the left are about 6-8 µm long and µm n dameter...40 Fgure 5-3 Corroson rate summary for 16 measurements ndcated by bo-analysts...4 Fgure 5-4 radar graph- Pearson product moment correlaton coeffcent...43 Fgure 5-5 correlatons and assocated p-values (orderng the varables)...43 Fgure 5-6 regresson modelng schema for the corroson rate...45 Fgure 6-1: Ppelne A1, profle, depth of cover, defect dstrbuton...55 Fgure 6-: ppelne A, profle, depth of cover, defect dstrbuton...55 Fgure 6-3: ppelne A3, profle, depth of cover, defect dstrbuton (PII)...56 Fgure 6-4: ppelne A1 defect dstrbuton wrt ppelne crcumference...57 Fgure 6-5: ppelne A defect dstrbuton wrt ppelne crcumference...57 Fgure 6-6 ppelne A3 defect dstrbuton wrt ppelne crcumference...57 Fgure 7-1: route map, geotechncal data...59 Fgure 7-: route map, geotechncal data- STEP Fgure 7-3: route map, geotechncal data- STEP...60 Fgure 7-4: Sol composton for the ppelne A Fgure 7-5: Sol composton for the ppelne A...61 Fgure 7-6: Sol composton for the ppelne A3...6 Fgure 7-7: Defect rate assocated for each possble sol composton...65 Fgure 7-8: ppelne A1, potentally defectve clusters, or clusters wth bad coatng...66 Fgure 7-9: ppelne A, potentally defectve clusters, or clusters wth bad coatng...66 Fgure 7-10: ppelne A3, potentally defectve clusters, or clusters wth bad coatng...66 Fgure 7-11: percentage dfference between sol composton vs. defect rate (ppelne A1)...68 Fgure 7-1: percentage dfference between sol composton vs. defect rate (ppelne A)...68 Fgure 7-13: percentage dfference between sol composton vs. defect rate (ppelne A3)...68 Fgure 7-14 defect rate for sectons where bad coatng was appled vs. sol types...69 Fgure 8-1: A1- no. of defects vs. water level per.5 km long sectons...73 Fgure 8- A- no. of defects vs. water level per.5 km long sectons...73 Fgure 8-3: A3- no. of defects vs. water level per.5 km long sectons...74 M.Sc. Thess Lech A. Grzelak

14 x A statstcal approach to determne the MIC rate of underground gas ppelnes Fgure 8-4: number of defects vs. coded groundwater level (ppelne A1)...76 Fgure 8-5: number of defects vs. coded groundwater level (ppelne A)...76 Fgure 8-6: number of defects vs. coded groundwater level (ppelne A3)...77 Delft Unversty of Technology

15 xv Introducton N.V. Nederlandse Gasune s the man gas-transportaton company n the Netherlands. Its gas ppelne network conssts of approxmately km of steel ppelnes (dameters from 4 to 48 nch) that was largely constructed n the perod of the md sxtes to early seventes. The network s splt nto a hgh pressure part (HTL) (5600 km, bar) and a medum pressure part (RTL) (600 km, 40 bar). The hgh pressure network s possble to nspect whereas the medum pressure part does not possess requred nspecton facltes. Untl 1999 Gasune had no ndcatons whatsoever that there was a corroson problem on one of ts ppelnes. Both regular Cathodc Protecton (CP) measurements as well as observatons durng excavatons or reroutngs ndcated that no sgnfcant corroson problem exsted. Nevertheless Gasune polcy was to verfy ppelne ntegrty perodcally by nspectng one of ts hgh pressure lnes on average once every 5 years snce The results of these nlne nspectons confrmed the exstng opnon that no corroson problem exsted. In-lne nspectons have been part of the verfcaton of ppelne ntegrty snce the late seventes n N.V. Nederlandse Gasune. The dscovery of external mcrobal corroson (MIC 1 ) n 1999 n one of the hgh pressure ppelnes changed the nspecton polcy from nspecton of a randomly selected ppelne once every 5 year to an nspecton program for the whole hgh pressure grd (approxmately km) to be completed n 10-1 years. Fgure 0-1: deteroratng gas ppelnes External Mcrobal Induced Corroson s a type of corroson where the corroson rate s nfluenced by the actvty of bactera, especally Sulfate Reducng Bactera (SRB). It can be found n many envronments. Wthn Gasune t s found as external corroson on gas ppelnes. The chemcal and mcrobal processes are complex and can therefore depend on many parameters. Based on experence, Gasune beleved that MIC s found n certan areas more than n others and that therefore the occurrence of MIC s related to sol type or other sol parameters. 1 Mcrobologcally Influenced Corroson M.Sc. Thess Lech A. Grzelak

16 xv A statstcal approach to determne the MIC rate of underground gas ppelnes MIC s ntated at locatons of dsbonded coatng (usually at feld appled coatng) when the bactera and also the ppelne surface are shelded from the cathodc protecton system. Even well mantaned cathodc protecton systems cannot protect aganst deterorated, dsbonded coatng. MIC s not only dangerous due to ncapacty of protectng, but also because of relatvely hgh corroson rate, whch s hgher than for galvanc corroson. TU Delft was commssoned by Gasune to make a statstcal analyss of the avalable data on three hgh pressure ppelnes where MIC was recognzed. Delft Unversty of Technology

17 xv The goals of the research The man ssue of the thess s the analyss of the MIC based on the data delvered by N.V. Nederlandse Gasune. One of the MIC nfluenced lnes was used to qualfy MFLpg (Magnetc Flux Leakage) from dfferent supplers. In the tme perod four dfferent MFL qualfcaton runs have taken place, resultng n 18 excavatons. After the fourth pgrun (5 years after the frst) Gasune decded to determne the corroson and defect rate and nvestgate nfluencng factors. The goal of ths thess s to dscuss to Mcrobologcally Induced Corroson and ts effects on the safety and mantenance ssues. The man objectves of Gasune are: Developng models descrbng the Mcrobal Induced Corroson and Defect rates Fndng a number of factors nfluencng both estmated rates. These models and results can be used for prortzatons of ppelnes for nspecton and determnaton of nspecton ntervals. The structure of corroson modelng s presented n the Fgure 0- below. Fgure 0-: corroson modelng schema Frst part of the schema ndcated by blue color refers to the frst secton of the thess where the corroson rate model s presented. The results from ths study are the nput for the part number two where the parameters nfluencng the mcrobologcally nduced corroson rate are nvestgated. The fnal secton shows the results of defect rate modelng of ppelne affected by MIC. Ths consttutes a detaled statstcal descrpton of a connecton between envronment measurements and the reported corroson events. The man research s drected to fnd factors (f there exst) that nfluence both: the corroson rate and the defect rate. The man assumptons n the analyss are: - Coatng condton s assumed unform and s not governng defect rate. - Pgrun feature type and statonng ndcatons are fully correct - Estmates of the corroson rate (prevous study) are certan - Nomnal wall thckness s assumed to be real Also called: Intellgent Pg, more detaled descrpton s presented n Chapter no. M.Sc. Thess Lech A. Grzelak

18 xv A statstcal approach to determne the MIC rate of underground gas ppelnes Outlne of the thess The thess conssts of three man parts. The frst secton ams to model the corroson rate. Frstly, Chapter 1 descrbes avalable corroson data and gves a small overvew on nspecton procedures and assocated nspecton tools. Later on, the data wll be used as an nput for corroson rate model. The model for corroson rate wll be ntroduced and dscussed n Chapter. Chapter 3 shows number of numercal approaches n order to get estmate of the corroson rate. The analyss s carred out startng from the smplest to more sophstcated models. The frst and the second approach are smply based on the unconstraned regresson analyss. The thrd model s based on the unbased measurements and pgs accuraces ntroduced prevously n Chapter. Second part of the thess begns wth Chapter 4 nvestgatng hstorcal data about the potental from the test-posts. Chapter no. 5 ncorporates the results from Part 1, onpotental analyss and bo-data n order to get number of parameters nfluencng the corroson rate. The last secton s dedcated to defect rate analyss. The secton starts wth ppelne characterstcs (Chapter 6) where general nformaton about gas ppelnes s presented. Chapter 7 shows procedures of collectng and analyzng the sol data from geotechncal surveys. The purpose of ths chapter s to assocate the defect rate of the ppelnes nduced by MIC wth sol compostons. The parameter whch s nvestgated n Chapter 8 s a groundwater step level data analyss. Chapter no. 9 combnes all the avalable sol data, ppelnes features and groundwater levels, and treats them as nputs for the regresson model descrbng the defect rate. Delft Unversty of Technology

19 Part I Corroson Rate Modelng M.Sc. Thess Lech A. Grzelak

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21 4 Chapter 1 1 Descrpton of the corroson data 1.1 Introducton In 1999 a 70 klometer long ppelne (36 nch) n the north of the Netherlands was nspected by MFL pg as part of the verfcaton program. The results of the pgrun ndcated, as an unpleasant surprse, 65 external corroson features but no nternal corroson. After a thorough defect assessment by the safety department of Gasune t was decded that 6 ndcatons had to be excavated and repared. The frst excavaton by the end of 1999 led to a second unpleasant surprse. At the excavaton the appearance of the corroson defect turned out to be totally dfferent from the few corroson defects that Gasune had experenced n the past. Analyss of the corroson products and the appearance of the defect led to the concluson that the corroson was nfluenced by sulphate reducng bactera (SRB s). Smlar experence was obtaned at three other excavatons. At the end the concluson was that at four out of the sx excavatons the corroson was Mcrobally Influenced Corroson (MIC). An addtonal excavaton n 000 showed also MIC. From the results Gasune concluded that t was no longer safe to assume that other lnes were free of ths corroson problem. Therefore the nspecton polcy was revsed. It was decded to start an n-lne nspecton program for all hgh pressure lnes to be completed n a tme frame of approxmately ten years. It was also decded that pg supplers had to qualfy before they could nspect Gasune ppelnes. Because of gas-transport reasons and because of the fact that some defects on the lne had been repared by clock sprng 3 or coatng repar and thus can be used as reference ponts for MFL pgs, the nspected lne was apponted as a qualfcaton lne. When startng up the nspecton program Gasune realzed that a relable corroson rate s paramount to determne a re-nspecton tme nterval for ts ppelnes. 3 Clock Sprng s a composte sleeve used to repar external defects n hgh-pressure ppelnes M.Sc. Thess Lech A. Grzelak

22 5 A statstcal approach to determne the MIC rate of underground gas ppelnes 1. Magnetc Flux Leakage (MFL)-pg In-lne nspectons are performed by so-called MFL-pgs also called ntellgent pgs that locate and characterze mechancal damage n ppelnes. It s a common approach to the management of external corroson n the ppelne ndustry. Inspectons followed by excavatons of extreme deep defects mnmze potental rsk. Fgure 1-1: small and large dameter MFL-pg 1.3 Matchng of the reported defects In order to compare reported defects from two pgruns the defects wll have to be matched. Ths can be done by usng the reported log-dstances from the pg, ppelne lengths, clock poston of the defect, and dstances to reference ponts lke welds, valves. Wthn Gasune ths process can be automatcally done wthn the PIMS 4 software. In ths software the matchng process vsualzaton of the defects on a ppelne segment s possble. Fgure 1- shows a typcal example of reported defects from two pgruns. Fgure 1-: vsualzaton of matched defects (defects 1 to 6 are from run 1 and 7 to 1 from run ) 4 Ppelne Integrty Management System Delft Unversty of Technology

23 6 As can be seen from Fgure 1- t s not always clear after the matchng of defects by the program, whch defects belong together (7 to 1 or 8 to 1?). In the program however the user has the possblty to defne certan areas around a defect. If a defect falls wthn such an area then the two defects are consdered to be the same defect. The advantage of the software s that when many defects are close together the user can optmze the area szes to get the best possble matchng of defects. Factors that complcate the matchng of defects are dfferences n termnology of supplers or dfferences n nterpretaton of defects: external corroson defect, nternal corroson defect, mll defect etc. Suppose that the software has matched a defect from run 1 to a defect from run. Suppler 1 calls the defect corroson whereas suppler dentfes t as mll defect. The queston s then f ths matched defect should be taken for the determnaton of the corroson rate. It was decded to work wth two scenaros to fnd out whether t was crtcal for the determnaton of the corroson rate f only the defects wth the same dentfcaton were used (only ndcated as External Corroson ) or also defects wth dfferent dentfcaton ( External Corroson and External metal loss, possble mll defect ). It turned out that the corroson rates that were calculated for both stuatons were almost the same. In order to keep the uncertantes n the process as small as possble t was decded to use only the matched defects wth the dentfcaton of External Corroson n all pgruns Number of matched defects In the matchng process dfferent categores of matchng defects orgnated: defects from run A that could not be matched or that could be matched only once (to B, C or D), twce (to B and C, C and D or B and D) or three tmes (to B, C and D). A smlar result was obtaned for the matchng of defects from the other runs. For the fnal data set used for the calculaton of corroson rate t was decded to use only the defects that had been detected by three or more pgs. Ths resulted n a data set of 5 matched defects wth a subdvson as ndcated below. A B C D A B C D Table 1: number of matched defects per combnaton of pgruns As can be seen from the table, the number of matchng defects wthn ths subset was smallest for the comparson of run A and B: only 30 defects matched there. Altogether a number of 9 defects were reported by all and only four supplers whereas 3 defects were reported by three supplers. 1.4 Avalable data For the analyss of the data two data sets are avalable: 1. reported defect dmensons from the pg suppler. defect dmensons as determned at the excavaton and repar of the defect M.Sc. Thess Lech A. Grzelak

24 7 A statstcal approach to determne the MIC rate of underground gas ppelnes Reported defects Although the clamed performance of the dfferent pgs s comparable (see Table ) the number of reported external corroson defects s dfferent per suppler as can be seen n the table below. Suppler Szng accuracy 5 Number of external corroson Date of nspecton A +/- 10% w.t. 65 October 1999 B +/- 10% w.t 7 Aprl 001 C +/- 10% w.t June 00 D +/- 10% w.t 441 March 004 Table : szng accuracy of the pgs and number of reported defects As t s generally known the process of defect recognton conssts of three steps: detecton of the defect, szng of the defect and dentfcaton of the defect. The experence of Gasune s that most of the dfferences between supplers arse from dfferences n dentfcaton. The dstncton between a mll defect and a corroson defect seems to be troublesome for the supplers n qute a number of stuatons. Ths accounts for part of the dfferences n the reported number of defects. Another explanaton for the dfference n numbers s tme related: n general the performance of pgs has mproved over the last 5 years and corroson defects that had a defect depth under the reportng threshold 5 years ago may have grown to a defect depth above the reportng threshold Excavaton data After the frst pgrun 7 excavatons have been performed, n whch 17 separate defects have been repared. All of the defects that have been repared were manually measured n the dtch by the usual gouges. The defects that have been repared by welded sleeves could not be used as reference ponts for the pgruns B, C and D whereas the defects that were repared by clock sprng or coatng could be used as reference ponts. Table 3 ndcates the number of reference ponts that have been detected by the dfferent supplers. Suppler Number of avalable reference ponts from excavaton A 17 B 9 C 11 D 10 Table 3: number of reference ponts per pgrun The fact that the number of reported reference defects vares between B, C and D s due to the fact that the appled POF- nteracton rules [1] lead to clusterng of defects. Dfferences n defect szes or dstances between defects wll nevtably lead to dfferent numbers of reported defects. 5 for defect depth of general corroson wth 80% confdence level (w.t. = wall thckness) 6 of whch 576 are below 10% of wall thckness Delft Unversty of Technology

25 8 Fgure 1-3: clock sprng nstallaton (left), gas ppelne excavaton (rght) Fgure 1-4 shows reported defect depths compared to the real metal loss measured at the excavaton (all pg measurements are oscllatng around a dashed lne- whch ndcates a lnear relatonshp between real metal loss, and pgs measurements). Fgure 1-4: reported vs. measured defect depth Because the excavatons were performed n a relatvely short tme perod after the pgrun, t s assumed that defect depth has not sgnfcantly changed between the tme of the pgrun and the tme of excavaton Measurement data from matched defects All the matched defects come from dfferent segments wth varyng wall thckness. In 10 cases the defects are from a segment wth a wall thckness of 11. [mm], whle the rest of the defects are from the ppe wth a wall thckness of 1.86 [mm] (These values are nomnal wall thcknesses). The measurement dataset of matched defects s presented n Fgure 1-5. M.Sc. Thess Lech A. Grzelak

26 9 A statstcal approach to determne the MIC rate of underground gas ppelnes Fgure 1-5: reported defect depth Snce the nspectons are chronologcally ordered (A, B, C and D), t s clear from the above fgure that some of the defects are mprovng n tme (corroson depth s decreasng), whch s physcally mpossble. Delft Unversty of Technology

27 10 Chapter Descrpton of the corroson rate model.1 Introducton Snce the corroson s reported by ntellgent pgs, t s very mportant to know what the accuracy of the pg s. Such nformaton can be obtaned from the calbraton data collected durng ppelne excavatons. Gven that all data gathered by pgs s not certan t s reasonable to combne the measurements wth error dstrbutons obtaned from the calbratng procedure. If the excavaton ndcates that for a certan pg the measurement error s sgnfcantly smaller than for the other pgs, then the model should also take ths nformaton nto account. Another demand whch model has to satsfy s to deal wth the mssng data f some of corroson spots haven t been regstered n all nspectons; so ths nformaton also needs to be taken nto account and to gve physcally unreadable estmates. Ths chapter shows ways of dealng wth uncertanty about measurements, estmatng reasonable corroson rate(s) (.e. non- decreasng n tme) and dealng wth the mssng corroson data.. Data calbraton The data measured at each pgrun can be calbrated by removng the bas. There can be dfferent reasons for ths bas n defect measurements. Two of the most mportant reasons are: a measurement error assocated wth the measurement technque of a MFL-pg, and an effect caused by the clusterng of defects. The result of clusterng of several ndvdual defects can be caused by that only the deepest ponts are compared they may not be related to the same ndvdual defect. The analyss starts wth the measurements calbraton for a possble bas. Ths s one of the most mportant actons because the calbraton nfluences all the collected measurements. M.Sc. Thess Lech A. Grzelak

28 11 A statstcal approach to determne the MIC rate of underground gas ppelnes Due to the small number of excavatons, t s dffcult to nvestgate how well a gven pg s calbrated for deeper or shallower defects. However, t s possble to check the pgs accuraces, assumng homogenety between defects for each pg. By comparng the calbraton results, a concluson on whch pg s the best, wth respect to a bas spread of the measurement errors can be obtaned. The calbraton algorthms are n the next paragraph...1 Data calbraton procedures Algorthm 1 (Calbraton data algorthm) In order to perform data calbraton we need to follow the procedure: take X as a vector of metal loss regstered by an ntellgent pg defne an actual metal loss vector Y defne the bas vector Z = X Y calculate the bas by takng the expectaton of Z ( EZ ) Ths procedure allows to measure (by means of the average value) how far s the regstered by a pg metal loss from actual the metal. The expectaton s equvalent to the measure of bas, and ndcates how pg measurements are consstent wth actual data. Algorthm (Measurement error dstrbuton algorthm) Ths algorthm presents the procedure for estmatng the dstrbuton of measurement error. Use the Calbraton algorthm and fnd EZ Defne the corrected (unbased) pg calbraton measurements as: X ' = X EZ Defne the resdual random varable ε = X ' Y Fnd the dstrbuton of ε (usng technques ntroduced n the background chapter- appendx A1) From Algorthm 1 and : EX ' = E( X EZ) = EX EZ = EX E( X Y ) = EX EX + EY = EY So, the expectaton of unbased pg equals the expected value of actual metal loss... Data calbraton results The calbraton procedure showed that all of the pgs have a bas. All the measurements requre removng the bas. The bas for all pgs dd not exceed a value of 0.6 [mm], and on average had a level of 0.1 [mm]. Two MFL-pgs led to underestmaton and two led to overestmaton of defect depth. Table 4 presents the results of the calbraton procedure appled to the excavaton dataset. nsp. no. of calbr. samp. bas [mm] concluson A overestmated B underestmated C underestmated D overestmated Table 4: calbraton results Delft Unversty of Technology

29 1 Removng the bas can be done by subtractng t from the measurements reported by the correspondng MFL pg. When all the measurements are unbased, the second stage of the pg calbraton can be ntated, namely- the measurement error analyss. Snce none of the MFL pgs reports measurements wthout error (see Fgure 1-4), all the pgs have ther own measurement error dstrbutons. The example of the measurement error hstogram for pg A wth the correspondng theoretcal curve s presented n Fgure -1. Fgure -1: error dstrbuton for pg A When the measurement error dstrbutons for all the pgs are known, the concluson about the pg s accuracy can be drawn. It depends on two factors. Frstly: on the level of the bas and secondly: on the standard devaton of the measurement error. The analyss showed that for all pgs, under the null hypothess, the measurement errors are normally dstrbuted cannot be rejected (a sgnfcance level s customarly chosen to be 0.05). The standard devatons of the measurement errors are tabulated and presented beneath n Table 5. nspecton dstrbuton 7 A N(0,0.93) B N(0,1.34) C N(0,0.77) D N(0,0.63) Table 5: standard devatons of the measurement errors The results from the table ndcate that pg D (the last nspecton) has the smallest spread of the measurement error. The worst one s pg B, whch has less than half the accuracy of D. Snce the uncertanty about the measurements reported by pgs s known, t s advsable to take ths nformaton nto account for corroson rate modelng. 7 N stands for a Normally dstrbuted random varable wth two parameters, mean and standard devaton M.Sc. Thess Lech A. Grzelak

30 13 A statstcal approach to determne the MIC rate of underground gas ppelnes..3 General model for data calbraton Let s assume that we have carred out n nspectons- done by n dfferent pgs. Each pg s based dependng on the regstered defects depth. Ths knd of stuaton requres specal treatment, whch s the man part of ths chapter. Proposed procedure shows ways to avod (reduce) the correlaton between reported defect depth and measurement error. Suppose, we take a certan pg for whch, measurements and actual metal loss can be presented as follows: Fgure -: example of defects clusterng Fgure - shows that for three dfferent defect populatons three dfferent bases can be specfed, and three assocated error devatons. Of course, the decson about combnng defects (clusterng 8 ) nto subpopulatons generally can be subjectve. However the choce of clusterng also can be done n mathematcal manner. Mathematcal tools that work wth ths problem are so called Clusterng algorthms. Lterature avalable on the topc of the clusterng s ntroduced n references [8], [9], [10] and [11]. Gven that th pg measurements are presented n Fgure - above, t s possble to recognze three dfferent defects groups: small defects (black dots) where the bas s negatve (pg gves lower values than actual loss) wth small standard devaton, second- where observatons oscllate around actual values but wth hgh spread, and thrd subpopulaton (blue dots) where the bas s postve. Ths observaton motvates to dstngush groups of shallow, mddle, and deep defects. Such groups should be calbrated separately. Presented stuaton, mght not the case of real measurement; but t s mportant to know that f such stuaton occurs then needs to be taken nto account n the modelng. Accordng to prevous notaton, we have n pgruns and each of them can be based for dfferent clusters of defects. Ths leads to more general procedure of data calbraton than the one ntroduced before. 8 The process of organzng objects nto groups whose members are smlar n some defned way Delft Unversty of Technology

31 14..4 General calbraton procedure: 1. Take calbraton data for pg where = 1, K, n. Check whether pg s measurements are homogenous, f not, then fnd number n of defects clusters (.e. subgroup of defects, where members are smlar n some way) a. For each j = 1, K,n apply the algorthm 1 and get the bas for group j. b. Remove the bas from all measurements obtaned by pg. c. Apply the algorthm and get the dstrbutons for ε, j where j = 1, K,n The effects of appled general procedure are: All measurements done by pgs can be calbrated accordng to pg accuracy for dfferent defects depth. n We have n dstrbutons functons of measurement error, = 1 whch wll be appled n order to estmate the corroson rate. In the prevous part t was checked that the measurement error dstrbutons for all pgruns are normally dstrbuted. In the general model, f both: the assumpton about the same populaton for all the errors and normal dstrbuton are satsfed then n order to estmate the corroson rate a lnear regresson can be appled. On the other hand the least squares errors approxmaton wthout mposed any constrans can produce best estmate whch for ex. ndcates decreasng corroson rate. Next paragraph presents the general corroson rate model and numercal results for descrbng corroson growth as a functonal dependence on tme (nspectons)..3 General corroson rate model Let s assume that accordng to dataset n dstngushable defects n tme were observed. Suppose that defect was observed n n nspectons. The task s to fnd the best functon of tme, whch descrbes the corroson growth for specfc defect. The frst dea s to apply lnear regresson to all observed values of defect. Ths s a reasonable guess, but has sgnfcant drawbacks: From the collected data t s clear that n many cases nspectons report the depths for whch the best lnear estmate s: o decreasng n tme- what s unacceptable o the slope of a functon s too hgh- t means that the corroson accordng to the functon grows too fast, and ndcates leakage- but such leakage n ppelne was not observed o corroson accordng to the best estmated functon starts before ppelne nstallaton or even ppelne producton The standard regresson estmaton can only be appled to the model f t s assumed that the errors are normal, come from the same dstrbuton and are uncorrelated. However, n the case when the calbraton procedure s appled, t s clear that the normalty mght not be the case; furthermore, t can happen that measurements error don t come from the same populaton (dstrbuton). M.Sc. Thess Lech A. Grzelak

32 15 A statstcal approach to determne the MIC rate of underground gas ppelnes The model that has none of ntroduced drawbacks and accordng to delvered data gves a functonal descrpton of the corroson rate s presented beneath. The dea behnd the model s to gve an estmate whch takes nto account nformaton about the measurement error dstrbuton for each specfc pg (f the case then also clusters for each pg). Frst, let s defne: 0 l + f : ( T j, α, K, α ) R - theoretcal model functon wth l+1 parameters, assocated wth th defect, T j - tme snce ppelne nstallaton at j th nspecton d -unbased depth of defect, measured at j th nspecton j w - nomnal wall thckness where th defect s observed m - total number of nspectons P, j - measurement error densty functon of defect at j th nspecton The functon f assocated wth th defect needs to satsfy followng optmzaton task: maxmze subject to : : L = m j= 1 P, j f 0 l ( f ( T j, α, K, α ) 0 l ( Tm, α, K, α ) w 1 0 l f ( 0, α, K, α ) f s d non decreasng The frst restrcton mposed on functon f says that a value of the functon at the last nspecton cannot be hgher than the ppelne wall thckness where defect was observed. The second condton rejects stuatons where corroson ntaton accordng to data s before ppelne nstallaton (f we want to fnd the corroson ntaton tme, we need to fnd 0 l a t, for such f ( t, α, K, α ) = 0.e. corroson level at ntaton tme s exactly equal to 1 0 l zero, hence t s equvalent wth f (0, α, K, α ) = t ). The thrd and last constrant says that the functon assocated wth defect s growth cannot be decreasng n tme..3.1 Example Suppose that: In three nspectons one defect was observed. Each tme, the measurements were done usng dfferent pgs. From calbraton procedure t s known that all three pgs have nonhomogenous measurement error.e. parameters (or dstrbuton) are dfferent for dfferent pgs. 0 Assumng lnearty of defect s growth, the model has to fnd such estmators of α and α for whch the Lkelhood functon L s maxmum. The functon: f = ˆ α ˆ + α t s a functon that descrbes the corroson growth n tme for specfc defect. The schema of ths procedure s presented below n Fgure -3., j 0 0 ) Delft Unversty of Technology

33 16 Fgure -3: corroson rate determnaton- general model It s clear that L gets the hgher value f the estmated lne s closer to real measurements. In the case when the lne goes through all observed measurements, then ths lne s the best, and the lkelhood s maxmal, hence ths approach agrees wth natural expectaton. Remark The optmzaton process can be performed by applyng technques ntroduced n a feld of optmzaton as multdmensonal constraned non- lnear programmng. The results presented n the report are obtaned by usng Matlab 9 optmzaton toolbox. Furthermore because of the computatonal complexty of ntroduced non-lnear task t s worth to transform the task by usng logarthm transformaton 10. The mplementaton of the formulated problem s presented n the appendces..3. Dealng wth mssng data Many of defects were observed only n three nspectons (whle total number of nspectons s four). The model assumes that f depth of certan defect was not reported durng nspecton, then the measurement error densty functon for ths defect s unform on the nterval bounded by the mnmum and maxmum observed defect s depth. Suppose that defect was not observed at thrd nspecton, then n optmzaton problem the measurement densty functon for unmeasured depth s P, 3 = 1[ mn depth of 'th defect,max depth of 'th defect]. Ths means that f a certan defect was not regstered, then the functon descrbng corroson growth s derved by usng only reported defects. The draft of such stuaton s presented on the Fgure Matlab (R14) 10 Any monotonc transformaton of a functon doesn t change ts extremes (ma1, mn). M.Sc. Thess Lech A. Grzelak

34 17 A statstcal approach to determne the MIC rate of underground gas ppelnes Fgure -4: corroson rate modelng- dealng wth mssng data.4 Conclusons The model presented n ths secton gves the corroson rate estmate when low number of the defect measurements s avalable. Very often standard regresson model doesn t gve relable and acceptable results; hence alternatve s requred. For many defects the regresson estmates are negatve or ndcate defect ntaton before ppelne nstallaton. The General corroson rate model solves all these drawbacks, and also takes nto account nformaton on pgs accuraces..5 Implementaton (CoroGas 1.0v) The theory ntroduced n ths chapter has been mplemented n CoroGas, software package developed by the author. Ths program analyzes the excavaton data, unbases the measurements, assesses the weghts for MFL-pgs and gves estmate of the corroson rate. The program has mplemented algorthms for predefnng the clusters for the calbraton. Fgure -5: calbraton data & optmzer wndow of CoroGas Appendx B descrbes CoroGas and all avalable functons. Delft Unversty of Technology

35 18 Chapter 3 3 Numercal results Ths chapter s mostly based on the artcle Determnaton of the corroson rate of MIC nfluenced ppelne usng 4 consecutve pgruns by Lech A. Grzelak & Gorgo G.J. Achterbosch publshed n Internatonal Ppelne Conference (IPC ) 3.1 Introducton Three dfferent approaches for corroson rate modelng wll be presented. The analyss s carred out startng from the smplest to more sophstcated models. The frst and the second approach are smply based on the unconstraned regresson analyss. The thrd and the last model, s based on the unbased measurements and pgs accuracy descrbed n the Chapter. 3. Approach 1 In the frst approach all the defects are pooled n 1 dataset and no corroson rate s calculated for ndvdual defects but only for the dataset as a whole. The frst approach starts wth verfcaton f the hypothess that the measurement errors for all MFL pgs are from the same populaton and are normally dstrbuted can be accepted. Ths was the case. Accordng to the maxmum lkelhood estmaton for the measurement error the parameters are 0 (mean) and 0.91 (for a standard devaton). If t s assumed that for all the errors, there s no correlaton between them, then the task of fndng a corroson rate assocated wth all the measurements s equvalent to a Gauss Markov regresson model. M.Sc. Thess Lech A. Grzelak

36 19 A statstcal approach to determne the MIC rate of underground gas ppelnes Let s defne necessary matrxes X and Y n followng way: ( T1 ) m1 1 X = 1 m 1 M ( T ) n m 1 n m Y = d ( T1 ) d( T ) L d( T ) d( T ) 1 1 where: [ ] T m 1 m 1 n 1 mn 1 n mn 1 1 m o d( T ) m d T d T d T m d T = m L 1 (,1) (,) (, 1) (, ) -a vector of m tmes unbased depths measured by pg at tme T, second ndex ndcates defect s number o n total number of nspectons o m - number of defects at th nspecton o m s total number of observed defects at n nspectons ( m = m + K+ mn o 1 1 L tmes m = m T T T 1 ) o ( T ) m x T T T T 1 = L where T s a tme of th nspecton m tmes If addtonally, t s assumed that the corroson rate s unform over tme (.e. corroson growth s lnear), then an applcaton of the Least Squares Error (LSE) method gves a lnear descrpton of the corroson growth n the followng form: y= ˆ α 0 + ˆ α1t and the 0 1 remanng ssue s to fnd the estmator [ ˆ ˆ ] T ˆ β = α α for the lnear functon. Standard calculatons gve that the estmators for unknown parameters are: ˆ α 0 =.4 and ˆ α 1 = Coeffcent ˆ α s equvalent to the measure of the corroson rate [mm/yr], so the LSE model estmated a corroson rate for the calbrated measurements of 0.1 [mm/yr] wth 95% confdence nterval [0.05, 0.0]. Delft Unversty of Technology

37 0 Fgure 3-1: defect depth n tme To check how well the model fts the data, a determnaton coeffcent s calculated. A goodness of ft measure resulted n R = Ths s poor because t ndcates low relatve predctve power of the model. Accordng to the model, the ntaton tme for corroson s 0 years [yrs snce ppelne nstallaton]. Even though the estmated parameters are n an acceptable range, ths approach has sgnfcant drawbacks: the model does not dstngush defects t does not take nto account that some of the defects are mprovng n tme (decreasng defect s depth whch s physcally mpossble), or for some the defects ntaton tme s before ppelne nstallaton the model assumes that all the defects have one corroson rate 3.3 Approach As was ponted out n the prevous secton, the frst approach has sgnfcant drawbacks. The second approach, proposes a way of dealng wth some of the enumerated dsadvantages. Lke before t s assumed that corroson growth s lnear n tme. The second model checks what the corroson rate s, f the defects are analyzed ndvdually.e. the model does not assume any more that there s only one corroson rate for all the defects but t calculates a corroson rate per defect. A smple regresson analyss appled to each unbased defect gves the followng graph. M.Sc. Thess Lech A. Grzelak

38 1 A statstcal approach to determne the MIC rate of underground gas ppelnes Fgure 3-: corroson rate dstrbuton Fgure 3-3: dstrbuton of ntaton tme From the hstogram presented n Fgure 3- t s clear that n many cases, smple regresson analyss appled to each defect results n negatve corroson rates. The mean corroson rate accordng to ths model s 0.16 [mm/yr] whch s close to the result obtaned before, however the 95% confdence nterval for the corroson rate s qute dfferent: [-0.31, 0.54]. The 95% confdence nterval comprses negatve values. The number of the defects ndcatng ether negatve corroson rate or corroson ntaton tme before the ppelne nstallaton s 16. One way of dealng wth a negatve corroson rate s to remove all the outlers from the dataset. However, such treatment s undesrable snce the dataset conssts of 30% bad defects. Further nvestgaton confrmed that the corroson rate follows a normal dstrbuton. The ntaton tme of the corroson s presented above, also n the form of a hstogram. The red bars n the pcture ndcate an ntaton tme outsde the nterval determned by the tme of ppelne nstallaton (t=0 [yr]) and the tme of the last nspecton (t=44.5 [yr]). A summary of the results obtaned from the second approach s presented n Table 6. results corroson rate nt. tme mean 0.16 [mm/yr] [yr] 95% conf. nt. Lower bound [mm/yr] [yr] Upper bound 0.54 [mm/yr] [yr] Table 6: corroson rate and ntaton tme for approach Stll the Least Squares Errors approach produces negatve corroson rates or ntaton tmes before ppelne nstallaton. Therefore an alternatve model for the presented models s presented: approach Approach 3 Ths approach s based on the General corroson rate model ntroduced n prevous chapter. The model takes nto account nformaton about the measurement error dstrbutons for each specfc pg and accordng to these dstrbutons assgns weghts to the measurements. The weghts are chosen n the followng way: a pg whch s accurate nfluences the fnal results stronger than a pg wth a lower level of accuracy. 11 Approach 3 s smplfed verson of General corroson rate model. Delft Unversty of Technology

39 Assumng that the functon assocated wth th defect s lnear n tme ( f ( t, α 0 0 1, α 1 ) = α + α t ), let s defne smplfed verson of general corroson rate model n followng way: f R + : ( T j, α 0 1, α ) - theoretcal lnear functon, assocated wth th defect (number of defects s 5) T j - tme snce ppelne nstallaton at j th nspecton d j -unbased depth of defect, measured at j th nspecton w - nomnal wall thckness where th defect was observed (two cases: 11. for 10 defects and 1.86 for 4 defects) m = 4 - total number of nspectons P, j - measurement error densty functon of defect 1 observed at j th nspecton The lkelhood estmaton s optmal when the followng functon s maxmzed: m 1 0 maxmze : L = P, j ( α T j + α d, j ) j = subject to : α T m + α w α / α 0 1 α 0 The results of the corroson rate obtaned by the model are presented n the Fgure 3-4. Fgure 3-4: corroson rate dstrbuton for approach 3 Fgure 3-5: dstrbuton of ntaton tme for approach 3 Because of the constrants the model s output s n harmony wth the physcal corroson propertes: none of the corroson rates are negatve. 1 A measurement error densty functon for each defect can depend on the cluster, from whch the defect comes. I.e. defects from one pg can have a few dstrbutons- one per ndvdual cluster, however, here n the analyss because of low number of calbratng data each pg has one dstrbuton M.Sc. Thess Lech A. Grzelak

40 3 A statstcal approach to determne the MIC rate of underground gas ppelnes In the other two approaches the corroson rate dstrbuton was normally dstrbuted, now t s qute dfferent there s no bass to reject the hypothess that the corroson rate s Beta dstrbuted (wth parameters 1.5 and 3.9). The hstogram of the ntaton tme s presented n the Fgure 3-5. The model says that all the ntaton tmes are n acceptable tme ntervals. However, 14 of the defects ntate at the tme of ppelne nstallaton- ths s ndcated wth a lghter blue color n the pcture above. The ntaton tme dstrbuton s a composton of two dstrbutons, dscrete n 0 and contnuous elsewhere. The contnuous part s also Beta dstrbuted (wth parameters 1.60 and 0.50). A small summary of the results of the corroson rate and the corroson ntaton tme s presented below. results corroson rate ntaton tme mean 0.4 [mm/yr].16 [yr] 95% conf. nt. Lower bound [yr] Upper bound [yr] Table 7: corroson rate and ntaton tme for approach 3 As s shown n the Table 7, all the results obtaned from the model are acceptable because they are physcally possble. The model even for a bad dataset gves reasonable and acceptable results. The results from Table 7 show that the model gves a corroson rate that s approxmately 0.05 hgher than the prevous ones. The reason for ths s the followng: n the prevous cases, the result of corroson rate was an outcome of all corroson rates (ncludng negatve values), whch resulted n a lower average value. 3.5 Depth nfluence on the corroson rate In order to nvestgate whether the defects depths have a sgnfcant nfluence on the corroson rate, the dataset of 5 unbased measurements s dvded nto two subsets. Frst, the weghted average for each defect was calculated. The weghts were assocated wth the measurement errors standard devatons.e. an accurate pg has the hghest weght etc. Then, the dataset was dvded nto two subsets, namely shallow and deep defects, n such a way that the MFL pg wth the lowest number of observatons (pg B) has an equal number of measurements n both sets. For both subsets, the General corroson rate model was appled. The mean corroson rate for deep defects turned out to be 0.5 [mm/y], and for shallower ones the rate s 0.3 [mm/yr]. However statstcally, there s no bass to reject the null hypothess that these mean values are the same. Hence t follows that statstcally the corroson rates for deep and shallow defects are not sgnfcantly dfferent. The hstograms of the rates for shallow and deep defects are presented beneath. Delft Unversty of Technology

41 4 Fgure 3-6: dstrbuton of corroson rates for shallow defect Fgure 3-7: dstrbuton of corroson rates for deep defects In both cases (shallow and deep defects) there s no bass to reject the hypothess that the corroson rates are from a beta dstrbuton. Although the number of measurements of pg B s equal n both subsets, the number of defects s not the same n these two sets. The number of shallow defects, accordng to the presented crteron, s 34. The rest (18 defects) are deep defects. Analyss showed that f the set of unbased measurements s dvded n such a way that the number of shallow and deep defects s equal n both subsets, t leads to the same concluson that there s no sgnfcant dfference n corroson rates. 3.6 Conclusons and recommendatons In order to determne relably the bas for a MFL-pg t s crucal to have multple reference defects n the ppelne for whch the dmensons are well known. These can then be used to calbrate the MFL reported values. For the descrbed ppelne the number of avalable reference ponts was lmted but stll made t possble to estmate the bas for every pg. Because the measurement uncertanty of the MFL-tool s domnant compared to the corroson growth n the tme perod between the pgruns, t s very dffcult to determne a relable corroson rate per defect. However, by assumng a smlar corroson process (MIC) for each defect, based on evaluaton of the MFL sgnals, hstorcal CP-measurements and results of excavatons, n combnaton wth the assumpton of a lnear corroson growth, t was possble to calculate a realstc corroson rate for ths ppelne. Dependng on the approach that was used a value for the average corroson rate of 0.1 mm/yr (approach 1), 0.16 mm/yr (approach ) or 0.4 mm/yr (approach 3) was obtaned. The numbers for the 95% upper bound values were respectvely 0.0 mm/yr, 0.54 and 0.6 mm/yr. The results from the frst two models are clearly underestmatng the corroson rate snce the fnal result s an average over postve and negatve corroson rates. The dea of the thrd approach s qute dfferent: the estmate of the corroson rate s an outcome of all the defects growths. Frstly, the model descrbes each defect separately as tme dependent functon. A functon that needs to satsfy mposed constrans derved from physcal features (.e. functon cannot be decreasng n tme etc.). When all these functons are known, then the nformaton about corroson rate assocated wth defects populaton s a product of all functons derved for all defects. The estmates of the thrd approach are nput data for the secton where nfluencng factors are nvestgated. M.Sc. Thess Lech A. Grzelak

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43 6 Part II Parameters nfluencng mcrobologcally nduced corroson rate M.Sc. Thess Lech A. Grzelak

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45 8 Chapter 4 4 Potental analyss 4.1 Introducton Cathodc protecton (CP) systems are used to protect bured steel ppelnes. The exceptons mght be nstances the ppelnes are nstalled n farly non- corrosve sol and where regulatons do not requre such CP systems. Accordng to regulatory agency requrements, a ppe-to-sol potental s to be at least mllvolts wth reference to a copper-copper sulfate reference electrode 13. More negatve potental protects more aganst galvanc corroson, but on the other hand too negatve potentals may damage the coatng protecton 14. The dea behnd cathodc protecton s to ensure a current flow towards the ppelne opposed to a corrodng current away from the ppelne. Ths chapter analyzes f there s any relatonshp between the potentals measured at test-posts and the corroson rate. Moreover, the results from ths chapter wll be appled n the regresson analyss n Chapter 5 (Mcrobal data analyss). Avalable potental dataset delvered by Gasune s not a set of potentals assocated wth real potentals at the coatng defects but so called on-potentals measured at test posts. On-potentals contan an IR-drop component. Ths s the potental drop n the sol between the locaton of the reference electrode (somewhere at ground level) and the steel/sol nterface at coatng defects. The IR-drop s caused by CP- and stray currents n the sol. These on-potentals are only an ndcaton of the general status of the Cathodc Protecton system. Usually on-potentals were collected durng a certan tme (about 5, 15, 60 mnutes) - durng ths tme maxmum and mnmum potental were recorded. 13 Crteron: NEN-EN The crtera are set to prevent corroson. Ths does not necessarly mean that corroson wll occur when the crtera are not met. It s of course not the case that a ppelne corrodes at 849 mv and does not corrode at 851 mv. The potentals are wth reference to a Cu/CuSO 4 reference electrode. M.Sc. Thess Lech A. Grzelak

46 9 A statstcal approach to determne the MIC rate of underground gas ppelnes On-potentals can vary contnuously due to e.g. nterference from other currents n the sol. These varatons are supermposed upon the CP potental of the ppelne. Because potentals are recorded over a certan tme-nterval (5 mnutes, 60 mnutes or 4 hours) and only mn/max values were recorded the exact on-potental s not always known. It s also not clear f values ndcated as max or mn were measured only 1% tme or a large part of tme. The assumpton mposed on the measurements s that varaton of the CP on- potentals n certan local tme nterval s lmted. To account for outlers n the measurements a smoothng procedure was appled. Fgure 4-1 cathodc protecton for a gas ppelne (left), voltage drops n a measurng crcut (rght) Fgure 4-: Cu/CuSO 4 reference electrode The data whch wll be analyzed n ths chapter was collected for the ppelne for whch the estmates of corroson rates for 5 dstngushable defects are avalable. Let s call ths ppelne A3. 4. Measurements and smoothng method In analyss one of the smoothng methods called movng averages was appled. The smoothng algorthm mnmzes local varablty of measurements, allowng to spot trends. The movng average s one of the smplest and oldest analytcal tools around. Some patterns and ndcators can be somewhat subjectve, where analysts may dsagree on f the pattern s truly formng or f there s a devaton that mght be an lluson. The movng average s more of a cut-and-dry approach to analyzng potental changes and predctng performance. The mathematcal formulae for movng average are presented n appendx A. The on-potental measurements were collected at 67 test posts randomly dstrbuted along the ppelne. It was possble to dstngush 30 dfferent dates when the Delft Unversty of Technology

47 30 measurements were collected. Pcture below presents how measurements from the test posts were dstrbuted over tme along the ppelne. Fgure 4-3: Test posts dstrbuton over tme From the Fgure, t s clear that number of test posts ncreased over tme from only 18 test posts n the 60 s to 67 today. Frstly, a grd whch presents on-potentals varablty wrt statonng and tme has to be defned. The dea s to reconstruct potentals gven partal avalable data. The grd wll consst of 30x67 ponts, where each pont wll present the measured average potental (snce only max and mn are measured the analyss wll be carred out wrt to average of these measurements). The uncertanty of the measurements over tme per test post s relatvely hgh- t s ndcated by hgh varaton of the potentals over a small perod of tme. The Fgure 4-4 below presents potentals at certan statonng before and after smoothng method appled (spam equal to 3 was appled). Fgure 4-4: on-potentals measurements and results of appled smoothng method M.Sc. Thess Lech A. Grzelak

48 31 A statstcal approach to determne the MIC rate of underground gas ppelnes Fgure 4-5: on-potentals before smoothng Fgure 4-6: on-potentals after smoothng Fgure 4-5 and Fgure 4-6 present how the appled smoothng procedure reduced the number of outlers n the dataset. The pattern of potental change over tme can be recognzed. Through out the analyss t s assumed that the change of potental between the measurements s lnear, t s sensble snce there s no addtonal nformaton avalable. 4.3 On-potental Grd constructon In order to defne the potental grd of the ppelne, all the measurements from the test posts are used. Suppose that: S 1, S, S3,..., Sr - statonng of the test posts, for each test post t s possble to defne a functon g : t, j V, j where t, j - ndcates the tme snce ppelne nstallaton at tme j at statonng S V, j - s a average potental at test post S at tme t, j In order to formulate ppe-to-sol potental grd wth respect to tme and statonng, followng procedure s defned: 1. Apply smoothng procedure for each test post (smoothng wth respect to tme), span of smoothng procedure should be chosen n a such way that the average potental from data corresponds to physcal potental phenomenon.e. local change of average potental cannot change too sharply.. Defne T ~ as ordered tmes of collected measurements (for all the test posts) T ~ = { t1,1, t1,,..., t,,..., tr tr } m,1,..., where l, mr, j, k, t, j tk, l and m - ndcates the last measurement recorded at statonng S. T ~ conssts of ordered dates of all collected measurements. Snce none of test posts was observed for all T ~, Delft Unversty of Technology

49 3 the nterpolaton for all the test posts s requred. Assumng lnearty between measurements, nterpolaton can be done n followng way: Suppose that two dfferent measurements for one test post were collected ~ V, j and V, k where k > j, then convex combnaton for V, l (for t, l T ) can be presented n followng way: l < k l > j V, l = ( 1 α ) V, j + αv, k where α = ( t, l t, j ) /( t, k t, j ), but stll, the nterpolaton can only be appled f t, j and t, k are defned Because of an ncreasng number of test posts over tme, t s possble to generate the measurements for a gven statonng (even between the test posts). Ths can be done by usng a lnear nterpolaton between test posts for each tme from T ~ ~. If one takes one test post then t j T V, j = ( 1 α ) V p, j + αvq, j where p and q ndcate the closest montored test post, where α = ( S, j S p, j ) /( Sq, j S p, j ), as before V, j can be calculated f V p, j and V q, j are defned. 4. In the case when nterpolaton cannot be carred out, because of mssng boundares, then the smplest way s to generate these data by usng lnear regresson approach appled to each test post. Beneath, the results of the appled technque are presented. In the smoothng technque, span 3 was chosen. Investgaton showed that changng the spam for movng average doesn't change sgnfcantly fnal results. Applcaton of the ntroduced technque produced the followng contour and 3D plot of the average potentals over the last 45 years along the ppelne. Fgure 4-7: on-potental contour plot 15 It s possble to generate mssng data- the procedure for that wll be ntroduced later on n text. M.Sc. Thess Lech A. Grzelak

50 33 A statstcal approach to determne the MIC rate of underground gas ppelnes Fgure 4-8: on-potental grd The Fgure 4-7 and Fgure 4-8 show the average potentals regstered at the test posts durng last 45 years. The contours present how potentals were changng over tme. Plot ndcates lack of the cathodc protecton for the frst ppelne klometers for a frst few years (the potental s sgnfcantly hgher than the one regstered after 0 years. From these fgures t looks lke CP ncreased up to 30 years then stablzed more or less for about ten years and then started to decrease. The plots show that n the frst 0 km secton of the ppelne the potental has sgnfcantly decreased after frst 0 years. If one compares obtaned potental wth places where 5 defects are dstrbuted, a blurry pattern can be recognzed. Fgure 4-9: 5 dstngushed defects and estmated corroson rates For the estmated on-potental grd a correlaton between corroson rates and average potentals has to be calculated. For each of the 5 dstngushed defects for the ppelne A3 the potentals can be easly obtaned usng technques ntroduced before.e. for each statonng t s easy to fnd neghborng test posts and nterpolate the potentals over tme. Snce the estmated corroson rate of the 5 defects s based on 4 nspecton and measurements are carred out wthn 5 years the average level of on-potentals should be taken only from that perod. The results are presented below. Delft Unversty of Technology

51 Correlaton between the corroson rate and average potentals recorded between frst and last nspecton Approach 1 Frst, let s defne: ~ V = V,, K, V,, where T 1 andt m ndcates tme of frst and last nspecton, [ ] T T1 Tm 1xl V, j - s on-potental at statonng at tme j t,, = 1,... n, n- number of the defects and l ndcates number of avalable measurements (real and nterpolated between frst and last nspecton) R r1,k, r n 1 = - vector of corroson rates for the correspondng defects [ ] T xn l [ ] T 1 ~ V = V1,K, V n 1xn where V = V ( k) l k= 1 The correlaton between corroson rate and average potental s defned as: ρ ( R, V ). Small summary of appled technques s tabulated below. The p-value 16 presented n the last column corresponds to the hypothess: H : ρ ( R, V ) 0 aganst H : ρ ( R, V ) 0 0 = No. of samples type of correlaton ρ 5 defects, l =61 1 p-value for the hypothess that correlaton s nsgnfcant Pearson Spearman Kendall Table 8: correlaton between the corroson rate and average on-potental recorded between frst and last nspecton The normalty condton for product moment correlaton s satsfed, for a chosen sgnfcance level α = the p-value suggests that there s no bass to reject the null hypothess that the correlaton s nsgnfcant. For a sgnfcance level 0. there are bass to reject the null hypothess. Overall concluson s that there s a weak postve correlaton between average potentals and the corroson rate of defects Approach Second approach presents another way of lookng at the connecton between corroson rate for the defects and average sol-to-ppe potentals. Suppose that we observe n dstngushable defects, and each of the defects has gven an estmated corroson rate r = 1,...,n. For each defect t s easy to fnd (from the generated grd) the average V potental vector [ ] T T1, Tm xl = V,,..., V where T 1 and T m ndcate tme of the frst and last 1 nspecton (pgrun). The total number of possble dfferent pars of defects s C n =. = 1 For each par of defects (,j) verfy the hypothess: n 1 16 Defntons and nterpretatons are ncluded nto appendx A M.Sc. Thess Lech A. Grzelak

52 35 A statstcal approach to determne the MIC rate of underground gas ppelnes l l 1 1 H 0 : ( V, k V j, k ) = 0 aganst H 1 : ( V, k V j, k ) 0 l k= 1 l k= 1 Under the assumpton that ( V V ) ~ N( ˆ, μ ˆ σ ) j - parameter μˆ s an unbased maxmum lkelhood estmator. The estmator for σ s assumed to be unknown. Under the normalty condton the cases a) and b) are counted: a) V > V j & r > rj or V < V j & r < rj b) V > V j & r < rj or V < V j & r > rj The frst pont a) s equvalent to concordance of corroson rate and average potental, and the second one to ther dscordance. In both cases there s a statstcally defendable dfference between averages of potentals. Results: 1. total number of pars for 5 defects s 136. n 1174 cases the null hypothess was rejected 3. n 564 the null hypothess was rejected and normalty condton was satsfed Concordance Dscordance Fgure 4-10: concordance and dscordance In 57 % cases hgher corroson rate s accompaned by hgher average potental (less negatve) and n 43% cases s the other way around. Ths approach also shows weak postve correlaton between average on-potentals and the corroson rate. 4.5 Correlaton between the corroson rate and potentals standard devaton recorded between frst and last nspecton. Frst, let s defne: ~ V = V, K V, where T 1 andt m ndcates tme of frst and last nspecton, [ ] T 1, Tm xl,, 1 V, j - s on-potental at statonng at tme t, j, = 1,... n, n- number of the defects and l ndcates number of avalable measurements (real and nterpolated between frst and last nspecton) l 1 ~ V = V ( k) = 1, K,n and R = [ r r ] T 1,K, n 1xn - corroson rates for n defects l k= 1 [ ] T l S S S S 1 ~ V = V1,K, Vn 1xn where V = ( V ( k) V ) l 1 k= 1 Delft Unversty of Technology

53 36 Then the correlaton between corroson rate and calculated standard devaton of S potental s defned as: ρ ( R, V ). ρ no. of samples Type of correlaton p-value for the hypothess that correlatons are nsgnfcant Pearson defects, Spearman l=61 Kendall Table 9: correlaton between the corroson rate and on-potental standard devaton recorded between frst and last nspecton Table 9 shows weak negatve correlaton between corroson rate and standard devaton of the on-potental. 4.6 Conclusons Corroson processes take tme and are therefore governed by a number of crcumstances. Unfortunately some nformaton s mssng: the measurements were made and only mn/max were recorded. The data used n ths chapter s not accurate. It s dffcult to say precsely n how on-potentals descrbe real ppelne potental. Introduced methodology ddn't gve defendable results.e. clear massage about the connecton between corroson rate and average on-potental. The analyss showed certan patterns of weak 17 correlatons between corroson rate for the 5 regstered defects along the A3 and level and on-potentals and ther varablty. The results from ths chapter wll be appled as an nput nto regresson n the next secton. Further analyss of the on-potentals has to be carred out n order to check how on-potentals are nteractng wth other varables potentally nfluencng the corroson rate. 17 read: statstcally nsgnfcant M.Sc. Thess Lech A. Grzelak

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55 38 Chapter 5 5 Mcrobal Data analyss 5.1 Introducton The dataset analyzed n ths chapter s delvered and collected by one company specalzng n bo-analyss, a company whch collected sol samples at the statonng of certan group of defects from the ppelne A3. The analyss of the collected sol samples was done n order to assess the crcumstances whch can nfluence the growth of bactera nvolved n corroson processes. Here, the bo-analyss s used as an nput for the corroson rate regresson model. The data collecton and analyss was carred out wth respect to qualtatve and quanttatve descrpton of the envronment where defects were reported. The defects under analyss (18 defects) were chosen from the set of 5 defects recognzed. For each of these defects lnear corroson (constant corroson rate) was assumed. The corroson rate for these defects was calculated usng corroson rate estmaton model descrbed n frst part of ths thess. The estmaton was based on measurements from four consecutve nspectons done by ntellgent pgs. The analyss wll be carred out n order to fnd the connecton between the envronment data and the corroson rate estmated usng the corroson rate model presented n Part 1 of the thess. 5. Mcrobologcally Influenced Corroson Accordng to chemsts: abundant n natural envronments Sulphate Reducng Bactera (SRB) are the most nfluental n MIC processes. SRB are anaerobc bactera utlzng sulfate as a termnal electron acceptor and organc substances as carbon sources. It s shown that although SRB are strctly anaerobc, some subpopulatons tolerate oxygen M.Sc. Thess Lech A. Grzelak

56 39 A statstcal approach to determne the MIC rate of underground gas ppelnes and are even able to grow at low oxygen concentratons. SRB has ablty to reduce sulfate produced carbonate whch neutralzes acds and sulfde, whch chemcally stablzes toxc metal ons as sold metal sulfdes. Experments showed that ph levels supposed to ncrease n presence of SRB metabolsm 18. The sol analyss from Texas to New Jersey has shown that number of bactera s lvng n the sol at or near protectve coatngs. In paper of Joseph L. Pkas [3], author suggests that f one compares two envronments wth the same sol type but one at or near to dtch and second undsturbed, then hgher number of SRB s expected to be n the frst envronment. J. O. Harrs [4] n hs notes says that snce condtons of the sol do not reman statc whether the sol s close to surface or near to a ppelne at the bottom of a dtch-mostly due to water fluctuatons- bacteral populatons n the sol consst of many types of dfferent speces. Moreover, the nterrelatonshp between dfferent types of bactera of mcroorgansms contrbutes to changes that occur n the sol. Fgure 5-1: mcrobologcally nfluenced corroson on gas ppelnes Mcroorgansms can be grouped nto few types presented n the table below. Prerequste Provder Knd of growth Energy source Lght Phototropc Chemcal substances Chemotrophc Carbon source CO Autotrophc Organc substances Heterotrophc Electron donor (to be oxdzed) Inorganc substances Lthotrophc Organc substances Organotrophc Oxygen Aerobc Electron acceptor (to be reduced) NO, NO 3 SO 4, CO Table 10: groups of mcroorgansms Anoxc Anaerobc Interestng result s that a very good place to lve for an anaerobc organsm s below an actve colony of aerobc organsms as these consume the oxygen and create anaerobc areas whch serve as habtats for the anaerobcs. As a result, anaerobc organsms lke SRB can be found next to aerobc organsms that protect anaerobc bactera whch can easly grow and multply. The oxdaton of sulfde, whch can be performed sulfur oxdzng bactera results n decreasng ph value s typcal example mportant for MIC. 18 P. Frank, UC Berkeley Department of Envronmental Scences Delft Unversty of Technology

57 40 Decreasng ph value s equvalent wth transformaton of weak acd n a strong one. Chemcal reacton of ths transformaton s shown as follows. S + O SO4 Over last decades many dfferent models have been proposed to explan the mechansms by whch SRB can nfluence the corroson of the steel. These models were concentrated on analyss based on cathodc depolarzaton by the enzyme hydrogenase, anodc depolarzaton, producton of corrosve ron sulphdes, release of exopolymers capable of bndng Fe - ons, sulphde-nduced stress- corroson crackng, and hydrogennduced crackng or blsterng. All the models showed that there s not only one predomnant factor nfluencng MIC and many dfferent factors are nvolved. Fgure 5-: mage of a sulphate-reducng bacteral culture wth a carbonate precptate, the bactera on the left are about 6-8 µm long and µm n dameter In order to confrm MIC t s essental to check presence of mcroorgansms by obtanng samples of the natural envronment surroundng the metal. 5.3 Dataset descrpton The dataset conssts of two knds of varables, ndependent whch are used as an nput to the model (the varables whch may nfluence the corroson rate) and dependent varable- often called varable of nterest or crteron varable whch s assocated wth the output whch s the corroson rate Independent varables The man bo-analyss of the samples was performed n order to gve Mult Crtera Analyss (MCA) for each specfc envronment. The MCA bascally gves score relatve to chance of gettng MIC corroson for specfc envronment. The formula for the score s based on fve factors: redox, avalablty of a carbon, avalablty of nutrton, degree of acdty (ph) and conductvty. The MCA formula s presented as follows: carbon MCA redox = 3 S + S + 1 S + 1 S + 1 S where: ndcates th measurement (place where defect was regstered) and scores S are assgned accordng to the table below. Scores are assgned by experts. nutr ph EC j M.Sc. Thess Lech A. Grzelak

58 41 A statstcal approach to determne the MIC rate of underground gas ppelnes Factor Classfcaton Score Aerobc 1 redox S carbon S nutr S ph S EC S Ntrate reducng 1 Iron reducng Sulphate reducng 3 Lack of methane 0-0 mg/l TOC mg/l TOC >40 mg/l TOC 3 N-tot < 1 mg/l P-tot < N-tot > 1 mg/l P-tot < 0.05 N-tot < 1 mg/l P-tot > 0.05 N-tot > 1 mg/l P-tot > ph > ph < μ S 1 > μ S 3 Table 11: factors and weghts for MCA score Other factors nvestgated and measured n the feld by bo-company are: Varable Notaton/unts Descrpton Amount of oxygen can ndcate exstence of anaerobc/ aerobc Oxygen bactera n sol, delvered data n half cases are beyond the oxygen [ mg / l] detecton lmt, so the oxygen varable wll be treated as a dummy varable 0. Redox redox potental [mv ] cond. conductvty [ μ S] ph [ ph ] TOC [ / l] Fe ron [ μ g / l] The redox potental s the reducton/ oxdaton potental of a compound measured under standard condtons aganst a standard reference half- cell. Conductvty s a measure of a materal s ablty to conduct an electrc current. ph s a measure of the actvty of hydrogen ons n a soluton and, therefore, ts acdty or alkalnty mg The amount of carbon bound n organc compounds. S1 S [ mg / l] S sulfate SO 4 [ mg / l] Methane [ μ g / l] SRBA SRB- A ] Iron s a chemcal element wth the symbol Fe (L.: Ferrum) and atomc number 6. In both cases, smlarly to oxygen measurements, approxmately half of the measurements are ndcated as beyond the detecton lmt, so the dummy varable s appled 1. The smplest hydrocarbon, methane, s a (natural) gas wth a chemcal formula of CH 4. [N Sulphate reducng bactera: type A In the measurements, boundares of the possble nterval of number of bactera were gven. In the analyss use mddle of ths nterval. SRBB SRB- B [N] Sulphate reducng bactera: type B water water level [m] D [m] depth of cover Water levels wth respect to the top of the ground- here the water (fluctuatons are not taken descrbed). 19 TOC- Total organc concentraton 0 A dummy varable s a numercal varable used n regresson analyss to represent subgroups of the sample n a study. In research desgn, a dummy varable s often used to dstngush dfferent treatment groups, n the smplest case takes values 0 or 1 1 Idem Delft Unversty of Technology

59 4 PN [m] ppelne wrt NAP level NAP [m] NAP level of the ground AP average of ppe-to-sol potental measured at the test posts [mv] reported between frst and last nspecton standard devaton of the on potentals measured between frst SP [mv] and the last nspecton WD=D-W [m] amount of water wrt the top of the ppelne Table 1 Mcrobal data descrpton 5.3. Dependent varable The analyss wll be based on the 16 defects for whch t was possble to assocate the corroson rate- only for 16 defects the detaled envronment data was delvered. Small summary of corroson rate data s presented below. Fgure 5-3 Corroson rate summary for 16 measurements ndcated by bo-analysts Mean Std corroson Upper 95% Lower 95% No. meas. corroson rate: rate: conf. nt. conf. nt Table 13: corroson rate summary for 16 measurements ndcated by bo-analysts 5.4 Correlaton analyss Snce all the measurements collected by bo-analysts were measured once, after the last pgrun, so t s mpossble to check the connecton between changes of defect s depths and change of the sol measurements (wrt e.g. groundwater fluctuatons, amount of oxygen etc.) However t s possble to check the connecton between estmated corroson rate and reported bo measurements. Frstly, the analyss wll be nvestgatng the correlaton between the corroson rate and all the varables. Due to low number of measurements the predctve model wll not be very relable. However t s enough to ndcate patterns and relatonshps. Table below presents correlatons and p-values assocated wth followng hypothess testng: The detaled analyss of the average ppe-to-sol potentals was ntroduced n Potental analyss chapter. M.Sc. Thess Lech A. Grzelak

60 43 A statstcal approach to determne the MIC rate of underground gas ppelnes H 0 : ρ ( X, Y ) = 0 aganst H1 : ρ ( X, Y ) 0 where: X- predctor varable, Y- crteron varable (corroson rate), Pearson, ρ S for Spearman and ρ K for Kendall correlaton coeffcents. ρ P - stands for Name no. of samp. ρ P p-value ρ S p-value ρ K p-value oxygen MCA redox cond ph TOC Fe methane SRBA SRBB water S S WD D PN AP SP NAP Table 14 correlaton between corroson rate and the varables, cell wth red background ndcate that normalty assumpton doesn't hold, yellow rows ndcate varables for whch the null hypothess was rejected for sgnfcance level 0. Graphcally, the correlatons between the predctor varables and crteron varable can be expressed n the form of radar graph. Each of the varables from the table corresponds to a ray n the graph below. The varable wth the hghest correlaton s plotted furthest from the center, and the varable wth the lowest respectvely closest to the center. AP SP oxygen MCS redox PN 0.3 cond. 0. D 0.1 ph 0 WD TOC S Fe S1 methane water SRB SRB1 Fgure 5-4 radar graph- Pearson product moment correlaton coeffcent Fgure 5-5 correlatons and assocated p-values (orderng the varables) The results are promsng, for sgnfcance level α =0. eght of varables are sgnfcantly correlated wth corroson rate, for a level 0.05 only two varables have sgnfcantly hgh correlaton. The most correlated wth corroson rate s ph (negatve correlaton) and as second one s redox potental (postve correlaton). If one looks only at two most Delft Unversty of Technology

61 44 correlated varables t s clear that the lowest corroson rate s obtaned for alkalne sol samples wth low redox potental. Interestng s that none of SRB has sgnfcant correlaton wth the corroson rate. If the sgnfcance level s changed to α =0., then addtonal 6 varables ndcate sgnfcant correlaton wth the rate. In four cases the correlaton s negatve: TOC, water level, depth of cover and standard devaton of on potental, and n two postve: ppelne wth respect to NAP and average on-potental. From the data one can observe that the best scenaro for a low corroson rate s when: oxygen s detectable, redox potental s low, sol s alkalne, hgh level of TOC, ppelne s dred (hgh values of W mean that water s deeper under the cover), ppelne s deep under the ground level, ppelne s low wrt. NAP level, the on-potental from rectfer s low (more negatve) and the standard devaton of the potental s relatvely hgh. The suggested best scenaro s based only on smple correlaton analyss; however, t doesn't gve quanttatve results, and t doesn't take nto account correlatons amongst the predctor varables. The key of the analyss s to fnd the predctor varables whch are sgnfcant for the multple regresson model. 5.5 Multple regresson analyss Ths secton s dedcated to the multple regresson model. Multple regresson s a statstcal technque whch allows predctng varables of nterest (sometmes called: dependent or crteron varables) on bass of scores of several other varables (these varables are customary called: ndependent or predctor varables). The man pont n the modelng s to explan the level of the varance on the bass of the level of one or more other varances. The regresson analyss should be based on the predctor varables that mght be (hghly) correlated wth the crteron varable, but not strongly correlated whch each other. In realty correlatons amongst the predctor varables are not unusual. Multcollnearty 3 can cause problems when tryng to fnd the relatve contrbuton of each predctor varable to the modelng. When there s a substantal multcollnearty n a regresson model, t s possble to have the full model account for a substantal amount of the varablty n the dependent varable wthout any tests of ts ndvdual parameters beng sgnfcant. One of the ways to avod ths problem s to apply so called stepwse regresson algorthms Analyss structure The analyss s focused on fndng a set of most nfluental predctor varables wrt corroson rate. Investgaton has to deal wth one very relevant ssue, namely: mssng data (two varables have mssng observatons). Because of low number of the collected measurements, the analyss has to be done n a way that the fnal result s based on number of measurements as large as possble. When the proper model s defned then ncluded varables should be ordered wth respect to mportance for the corroson rate. Fgure 5-6 below presents general dea of the regresson modelng of the corroson rate. 3 Multcollnearty (collnearty)- the term s used to descrbe the stuaton when a hgh correlaton s detected between two or more predctor varables. M.Sc. Thess Lech A. Grzelak

62 45 A statstcal approach to determne the MIC rate of underground gas ppelnes Fgure 5-6 regresson modelng schema for the corroson rate 5.5. Mssng data Two varables SRB-A and SRB-B have mssng observatons at the statonngs 5 [m] and 451 [m]. Due to these mssng data, frstly the analyss has to verfy f these two varables are sgnfcant for the analyss. If they are not sgnfcant then the problem wth mssng data for these two varables doesn t exst anymore (.e. then these two varables can be easly removed and the study can be carred out for all the observatons), f they are sgnfcant, then mssng observatons have to be reproduced or removed (the mssng data for one of the varables would mply removng the correspondng values for all the other nput varables). In order to check f SRB-A and SRB-B are sgnfcant, let s o remove the mssng observatons for all the varables (total number of remanng observatons s now 14) o n order to get a number of the most relevant parameters apply stepwse regresson 4. Stepwse regresson wll be appled to the followng varables (each of the varables has 14 observatons): o Y- varable of nterest- defect rate [mm/yr] o X - ndependent varables: 1 MCA 5 ph 9 S 13 water 17 SP oxygen 6 TOC 10 methane 14 depth 18 WD 3 redox 7 Fe 11 SRBA 15 PN 19 NAP 4 cond 8 S1 1 SRBB 16 AP o n s assocated wth the number of varables whch s equal to 19 o X X j -, j N where j product of centered ndependent varables The tables below descrbe followng hypothess testng: o t-statstcs and p-value for t-test are assocated wth followng hypothess testng: H : ˆ 0 β = 0 aganst alternatve H : ˆ 1 β 0 o F-statstcs and p-value for F-test are assocated wth: 4 See chapter- Analyss methods and nterpretaton, appendx A Delft Unversty of Technology

63 46 H : ˆ ˆ ˆ 0 β 0 = β1 = K = β n = 0 aganst alternatve H : ˆ 1 β 0 Two man models wll be consdered: Model wthout nteractons between varables Model 1 Varables And model statstcs Y β + β X β + ε = n X n Coeffcents βˆ Std. Error t- statstcs p-value for t-test (Constant) ε 5 ph Depth of cover Table 15: parameters and assocated statstcs MODEL R Adjusted R F statstcs Statstcs p-value for F test Table 16: regresson model standard descrpton Model wth nteractons between varables Model Y = β0 + β1x β n X n + β n+ 1X β n X β X X β X X β X X + β 1, n 1 n,3 3, j j n + β X n 1, n 1, n 1 X1X +... X + ε n Varables Coeffcents βˆ Std. Error t- statstcs p-value for t-test (Constant) ε PN*water ε PN ε Oxygen*pH ε MCA*MCA ε Table 17: parameters and assocated statstcs And the model statstcs: MODEL R Adjusted R F statstcs Statstcs p-value for F test ε Table 18: regresson model standard descrpton 5 ε stands for number less than M.Sc. Thess Lech A. Grzelak

64 47 A statstcal approach to determne the MIC rate of underground gas ppelnes Conclusons (based on 14 measurements): From the appled stepwse regresson to the set of 19 varables wth 14 observatons we have that: 1. For the model wthout nteractons: o There s a negatve correlaton between ph and the corroson rate o o There s a negatve correlaton between depth of cover and corroson rate Both SRB-A and SRB-B are nsgnfcant for the corroson rate modelng, hence can be easly removed. For the model wth nteractons: o Also n ths case analyss dd show that both SRB-A and SRB-B are nsgnfcant (even when nteractng wth other varables) o Only one man effect s sgnfcantly affectng the corroson rate- ppelne wrt. NAP For both models R and adjusted R ndcate that the corroson rate s qute well descrbed by the proposed models. Analyss showed that errors from the both models are normally dstrbuted and uncorrelated. Hgh adjusted R ndcated well defned model; however the number of the measurements s not hgh enough to make such concluson Stepwse regresson for ncluded varables In the prevous subsecton t was shown that SRB-A and SRB-B can be removed from the analyss snce they are nsgnfcant for the corroson rate modelng. Here, the analyss wll be based on the remanng varables. The study wll be based on two models ntroduced before: model 1 (wthout nteractons) model (wth nteractons). These two models are appled to the followng dataset: o Y- varable of nterest- defect rate [mm/yr] o X - ndependent varables = 1,..,17 : 1 MCA 5 ph 9 S 13 PN 17 NAP oxygen 6 TOC 10 methane 14 AP 3 redox 7 Fe 11 water 15 SP 4 cond 8 S1 1 depth 16 WD Model wthout nteractons Suppose that the corroson rate s modeled only by man effects- accordng to the Model 1. Stepwse regresson resulted that only one man effect may be an nfluental parameter for the corroson rate. Varables βˆ Coeffcents Std. Error t- statstcs p-value for t-test (Constant) ε PN Table 19: parameters and assocated statstcs (model wthout nteractons) Delft Unversty of Technology

65 48 The determnaton coeffcent- R from the Table 0 shows model ft. Ths means that Model 1 may be too smple to gve relable estmate of the parameters nfluencng the corroson rate. MODEL R Adjusted R F statstcs Statstcs p-value for F test Table 0: regresson model standard descrpton (model wthout nteractons) Model wth nteractons o The model presents much more complcated stuaton, where except man effects (17 varables), all the possble combnatons (136 varables). Stepwse regresson gave the followng results. Varables βˆ Coeffcents Std. Error t- statstcs p-value for t-test (Constant) ε Redox*water ε PN ε Oxygen*pH ε TOC*PN ε MCA*MCA ε Methane*SP ε Table 1: parameters and assocated statstcs (model wth nteractons) MODEL R Adjusted R F statstcs Statstcs p-value for F test ε Table : regresson model standard descrpton (model wth nteractons) Now, varables ncluded n the model are dfferent than for the case wthout nteractons. The fnal model (consstng only of sgnfcant varables) ncludes 6 varables (one man effect and 5 nteracton effects). Hgh value of R ndcates very good ft. It can, however ndcate over-ft because of lmted number of the measurements. As before both models the errors are normally dstrbuted and uncorrelated. 5.6 Senstvty analyss of the parameters nfluencng the corroson rate A senstvty analyss s a process of nvestgatng nfluences of model nputs on outputs. If a small change n a parameter results n relatvely larger changes n the outcomes, then the outcomes are sad to be senstve to that parameter. Ths may mean that the parameter has to be determned very accurately. Bascally, a senstvty analyss s a M.Sc. Thess Lech A. Grzelak

66 49 A statstcal approach to determne the MIC rate of underground gas ppelnes study of how the varaton n the output of a model can be apportoned, qualtatvely or quanttatvely, to dfferent sources of varaton of the nput. One of the most popular and easest senstvty methods s so called Correlaton Rato (CR) 6 whch detaled s presented n the appendx A1. Accordng to ths method certan the level of the polynomal of E ( Y X ) has to be assumed. If we calculate the CR for all varables, then as before t s possble to order the varables accordng to the correlaton rato wrt mportance. The results of the appled technque are followng: Order Predcted varable Correlaton Degree of polynomal for rato condtonal expectaton 1 PN Redox * W TOC* PN Oxygen * PH Methane * SP MCA* MCA Table 3 senstvty analyss of the sgnfcant parameters All the varables n the Table 3 are ordered wth respect to mportance for the model. Appled senstvty analyss showed that the most nfluental/ mportant for the corroson rate modelng s varable- ppelne wrt nap level, then nteractng redox potental wth groundwater step level etc. 6 See appendx A1- Analyss methods and nterpretaton Delft Unversty of Technology

67 Conclusons and recommendatons The analyss showed that the most relevant (statstcally sgnfcant) parameters for the corroson rate are: Ppelne wth respect to NAP level (postve correlaton) Interactons between: o Redox wth water level (negatve correlaton) o TOC wth Ppelne wrt NAP level (postve correlaton) o Oxygen wth ph (postve correlaton) o Methane wth standard devaton of on-potentals (postve correlaton) o MCA wth MCA (negatve correlatons) The presented varables are ordered wth respect to level of the correlaton wth the corroson rate (accordng to senstvty analyss).e. ppelne wrt NAP level s the most mportant, second s nteracton between redox and water level etc. The analyss showed that number of sulphate reducng bactera of type A and B (SRB-A and SRB-B) are nsgnfcant for the analyss. The study proved that there s not only one predomnant factor nfluencng MIC and many dfferent nteractng parameters are nvolved. It was shown that number of observatons strongly nfluences the number of statstcally sgnfcant varables. Dependng on the approach dfferent sets of varables are mportant for the model. Frst approach wth only 14 observatons resulted n two models wth and wthout nteractons. Frst one conssted of two man effects: depth of cover and ph level, and the second one wth one man effect: ppelne wrt NAP level and remanng nteractons: ppelne wrt NAP nteractng wth water level, oxygen wth ph and MCA wth MCA. Both models showed common three parameters as the most mportant- ppelne wrt NAP level, oxygen nteractng wth ph and MCA square. Because of hgh measurement error addtonal observatons are requred. The models wth nteractons llustrate very good ft to the real measurements. Because of lack of the measurements ths perfect ft may ndcate exstng overftng problem whch can be reduced by supportng the model wth hgher number of the envronmental measurements. Recommendatons In the study t was assumed that estmates of the corroson rate from frst secton are certan. Ths assumpton s unlkely to be realstc. The future nvestgatons should be also amed to take the estmate errors nto account. Crucal n the modelng s to have large and accurate dataset. The analyss was based on certan number of possble nfluencng parameters; however those parameters do not exhaust all the possble nfluencng factors- ex. lack of data about groundwater fluctuatons etc. Some of the defects ndcatng decreasng corroson rate, the corroson rate model mposed certan number of constrans, t s lkely that ths phenomenon came out as a result of the clusterng procedure. As a consequence assocaton defect wth the envronment parameters doesn t guarantee good results. The clusterng/matchng defects and assocated errors should be deeper nvestgated. Poor dataset s strongly nfluenced by unusual of observatons; snce mssng data occur t s mportant to know reasons of that- perhaps the defects are n unusual envronment. Non-lnear relatonshps should be nvestgated. M.Sc. Thess Lech A. Grzelak

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69 5 Part III Parameters nfluencng mcrobologcally nduced defect rate M.Sc. Thess Lech A. Grzelak

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71 54 Chapter 6 6 Ppelne characterstcs In ths chapter three hgh-pressure ppelnes: A1, A and A3 are under the analyss. The nvestgatons n the prevous two sectons concerned only the ppelne A3 for whch the set of 5 dstngushable defects was collected. Snce ths secton s dedcated to defect rate modelng, t s not requred any more to be restrcted to one ppelne (for whch t was possble to gve an estmate of the corroson). Two addtonal hgh pressure ppes were chosen accordng to followng features: the exstence of MIC recorded durng the excavatons, -the age of all three ppelnes -the coatng and appled technology. 6.1 Defect dstrbuton The pctures presented below show the ppelnes profle, depth of cover and defects dstrbuted along the ppelnes. In the cases of A1 and A number of defects s much lower than for A3, although the ppelnes were nstalled about the same tme n the 60s. In all three cases the nstallaton customs and appled coatng (btumen) were generally the same. Ths may ndcate that the dfference between the numbers of defects can be caused by envronmental factors. Also n the case of parallel ppelnes A1 and A, the profles are very smlar; however number of the defects for A s sgnfcantly hgher. The analyss starts wth verfyng the hypothess about defects random (unform) dstrbuton along the ppelne. In all three cases there are statstcal bass to reject the hypothess that the defects are unformly dstrbuted along the ppelne. M.Sc. Thess Lech A. Grzelak

72 55 A statstcal approach to determne the MIC rate of underground gas ppelnes Ppelne A1 Fgure 6-1: Ppelne A1, profle, depth of cover, defect dstrbuton Ppelne A Fgure 6-: ppelne A, profle, depth of cover, defect dstrbuton Delft Unversty of Technology

73 56 Ppelne A3 Fgure 6-3: ppelne A3, profle, depth of cover, defect dstrbuton (PII) 6. Depth of cover Fgures presented above don t depct that there s any connecton between the defects exstence and depth of cover. Moreover for the smlar ppelnes profles of A1 and A the pattern of the defects dstrbutons s smlar wrt number of defects and ther statonngs. 6.3 Ppelne elevaton For the ppelne A3 the profle ndcates that the ppelne s lad n lowland (max elevaton s about 5-6 meters); whereas for A1 and A ppelne profles changes about 0 meters wthn few hundred meters. The dstrbuton of the defects may be caused by the groundwater levels (ppelne whch s hgher wrt NAP level s less lkely to be n wet envronment than ppelne closer to the reference NAP level). Graphs below present how the defects are dstrbuted wrt ppelne crcumference. In all the cases most of the defects are concentrated n the bottom of the ppelne. It s not certan that three ppelnes under analyss are nduced by MIC; however the number of excavatons showed that ths s really the case. Moreover, most of excavated defects wth MIC were on the bottom of the ppelne. M.Sc. Thess Lech A. Grzelak

74 57 A statstcal approach to determne the MIC rate of underground gas ppelnes Ppelne A1 Ppelne A Fgure 6-4: ppelne A1 defect dstrbuton wrt ppelne crcumference Fgure 6-5: ppelne A defect dstrbuton wrt ppelne crcumference Ppelne A3 Fgure 6-6 ppelne A3 defect dstrbuton wrt ppelne crcumference Delft Unversty of Technology

75 58 Chapter 7 7 Sol data analyss 7.1 Introducton Ths chapter s focused on fndng the relaton between the sol composton and corroson defect rate for the hgh pressure underground gas ppelnes. The sol data analyss s performed on three ppelnes where mcrobologcally nfluenced corroson was detected. In all three cases the sol data was collected from a geotechncal surveys performed before ppelnes constructon. 7. Descrpton of avalable dataset Durng nspectons, an ntellgent pg reports defects and assocated defects statonng. Due to technologcal drawbacks the statonng of defects and defect feature type are not accurate. The analyss n ths chapter s amed at statstcally sgnfcant number of envronmental parameters nfluencng the defect rate. Hence the data about envronment has to be ncorporated. The most relable nformaton about the ppelne natural envronment s one obtaned from ppelne geotechncal surveys and presented on route maps. The data about the ppelne sol type was collected before ppelne nstallaton. Each route map represents certan part of the ppelne route. Usually the length of the route maps s about km. These maps present ground elevaton (ref. NAP) where the ppelne s placed, but not the ppelne s profle. A ppelne s profle s avalable on PMS 7 so match can be easly done. Each map presents about 5-5 sol samples spread wthn a route map. In most cases t s mpossble to fnd exact statonng of the sol samples wthn a map 8. However the approxmaton can be done based on the fact that each map s spted n few parts (usually 5-10) and the statonngs of the boundares are gven. 7 Ppelne Integrty Management System 8 Exact statonng of the measurements was gven only for the ppelne A1 and A3 M.Sc. Thess Lech A. Grzelak

76 59 A statstcal approach to determne the MIC rate of underground gas ppelnes The statonngs of the sol samples collected from the maps are not certan. The man reasons of the uncertanty are followng: re-routng of the ppelne.e. n some places, because of the nfrastructure, some ppelnes had to be rerouted the startng pont for a pg and zero meter of the frst route map are not the same so the calbraton s always requred Each of collected samples presented n the route maps s n the form where sol layers can be dstngushed. However, accordng to the nstallaton customs of the 60s a sol layer durng backfll of the ppelne trench were mxed, hence there s no pont of analyzng nfluence of the sol layers on corroson. There may stll be an nfluence of sol layers, but due to lack of exact data such analyss won t be carred out Sol data collecton Essental n a defect rate modelng s to specfy envronments along the ppelne. As t was mentoned, each of the route maps shows certan number of measurements. Snce the sol samples are dstrbuted along the ppelne, certan assumptons about the envronments between the measurements have to be ntroduced. Suppose that at a certan part of the ppelne two sol samples were collected. The assumpton that requres to be mposed s about the sol type n between the collected measurements. If an ntellgent pg reports a defect somewhere n between the collected samples, then the problem s to decde n whch envronment defect s observed. Fgure 7-1, Fgure 7- and Fgure 7-3 below present example of proposed procedure. Each of pctures presents a route map, where the samples and some nner statonngs (not of sol sample) are reported. Each route map s dvded nto sectons for whch she statonng s known. Along each secton sol samples are presented. The problem s that the statonng of the samples s unknown. Fgure 7-1: route map, geotechncal data Step 1 Snce the statonng of the measurements s unknown wthn the sectons, so let s assume that the measurements are equal-dstance dstrbuted wthn each secton. And so, for the frst secton 0 [m] [m] the dstance between the measurements s equal 9 Sol layers from the route maps do not show today s layers state. Delft Unversty of Technology

77 to c, and for example for the secton two [m] [m] the dstance s equal to b. Snce now, each of the measurements wll have specfed statonng (before ths was unknown). 60 Fgure 7-: route map, geotechncal data- STEP 1 Step Second and the fnal step, defnes the envronments (also called sol clusters). Snce the statonngs of the measurements are specfed (at step 1) the sectons have to be combned. Sequentally, to get the envronment map of the whole ppelne the boundary measurements from each secton have to be defned as well. In ths procedure, t s assumed that for each two neghborng sectons the boundary between the closest samples for these sectons s n the mddle of them. On the schema the clusters are ndcated by dfferent colors. Fgure 7-3: route map, geotechncal data- STEP Applcaton of the presented procedure defnes sol type for any gven statonng. Because of the lack of quanttatve measurements t s mpossble to avod sharp boundares between the sectons/clusters. The number of measurements vares from one ppelne to another; small summary of avalable data s presented n the Table 4 below. M.Sc. Thess Lech A. Grzelak

78 61 A statstcal approach to determne the MIC rate of underground gas ppelnes ppelne no. of sol measurements no. of route maps length of the ppelne average cluster length number of measurements per km A [km] 80 [m] 3.5 A [km] 100 [m] 10 A [km] 40 [m] 4 Table 4: general ppelnes descrpton Frst glance at the Table 4 shows that the hghest accuracy for sol samples s obtaned for the ppelne A average cluster length s 100 meters and t s more than two tmes less than for other ppelnes. The accuracy of the obtaned results for a ppelne A s much hgher than for the others. The data avalable, doesn't allow analyzng the sol samples quanttatvely.e. t s mpossble to say f n the sol sample s more one component or another. Remark The route maps are connected by usng the same methodology as for combnng the secton wthn a route map. So for step the route map boundares are gnored. Introduced procedure allows descrbng a sol composton for each of the ppelnes usng the data presented by geotechncal data from the route maps. Pctures below show results of appled algorthm. Sol type ppelne A1 Sol type ppelne A Fgure 7-4: Sol composton for the ppelne A1 Fgure 7-5: Sol composton for the ppelne A Delft Unversty of Technology

79 6 Sol type ppelne A3 Fgure 7-6: Sol composton for the ppelne A3 All three pctures Fgure 7-4, Fgure 7-5, and Fgure 7-6 present both: defects dstrbuton and exstence of the each sol component 30. The top plot presents the statonng of the defects reported by an ntellgent pg and the depth of the defects. Each of the pctures beneath s assocated wth the sol component. Red color ndcates exstence of peat, blue of sand, lght green of clay and lght blue loam. Each of the vertcal lnes ndcates a sample cluster (defned before) where presence of peat, sand, loam or clay was ponted out. The fgures llustrate that almost whole the ppelne s lad n sand mxed wth other elements. It reasonable to conclude that for the ppelnes A1 and A the most of the defects are n the area where sol type s a mxture of sand and peat. For the ppelne A3 there s no clear pattern of relatonshps. Another outlne s that n the mddle of the ppelnes A and A1 hgh concentraton of clay and loam wth sand s assocated wth relatvely low number of defects. From the collected data t s clear that four presented sol types do not exhaust all the possbltes. All the possble types of the sol whch can be observed solely from the geotechncal data are: peat, sand, clay, loam, peat-sand, peat-clay, peat-loam, sandclay, sand-loam, clay-loam, peat-sand-clay, peat-sand-loam, peat-clay-loam, sand-clayloam and the last one peat-sand-clay-loam. Each of the sol types has to be analyzed wrt nfluence on the defect rate. Remarks Ppelne A: mssng sol data for statonng 11m- 673 m Ppelne A3: mssng sol data for the frst 3135 m- ths mssng data s recovered from the A1 whch s parallel to A3 7.4 Sol type nfluence on defect rate The man pont of the study n ths subchapter s to check what the defect rate for whole the ppelne s and to verfy f the defect rate depends on the sol type. The plan s to 30 Man components are: sand, peat, clay and loam. The sol samples conssts ether from man components or components combnatons. M.Sc. Thess Lech A. Grzelak

80 63 A statstcal approach to determne the MIC rate of underground gas ppelnes count the number of the defect assocated wth each possble envronment (sol type) and dvde ths by the total length where each specfc sol type was observed. The overall defect rate for the ppelnes s presented beneath n the Table 5. ppelne no. of defects length of the ppelne [km] Overall defect rate [def/km] A A A Table 5 Overall defect rate for the ppelnes Two dfferent approaches on defects rate wrt sol type are presented beneath. Each of the approaches presents dfferent pont of vew on modelng Defect rate- Approach 1 The frst approach assocates the defect rate wth each possble sol type. The assumpton whch s gong to be mposed on the analyss s of followng form: Assumpton Assume that qualty of a coatng of any ppelne s deterorated at the same level.e. condton of t s unform along the whole ppelne length. Delft Unversty of Technology

81 Table 6 shows the results for all three ppelnes. Sol type Length of ppelne exposed to the sol type Percentage of ppelne exposed to the sol type Ppelne A1 [km] [%] Number of defects Defect RATE [no. of defects per km] Peat Sand Clay Loam Peat- Sand Peat- Clay Peat- Loam Sand- Clay Sand- Loam Clay- Loam Peat- Clay- Loam Peat- Sand- Clay Peat- Sand- Loam Sand- Clay- Loam Peat- Sand- Clay- Loam TOTAL 83.9 [km] 100% 9 defects - 64 Ppelne A Ppelne A3 Peat Sand Clay Loam Peat- Sand Peat- Clay Peat- Loam Sand- Clay Sand- Loam Clay- Loam Peat- Clay- Loam Peat- Sand- Clay Peat- Sand- Loam Sand- Clay- Loam Peat- Sand- Clay- Loam TOTAL 84.5 [km] 100% 37 defects 31 - Peat Sand Clay Loam Peat- Sand Peat- Clay Peat- Loam Sand- Clay Sand- Loam Clay- Loam Peat- Clay- Loam Peat- Sand- Clay Peat- Sand- Loam Sand- Clay- Loam Peat- Sand- Clay- Loam TOTAL 68.6 [km] 100% 656 defects - Table 6 defect rate for each specfc sol type 31 Number of defects n the envronment where sol data s mssng s 30. M.Sc. Thess Lech A. Grzelak

82 65 A statstcal approach to determne the MIC rate of underground gas ppelnes Fgure 7-7: Defect rate assocated for each possble sol composton Fgure 7-7 presents the results from the Table 6. Due to hgh number of the defects dstrbuted along the ppelne A3 the defect rate for each of the sol types s much hgher than for the remanng ppelnes. In four cases: sand, peat-sand, sand-clay, and peatsand-clay- loam both ppelnes A1 and A have postve defect rate. The hghest defect rate s obtaned: A1 for peat-sand-clay-loam, A for peat-sand-loam, A3 for sand-clay. In classfy and check f the estmated defects rates are relable or not, t s requred s to test how the sol type s dstrbuted n potentally defectve and no-defectve envronment. Suppose that the sol type A occurs n two dfferent parts of the ppelne, but defects are only observable n one of them (the second has none of defects). Then the defect rate won t be a relable tool to say whch sol type s more lkely to be more defectve. The addtonal nformaton that has to be delvered s the nformaton about changes of percentage content of the sol type A where the defects are observable and where they are not. To check how much of each sol type s n potentally defectve envronment and how much s not, the assumpton about a potentally defectve envronment has to be mpressed. Suppose that defect j was regstered at a certan statonng S j [km]. The envronment for ths defect s defned as the area surroundng the defects wthn predefned radus r L = S r, S r. The radus surroundng [km] and thus the envronment for defect s [ ] Delft Unversty of Technology j j j + the defects s chosen to be equal to 50 [m]. Ths length s motvated by average length of the cluster for all the ppelnes. The total length of the envronments where all the defects were regstered s then defned n followng way: If defects clusters L j are not dsjont then total length of potentally defectve envronment s: T = L1 + L Ln L1L L1 L3... L1 Ln LL3... LLn Ln 1Ln + L1 LL ± L1 L... Ln ± next to the last terms means: - f n s even, and + f n s odd. Table 7 shows length of potentally defectve and not defectve envronments accordng to ntroduced method. ppelne Potentally not total length of the Potentally defectve detectve ppelne [km] envronment [km] envronment [km] A A A Table 7: descrpton of the potentally defectve and not defectve envronments

83 66 Ppelne- A1 Fgure 7-8: ppelne A1, potentally defectve clusters, or clusters wth bad coatng Ppelne- A Fgure 7-9: ppelne A, potentally defectve clusters, or clusters wth bad coatng Ppelne- A3 Fgure 7-10: ppelne A3, potentally defectve clusters, or clusters wth bad coatng In order to compare the dfference n sol composton between potentally defectve and not defectve envronment, let s defne: D - potentally defectve area (on the Fgure 7-8, Fgure 7-9 and Fgure 7-10 D t s a set of ntervals ndcated by red bars) c D - area defned as area complementary to D M.Sc. Thess Lech A. Grzelak

84 67 A statstcal approach to determne the MIC rate of underground gas ppelnes A - sol type A, then: c A D A D Dfference = c D D Table 8 below shows percentage content of these two envronments: Ppelne- A1 Ppelne- A Sol type DEFECTIVE- Envronment NOT DEFECTIVE- Envronment Dfference % [km] % [km] % Peat Sand Clay Loam Peat- Sand Peat- Clay Peat- Loam Sand- Clay Sand- Loam Clay- Loam Peat- Clay- Loam Peat- Sand- Clay Peat- Sand- Loam Sand- Clay- Loam Peat- Sand- Clay- Loam TOTAL Peat Sand Clay Loam Peat- Sand Peat- Clay Peat- Loam Sand- Clay Sand- Loam Clay- Loam Peat- Clay- Loam Peat- Sand- Clay Peat- Sand- Loam Sand- Clay- Loam Peat- Sand- Clay- Loam TOTAL Ppelne- A3 Peat Sand Clay Loam Peat- Sand Peat- Clay Peat- Loam Sand- Clay Sand- Loam Clay- Loam Peat- Clay- Loam Peat- Sand- Clay Peat- Sand- Loam Sand- Clay- Loam Peat- Sand- Clay- Loam TOTAL Table 8: comparson of potentally defectve and not defectve envronments Delft Unversty of Technology

85 68 Ppelne- A1 Fgure 7-11: percentage dfference between sol composton vs. defect rate (ppelne A1) Ppelne- A Fgure 7-1: percentage dfference between sol composton vs. defect rate (ppelne A) Ppelne- A3 Fgure 7-13: percentage dfference between sol composton vs. defect rate (ppelne A3) The Fgures Fgure 7-11, Fgure 7-1 and Fgure 7-13 show the relaton between dfferences n sol composton for potentally defectve and not defectve envronments. Other sol types ndcates all the remanng sol types. Accordng to the establshed technque, the general concluson s that t s dffcult to fnd clear pattern combnng data from all the ppelnes. However, certan matches are vsble: for both A1 and A there s much more peat-sand n potentally defectve envronments and much more sand and sand-loam n potentally not-defectve envronment. There s no clear match between A3 and the others. M.Sc. Thess Lech A. Grzelak

86 69 A statstcal approach to determne the MIC rate of underground gas ppelnes 7.4. Defect rate- Approach Before t was assumed that envronment wthout defects s potentally not corrosve envronment. Ths mght be not the case. Many of excavatons showed that exstence of mcrobal corroson was assocated wth coatng damage. Here, dfferent way of lookng at the defect rate estmaton s presented. In the prevous secton unform coatng along the ppelne was assumed. Now, the assumpton s followng: Assumpton Assume that the ppelne conssts of two dfferent qualtes of the coatng good (wthout defects) and bad (wth defects). Bad coatng of the ppelne s defned n the same way as for potentally defectve envronment from prevous secton. The second approach tres analyzng the ppelnes defect rate only n the sectons where bad coatng was appled.. Below n the Table 9, summary of the results s presented. Sol type Ppelne A1 Ppelne A Ppelne A3 Exposure of bad coatng Def. rate for bad coatng Exposure of bad coatng Def. rate for bad coatng Exposure of bad coatng Def. rate for bad coatng [km] [def/km] [km] [def/km] [km] [def/km] Peat Sand Clay Loam Peat- Sand Peat- Clay Peat- Loam Sand- Clay Sand- Loam Clay- Loam Peat- Clay- Loam Peat- Sand- Clay Peat- Sand- Loam Sand- Clay- Loam Peat- Sand- Clay- Loam TOTAL Table 9: defect rate for sectons where bad coatng was appled Fgure 7-14 defect rate for sectons where bad coatng was appled vs. sol types Delft Unversty of Technology

87 The hghest defect rate for ppelne A1 s obtaned for sand, A sand-clay-loam and A3 for sand-clay Correlaton analyss Technques ntroduced before allow checkng what the correlatons between all the ppelnes wrt sol types and corroson defects are Correlaton between sol exposures Let s defne: X A1, X A, X A3 - percentage of the ppelne exposed to every sol type for the ppelnes A1, A and A3, ( th coordnate of the vector X A... descrbes percentage of the ppelne exposed to th sol type, number of s s 13 3 ) Ppelne ρ (Spearman corr) p-value ρ P (Pearson corr) 33 p-value S A1-A A1-A A-A Table 30: correlaton between sol compostons of the ppelnes p-value n the table s assocated wth followng null hypothess H 0 : ρ ( X, X j ) = 0 aganst H1 : ρ ( X, X j ) Correlaton between defect rates wrt sol types Let s defne: X A1, X A, X A3 - -defect rate vectors for the ppelnes A1, A and A3 ( th coordnate of the vector X A... descrbes defect rate for th sol type, number of s s 13 3 ) Ppelne ρ (Spearman corr) p-value ρ P (Pearson corr) 33 p-value S A1-A A1-A A-A Table 31: correlaton between defect rates of the sol types for ppelnes Correlaton between defect rates wrt sol types where bad coatng was assumed X, X A, X A3 - -defect rate vectors for the ppelnes A1, A and A3 for the areas of the ppelnes where bad coatng was assumed ( th coordnate of the vector X A... descrbes defect rate for th sol type wthn the ppelne where bad coatng was used, number of s s 13 3 ) A1 Ppelne ρ P (Pearson corr) 33 p-value ρ S (Spearman corr) p-value A1-A A1-A A-A Table 3: correlaton between defect rates of the sol types for ppelnes, n the clusters where bad coatng was appled 70 3 The number of all possble component combnatons s 15; however peat-loam and clay-loam are not regstered at all. 33 Normalty condton for Pearson correlaton coeffcent s satsfed M.Sc. Thess Lech A. Grzelak

88 71 A statstcal approach to determne the MIC rate of underground gas ppelnes 7.5 Conclusons and recommendatons The analyss n ths chapter was based on three hgh pressure ppelnes for whch excavatons showed exstence of MIC. Conclusons Overall conclusons from presented approaches are followng. there s no sgnfcant correlaton between defect rates and sol types there s no sgnfcant correlaton between defect rates and sol types n the places where bad coatng was assumed the hghest defect rates are o all the ppelne ppelne A1: n mxture of peat-sand-clay-loam (3.75 [def/km] t s about 41% of all the corroson defect rates) ppelne A: n mxture of peat-sand-loam (8.46 [def/km] -46% of total corroson defect rates) o ppelne A3: n mxture of sand-clay (35.5 [def/km]- 6%) ppelne n the area wth bad coatng was assumed ppelne A1: n sand (6.0 [def/km]- 7%) ppelne A: n mxture of peat-sand-loam (.75 [def/km]- 30%) ppelne A3: n mxture of sand-clay ( [def/km]- 7%) For both the ppelnes A1 and A there s much more peat-sand, and peat-sandclay-loam and much less sand and sand-loam n the areas where bad coatng was assumed Ppelne A3 doesn't show any pattern wrt the other ppelnes An explanaton for lack of correlatons (hgh uncertanty) may be due to hgh uncertanty of the sol measurements lack of data on coatng qualty- certan assumptons about good and bad coatng had to be mposed lack of data about quanttatve amount of sol components n the sol samples Recommendatons: The analyss was based on three ppelnes, f ths s the case, all the ppelnes where MIC was reported should be analyzed n order to get better pattern. Deeper nvestgaton of assumpton about good and bad coatng s requred. Delft Unversty of Technology

89 7 Chapter 8 8 Water table analyss 8.1 Introducton Ths chapter s dedcated to groundwater table (level) analyss. The data on water levels were collected from the sol map of the Netherlands. The maps show the average mn and max ground water levels. Unfortunately, the data are not accurate, manly because of followng features: only rough estmate s presented (groundwater level step classfcaton data), maps do not present exact water level for a specfed statonng. The data were not delvered n a dgtal form but were collected drectly from the water table map by the author. The legend n the maps s followng: Groundwater level step classfcaton data Groundwater step I II III IV V VI VII Average of the hghest groundwater level n cm below - - <40 >40 < >80 the ground Average of the lowest groundwater level n cm below < >10 >10 >10 the ground Table 33: ground water step levels (legend) The data was collected smply by projectng the ppelne profle on the groundwater level map and the values of groundwater step levels were collected. The data shows that t s qute dffcult to ndcate precsely what the groundwater level for a gven statonng s. Ths s mostly due to ppelne shape (t s dffcult to calculate ppelne s length usng only the map), and the map tself (maps are constructed based on contour plots). M.Sc. Thess Lech A. Grzelak

90 73 A statstcal approach to determne the MIC rate of underground gas ppelnes For smplfcaton the ppelnes were dvded nto.5 [km] long sectons. For each of the secton average groundwater class was calculated. The analyss s amed to check the relatonshp between the number of defects and the water level for the assocated sectons. 8. Approach 1 The fgures below: Fgure 8-1, Fgure 8- and Fgure 8-3 consst of two parts. The plot on the top shows the number of defects per each segment of length.5 [km], and the bottom plot presents the groundwater step level for the correspondng secton. From the avalable maps t was dffcult to assocate groundwater step levels for narrower sectons..ppelne A1 Fgure 8-1: A1- no. of defects vs. water level per.5 km long sectons Ppelne- A Fgure 8- A- no. of defects vs. water level per.5 km long sectons Delft Unversty of Technology

91 74 Ppelne A3 Fgure 8-3: A3- no. of defects vs. water level per.5 km long sectons Table 34 shows summary of avalable groundwater level data. Ppelne Number of samples Mnmum Maxmum Average A A A Table 34: groundwater step level summary per each secton of.5 km Number of Std. Ppelne Mnmum Maxmum Mean samples Devaton A A A Table 35 defect rate summary- per each secton of.5 km Fgure 8-1 and Fgure 8- show that n the places where groundwater level s hgh (t s ndcated by low step number) the number of the defects s also hgh. In order to verfy ths pattern for all the ppelnes let s defne: X- defect rate per each secton of.5 [km] Y- average groundwater step level for each.5 [km] long secton The hypothess that has to be tested s of the followng form: H 0 : ρ ( X, Y ) = 0 aganst H1 : ρ ( X, Y ) 0 Table 36 below shows the results for correlatons: ρ P - Pearson correlaton, ρ S - Spearman correlaton and ρ K - Kendall correlaton, and also assocated p-values for all the ppelnes. M.Sc. Thess Lech A. Grzelak

92 75 A statstcal approach to determne the MIC rate of underground gas ppelnes Ppelne ρ P p-value ρ S p-value ρ K p-value A A A Table 36 correlaton between defect rate and water level for sectons of.5 km The results presented n the table confrm the suspcon. The groundwater step level for the ppelnes A1 and A s negatvely correlated wth the defect rate. However ths correlaton s not statstcally sgnfcant. For A3 there s no sgnfcant relatonshp between the defect rate and groundwater step level. Overall concluson s followng: accordng to the ntroduced methodology there s weak correlaton between defect rate and groundwater step level. 8.3 Approach For the ppelnes A1 and A t seems that there are many areas where zero or only one defect was reported. So the ppelne wll be analyzed wth respect to these sectons. The frst feature wll be ndcated as presence of the defects, and the second by defects absence. Let s defne: X- groundwater step level vector for area where only 0 or 1 defect was observed Y- groundwater step level vector for area where more than 1 defects were observed The task s to check f the dfference between averages of groundwater levels for defned varables s statstcally sgnfcant. So defne the hypothess as: H : X Y = 0 X = Y H : X 0 aganst the alternatve 1 Y ppelne Varables df t-statstc p-value for t- test Lower 95% bound (for dfference) Upper 95% bound (for dfference) A1 X-Y A X-Y Table 37: statstcs and confdence bounds for estmate (ppelne A1 and A) The results show that for a sgnfcance level α = the null hypothess cannot be rejected. It means that averages of groundwater step levels of the envronments wth and wthout defects are not statstcally dfferent. Beneath n the Table 38 small descrpton of the groundwater levels for defectve and no defectve envronments s presented. ppelne A1 A Descrptve Statstc Number of samples Mn. groundwater step Max. groundwater step Mean groundwater step X Y X Y Table 38: descrptve statstcs (ppelnes A1 and A) 34 The analyss showed that the normalty assumpton for the varable number of defects per cluster s not satsfed. Delft Unversty of Technology

93 76 The results from the presented dea dd show nsgnfcant relatonshps. 8.4 Approach 3 Next approach whch can be appled to the water table analyss s carred out by codng the groundwater steps levels. The Table 33 shows that frst fve steps (I, II, III, IV, V) can be assocated wth hgh groundwater level (and coded as 0) and other two (VI and VII) wth low groundwater level (and coded as 1). The Fgure 8-4, Fgure 8-5, and Fgure 8-6 below present how coded average groundwater step levels are assocated wth the number of defects for each.5 km long clusters..ppelne A1 Fgure 8-4: number of defects vs. coded groundwater level (ppelne A1) Ppelne- A Fgure 8-5: number of defects vs. coded groundwater level (ppelne A) M.Sc. Thess Lech A. Grzelak

94 77 A statstcal approach to determne the MIC rate of underground gas ppelnes Ppelne A3 Fgure 8-6: number of defects vs. coded groundwater level (ppelne A3) The hypothess whch has to be verfed s followng: the average number of defects for average groundwater step level coded by 0 and 1 s sgnfcantly dfferent aganst alternatve that t s not. Necessary defntons are followng: X- number of defects where average groundwater step level s coded as 0 Y- number of defects where average groundwater step level s coded as 1 A mathematcal formulaton of hypothess s followng: H 0 : X Y = 0 X = Y aganst the alternatve H1 : X Y ppelne Varables df t-statstc p-value for t- test Lower 95% bound (for dfference) Upper 95% bound (for dfference) A1 X&Y A X&Y A3 X&Y Table 39: statstcs and confdence bounds for estmate (ppelnes A1, A and A3) ppelne A1 A A3 Descrptve Statstc Number of samples Mn. no. of defects Max. no. of defects Average no. of defects X Y X Y X Y Table 40: descrptve statstcs (ppelnes A1, A and A3) The Table 39 and Table 40 show that for A1 and A the average number of defects where groundwater step level s 0 (groundwater level s hgh) s hgher than for the area where groundwater step level s 1 (groundwater s relatvely lower); however ths dfference s not statstcally sgnfcant 35. In the case of A3 the there s no evdence to dstngush between the numbers of defects for coded groundwater levels. Also ths approach showed weak correlaton between groundwater level and defect rate. 35 Ths s due to hgh varablty of the number of defects for groundwater step level coded as 0. Delft Unversty of Technology

95 Conclusons and recommendatons The analyss ddn't show that the groundwater levels n a statstcally sgnfcant way nfluence the number of defects. The results may be not accurate snce the ppelnes were dvded n sectons of.5 [km]. However, the ppelne dvson n smaller segments s a challengng task snce the data s not avalable n an electronc form, but avalable only drectly from the maps. Analyss showed that for ppelnes A1 and A hgher groundwater level s postvely correlated wth the number of defects (more water - more defects); however these results accordng to statstcal evdence are not strong enough. The Table 41 below shows small overall summary of the results. Ppelne Total number of defects average groundwater step level A A 67 5 A Table 41: defects and average groundwater level All three ppelnes were nstalled n the 60s however the number of defects for each of them s very dfferent. Comparson of the number of defects and the average groundwater step level shows that the hghest number of defects for the A3 s assocated wth the hghest groundwater level (lowest groundwater step level). Recommendatons Deeper nvestgaton of the groundwater levels s requred. Because of the ppelne profle t was too dffcult to collect the groundwater level for any gven ppelne statonng. Precse approxmate of the groundwater level for any gven statonng would sgnfcantly ncrease the accuracy of the results. M.Sc. Thess Lech A. Grzelak

96

97 80 Chapter 9 9 Factors nfluencng defect rate 9.1 Introducton Ths chapter proposes the way of the defect rate modelng gven all avalable measurements. Factors whch are avalable are: groundwater level, sol types, NAP level of the ground, NAP level of the ppelne and depth of cover The methodology used n ths chapter s based on the regresson analyss where the dependent varable s defect rate and ndependent varables are all the factors whch may nfluence the defect rate. The regresson analyss s based on a number of observatons whch descrbe the varable of nterest. In order to defne such observatons a ppelne dscretzaton s requred. The queston whch has to be answered s n how long segments the ppelne should be dvded. Here, the same as n the prevous chapter water table analyss -- the ppelne wll be dvded n.5 [km] long sectons (accordng to the groundwater level measurements). In ths study two general approaches wll be presented. Frst, let's defne two models. model wthout nteractons Model 3 Y + β X = n X n β β + ε model wth nteractons up to second degree M.Sc. Thess Lech A. Grzelak

98 81 A statstcal approach to determne the MIC rate of underground gas ppelnes Model 4 Y = β0 + β1x β n X n + β n+ 1X β n X β X X β X X β X X + β 1, n 1 n,3 3, j j n + β X n 1, n 1, n 1 X1X +... X + ε n where: o Y- varable of nterest- defect rate [no. of defects per km] calculated for each secton of.5 [km] o X - ndependent varables: peat, sand, clay, loam, peat-sand, peat-clay, peat-loam, sand-clay, sand-loam, clay-loam, peat-clay-loam, peat-sand-clay, peat-sandloam, sand-clay-loam, peat-sand-clay-loam whch are descrbed as percentage content of each secton for each.5 [km] (each value says percentage of each sol type s n each secton ) groundwater step level (calculated as average for each secton of.5 [km]) NAP level of the ground (average for each secton) NAP level of the ppelne (average for each secton) depth of cover (average for each secton) o n s assocated wth the number of varables whch s equal to 19 (accordng to number of avalable varables) 9. Model wthout nteractons For the defned Model 3, n order to get the best possble combnaton of parameters whch descrbe defect rate a stepwse regresson 36 s appled. The Model 3 s concerns each of the ppelnes to check whch parameters for each ppelne are sgnfcantly nfluencng the defect rate. Moreover, the model also s used to the ppelnes combnatons. The dea behnd such combnatons s to fnd the common parameters nfluencng the defect rate for all the ppelnes. Table 4 and Table 43 below show standard statstcs of the estmated coeffcents. Stepwse regresson output s a set of statstcally sgnfcant varables nfluencng crteron varable whch s defect rate. All the varables not ncluded n the tables are nsgnfcant n the modelng. For the ppelne A1 there are no sgnfcant parameters descrbng the defect rate. For the A the most relevant varables are percentage amount of peat-sand and peat, and for A3, sand-clay and sand-loam. For A1 combned wth A the most relevant s peatsand (t means that peat-sand s a common factor nfluencng the defect rate for these two ppelnes). And for combnaton of three ppelnes A1, A and A3 the most relevant varables are percentage amount of sand-loam and NAP level of the ground (these varables are common varables for all the three ppelnes). 36 See chapter: analyss methods and nterpretaton- appendx A Delft Unversty of Technology

99 8 A1 37 The parameters whch are sgnfcant for the defect rate modelng Coeffcents βˆ None of parameters s ncluded Std. Error t- stat. p-value for t-stat. 95% Confdence β Interval for L. U. Bound Bound (Constant) A 38 Peat-sand ε peat (Constant) ε A3 39 Sand-clay Sand-loam (Constant) A1-A 40 Peat-Sand (Constant) ε A1-A-A3 41 NAP level of the ground ε Sand-loam Table 4 Stepwse regresson estmates (model wthout nteractons) The Table 43 below assocates the estmated models and the multple correlaton coeffcent ( R ), whch descrbes level at whch the varance of dependent varable s descrbed. In all the cases R s relatvely low- t doesn't exceed the level of 40%. Ths can be caused by uncertanty about the measurements or by the varables whch are relevant but were not ncluded n the model. PIPELINE Number of obs. Statstcs R Adjusted R F statstcs p-value for F test A A A A1-A A1-A-A ε 4 Table 43: standard model statstcs (model wthout nteractons) 37 A1: The followng varables are constants or have mssng correlatons, so wll be deleted from the analyss: peat-loam, clay-loam 38 A: The followng varables are constants or have mssng correlatons, so wll be deleted from the analyss: Clay, Loam, peat-clay, peat-loam, sand-clay, clay-loam, peat-clay-loam, and peat-sand-clay. 39 A3: The followng varables are constants or have mssng correlatons, so wll be deleted from the analyss: peat-loam, clay-loam, peat-clay-loam 40 A1-A: The followng varables are constants or have mssng correlatons, so wll be deleted from the analyss: peat-loam, clay-loam 41 A1-A-A3: The followng varables are constants or have mssng correlatons, so wll be deleted from the analyss: peat-loam and clay-loam 4 ε stands for number smaller than M.Sc. Thess Lech A. Grzelak

100 83 A statstcal approach to determne the MIC rate of underground gas ppelnes 9.3 Model wth nteractons Second model s a model wth nteractons; t means that except all man effects, also nteractons between effects are taken nto account. Ths s dea s motvated by a well known fact that MIC corroson s not only nfluenced by closed number of man effects. A1 The parameters whch are sgnfcant for the defect rate modelng Coeffcents βˆ Std. Error t- stat. p-value for t-stat. 95% Confdence Interval for β L. Bound U. Bound (Constant) Depth of cover * depth of cover ε (sand-clay-loam) * depth of cover ε Sand * Sand ε ε (Constant) A Peat*(NAP level of the ppelne) Peat-sand (peat-sand) * (NAP level of the ground) (Constant) ε A3 (sand-clay) * (sand-clay-loam) Sand * sand (peat-clay-sand-loam) * (peat-claysand-loam) A1-A (Constant) ε (Peat-sand) * (groundwater step level) ε Peat-sand ε (Constant) ε NAP level of the ground ε A1-A-A3 Sand-loam (Sand-loam) * (NAP level of the ppelne) (Peat-sand ) * (groundwater step level) Sand * sand ε Table 44 Stepwse regresson estmates (model wth nteractons) The Table 44 above shows whch of the varables from the Model 4 are sgnfcant n the defect rate modelng. Interestng s that for the most of the cases (the ppelnes and ppelne combnatons) the most relevant varables ncluded n the model are nteractons. Only n two cases man effects were ncluded n the model. The Table 45 below shows that now R s hgher than before for model wth only man effects. However, the R stll s very low- what ndcates poor correlaton. Delft Unversty of Technology

101 84 PIPELINE Number of obs. R Adjusted R F statstcs Statstcs p-value for F test A ε A ε A ε A1-A ε A1-A-A ε Table 45: standard model statstcs (model wth nteractons) 9.4 Conclusons and recommendatons The analyss showed that much more relevant for the modelng the defect rate s lookng at nteractons than on man effects. Accordng to the ntroduced methodology the sgnfcant parameters (wth strong correlaton) whch descrbe the defect rate are: For the ppelne A1: o sand (nsgnfcant negatve correlaton) o nteractons between sand-clay-loam wth depth of cover (nsgnfcant negatve correlaton) depth of cover wth depth of cover ( md. postve correlaton wth the defect rate) For the ppelne A: o Interactons between: peat and NAP level of the ppelne ( nsgnfcant postve correlaton) peat-sand and NAP level of the ground (md. negatve correlaton) o Peat-sand (md. postve correlaton) For the ppelne A3: o Interactons between: Sand-clay wth sand-clay-loam (md. negatve correlaton) Sand wth sand (md. negatve correlaton) Peat-clay-sand-loam wth peat-clay-sand-loam (nsgnfcant negatve correlaton) Common factors for A1 and A o Peat- sand (md. postve correlaton) o Interacton between Peat-sand wth peat-sand-loam (md. postve correlaton) Common factors for A1, A and A3 o NAP level of the ground (md. negatve correlaton) o Sand-loam ( nsgnfcant correlaton) o Interactons between: Sand-loam wth NAP level of the ppelne (weak negatve correlaton) Peat-sand wth groundwater step level (weak postve correlaton) Sand wth sand (nsgnfcant negatve correlaton) M.Sc. Thess Lech A. Grzelak

102 85 A statstcal approach to determne the MIC rate of underground gas ppelnes Recommendatons The analyss showed that the parameters ncluded nto regresson model do not fully descrbe defect rate - ths s ndcated by low R. Two man reasons whch have to be deeper nvestgated are: o measurement error- there s a lot of uncertanty n the data, the uncertanty can be reduced by analyzng other ppelnes where MIC was reported n order to get more general common factors nfluencng MIC defect rate o other nfluencng factors whch were not taken nto account (ths should be verfed as well) o ppelnes were dvded n.5 km long sectons accordng to groundwater data, further nvestgaton should be amng at gettng more narrower sectons Delft Unversty of Technology

103 86 Chapter Conclusons The thess conducts an analyss on corroson modelng for underground gas ppelnes n the Netherlands. Each of the dvsons ncorporates certan knowledge about corroson. Furthermore all the parts combned together delver nformaton about the whole process of corroson rate/defect modelng. Frst part showed the procedure of the corroson rate modelng when low number of nspectons s avalable. Implemented model shows that n order to determne relable uncertanty and bas about MFL-pgs t s very mportant to have multple reference defects n the ppelne, defects for whch the real dmensons are well known (ex. from excavatons). Because of the measurement uncertanty of the MFL-tool t domnant compared to the corroson growth n the tme perod between the pgruns, t s very dffcult to determne a relable corroson rate per defect. Model proposed n frst secton showed the way to calbrate all the nspectng pgs and how to derve physcally acceptable functonal descrpton of corroson growth. Assumng constant corroson rate the model estmated average corroson rate of level 0.4 [mm/yr] wth upper bound value of 0.6 [mm/yr]. The assumpton that the corroson process s lnear was underpnned by the analyses of two subsets: deep and shallow defects. For both subsets a smlar average corroson rate was calculated: 0.3 mm/yr and 0.5 mm/yr. The dfference s not sgnfcant. Snce the corroson rates have been determned, Gasune wll have to decde how to use these values for the calculaton of a re-nspecton nterval for ths lne. The calculaton wll probably be done n a determnstc way for every defect, takng nto account measurement uncertantes and the uncertanty of the corroson rate. Second secton of the thess ncorporated results from the prevous one n order to fnd factors nfluencng the corroson rate. The analyss was performed based partally on bo-assessed measurements and partally usng ppelne ntegrty management system. M.Sc. Thess Lech A. Grzelak

104 87 A statstcal approach to determne the MIC rate of underground gas ppelnes The study based on regresson analyss showed mportance of havng addtonal measurements. It has been shown that much more nfluencng for the corroson rate are nteractons between the varables than man effects. For ncorporated set of 19 varables, two of them (SRB-A and SRB-B) were nsgnfcant and due to mssng data were removed from further analyss. Accordng to appled senstvty measures the most nfluental for the corroson rate s varable descrbng ppelne wrt NAP level (postve correlaton), then accordngly to mportance: nteractons between redox and water level (negatve correlaton), TOC and ppelne wrt NAP level (postve correlaton), oxygen and ph (postve correlaton), methane and SP (postve correlaton) and fnal one MCA squared (negatve correlaton). The number of avalable observatons plays crucal role n the modelng. Most analysts recommend that one should have at least 10 to 0 tmes as many observatons as one has varables, otherwse the estmates of the regresson are probably very unstable and unlkely to replcate f one were to do the study over. In the study number of ncluded varables was 5 and 16 was a number observatons. So, clearly fnal results cannot be used as relable predctve tool. Thrd and the last secton demonstrated the deas of defect rate modelng for a MIC nfluenced ppelnes. Three ppelnes affected by MIC were analyzed. Frstly, the sol type analyss was performed. Two of the ppelnes A1 and A showed that n the areas where bad coatng was assumed s much more peat-sand, and peat-sand-clay-loam and much less sand and sand-loam. Thrd ppelne A3 ddn't show any sgnfcant pattern snce the a whole ppelne s affected by corroson. Appled correlaton analyss ddn't show sgnfcant correlatons between sol types and defect rates for all the ppelnes. Because of lack relable measurements also groundwater analyss ddn't show any sgnfcant correlatons, between groundwater levels and number of defects. However, all the data combned together and appled to regresson analyss showed certan patterns. Smlarly lke for the corroson rate, the defect rate s much stronger nfluenced by nteractons than by man effects. The defect rate modeled n ths chapter showed that common factors nfluencng all three ppelnes, A1, A and A3 are: NAP level of the ground (negatve correlaton), and sand mxed wth loam (postve correlaton) then nteractons between: mxture of sand-loam and NAP level of the ground (negatve correlaton), mxture of peat-sand and groundwater step level (postve correlaton) and sand squared (negatve correlaton). It was showed that ppelne A3 s much more dfferent from the others. The parameters nfluencng the defect rate for both remanng ppelnes A1 and A are: peat mxed wth sand (postve correlaton) and nteracton between mxture peat-sand and groundwater step level (postve correlaton). Delft Unversty of Technology

105 88 Bblography [1] POF specfcatons [] Worthngham R.G., Fenyves L.L., Morrson T.B., Desjardns G. J., Analyss of corroson rates on a gas transmsson ppelne, NACExpo/Conference 00 Techncal Paper [3] Bhata A., Mangat N.S, Morrson T, Estmaton of measurement errors. [4] Desjardns G., Corroson rate and severty results from n-lne-inspecton data, ACE Corroson 001 paper 0164 [5] Worthngham R.G., Morrson T, Mangat N.S., Desjardns G., Bayesan estmates of measurement errors for In-lne nspecton and feld tools, IPC [6] Fenyves L, Mller S., Determnng corroson growth, Ppelne and Gas technology October 005, page -30. [7] Grzelak L. A., Achterbosch G. G. J., Determnaton of the corroson rate of a MIC nfluenced ppelne usng 4 consecutve pgruns, Internatonal Ppelne Conference, IPC [8] Bjen, E.J., Cluster Analyss, Tlburg Unversty Press 1973 [9] Späth, H., Cluster Analyss Algorthms for Data Reducton and Classfcaton of Objects, Chchster, Horwood, 1980 [10] Hartgan, J.A., Clusterng Algorthms, New Your, Wley, 1985 [11] Clason, R. Fndng Clusters: An Applcaton of the Dstance Concept, Aprl 1990 [1] Edwards, A.L, An Introducton to Lnear Regresson and Correlaton, second edton, Unversty of Washngton, 1984 [13] McCullagh, P., Nelderd J.A. Generalzed Lnear Models, second edton, Department of Statstcs, Unversty of Chcago, Department of Mathematcs, Imperal College of Scence and Technology, London, 1989 [14] Mles, J., Shevln M., Applyng Regresson & Correlaton, SAGE Publcatons, 001 [15] Muhlbauer, W.K., Ppelne Rsk Management Manual- A Systematc Approach to Loss Preventon and Rsk Assessment, Gulf Publshng Company, 199 [16] Krysck, W. Bartos, J., Dyczka W., Krolkowska, K., Waslewsk, M. Rachunek Prawdopodobenstwa I Statystyka Matematyczna w Zadanach-, Wydawnctwo Naukowe PWN, Warszawa 000 M.Sc. Thess Lech A. Grzelak

106 89 A statstcal approach to determne the MIC rate of underground gas ppelnes [17] Dzechcarz, J., Ekonometra, Wydawnctwo Akadem Ekonomcznej, Wroclaw 003 [18] Varaya, P.P., Lecture Notes on Optmzaton, Berkeley, Calforna 1998 (onlne avalable) [19] Han, S.P., A Globally Convergent Method For Nonlnear Programmng, Journal of Optmzaton Theory and Applcatons, Vol., p.97, 1977 [0] Powell, M.J.D., A Fast Algorthm for Nonlnear Constraned Optmzaton Calculus, Numercal Analyss, ed. G.A. Watson, Lecture Notes n Mathematcs, Vol. 630, Sprnger Verlag, 1978 [1] Cegelsk, A. Programowane nelnowe (lecture notes), 004/005, [] Lewandowsk D. Gas ppelnes corroson data analyss and related topcs, Delft, The Netherlands, 00 [3] Joseph L. Pkas, Case Hstores of External Mcrobologcally Influenced Corroson Underneath Dsbonded Coatngs, The NACE Internatonal Annual Conference and Exposton 1996, paper no. 198 [4] Harrs, Dr. J. O., Sol Bectera and Corroson, Appalachan Underground Short Course (1961) Delft Unversty of Technology

107 90 Appendx A A: Analyss methods and nterpretaton Defnton 1 (Random Varable) Let F be a σ algebra and Ω the probablty space. A functon X ( Ω, F) R where k R, the condton A k F satsfes. : s a random varable f for every subset = { ω : X ( ω) k} Defnton (The Lkelhood Functon) Let X = X, K, X ) be a random vector (random varable on n components) and ( 1 n { (x θ ) :θ Θ} f X a statstcal model parameterzed by θ = ( θ 1, K, θ k ), the parameter vector on the parameter space Θ. The lkelhood functon s a map L : Θ [0,1] R gven L( θ X ) = f X ( x θ ). The lkelhood functon s functonally the same n form as a probablty densty functon (the emphass s changed from X to the θ ). Defnton 3 (A maxmal Lkelhood estmate) The parameter θˆ for such L( ˆ θ X ) L( θ X ) θ Θ s called a maxmal lkelhood estmate (MLE) of θ. Remark 1 Many of densty functons are smooth functons (exponental), hence t s very comfortable to transform them to the log-lkelhood functon (any strctly monotonc transformaton preserves functon s extremes). Defnton 4 (The Normal dstrbuton) A k M.Sc. Thess Lech A. Grzelak

108 91 A statstcal approach to determne the MIC rate of underground gas ppelnes Delft Unversty of Technology We say that X s normally dstrbuted random varable wth mean μ and standard devaton σ f ts dstrbuton functon s followng: dx e t F t x = ) ( 1 ), ; ( σ μ πσ σ μ The densty curve s symmetrcal, centered about ts mean, wth ts spread determned by ts standard devaton. Maxmal Lkelhood estmaton of parameters Suppose that ( ) X n X X,, 1 K = s random vector and n X X,, 1 K are..d. (ndependently and dentcally dstrbuted), normally dstrbuted random varables wth the expectaton μ and varance σ. In order to fnd estmators of unknown parameter ), ( σ μ θ = we apply the maxmal lkelhood estmate method. 1 ) ( / ) ( 1 1 ) ( 1 1 ) ( ) ( ) ( σ μ σ μ π σ πσ θ θ θ = = = = = = = n X n n x n n X d X e e X f X f X L The log-lkelhood functon s: ) ln( ) ln( ) ( 1 )) ( log( ) ( 1 π σ μ σ θ θ n n X X L X l n = = = In order to fnd extremes we compute gradent ' ) ( θ θ X l what gves: ( ) = = n X X l 1 1 ) ( μ σ μ θ and ( ) ) ( σ μ σ σ θ n X X l n = = Settng 0 ) ( ' = θ θ X l (frst order condton) we get: = = n X n 1 1 μ and ( ) = = n X n 1 1 μ σ hence we get that: ) ˆ) ( 1, 1 ( ) ˆ ˆ, ( ˆ 1 1 = = = = n n X n X n μ σ μ θ Fnally, to know whether θˆ s ndeed the MLE we need to check that second order dervatves are negatve. Estmated parameter ) ˆ ˆ, ( ˆ σ μ θ = s ndeed the MLE estmator. Defnton 5 (The Beta dstrbuton) We say that X s standard beta dstrbuted a random varable wth parameters α and β f ts dstrbuton functon s followng:

109 9 F( t; α, β ) = where: t 0 β 1 (1 x) x B( α, β ) α 1 dx Γ( α) Γ( β ) Beta functon: B ( α, β ) = and where x dx Γ( α + β ) e x α 1 Γ( α) = s a gamma functon. 0 Maxmal Lkelhood estmaton of parameters Usng the same procedure as before we can easly derve MLE estmator ˆ θ = ( ˆ, α ˆ) β for the standard Beta dstrbuton, the results are shown below: X (1 X ) ˆ α = X 1 n 1/ ( ) and ˆ β = ( 1 X ) n X X = 1 where X -customary means mean. X (1 X ) 1/ n = 1 1 n ( ) X X Defnton 6 (The Gamma dstrbuton) We say that X s gamma dstrbuted a random varable wth parameters α and β f ther dstrbutons functon s followng: t β α β 1 α F( t; α, β ) = x e x dx Γ( β ) 0 where Γ(α ) s a gamma functon ntroduced n prevous defnton. Maxmal Lkelhood estmaton of parameters Also n a case of a Gamma dstrbuton we can easly derve MLE estmator of unknown parameter ˆ θ = ( ˆ, α ˆ β ), the results s followng: X ˆα = and ˆ β = n 1/ n ( X X ) 1/ n = 1 n = 1 X ( X X ) Defnton 7 (Expected value) If X s a random varable defned on a probablty space ( Ω, F, P) then the expected value of X (denoted as EX) s defned n followng way: EX = XdP Ω where ntegral s n the meanng of Lebesgue. In case when random varable X admts a probablty densty functon f (x) then the expected value s: + EX = xf ( x) dx M.Sc. Thess Lech A. Grzelak

110 93 A statstcal approach to determne the MIC rate of underground gas ppelnes When X s a dscrete random varable wth values probabltes p 1, K, pn then: EX = n = 1 x p x, Kx 1 n and correspondng Defnton 8 (P-value) The p-value s the probablty that sample could have been drawn from the populaton(s) beng tested gven the assumpton that the null hypothess s true. A p-value of 0., for example, ndcates that we would have a 0% chance of drawng the sample beng tested f the null hypothess was actually true. Defnton 9 (Kolmogorov- Smrnov two samples test) The Kolmogorov-Smrnov two-sample test s a test of the null hypothess that two ndependent samples have been drawn from the same populaton (or from populatons wth the same dstrbuton). The test uses the maxmal dfference between cumulatve frequency dstrbutons of two samples as the test statstc. The man dea behnd the test s to compare the proporton of the values less than certan level x between two sample sets. The test checks what the maxmal dfference between proportons s. The test doesn t requre that samples are the same sze. Accordng to Kolmogorov- Smrnov n1n the test s reasonably accurate for sample szes n 1 and n when 4. n1 + n The procedure s followng: for random vectors X 1 and X wth respectvely number of samples n 1 and n x D = max{ FX ( x) F ( x) } 1 X The usual way of carryng out the two-sample test s to compute the p-value drectly from the test statstc, wth no need to compare t to a crtcal value. Ths s an example presented n Nonparametrc Statstcal Methods by Hollander & Wolfe. So, the dea s to Compute the asymptotc p-value approxmaton and accept or reject the null hypothess on the bass of the p-value. The drect formula for a two sded test p-value s expressed n the followng way: p value = n1n n = ( n + n ) 1 k = 1 ( 1) k 1 e λ k where λ = max(( n / n) D,0) and Defnton 10 (Kolmogorov- Smrnov - goodness of ft test) The Kolmogorov- Smrnov test (K-S test) can be used to decde whether a sample comes from a populaton wth specfed dstrbuton. An attractve feature of ths test s that the dstrbuton of the K-S test statstc tself does not depend on the underlyng cumulatve dstrbuton functon beng tested. Another advantage s that t s an exact test (the ch-square goodness-of-ft test depends on an adequate sample sze for the approxmatons to be vald). Despte these advantages, the K-S test has several mportant lmtatons: 1. It only apples to contnuous dstrbutons. Delft Unversty of Technology

111 94. It tends to be more senstve near the center of the dstrbuton than at the tals. The K-S test procedure of testng s followng: 1. Specfy dstrbuton F0 - assocated wth theoretcal dstrbuton functon. Order samples n non-decreasng manner 3. 1 Calculate D = max max F0 ( X ),max F0 ( X ) 1 n n 1 n n 4. Calculate crtcal and p-value a) for small samples (less than 0), we use the drect values from tables b) for sample sze larger than 0 Mller s formula s appled: C = 0.5ln( α) / n c) the drect formula for p-value s gven by the same formula as n defnton no. 9 Defnton 11.1 (Product moment correlaton) The correlaton ρ X, Y between two random varables X and Y wth expected values μ X and μ Y respectvely, and standard devatons σ X and σ Y s defned as: cov( X, Y ) E(( X μ X )( Y μy )) ρ X, Y = =. σ Xσ Y σ Xσ Y The correlaton s the measure whch takes values from [-1,1] and s assocated wth the strength of the relatonshp between two varables. If correlaton coeffcent ρ takes value 1- then t means that there s a perfect lnear relatonshp, n case when ρ = 1 the perfect lnear relatonshp s negatve, but when ρ = 0 - then there s no relatonshp between varables. Test of Pearson's correlaton Let suppose that we have already computed a correlaton coeffcent ρ. Now, we would lke to verfy the hypothess that: H 0 : ρ = 0 aganst H 1 : ρ <> 0 Frst, we need to calculate the probablty of obtanng a statstc as dfferent from or more dfferent from the parameter specfed n the null hypothess as the statstc obtaned n the experment. The probablty value s computed assumng the null hypothess s true. If the probablty value s below the sgnfcance level then the null hypothess s rejected. To get a p-value we have to calculate: n t = ρ where n s number of samples, now the analyst has to check In t- 1 ρ Table the probablty value for a t, and to compare to sgnfcance level α. If the p-value s less than sgnfcance level, then the correlaton s sgnfcant. Defnton 11. (Spearman s rho correlaton coeffcent) General dea of rho correlaton coeffcent can be expressed n followng way: nstead of quanttatve measures on each of n pars of varables, we assgn ranks a on the frst varable (populaton characterstc) and a set of rankngs b on the second one. Each of sets a,..., a } and b,..., b } s some perturbaton of the ntegers 1,,,n. The { n { n M.Sc. Thess Lech A. Grzelak

112 95 A statstcal approach to determne the MIC rate of underground gas ppelnes Spearman correlaton coeffcent can be expressed as correlaton between ranks nstead of observatons X and Y n followng way: ρ = n = 1 ( X X )( Y n n ( X X ) = 1 = 1 Y ) ( Y Y ) ρ = n = 1 ( a a)( b n n ( a a) = 1 = 1 A basc algebra calculus wll show that above formula can be reduced to: n 6 ρ = 1 d where d = a b ( n 1) n = 1 Defnton 11.3 (Kendall s tau correlaton coeffcent) b ) ( b b ) Let s defne varables: P and -Q, whch corresponds to number of postve scores (concordant), and negatves (dscordant) respectvely. A lnear relatonshp of varables s defned n followng way: τ = ( P + Q) / S where S s maxmal avalable postve score. If we consder two rankngs of exactly n components, then basc calculus gves that, number of possble pars s n(n-1)/. Hence Kendall s τ coeffcent has the followng form: S S τ = = but C n( n 1) n 4P + C n hence τ = 1 n( n 1) P Q = Defnton 1 (Quantles) Let s take random varable X. A p quantle s such an q that P( X q p ) = p Defnton 13 (correlaton rato) Correlaton rato n statstcs s a measure of the relatonshp between the statstcal varablty wthn ndvdual categores and the dsperson of whole populaton or sample. The am s to order the varables X 1, K, X n (ncluded n the model) accordng to nfluence on the crteron varable. The quantty of nterest s E( Var( Yˆ X )) 43. Snce the equaton: E( Var( Y X )) + Var( E( Y X )) = Var( Y ) s well known f follows that E( Var( Y X )) 0 ndcates hgher mportance of the varable X. As a consequence the correlaton rato for the varable th s defned as: Var( E( Y X )) CR = Var Y ( ) 43 The quantty that should be consdered s ˆ * Var ( Y X = x ) (change of predctor varable f one quantty s * sad to be constant), however snce x s unknown, the dea s to calculate change of the varance overall the values of X Delft Unversty of Technology

113 96 Intutvely: hgher value of the CR follows hgher the share of the varance decomposton of the varable X. Method 1 Lnear Regresson The mportant task n statstcs s concerned wth determnng functonal relatonshps between a gven set of varables. Lnear least squares error technque s very mportant and applcable modelng method. Bascally, the method gves the estmators for predefned functon n order to get a functon, whch s best, ftted to the data n the meanng of least squares errors. Standard assumptons of the multple regressons are: 1. The model s defned as Y = Xβ + ε, where Y s the vector of outputs and X s a matrx of covarates. The number of observatons s grater than number of parameters to be estmated 3. Eε = 0 Var ( ε ) = σ Cov( ε, ε j ) = 0 for j The dea to get the best estmator of β we need to solve optmzaton problem: mn β Y Xβ = mn ε. β Smple calculatons show that the optmal estmator of β s expressed n followng way: ˆ T 1 T β = ( X X ) X Y Sgnfcance of estmated coeffcents Important fact n lnear regresson s that, we are allowed to check whether estmated parameter vector βˆ can be neglected or not. To test t we need to verfy the null hypothess: H : ˆ 0 β = 0 aganst alternatve: H1 : β such that β 0 To verfy hypothess above we need to calculate p-value for gven sgnfcance level α. The p-value assocated wth ntroduced hypothess s: p value = ˆ T T β X Y Y T ( ˆ T Y Y βx Y )( p 1) /( n p) ~ F( p 1, n p) For ntroduced p-value we have bass to reject null hypothess when p value > F( p 1, n 1,1 α) otherwse we do not have such a bass. (The n s a number of samples; p s a number of columns n covarate matrx X; and F s F- dstrbuton 44 ). Calculaton of confdence nterval estmates for ndvdual coeffcents The ( 1 α) confdence nterval for estmated parameter βˆ s gven by: ˆ α β t( n p,1 ) where s = /( n p) dagonal of matrx ( X n ( ) T = 1 X T X 1 ˆ α T X ), s, β + t( n p,1 ) ( X X ) 1, 1 ( Y Yˆ ) ; Y ˆ = X βˆ ; where ( X ( ) 1 T X ) s 1, s th value from ; and where t- s t-student 45 dstrbuton functon 44 F-dstrbuton- Fsher-Snedecor Dstrbuton 45 t-student dstrbuton- Gosset s dstrbuton M.Sc. Thess Lech A. Grzelak

114 97 A statstcal approach to determne the MIC rate of underground gas ppelnes Goodness of ft measure To check whether created model s well ftted to data we ntroduce ft measure: T ˆ ( Y Y ) ( Y Y ) R = whch takes values from [0,1]: T ( Y Y ) ( Y Y )( Yˆ T Y ) ( Yˆ Y ) 1 f the model s perfect, and 0 when model s badly ftted Method (stepwse regresson) Stepwse regresson s a model-buldng procedure that attempts to maxmze the amount of varance possble to explan n dependent varable whle smultaneously mnmzng the number of ndependent varables n the statstcal model. The stepwse regresson s typcally used when a large number of predctor varables are avalable whle the best combnaton of varables to predct the value of the crteron s wanted. It s desgned to gve a model that predcts as much varablty as possble wth the smallest number of parameters. The stepwse regresson should be nterpreted cautously or avoded entrely when tryng to understand theoretcal relaton. It makes ts selecton based purely on the amount of varance that varables can explan wthout any consderaton of causal or logcal prorty. As the consequence ndependent varables chosen through a stepwse regresson are not guaranteed to be the most mportant factors affectng the crteron varable. A theoretcally meanngful varable that explans a larger amount of varablty n the crteron varable could be excluded f t also happens to cause changes n other ndependent varables, because t would be collnear wth those varables. Addtonally, stepwse regresson attempts to maxmze the predctve ablty for the predctor varables n the one specfc sample that was collected. Its selectons wll therefore be affected by any relatons that happen to appear due to chance alone. If t s mpossble to come up wth a theoretcal explanaton for an observed relaton between predctor and crteron varables t may just be an artfact only found the partcular collected sample. There s one crcumstance under whch stepwse regresson should be used at most: when the most mportant am s to determne the best predctve model, wthout nterestng n drawng nferences about the relatons n the data. Stepwse regresson conssts of the two steps: frst start wth a smple model and gradually add ndependent varables to t untl any sgnfcant mprovement s not made.e. mnmze probablty that all the factors are equal to zero, but drop varables whch become no longer "sgnfcant" after ntroducton of new varables. In other words check the old set of ndependent varables each tme a new one s added to the model to make sure that they are stll sgnfcant. Secondly f t turns out that a predctor varable ncluded n an earler step s no longer makng a sgnfcant contrbuton to the predcton of the dependent varable, then the varable s dropped from the model. The algorthm of the stepwse regresson can be presented n the followng way. Suppose that and one crteron varable Y. Notaton used n the procedure s followng: Notaton -n s number of ndependent varables -Y s crteron varable, ex. corroson rate [mm/yr] or defect rate [def/km] X ndcates the th varable where = 1,..., n - - 1,... n R - R square coeffcent for the model: Y β + β X β + ε = n X n Delft Unversty of Technology

115 98 Step wse algorthm Step 1 - m = 1 - defne n models of the form: Y = β 0 + β1x + ε Step - take the varable (model) k for whch: max R and p-value for the hypothess: H 0 : β k = 0 aganst H 1 : B k 0 s less than sgnfcance level α Step 3 -for the remanng n-m varables defne n-m models of the form: Y = β 0 + β1x k + β X l + ε, wherel = 1,.. k 1, k + 1,.. n, and calculate R k,l and check the value for dfference between R squares: R = R k, j Rk Step 4 -defne a new regresson model for whch R takes maxmum value and s statstcally sgnfcant 46 Step 5 -recalculate the p-values for the t-test for all the varables n the model and check f all are sgnfcant, f any of them s nsgnfcant then should be removed and step 3 should be appled one more tme to the model consstng only of sgnfcant varable. Steps 3,4 and 5 have to be repeated sequentally, the new model addng/droppng the varables has to be fnshed when addng or droppng the varables wont mprove the determnaton coeffcent (R square) sgnfcantly and all the varables n fnal model wll be sgnfcant. k k 46 In order to check f dfference between R squares for two models s sgnfcant hypothess testng based on F statstcs has to be appled- see appendx M.Sc. Thess Lech A. Grzelak

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117 100 Appendx B B: Tutoral fle for CoroGas Author: Lech A. Grzelak, program created for N. V. Netherlandse Gasune The program does all the theory ntroduced n part one of the thess. Ths tutoral conssts of followng parts: a) Data calbraton b) Corroson rate modelng c) How to transfer the fles nto *.dat type? d) Fles constructon e) Examples f) Avalable datasets descrptons a) Data Calbraton The frst step of usng the program A statstcal approach to determne the corroson rate modelng of the underground gas ppelnes s to open the program n the Matlab envronment. In order to ntate the program the user has to open the matlab fle: fle/open/start.m as t s shown on the screenshot below. M.Sc. Thess Lech A. Grzelak

118 101 A statstcal approach to determne the MIC rate of underground gas ppelnes When the fle start.m s opened then the ntaton of the program can be done n two ways. The frst way s easy - press the run button on the top of the new opened wndow as t s shown below. RUN The second way to start the program s to tape the command start.m n the command wndow. If the ntaton process s successful then the user should see the graphcal user nterface wndow. Delft Unversty of Technology

119 10 On the bottom of the screen are presented avalable optons. The frst one s LOAD whch allows loadng prevously prepared excavaton dataset, and s supposed to conssts of calbraton data. The detaled descrpton of the fle wll be presented later on n the text. The Second opton NEW SET gves the opportunty to create user s calbraton dataset (opton under constructon), the thrd s MODEL whch goes drectly to corroson rate modelng part wthout calbratng the data, the forth and the last opton s EXIT whch termnates the program Suppose that user wants to LOAD prevously prepared dataset. Frst a proper path to the calbraton data has to be ndcated n the load wndow. The program recognzes the *.dat fle (later n text t s shown how to create ths type of fles). M.Sc. Thess Lech A. Grzelak

120 103 A statstcal approach to determne the MIC rate of underground gas ppelnes When the proper calbraton dataset s chosen then the program mmedately analyzes the dataset and proceeds to the second stage. The new wndow conssts of a few parts - the block on the left called INSPECTIONS, CLUSTERING PROCEDURE, fgures of results, and TOOLBOX on the bottom of the wndow. INSPECTIONS - allows user to have a vew on the loaded dataset and also gves the opportunty to look at the graphcal representaton of the dataset (n the fgures on the rght). A clck on the button data vew, ntates data vew. Delft Unversty of Technology

121 104 The new wndow presents dataset loaded for calbraton. The frst column corresponds to the defects measured at the excavaton by an nspector. The other columns are assocated wth the measurements reported by ntellgent pgs. The empty slots ndcate mssng data caused ether by defect reparaton (n current case the defects were repared after tuboscope nspecton) or by clusterng procedure. All the measurements are presented n mllmeters. The second avalable opton n ths part s to show the calbraton graphcally. When user chooses any of the check-boxes correspondng to pgs-nspectors, then the program updates two graphcs. The frst one shows relaton between the pg measurements and the excavaton data, and the second relaton between the pg measurements and the measurement errors (.e. all data are treated as exactly one cluster). CLUSTERING PROCEDURE - Ths part gves an opportunty to calbrate dataset accordng to a gven data (clusters). Frstly user has to use the so called popup-menu and choose a pg for calbraton. M.Sc. Thess Lech A. Grzelak

122 105 A statstcal approach to determne the MIC rate of underground gas ppelnes When a certan pg s chosen, then mmedately the followng results appear: o Number of clusters - number of predefned clusters o Clusters - bounds of the nterval where pg measurements were reported (nterval s boundares are extended by 0.1 [mm]) o No. - number of defects regstered durng excavaton and assocated wth pg measurements (wthn the cluster) o Bas - corresponds to the mean of the dfference between the pg measurements and measurements from the excavaton (see the A statstcal approach to determne the corroson rate of underground gas ppelnes report for detals) o Std - standard devaton of the measurement error o N-p-V s the p-value assocated wth the followng hypothess: H : Measurement errors are from normal dstrbuton o o 0 H 1 : Measurement errors are not from normal dstrbuton Rho-p-V - s the p-value assocated wth the followng hypothess: H 0 : The Spearman s correlaton between measurement errors and pg measurement s 0 H1 : The Spearman s correlaton between measurement errors and pg measurement s not 0 ERR. DISTR - gves the p-value, standard devaton and mean for the measurement error. In the case where the user does not generate addtonal clusters then the p-value correspondng to Kolmogorov Smrnov test should be the same as gven n the column N-p-V Delft Unversty of Technology

123 106 When user defnes the number of clusters ex. accordng to the technques ntroduced n the report A statstcal approach to determne the corroson rate of underground gas ppelnes 47 then, he s also oblgated to defne boundares of these ntervals. In order to generate these ntervals (clusters) user has to fll n the number of clusters and press the button GENERATE. The maxmal number of ntervals s 6.e. the program analyzes up to 6 clusters per one pg. When all the boundares for clusters are well defned (cover the entre doman, and are not ntersectng), then n order to perform the analyss the ANALYZE button has to be pressed. After few seconds a computer should gve the answer n the form presented on the pcture below. Each row and calculated values correspond to a predefned cluster. If for the Rho-p-V the background color becomes red- then user should have a look closer at the cluster. Red background ndcates that p-value for the Spearman s (rank) correlaton hypothess that measurement error s uncorrelated wth the measurements s less than sgnfcance level Informaton avalable on the screen (yellow rectangle) s the verfcaton of the null hypothess that errors whch come from clusters come from the same populaton (dstrbuton). The HISTOGRAM produces the hstogram of the unbased errors (f the hypothess about the errors comng from the same dstrbuton s not rejected). 47 Report done by Lech A. Grzelak, for Gasune Research Department M.Sc. Thess Lech A. Grzelak

124 107 A statstcal approach to determne the MIC rate of underground gas ppelnes Another opton avalable n the clusterng procedure s to have a vew on the errors assocated wth the predefned clusters. As t s presented on the pcture below; f user clcks on the check box assocated wth the analyzed cluster, then the error from the cluster s presented lke on the pcture on the rght the errors wthout mposed clusters are as a background. Delft Unversty of Technology

125 108 When user s satsfed wth the clusters, then such decson has to be ndcated by pressng the button ACCEPT on the bottom of the wndow. The button ACCEPT saves chosen clusters nto the memory and ntates the REPORT n the mddle of the wndow and gves access to the button VIEW RESULTS. The columns n the new opened wndow are: o No - ndex of the nspecton o name - name of the nspectng pg o clust. - number of predefned clusters o K-S - takes values YES/NO, whch ndcate whether measurement errors come from the same populaton accordng to Kolmogorov- Smrnov test o o std - measurement error standard devaton (f the K-S s YES ) Corr. - takes values B/OK- ndcates whether measurement errors are correlated wth pg measurements, the B ndcates BAD - correlaton and OK - no correlaton. The correlaton used n the software s the Spearman s rank correlaton The button VIEW RESULTS allows to see how the accepted clustered are defned. The frst column corresponds to the name of the ntellgent pg, the second ndcates number of predefned clusters. Column number three shows current cluster, and then respectvely are: cluster s boundares, bas assocated wth ths cluster and the standard devaton of the errors. Ths nformaton s collected and later on wll be appled n the second stage of the analyss done by the program - namely n the corroson rate modelng. When the clusterng procedure s performed for all the pgs, then n order to save the results user has to press the SAVE TO FILE button. M.Sc. Thess Lech A. Grzelak

126 109 A statstcal approach to determne the MIC rate of underground gas ppelnes To leave the vew wndow of the defned cluster press the button Close wndow. It s also possble to load the clusters and calbraton data from prevous sesson and update them. The opton whch allows ths s the button LOAD & UPDATE. b) Corroson Rate modelng When the calbraton procedure s carred out successfully, then the button MODEL gves access to the second part of the program. The optons/buttons avalable at ths stage are: o LOAD MEASUREMENTS - user s requred to gve the path to the fle wth the measurements collected by ntellgent pgs o LOAD CALIBRATION - when after the calbraton procedure the clusters are defned and saved to the fle then user has to ndcate accordng to whch fle the calbraton of the measurements has to be performed o WALL THICKNESS - loads the column vector of the nomnal wall thcknesses, each row of the vector has to correspond to nomnal wall thckness where defects were observed at ths stage program doesn't take nto account the uncertanty about wall thckness o DATES OF INSPECTIONS - accordng to the data each nspecton was performed at certan tme, ths opton loads the dates of performed nspectons- the rows of ths vector correspond to the columns of the measurements o CALIBRATION - when all requred data are loaded then ths button performs the calbraton procedure and gves access to the SIMULATION Delft Unversty of Technology

127 110 When the data s loaded then by pressng the button VIEW user can have a look on the loaded datasets. The button CALIBRATION calbrates the data and gves some results of the calbraton o # ERR DISTRIB - the number of the dstrbutons (sum of dstrbutons assocated wth all clusters for all pgruns) o INSPECTIONS - number of nspectons o MISSING DATA - number of mssng data n the whole measurement dataset o NUMBER OF DEFECTS - number of defects observed durng nspectons o PREDEFINE STD FOR ALL MEASUREMENTS - when user types any postve value n the STD box, then the weghts for the nspectons are equal (all pgs have the same weght) o LOAD RESULTS - gves the possblty to load prevously saved Corroson modelng sesson M.Sc. Thess Lech A. Grzelak

128 111 A statstcal approach to determne the MIC rate of underground gas ppelnes When the datasets are loaded, then the button SIMULATION ntates the smulaton. The smulaton produces the optmal soluton. Tests show that estmaton for one sngle defect observed at four nspectons requres about 15 seconds to obtan the optmal functon. When the smulaton s complete, after a few seconds a new screen should appear. Delft Unversty of Technology

129 11 On the top of the screen s so called CORROSION RATE STATISTICS, whch conssts of basc statstc measures lke o corroson rate (mean corroson rate from all rates), o 95% upper bound of the corroson rate o 95% lower bound of the corroson rate o Max corroson rate o Mn corroson rate o mean ntal tme of corroson ntaton o nt. At t=0 - ndcates how many defects start at tme 0 (at ppelne nstallaton tme) o mean ntal tme>0 - mean ntal tme of defects ntatng after ppelne nstallaton On the rght sde, the plot presents the metal loss as a lnear functon of tme for all defects together wth the measurements (blue stars). The mddle part presents two hstograms. Frst of them corresponds to the dstrbuton of the corroson rate, and the second one to the ntaton tme of corroson per defect. There s a small summary of ftted dstrbutons beneath the hstograms -- on the left hand sde all avalable dstrbutons wth assocated p-values, and on the rght dstrbutons wth estmated lkelhood parameters for whch the p-value s the hghest. After the smulaton three new optons become avalable. The frst of them s SAVE RESULTS - whch allows exportng the results to a *.dat fle. The next s DEFECTS & ESTIMATE, whch allows to have a closer look at the estmated rates. Ths opton allows to check what s the corroson rate for each defect and to compare estmated curve wth the Least Squares approach. M.Sc. Thess Lech A. Grzelak

130 113 A statstcal approach to determne the MIC rate of underground gas ppelnes The thrd opton s EXPORT RESULTS TO EXCEL - exports the results to an Excel fle *.xls. If the user decdes to export results to Excel, then the exported fle conssts of eght columns where each row corresponds to one defect. Delft Unversty of Technology

131 114 The frst column n the exported Excel fle s the number of analyzed defects. The second and thrd column correspond to the ntal guess gven to the optmzer, the fourth column gves the optmal value of the functon log L where the L s the Lkelhood functon presented n the report. Column ndcated as E corresponds to the corroson rate of estmated defect, the next s the ntercept of the lnear functon. Column G gves the tme when the corroson process has ntated. The last column corresponds to the depth of the defect at the last nspecton. c) How to transform a data to the *.dat fle type? As t was presented prevously n the tutoral, the type of fles whch the program works wth s the type wth extenson *.dat. In order to make such a fle, user has to apply fle converter. The procedure s followng: 1. Defne the dataset n matlab and save t as *.mat fle- for example create a varable called FILE and save t usng command save FILE.mat FILE Remark: Mssng data should be ndcated as 0 (zero) M.Sc. Thess Lech A. Grzelak

132 115 A statstcal approach to determne the MIC rate of underground gas ppelnes. open the fle savetofile_generator.m, and n the 5 and 6 lne wrte load FILE.mat and WT=FILE 3. run the program and choose the name of *.dat fle, snce now the FILE wll be transformed to the *.dat fle Delft Unversty of Technology

133 116 d) Fles constructon Each fle n the program has hs own constructon, and so: excavaton measurements conssts of n+1 columns (where n s the number of nspectons), the frst column corresponds to data recorded from the excavaton, all the rest correspond to nspectons, unts are mllmeters measurement set - has exactly n columns and m rows, m s the number of observed defects and n s the number of nspectons, unts are mllmeters dates of nspectons - s the column vector consstng of dates of the nspectons, all the dates are counted snce ppelne nstallaton (ppelne nstallaton s at tme t=0) unts are years wall thckness - s the column vector, where each row corresponds to each defect, values are presented as nomnal wall thckness n mllmeters 5 Examples Suppose that we have carred out 5 nspectons, the excavaton was done after frst pgrun. The calbratng dataset s followng: Excavaton Insp 1 Insp Insp 3 Insp 4 Insp Table 46: excavaton dataset [mm] When the dataset s prepared we need to convert t nto *.dat fle, n the manner explaned before. And so: 1. We type the data n matlab and save t as E.mat M.Sc. Thess Lech A. Grzelak

134 117 A statstcal approach to determne the MIC rate of underground gas ppelnes. Second step s to convert the dataset to the E.dat fle; hence we change the code n the fle savetofile_generator.mat and turn the program on. The hypothetcal measurements reported by the pgs durng nspectons are followng: Defect no. Insp 1 Insp Insp 3 Insp 4 Insp Table 47: example of the measurements The theoretcal ppelne was constructed n So the dates of nspectons are followng: Delft Unversty of Technology

135 118 Now, let s save the varable D (dates of nspecton) as D.dat and the measurements M as M.dat,. For the nomnal wall thckness we assume: defect no. Nomnal wall thckness Table 48: The nomnal wall thckness In ths case the wall thckness vector we denote as W and perform the same transformaton procedure as for the others. Now, when all necessary data are collected, typed, and transformed nto *.dat type fle, so the analyss can be carred out. 1. Frst: run the program and load the excavaton data (here E.dat fle) M.Sc. Thess Lech A. Grzelak

136 119 A statstcal approach to determne the MIC rate of underground gas ppelnes. When excavaton data s loaded then program mmedately goes to the calbraton stage, for smplcty defne only one cluster for each nspecton (one cluster contans all the measurements). So for each nspecton generate exactly one cluster and confrm by pressng GENERATE, ANALYZE and ACCEPT buttons. a. Choose nspecton Delft Unversty of Technology

137 10 b. Generate one cluster c. Analyze the clusters d. Accept the results M.Sc. Thess Lech A. Grzelak

138 11 A statstcal approach to determne the MIC rate of underground gas ppelnes e. Save to fle (red button) as CAL.dat n ths case no fle transformaton s requred. Matlab mmedately saves the results as the *.dat fle. 3. When the calbraton procedure s performed, then go to the second stagenamely to the MODEL. Frstly, all measurements should be loaded (wall thckness (W), measurements (M), calbraton data (CAL), dates of nspectons (D)) and the CALIBRATION button should be pressed. Delft Unversty of Technology

139 1 A small summary shows that from the calbraton procedure we have exactly 5 dstrbutons (one dstrbuton per nspecton), the number of defects s 6, the number of mssng data s zero. Now we can proceed and press button SIMULATION. After a few second the results should appear. M.Sc. Thess Lech A. Grzelak

140 13 A statstcal approach to determne the MIC rate of underground gas ppelnes All the fles are attached to the delvered program, hence the fles converson s not requred. e) Avalable DATABASES/ EXAMPLES 1. The database based on the real data collected by Gasune, Inspectons done by 4 pgs, one excavaton done after the frst nspecton Fles: excavatondataset.dat - conssts of excavated metal loss depths and data necessary for calbraton measurements.dat - measurements done durng 4 nspectons- 5 reported defects dateofnspectons.dat - conssts of data concernng nformaton about carred out dates of nspecton WALLthckness.dat - vector of wall thckness for where 5 defects were reported 1pg1dstrbutonCLUSTER.dat - result of calbraton procedureeach pg has exactly one cluster- measurement error dstrbutoncan be appled n the MODEL 1pgdstrbutonsCLUSTER.dat - dem, but now each pg has clusters- dstrbutons. Example presented n the tutoral M.dat - measurements D.dat - dates of nspectons E.dat - calbratng dataset- excavaton dataset W.dat -wall thckness fle CAL.dat result of calbraton procedure- each pg exactly one dstrbuton Delft Unversty of Technology