Applied Science, Faculty of
Engineering, School of (Okanagan)
DSpace
UBCO
Dey, Sandip
Doctor of Philosophy - PhD
University of British Columbia
This thesis entails developing analysis methods for evaluating seismic risk to oil and gas pipelines due to permanent ground deformation resulting from fault rupture. To address this, three modules are identified as core part of this research. Firstly, a finite element (FE) model for oil and gas pipeline crossing a seismic induced fault line is developed. Subsequently, uncertainty in input parameters is introduced in the developed FE model and analyzed in a multi-fidelity approach. Finally, probabilistic regional ground deformation due to fault rupture is assessed and integrated with the pipe-soil structural model. A detailed FE analysis technique to study the behavior of buried continuous pipelines crossing fault movements is developed and established with appropriate evaluations. A non-linear sand constitutive model is adopted and implemented in commercial FE package ABAQUS. The adopted material model is first evaluated using available tri-axial test results. This material model is thereafter suitably calibrated for a large-scale test based on direct shear soil test data for that experiment. The pipe-soil FE model is then evaluated against full-scale experimental results. Subsequently, structural response of a buried continuous steel pipeline undergoing fault rupture deformation is studied in a systematic manner. A detailed and efficient analysis framework for design of buried continuous steel pipelines crossing faults is proposed and explained with a case study using Taguchi method for design of experiments. Further to this, uncertainty in input parameters, such as pipe diameter, pipe wall thickness, young’s modulus, yield strength, ultimate strength, sand confining pressure, sand relative density, operating temperature and operating pressure is considered using a multi-fidelity approach which relies on analyses results from a small number of detailed structural analyses and large numbers of simplified structural analyses. The approach efficiently relies on both types of results to predict structural performance accurately considering uncertainty and variability in geometrical and material properties. Significant computational cost savings is achieved where by 500 analyses are completed using 15 minutes computational time low-fidelity (LF) models instead of 48 hours computational time high-fidelity (HF) models with a 3.20GHz eight-core processor with 32 GB RAM for 500 analyses. Finally, extension of the structural analysis framework to a regional scale is completed. This is done by calculating fault rupture induced regional ground displacements from available analytical techniques using probabilistic approaches and thereafter integrating it with the developed structural analysis uncertainty quantification framework.
Seismic Risk Assessment of EnergyPipelines under Permanent GroundDeformation due to Fault RupturebySandip DeyB.E., Indian Institute of Engineering Science and Technology Shibpur, 2012M.A.Sc., Concordia University Montreal, 2014A THESIS SUBMITTED IN PARTIAL FULFILLMENT OFTHE REQUIREMENTS FOR THE DEGREE OFDOCTOR OF PHILOSOPHYinThe College of Graduate Studies(Civil Engineering)THE UNIVERSITY OF BRITISH COLUMBIA(Okanagan)November 2021© Sandip Dey, 2021The following individuals certify that they have read, and recommendto the College of Graduate Studies for acceptance, a thesis/dissertation en-titled:Seismic Risk Assessment of Energy Pipelines under Permanent GroundDeformation due to Fault Rupturesubmitted by Sandip Dey in partial fulfilment of the requirements of thedegree of Doctor of PhilosophySolomon Tesfamariam, School of EngineeringSupervisorKatsuichiro Goda, Western UniversitySupervisory Committee MemberAbbas Milani, School of EngineeringSupervisory Committee MemberDimitry SediakoUniversity ExaminerIan MooreExternal ExamineriiAbstractThis thesis entails developing analysis methods for evaluating seismicrisk to oil and gas pipelines due to permanent ground deformation resultingfrom fault rupture. To address this, three modules are identified as corepart of this research. Firstly, a finite element (FE) model for oil and gaspipeline crossing a seismic induced fault line is developed. Subsequently,uncertainty in input parameters is introduced in the developed FE modeland analyzed in a multi-fidelity approach. Finally, probabilistic regionalground deformation due to fault rupture is assessed and integrated with thepipe-soil structural model.A detailed FE analysis technique to study the behavior of buried contin-uous pipelines crossing fault movements is developed and established withappropriate evaluations. A non-linear sand constitutive model is adoptedand implemented in commercial FE package ABAQUS. The adopted ma-terial model is first evaluated using available tri-axial test results. Thismaterial model is thereafter suitably calibrated for a large-scale test basedon direct shear soil test data for that experiment. The pipe-soil FE modelis then evaluated against full-scale experimental results.Subsequently, structural response of a buried continuous steel pipelineundergoing fault rupture deformation is studied in a systematic manner.A detailed and efficient analysis framework for design of buried continuoussteel pipelines crossing faults is proposed and explained with a case studyusing Taguchi method for design of experiments.Further to this, uncertainty in input parameters, such as pipe diame-ter, pipe wall thickness, young’s modulus, yield strength, ultimate strength,sand confining pressure, sand relative density, operating temperature andoperating pressure is considered using a multi-fidelity approach which relieson analyses results from a small number of detailed structural analyses andlarge numbers of simplified structural analyses. The approach efficientlyrelies on both types of results to predict structural performance accuratelyconsidering uncertainty and variability in geometrical and material proper-ties. Significant computational cost savings is achieved where by 500 anal-yses are completed using 15 minutes computational time low-fidelity (LF)iiimodels instead of 48 hours computational time high-fidelity (HF) modelswith a 3.20GHz eight-core processor with 32 GB RAM for 500 analyses.Finally, extension of the structural analysis framework to a regional scaleis completed. This is done by calculating fault rupture induced regionalground displacements from available analytical techniques using probabilis-tic approaches and thereafter integrating it with the developed structuralanalysis uncertainty quantification framework.ivLay SummaryBuried continuous energy pipelines traversing seismically active regionsare vulnerable to failure due to permanent ground deformation resultingfrom fault rupture. Hence, it is important to develop efficient and reliablemethods to characterise seismic risks to these pipelines arising from perma-nent ground deformations due to fault ruptures. To address this, an exper-imentally evaluated FE pipe-soil model to study pipeline response due toground deformation loads with consideration to associated uncertainties ininfluencing geometric and material factors have been developed. To ensureefficiency of the developed method, a multi-fidelity approach using Gaussianprocesses is introduced. A multi-fidelity approach relies on the accuracy ofdetailed HF models and the efficiency of simplified LF models to predictreasonably accurate results in an efficient manner. Finally, this developedmethod is integrated with available probabilistic regional ground deforma-tion assessment approaches for regional seismic risk assessment of a pipelinenetwork.vPrefaceI, Sandip Dey, prepared all the seven chapters of this dissertation un-der the supervision of Professor Solomon Tesfamariam. This dissertationis prepared based on a compilation of four published peer-reviewed jour-nal papers (Chapters 3, 4), one submitted journal paper (Chapter 6) andone paper under preparation (Chapter 5). The author of this dissertationconducted the literature review, performed all analyses, prepared surrogateand numerical models of the studied pipelines, processed the data, inter-preted the results, and wrote all the chapters (manuscripts). ProfessorsSolomon Tesfamariam, Katsuichiro Goda and Souvik Chakraborty offeredtechnical assistance, reviewed the manuscripts and provided critical feedbackto improve the manuscripts. Professor Solomon Tesfamariam facilitated thepartial funding for this research.Published papers1. S. Dey, S. Chakraborty, S. Tesfamariam, Multi-fidelity approach foruncertainty quantification of buried pipeline response undergoing fault rup-ture displacements in sand, Computers and Geotechnics 136 (2021) 104197.2. S. Dey, S. Chakraborty, S. Tesfamariam, Structural performance ofburied pipeline undergoing strike-slip fault rupture in 3D using a non-linearsand model, Soil Dynamics and Earthquake Engineering 135 (November2019) (2020) 106180.Papers submitted1. S. Dey, S. Tesfamariam, Design of buried pipeline undergoing faultrupture in sand using Taguchi design of experiments, Soil Dynamics andEarthquake Engineering.viPAPERS UNDER PREPARATIONPapers under preparation1. Regional probabilistic seismic risk analysis of buried pipelines dueto permanent ground deformation hazard for Victoria, British Columbia,Canada.viiTable of ContentsAbstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iiiLay Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . vPreface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . viPublished papers . . . . . . . . . . . . . . . . . . . . . . . . . . . . viPapers submitted . . . . . . . . . . . . . . . . . . . . . . . . . . . . viPapers under preparation . . . . . . . . . . . . . . . . . . . . . . . viiTable of Contents . . . . . . . . . . . . . . . . . . . . . . . . . . . viiiList of Tables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xiList of Figures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xiiAcknowledgements . . . . . . . . . . . . . . . . . . . . . . . . . .xviiDedication . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .xviiiChapter 1: Introduction . . . . . . . . . . . . . . . . . . . . . . . 11.1 Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.2 Research Objectives . . . . . . . . . . . . . . . . . . . . . . . 31.3 Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41.4 Thesis Organisation . . . . . . . . . . . . . . . . . . . . . . . 6Chapter 2: Literature Review . . . . . . . . . . . . . . . . . . . 82.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82.2 Pipeline-fault rupture analysis . . . . . . . . . . . . . . . . . . 82.3 Uncertainty quantification and uncertainty propagation of buriedpipeline response . . . . . . . . . . . . . . . . . . . . . . . . . 112.4 Multi-fidelity models . . . . . . . . . . . . . . . . . . . . . . . 14viiiTABLE OF CONTENTS2.5 Probabilistic ground deformation due to fault rupture and theLeech River Fault Zone . . . . . . . . . . . . . . . . . . . . . 212.6 Probabilistic pipeline strain risk analysis . . . . . . . . . . . . 23Chapter 3: Finite Element Modelling and Evaluation of BuriedPipeline Response undergoing Fault Rupture De-formations . . . . . . . . . . . . . . . . . . . . . . . . 253.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 253.2 Physical model of buried pipeline . . . . . . . . . . . . . . . . 263.3 Numerical modelling . . . . . . . . . . . . . . . . . . . . . . . 283.3.1 FE Modelling details . . . . . . . . . . . . . . . . . . . 283.3.2 Soil material model . . . . . . . . . . . . . . . . . . . . 303.3.3 Pipe material model . . . . . . . . . . . . . . . . . . . 343.3.4 Soil-pipe interaction modelling . . . . . . . . . . . . . 343.3.5 Solution procedure . . . . . . . . . . . . . . . . . . . . 353.4 Results and discussion . . . . . . . . . . . . . . . . . . . . . . 363.4.1 Evaluation of material model . . . . . . . . . . . . . . 363.4.2 Evaluation of full-scale FE model . . . . . . . . . . . . 373.5 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43Chapter 4: Multi-fidelity approach for Uncertainty Quantifi-cation of Buried Pipeline Response undergoingFault Rupture Displacements in Sand . . . . . . . 474.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 474.2 Problem Statement . . . . . . . . . . . . . . . . . . . . . . . . 484.2.1 High-fidelity FE model . . . . . . . . . . . . . . . . . . 494.2.2 Low-Fidelity FE model . . . . . . . . . . . . . . . . . 514.3 Uncertainty quantification using multi-fidelity surrogate . . . 534.3.1 Multi-fidelity Gaussian process . . . . . . . . . . . . . 544.3.2 Uncertainty quantification using MF-GP . . . . . . . . 554.4 Results and discussion . . . . . . . . . . . . . . . . . . . . . . 564.4.1 Cross-evaluation test . . . . . . . . . . . . . . . . . . . 574.4.2 Sensitivity analysis and generation of fragility curves . 584.5 Summary and future work . . . . . . . . . . . . . . . . . . . . 60Chapter 5: Regional Probabilistic Seismic Risk Analysis ofBuried Pipelines due to Permanent Ground De-formation Hazard for Victoria, British Columbia,Canada . . . . . . . . . . . . . . . . . . . . . . . . . 815.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81ixTABLE OF CONTENTS5.2 Integrated risk assessment method for buried pipelines . . . . 825.3 Probabilistic ground deformation hazard assessment for Vic-toria, British Columbia . . . . . . . . . . . . . . . . . . . . . . 845.3.1 Leech River Valley Fault Zone (LRVFZ) Source Char-acterization and Rupture Occurrence . . . . . . . . . . 845.3.2 Probabilistic Ground Deformation Hazard Assessment 855.4 Finite Element Models of Buried Pipelines . . . . . . . . . . . 865.4.1 Variation in Model Parameters . . . . . . . . . . . . . 865.4.2 High-Fidelity FE Model . . . . . . . . . . . . . . . . . 875.4.3 Low-Fidelity FE Model . . . . . . . . . . . . . . . . . 885.5 Uncertainty Quantification using Multi-Fidelity Surrogate Mod-elling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 885.6 Results and Discussion . . . . . . . . . . . . . . . . . . . . . . 895.6.1 Probabilistic Ground Deformation Hazard Estimationfor LRFZ . . . . . . . . . . . . . . . . . . . . . . . . . 905.6.2 Cross-evaluation and uncertainty quantification of pipelineresponse . . . . . . . . . . . . . . . . . . . . . . . . . . 905.7 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91Chapter 6: Design of Buried Pipeline undergoing Fault Rup-ture Deformations in Sand using Taguchi Designof Experiments . . . . . . . . . . . . . . . . . . . . . 1046.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1046.2 Problem statement . . . . . . . . . . . . . . . . . . . . . . . . 1056.3 Taguchi’s Method for Sensitivity Analysis . . . . . . . . . . . 1066.4 Case study . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1116.4.1 Design of experiment using Taguchi method . . . . . . 1116.4.2 Analysis of results using Taguchi method and devel-opment of envelope strain demand curves . . . . . . . 1126.5 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 119Chapter 7: Summary and Conclusions . . . . . . . . . . . . . . 1237.1 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1237.2 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 125Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 126xList of TablesTable 2.1 Representative summary of different studies on buriedpipeline undergoing fault rupture. . . . . . . . . . . . 12Table 3.1 Performance of the proposed approach in predictingthe east spring line axial strain (under tension) underdisplacements 300 mm, 600 mm and 900 mm . . . . . 41Table 3.2 Peak tensile strains for varying mesh sizes of the pipelinefor displacements 300 mm, 600 mm and 900 mm . . . 43Table 4.1 Compressive strain percent generated for HF modelanalysis 1-3 for varying fault displacement and meshsize . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51Table 4.2 Coefficient of determination for surrogate models trainedwith only 30 HF models, only 500 LF models and com-bination of 30 HF and 500 LF multi-fidelity models . . 57Table 4.3 Coefficient of determination for multi-fidelity modelstrained with varying number of HF models and con-stant 500 LF models . . . . . . . . . . . . . . . . . . . 57Table 6.1 Matrix 1 . . . . . . . . . . . . . . . . . . . . . . . . . . 108Table 6.2 Matrix 2 . . . . . . . . . . . . . . . . . . . . . . . . . . 109Table 6.3 Matrix 3 . . . . . . . . . . . . . . . . . . . . . . . . . . 109Table 6.4 Matrix 4 . . . . . . . . . . . . . . . . . . . . . . . . . . 110Table 6.5 Taguchi design of experiment for the problem underconsideration . . . . . . . . . . . . . . . . . . . . . . . 119Table 6.6 ANOVA analysis results for axial tensile strains . . . . 120Table 6.7 ANOVA analysis results for axial compressive strains . 122xiList of FiguresFigure 1.1 Thesis chapters flowchart . . . . . . . . . . . . . . . . 7Figure 2.1 Literature Review Flowchart . . . . . . . . . . . . . . 9Figure 3.1 Pipeline-fault schematic diagram . . . . . . . . . . . . 27Figure 3.2 Pipeline-fault isometric mesh . . . . . . . . . . . . . . 29Figure 3.3 Finite element model mesh . . . . . . . . . . . . . . . 30Figure 3.4 Mobilized friction angle and dilation angle . . . . . . 33Figure 3.5 Mobilized friction angle and dilation angle for varyingrelative density . . . . . . . . . . . . . . . . . . . . . . 34Figure 3.6 Poly ethylene stress strain curve . . . . . . . . . . . . 35Figure 3.7 Triaxial test single axis-symmetric element . . . . . . 36Figure 3.8 Evaluation of material model considered in this study 37Figure 3.9 Effect of relative density on the stress strain behaviorof sand . . . . . . . . . . . . . . . . . . . . . . . . . . 38Figure 3.10 Failure plane computed using the proposed FE model 39Figure 3.11 Plastic shear strain (FV1) at different locations under300 mm displacement – (a) 2 m from fault plane onthe moving soil mass (b) 2 m from fault plane on thestationary soil mass (c) vicinity of the fault plane onmoving soil mass, and (d) vicinity of the fault planeon static soil mass. . . . . . . . . . . . . . . . . . . . 40Figure 3.12 Strain profile (tensile) under a displacement of (a) 300mm, (b) 600 mm and (c) 900 mm obtained using cur-rent study, interface model, coupled model and full-scale experimental test [1] (d) Bending strain profilefrom current FE analysis. . . . . . . . . . . . . . . . . 41xiiLIST OF FIGURESFigure 3.13 Strain profile (under compression) with a displace-ment of (a) 300 mm and (b) 600 mm obtained usingthe proposed approach, beam-element model and full-scale experimental test [2] (c) Bending strain profilefrom current FE analysis. . . . . . . . . . . . . . . . . 42Figure 3.14 Deformation of the pipe under compression and ten-sion at a displacement of (a) 300 mm (compression),(b) 600 mm (compression), (c) 300 mm (tension), (d)600 mm (tension), (e) 900 mm (tension). . . . . . . . 45Figure 3.15 Axial strain of the pipe under compression and ten-sion at a displacement of (a) 300 mm (compression),(b) 600 mm (compression), (c) 300 mm (tension), (d)600 mm (tension), (e) 900 mm (tension). . . . . . . . 46Figure 4.1 HF finite element model mesh . . . . . . . . . . . . . 49Figure 4.2 Non-linear soil constitutive relation . . . . . . . . . . 50Figure 4.3 Non-linear variation of (a) mobilized friction angleand (b) mobilized dilation angle with plastic shearstrain used in HF soil material models for analyses 1to 10 among a total of 30 analyses . . . . . . . . . . . 62Figure 4.4 Non-linear variation of (a) mobilized friction angleand (b) mobilized dilation angle with plastic shearstrain used in HF soil material models for analyses11 to 20 among a total of 30 analyses . . . . . . . . . 63Figure 4.5 Non-linear variation of (a) mobilized friction angleand (b) mobilized dilation angle with plastic shearstrain used in HF soil material models for analyses21 to 30 among a total of 30 analyses . . . . . . . . . 64Figure 4.6 Schematic diagram of LF pipe-soil model . . . . . . . 65Figure 4.7 MF-GP uncertainty quantification flowchart . . . . . 65Figure 4.8 HF pipe-soil model: pipe von Mises stress plot andsoil plastic shear strain plot . . . . . . . . . . . . . . . 66Figure 4.9 Peak compressive strain percent generated in the pipeat various instants with increasing fault displacements:(a) analysis 1 - 5 (b) analysis 6 - 10 . . . . . . . . . . 67Figure 4.10 Peak compressive strain percent generated in the pipeat various instants with increasing fault displacements:(a) analysis 11 - 15 (b) analysis 16 - 20 . . . . . . . . 68xiiiLIST OF FIGURESFigure 4.11 Peak compressive strain percent generated in the pipeat various instants with increasing fault displacements:(a) analysis 21 - 25 (b) analysis 26 - 30 . . . . . . . . 69Figure 4.12 Cross-evaluation data (peak compressive strain per-cent) at 25 mm fault displacement for multi-fidelitymodel trained with both HF and LF data . . . . . . . 70Figure 4.13 Cross-evaluation data (peak compressive strain per-cent) at 25 mm fault displacement for multi-fidelitymodel trained with only HF or LF data . . . . . . . . 71Figure 4.14 Cross-evaluation data (peak compressive strain per-cent) at 100 mm fault displacement for multi-fidelitymodel trained with both HF and LF data . . . . . . . 72Figure 4.15 Cross-evaluation data (peak compressive strain per-cent) at 100 mm fault displacement for multi-fidelitymodel trained with only HF or LF data . . . . . . . . 73Figure 4.16 Cross-evaluation data (peak compressive strain per-cent) at 300 mm fault displacement for multi-fidelitymodel trained with both HF and LF data . . . . . . . 74Figure 4.17 Cross-evaluation data (peak compressive strain per-cent) at 300 mm fault displacement for multi-fidelitymodel trained with only HF or LF data . . . . . . . . 75Figure 4.18 Sensitivity of input variables at 25 mm, 100 mm and300 mm fault displacements . . . . . . . . . . . . . . 76Figure 4.19 Peak compressive strain predictions at (a) 25 mmfault displacement and (b) 100 mm fault displacement 77Figure 4.20 Peak compressive strain predictions at 300 mm faultdisplacement . . . . . . . . . . . . . . . . . . . . . . . 78Figure 4.21 Peak compressive strain predictions at (a) 25 mmfault displacement and (b) 100 mm fault displacement 79Figure 4.22 Fragility curve for pipeline failure under compression 80Figure 5.1 Integration of regional probabilistic ground deforma-tion with pipeline response uncertainty quantification 93Figure 5.2 Schematic diagram of FE model displacement bound-ary conditions derivations from probabilistic grounddeformation analyses . . . . . . . . . . . . . . . . . . 94Figure 5.3 LRFZ finite fault model and sub-fault discretizationalong with [3] fault scarp information . . . . . . . . . 94xivLIST OF FIGURESFigure 5.4 Sample critical rupture scenario slip distribution andregional ground deformation profiles in the East-west,North-south and Up-down direction in the City ofVictoria . . . . . . . . . . . . . . . . . . . . . . . . . . 95Figure 5.5 Sample critical rupture scenario slip distribution andregional ground deformation profiles in the East-west,North-south and Up-down direction in the City ofVictoria . . . . . . . . . . . . . . . . . . . . . . . . . . 96Figure 5.6 City of Victoria buried gas pipeline network underconsideration . . . . . . . . . . . . . . . . . . . . . . 97Figure 5.7 Cross-evaluation of MF-GP surrogate model for (a)peak axial compressive strains and (b) peak axial ten-sile strains . . . . . . . . . . . . . . . . . . . . . . . . 98Figure 5.8 Histograms of the simulated differential ground de-formations in the (a) Up-Down and (b) North-Southdirection at the location of interest . . . . . . . . . . 99Figure 5.9 Histograms of the simulated differential ground de-formations in the East-West direction at the locationof interest . . . . . . . . . . . . . . . . . . . . . . . . 100Figure 5.10 Variation of predicted peak axial strains with themost sensitive parameters at the location of interest(a) compressive strain vs relative density (b) tensilestrain vs outer diameter . . . . . . . . . . . . . . . . . 101Figure 5.11 Histograms of the pipeline peak axial (a) tensile strainsand (b) compressive strains generated at the locationof interest . . . . . . . . . . . . . . . . . . . . . . . . 102Figure 5.12 Fragility curve for pipeline damage using [4] strainlimits . . . . . . . . . . . . . . . . . . . . . . . . . . . 103Figure 6.1 Analysis methodology using Taguchi design of exper-iments . . . . . . . . . . . . . . . . . . . . . . . . . . 107Figure 6.2 Plastic shear strain in sand corresponding to analysis7 (a) at fault side 1 in the vicinity of the fault (b) atfault side 1, 5 m away from fault (c) at fault side 2 inthe vicinity of the fault (d) at fault side 2, 5 m awayfrom fault; analysis 8 (e) at fault side 1 in the vicinityof the fault (f) at fault side 1, 5 m away from fault(g) at fault side 2 in the vicinity of the fault (h) atfault side 2, 5 m away from fault . . . . . . . . . . . . 114xvLIST OF FIGURESFigure 6.3 Plastic shear strain in sand corresponding to analysis12 (a) at fault side 1 in the vicinity of the fault (b)at fault side 1, 5 m away from fault (c) at fault side2 in the vicinity of the fault (d) at fault side 2, 5 maway from fault; analysis 11 (e) 5 m away from fault(f) at the vicinity of fault . . . . . . . . . . . . . . . . 115Figure 6.4 Influence of fault-pipe crossing angle on the peak axialstrains generated at 300 mm and 750 mm fault dis-placements: (a) tensile strain (b) compressive strain;Influence of faulting type: (c) tensile strain (d) com-pressive strain; Influence of pipe wall thickness: (e)tensile strain (f) compressive strain . . . . . . . . . . 116Figure 6.5 Influence of steel grade on the peak axial strains gen-erated at 300 mm and 750 mm fault displacements:(a) tensile strain (b) compressive strain; Influenceof soil type level: (a) tensile strain (b) compressivestrain; Influence of temperature: (a) tensile strain (b)compressive strain; Influence of pressure: (a) tensilestrain (b) compressive strain . . . . . . . . . . . . . . 117Figure 6.6 Peak strain envelope as a function of fault displace-ment: (a) tensile (b) compressive . . . . . . . . . . . . 118xviAcknowledgementsI would like to express my sincere gratitude and appreciation to Dr.Solomon Tesfamariam for his guidance throughout the course of my doc-toral studies. I also acknowledge the financial support facilitated by Dr.Tesfamariam through the Natural Sciences an Engineering Research Coun-cil of Canada (RGPIN-2019–05584) under the Discovery Grant program.I would also like to thank Dr. Katsuichiro Goda for his help and guid-ance in regards to probabilistic regional ground deformation assessment.The probabilistic ground deformation approach used in this thesis has beenprovided by Dr. Katsuichiro Goda. I would also like to thank Dr. SouvikChakraborty for his help and guidance in regards to uncertainty quantifica-tion. The initial conceptualization of this multi-fidelity uncertainty quan-tification work has been suggested by Dr. Souvik Chakraborty. I wouldalso like to thank Dr. Abbas Milani for his valuable comments on enrichingvarious parts of the thesis. Additionally, I would like to thank Dr. DimitrySediako and Dr. Ian Moore for their comments on enhancing various partsof the thesis.I would also like to thank Dr. Tesfamariam research group membersSourav Das and Kostas Skandalos for having discussions on various technicalaspects of my research.xviiDedicationDedicated to my parents.xviiiChapter 1Introduction1.1 MotivationBuried continuous energy pipelines are important structures transport-ing oil, gas and other petroleum byproducts over large distances. Energypipelines can be classified primarily into major transmission pipelines andsecondary distribution networks. These pipelines are generally designed fortheir operating condition loads, such as operating temperature and internalpressure. However, in addition to these loads, these pipelines are also facedwith the risk of failure due to seismic activity in seismically active regions.Buried continuous pipelines are especially vulnerable to failure due to faultrupture induced permanent ground deformations during earthquakes.A review of literature indicates that for buried pipelines, seismic hazardsare primarily due to wave propagation and permanent ground deformation.Damage due to permanent ground deformations are found typically in local-ized areas with high tendency and degree of damage whereas, damage dueto seismic wave propagation are found to be distributed over much widerzones with substantially lower chances and degree of damage [5]. In gen-eral, continuous welded pipelines are more vulnerable to permanent grounddeformation hazard in comparison to wave propagation hazard. Majorityof research works in earthquake engineering has been on the behaviour ofstructures under seismic wave propagation, where as, significantly less at-tention has been paid on the behavior of structures under permanent grounddeformation due to tectonic fault rupture. The reason being, this hazard isobserved in relatively localized areas and poses a threat when the rupturepropagates to the surface.Traditionally, seismic risk assessment of buried pipeline system has in-volved estimation of damage in the pipelines using empirical correlations be-tween observed damage to seismic wave propagation and permanent grounddeformation data. These empirical correlations have been derived by a num-ber of authors based on historically collected data. These data collectedfrom historical seismic events range in various locations and time. Perma-nent ground deformations can be localized, such as due to a fault movement11.1. MOTIVATIONor spatially extended; for example, due to soil liquefaction over a region.A number of empirical relationships exist in literature to quantify faultdisplacements based on the earthquake moment magnitude derived fromhistorical seismic data. Additionally, numerous empirical relationships areavailable to predict magnitude and extent of extended permanent grounddeformations, such as landslides, liquefaction and settlement.Buried pipelines can be subjected to both transient ground deformationsand permanent ground deformations in the event of an earthquake. Tran-sient ground deformations are caused by passage of seismic waves whereas,permanent ground deformations are a result of surface rupture or effects,such as liquefaction and landslides. The impact of earthquake on buriedinfrastructure can vary. Damages due to transient effects are felt over widegeographical areas and damages tend to be widespread. On the other hand,surface ruptures coinciding with buried pipelines result in significantly highpipeline damages but in localised areas. The response of buried pipelinesvary from above ground structures due to the soil support available to bal-ance the inertia forces in contrast to above ground structures where thestructure must bear the loads. Damages in continuous steel pipeline due totransient ground deformations has been reported for the Michoacan earth-quake in 1985 [6]. Damages to continuous steel pipelines due to permanentground deformation on numerous occasions have been reported for the SanFernando earthquake in 1971 [7]. Numerous pipeline failures are observedin the 1971 San Fernando earthquake, such as flexural failure and shearfailure. Buckling failure are observed mostly in pipelines crossing faults.Extensive damage in buried pipelines due to landslides are observed in the1964 Alaska earthquake. Large numbers of damages to buried pipelines dueto earthquakes in Japan with magnitude greater than 7 since 1920 has beenattributed to direct seismic shaking [8].Several methods to assess seismic risk to buried pipelines due to groundshaking have been proposed in the past by [9–16]. Seismic risk analysisapproaches for buried pipelines faced with fault rupture hazard has beenproposed by [17–19]. However, there are some major limitations of theseworks. Firstly, simplified pipe-soil models are used which are not capableof predicting the true behavior of the pipeline. Additionally, probabilisticground deformation hazard is computed using available regression models.Moreover, the methods use empirical fragility functions for pipeline damageto characterize seismic risk. Finally, these approaches are suitable for singlepipeline fault crossing sites and not for large regions.Earthquakes result due to release of accumulated stress at tectonic plateboundaries. Fault rupture is associated with release of excess stress and21.2. RESEARCH OBJECTIVESstrain energy when the rock can no-more resist them. Faults can be primar-ily classified as ‘strike-slip’ and ‘dip-slip’. ‘Strike-slip’ fault rupture involvesprimarily movement in the horizontal direction. ‘Dip-slip’ fault ruptureinvolves movement along the dip of the fault-plane. Under ‘dip-slip’ move-ment, when the horizontal movement is compression, it is called ‘reversefaulting’ whereas, when the horizontal movement is tensile, it is called ‘nor-mal faulting’. The pipeline structural response to surface rupture is depen-dent on the orientation of the pipeline in respect to the rupture. Failuremode due to axial compression may include buckling and failure mode dueto axial tension may result in rupture. ‘Strike-slip’ faulting leads to symmet-rical loading on both sides of the pipeline, whereas, ‘dip-slip’ faulting leadsto unsymmetrical loading on both sides of the pipeline. Generally, ‘dip-slip’faulting is more damaging than ‘strike-slip’ ones because the bearing loadon a buried structure moving downward in soil is greater than the load tomove it in the lateral direction.1.2 Research ObjectivesLiterature review suggests that, there is lack in availability of clear guide-lines and methodologies for assessing risk to buried continuous pipelinesdue to permanent ground deformations arising from fault ruptures. Thisbroad problem covers methods for appropriately modelling pipe-soil struc-tural model to evaluate pipeline structural response including implemen-tation of suitable soil constitutive models, consideration to associated un-certainties in the structural model inputs, considerations to efficiency withrespect to computational time and effort to make the process usable for re-alistic purposes and integration of the pipeline behavior with state of theart probabilistic methods for evaluating ground deformations arising due tofault rupture over a region.The overall objective of this research is to develop an improved and ro-bust method for seismic risk assessment of buried continuous energy pipelinessubjected to fault rupture permanent ground deformations, considering un-certainty at various levels. The specific objectives of this research are asfollows:1. Develop an experimentally evaluated model for soil non-linear ma-terial property. This includes modifying the elastic-perfectly plastic Mohr-Coulomb model in ABAQUS and introducing suitable constitutive relationsfor mobilized friction angle and dilation angle with varying plastic shearstrain. Thereafter, the objective is to evaluate the modified Mohr-Coulomb31.3. OVERVIEWmodel with a sand triaxial test.Develop an experimentally evaluated detailed 3D FE model of pipelineundergoing fault rupture deformations. This includes generating suitablesand constitutive relations with respect to the large scale test and developinga detailed FE model to simulate large scale experimental arrangement andevaluating the model with experimental results.2. Develop a multi-fidelity-based uncertainty quantification approach forburied pipeline undergoing fault rupture deformations.3. Develop a method for determination of target strain envelope curvesand perform sensitivity analysis using Taguchi design of experiments forpipelines undergoing permanent ground deformations.4. To estimate probabilistic ground deformation hazard due to faultrupture at a given region using available state of the art techniques andintegrate it with a pipeline response uncertainty quantification framework todetermine pipeline seismic risk as a result of permanent ground deformation.1.3 OverviewFirstly, nonlinear structural response of a buried continuous pipeline un-dergoing strike-slip fault rupture, i.e., where soil masses get displaced inthe horizontal plane along a fault line, is studied in a detailed manner.A detailed analysis technique employing ABAQUS/Standard with implicitformulation to study the behavior of buried continuous pipelines crossingfault movements is proposed and established with suitable evaluations. Athree-dimensional nonlinear FE model including both material and geomet-ric non-linearity is used for this study. Firstly, a non-linear sand constitutivemodel is adopted and implemented in commercial FE package ABAQUS.The adopted material model is evaluated with available experimental tri-axial test results. This material model is thereafter suitably calibrated for alarge-scale test based on direct shear soil test data for that experiment. Thestudy identified important soil strength parameters from direct shear soiltests conducted for the large-scale test program and converted them suit-ably with respect to the adopted sand constitutive model. The FE model isthen evaluated against full-scale experimental results.Subsequently, nonlinear structural response of buried continuous steelpipeline undergoing fault rupture deformation is studied in a systematicmanner. A detailed and efficient analysis framework for design of buriedcontinuous steel pipelines crossing faults is proposed and illustrated with acase study. This analysis framework considers a number of levels for each41.3. OVERVIEWof the factors (input parameters) thereby allowing to predict peak strainexpected as a function of fault displacement for a range of variation of theinput parameters. A three-dimensional nonlinear FE model including bothmaterial and geometric non-linearity previously developed and evaluatedis used for this study. Taguchi method for design of experiments is thenemployed to evaluate the structural performance of buried pipelines. It isalso used to identify the influence of different parameters on the structuralbehavior of buried pipelines. A case study involving NPS 24 steel pipelinewith a realistic maximum operating internal pressure of 9.1 MPa is alsocarried out. Various realistic steel grades are selected for the study, suchas Gr 290, Gr 386 and Gr 483. The method presented here is suitable forpipeline strain hazard analysis, applicable for a single site.A further extension to the above deterministic analysis approach is to in-clude uncertainty in the input parameters. One of the simplest methods foruncertainty quantification and propagation is the Monte Carlo simulation(MCS). The general process involves identifying sets of random and fixedvariables, generating realizations corresponding to the random variables andthen analysing each case using deterministic FE method. The results ob-tained from the multiple runs of the FE solver are used for computing theprobability distribution and statistical moments of the stochastic output. Alarge number of FE runs are needed for this method to converge makingit infeasible for solving actual problems. An alternative to the simulationbased approaches discussed above are the surrogate based approaches. Theprimary idea here is to train a machine learning model to act as a surrogateto the computationally expensive FE solver. MCS is then performed by us-ing the trained surrogate model. However, even the most efficient surrogatemodels require hundreds of FE runs for generating the necessary trainingdata. This is problematic for cases, such as the 3D pipe soil interactionproblem undergoing fault displacement as the nonlinear 3D FE solver isextremely expensive.Given the limitations with the simulation based and surrogate basedapproaches discussed above, a special type of surrogate model, referred toas the multi-fidelity (MF) surrogate for uncertainty quantification of buriedpipeline undergoing strike slip fault rupture in sand is proposed. In MFsurrogate models, the training data is generated from two solvers – a com-putationally expensive HF solver that yields highly accurate results and acomputationally efficient LF solver that yields results of reasonable accu-racy. The idea is to have a small number of training samples from the HFsolver (in the order of tens) and significant number of training samples fromthe LF solver (in the order of hundreds). Since the LF solver is relatively51.4. THESIS ORGANISATIONefficient, the data generating process in this case is relatively efficient. Asthe LF solver, the beam element pipe along with bi-linear soil springs modelhave been used. The 3D model developed in the authors’ previous studyis used as the HF solver. Among different MF surrogates available in theliterature, we have used the MF Gaussian process (MF-GP) also known asco-Kriging for the current study.A novel method to estimate regional seismic risk to buried continuouspipelines probabilistically, due to earthquake-induced permanent ground de-formations is proposed. The seismic risk assessment method is illustrated forburied gas pipelines in the City of Victoria considering seismic risk from theLeech River Valley Fault Zone (LRVFZ). The method considers uncertaintyon earthquake rupture, soil properties, pipe geometry as well as operatingconditions. It utilizes experimentally evaluated complex pipe-soil numericalmodels in an efficient manner. Major improvements of this method over ex-isting studies are highlighted here. Firstly, stochastic source modelling andanalytical Okada solutions are used in this study to generate regional grounddeformation, probabilistically. Previous studies used regression equations todefine probabilistic ground deformations along a fault. Secondly, in the cur-rent study complex experimentally evaluated pipe-soil FE models are em-ployed including non-linear soil constitutive relations. Earlier investigationsused simple soil spring – beam element pipe models to evaluate pipelineresponse. Finally, the current approach uses multi-fidelity Gaussian pro-cesses to ensure efficiency and limit required computational resource. Thisis required since a large number of runs are necessary for uncertainty quan-tification exercises. It is also necessary due to the computationally expensivenature of complex pipe-soil FE models used.1.4 Thesis OrganisationThe thesis is written based on manuscripts published/submitted in var-ious journals. In chapter 1, introduction provides the research objectivesand a brief background of the problem addressed here. A detailed literaturereview on various aspects of the research has been presented in chapter 2.Details on the developed pipe-soil FE model to study pipeline behavior un-der ground deformation loads are presented in chapter 3. A multi-fidelitybased approach for uncertainty quantification on the above problem anddevelopment of associated fragility curves is presented in chapter 4. In-tegration of the developed uncertainty quantification model with regionalprobabilistic ground deformation approaches is presented in chapter 5. A61.4. THESIS ORGANISATIONTaguchi design of experiments based sensitivity analysis and strain envelopecurve development approach on the above problem is presented in chapter 6.Concluding remarks are made in chapter 7.Chapter 1: IntroductionChapter 2: Literature ReviewChapter 3: Finite Element Modelling and Evaluation of Buried Pipeline Response undergoing Fault Rupture DeformationsChapter 4: Multi-fidelity approach for Uncertainty Quantification of Buried Pipeline Response undergoing Fault Rupture Displacements in SandChapter 5: Regional Probabilistic Seismic Risk Analysis of Buried Pipelines due to Permanent Ground Deformation Hazard for Victoria, British Columbia, CanadaChapter 6: Design of Buried Pipeline undergoing Fault Rupture Deformations in Sand using Taguchi Design of ExperimentsChapter 7: Summary and ConclusionsRegional Seismic Risk Assessment of Buried PipelinesFigure 1.1: Thesis chapters flowchart7Chapter 2Literature Review2.1 IntroductionThe literature review conducted can be classified as shown in Figure 2.1.The objective of the literature survey conducted is to address the abovesubject under a number of different modules. The areas covered in the lit-erature review include methods to analyze the structural performance of aburied pipeline undergoing fault rupture deformations, experimental teststo study pipeline behavior under fault rupture, soil (sand) constitutive re-lations to appropriately model non-linear behavior, available approaches foruncertainty quantification, multi-fidelity methods for uncertainty quantifi-cation and probabilistic approaches to estimate ground deformation due tofault rupture on a regional scale.2.2 Pipeline-fault rupture analysisEnergy pipelines are an important element of oil and gas infrastructurewhich comprise of major transmission pipelines and distribution pipelinenetwork. Energy pipelines play a vital role in regional, national and inter-national economy linking geopolitical matters and matters of global security[20]. Oil and gas pipelines are linear structures of high importance and fail-ure in high pressure oil and gas pipeline can lead to severe consequences,such as contents spill leading to ecological disaster, explosion leading to fire,severe economic losses and loss of lives [21]. Buried energy pipelines trav-elling across seismically active regions are faced with an inherent risk dueto seismic activity. This make seismic risk assessment of buried pipelinesimportant to safeguard against possible seismic hazard. International designcodes and available guidelines for energy pipelines, such as [22], [23], [24],[25], [26], [27] provide pipeline strain limits; however, they do not provideclear and detailed guidelines on predicting the strain demands for pipelines.Numerical modelling is recommended in some guidelines but only in a verygeneric sense. Additionally, none of these guidelines provide methods to re-82.2. PIPELINE-FAULT RUPTURE ANALYSISRisk Assessment of Buried Continuous Pipelines due to Fault RupturePipeline-Fault Rupture AnalysesMulti-fidelity based Uncertainty Quantification of Pipeline-fault Structural SystemProbabilistic Fault Displacement HazardAnalytical MethodsSimplified Numerical ModelsComplex 3D Numerical ModelsSoil Constitutive RelationsApproaches for Uncertainty QuantificationFactors contributing to uncertainty in a pipe-soil problemApproaches for estimating ground displacement due to Fault Rupture ProbabilisticallyFigure 2.1: Literature Review Flowchartliably use available soil constitutive relations and apply to site specific soildata to complete a detailed fault rupture-pipeline FE analysis. A represen-tative summary of different studies on this topic are provided in Table 2.1.Earthquakes are associated with fault ruptures, such as normal fault-ing, strike-slip faulting and reverse faulting as a result of bedrock thrusting.These ruptures propagate towards ground surface resulting in surface rup-tures. These surface ruptures pose a serious threat towards the structuralintegrity of buried pipelines, along with other infrastructure. The responseof buried pipelines to surface rupture has been studied extensively in nu-merous research investigations. Based on the method adopted, the availablestudies can be broadly classified into three categories – analytical and/orsemi-analytical studies, experimental studies and numerical studies. In an-alytical and/or semi-analytical methods, simplified models are utilized toanalyze the problem at hand ([28], [29]). The analytical and semi-analyticalmodels-based approaches are generally based on beam on elastic foundationtheory employing bi-linear steel material which are an approximate repre-sentation of the materials involved. These methods consider the pipe tobe under tension and bending which covers only a partial loading scenarioof the problem and are not accurate enough partly, due to their inability92.2. PIPELINE-FAULT RUPTURE ANALYSISto capture ovalities or local buckling. Attempts to improve the simplifiedmodels can also be found in the literature ([30], [31], [32], [33], [34]).Experimental studies to identify the effect of fault ruptures on buriedpipelines can also be found in literature. [35] carried out centrifuge tests toinvestigate the effect of moisture content, strain rate, burial depth and pipediameter on the pipeline. [36] studied effect of normal and strike-slip faultingand the effect of fault crossing angle on the generated pipeline strains. [37]conducted centrifuge tests to examine one positive pipe-fault crossing angleand another negative pipe-fault crossing angle. [38] performed centrifugetests to study the effect of burial depth and pipe diameter. Full-scale studiesfor identifying the behavior of buried pipelines can also be found in literature([39], [40], [41], [42], [43], [44]).[45] carried out numerical studies where the beam element is used tomodel the pipe and bi-linear springs are used to model the soil. This modelis used to simulate a pipeline damage due to faulting observed in Izmit(Turkey). Use of similar simple numerical models is also reported in [46]and [17]. Another analytical approach to model pipe-soil interaction includethat of by [47] where a shell pipe supported by spring elements simulatingthe soil is considered. Unfortunately, numerical models considering the pipeas a beam and surrounding soil as bi-linear springs supporting the pipe inthree mutually perpendicular directions are incapable of capturing the truethree-dimensional nature of this problem due to its simplistic nature [26].Therefore, researchers now-a-days are inclined towards using more detailedthree-dimensional continuum pipe-soil interaction models for this problem[1, 48–55]. The simplified beam element pipeline model supported by soilsprings in three mutually perpendicular directions, is computationally inex-pensive due to the absence of contact simulation, due to the involvement ofmuch lesser number of elements and simplified formulations. However, thesoil springs can only approximately model the soil response due to its springstiffnesses acting in three mutually perpendicular directions. These stiff-nesses have been obtained for axial, lateral, vertically up and vertically downmovements of a pipeline and are not applicable for the complex pipe-soilmovement that takes place during a fault rupture. There do exist beam ele-ment with additional degree of freedom for capturing complex phenomenon,such as ovalization. However, even such advanced beam elements can onlybe associated with soil springs and hence, the above-mentioned problempersists.A 3D continuum soil and shell element pipeline model, on the otherhand, is capable to simulating the true nature of pipe-soil interaction alongwith associated local buckling and ovalization of the pipeline. However, one102.3. UNCERTAINTY QUANTIFICATION AND UNCERTAINTY PROPAGATION OF BURIED PIPELINE RESPONSEmust bear in mind that the computational cost associated with such modelis significantly more as compared to the beam-element based models. Oneof the primary challenges associated with three-dimensional pipe-soil inter-action model is development of material model for soil. The most popularmaterial model for sand is the Mohr-Coulomb criteria. In this model, thestress-strain relation is represented by using two parameters namely, cohe-sion and internal friction angles. The stress-strain behavior of dense sandcan be grossly defined as elastic zone, pre-peak plastic zone ([56]), rangingfrom the initial yield surface up to the failure surface and the post peaksoftening zone, ranging from failure to higher deformations. Some of themost comprehensive and simple to use equations to define this behavior areprovided by [57] and [58] among others. [58] conducted a series of triaxialtests of sands with varying relative densities and varying confining pressuresto propose these equations. [59] and [60] provide expressions to estimateelastic modulus of soils. [61] proposed a closed-form relation between thepeak and critical internal angle of friction. Other works on sand constitutivemodel includes that of by [62], [63], [64], [65], [66] and [67].In regards to the current study, [1], [53] and [55] had analyzed pipelineresponses under fault movements using 3D soil continuum model but onlywith linear variation of friction angle and dilation angle as a function ofplastic deviatoric strain. [53] considered an idealized dense sand for theirstudy. [68] had successfully used non-linear soil strength with varying plasticshear strain to analyze 2D pipe-soil problems.2.3 Uncertainty quantification and uncertaintypropagation of buried pipeline responseAn important aspect associated with the problem of pipelines under faultmovement is the presence of uncertainties. For example, there is an inherentvariability in the pipeline geometry and material strength due to the detailsinvolved in the production process. Additionally, soil properties are char-acterised by significant variability over location and time. The operatingconditions in the pipeline, such as internal pressure and temperature alsovaries over time. All these variability leads to significant uncertainties inthe prediction of structural performance of a pipeline. To account for theseuncertainties, design engineers generally use a safety factor over a deter-ministic analysis design. However, considering a safety factor may result inunsafe design or in some cases over conservative design, resulting in highercosts. Therefore, for a safe and economical design, it is important to track112.3. UNCERTAINTY QUANTIFICATION AND UNCERTAINTY PROPAGATION OF BURIED PIPELINE RESPONSETable 2.1: Representative summary of different studies on buried pipelineundergoing fault rupture.ReferenceStudy Type Variation of Important Parameters StudiedAnalytical ExperimentalNumerical FaultTypeCrossingAnglePipeCross-sectionSoilTypeInternalPressureTemperaturePipe Materialand Grade1D 3D[28] *[29] *[30] *[31] * *[32] * *[33] * *[34] * * * * * *[35] * * *[36] * * *[37] * *[38] * *[39] * * * * *[41] * * *[40] * * * * *[42] * *[43] * * * * *[44] * *[45] * *[46] * * * * *[17] * * *[48] * * * * * *[49] * *[50] * * * * *[1] * * * *[52] *[53] * *[54] * *[55] * * * * *the propagation of uncertainties from the input variables to the output.The literature on uncertainty quantification and uncertainty propagationin buried pipelines is quite sparse; this is primarily because of the compu-tational cost associated with solving an uncertainty propagation problem.[69] considered variability of input parameters and used the response sur-face methodology to perform reliability analysis of a pipe-soil problem. [70]studied pipeline upheaval buckling using random variables for initial imper-fection and pipe-soil interactions. Effect of spatial variability in pipe embed-ment depth, which in turn results in variable lateral resistance in sub-seapipelines is studied by [71]. Reliability analysis of buried pipeline based onvariability of traffic load is conducted by [72]. Upheaval buckling studies ofpipeline considering variability in inputs using artificial neural network ap-proaches is conducted by [73]. Studies which focused on studying variabilityof soil properties in pipe-soil interaction problems include that of by [74],[75] and [76]. Other probabilistic pipe-soil studies include [77], [78], [79],[80] and [81]. However, none of the studies discussed above deal with buriedpipelines undergoing fault rupture displacements.The only work in this domain is conducted by [18]. This work considers122.3. UNCERTAINTY QUANTIFICATION AND UNCERTAINTY PROPAGATION OF BURIED PIPELINE RESPONSEuncertainties in defining the fault displacement hazard but is faced with anumber of limitations, such as a simple approximate beam type pipe modelsupported by soil springs for structural analysis. Additionally, the study didnot consider uncertainties with respect to important influencing parameters,such as pipeline geometric and material properties, soil material properties,operating conditions and pipeline-fault crossing angle. The objective of thecurrent study is to address the limitations discussed above by conductinga thorough study to investigate the effect of input uncertainties on the re-sponse.The simplest method for uncertainty quantification and propagation isthe Monte Carlo simulation (MCS) ([82] and [83]). The general process in-volves identifying sets of random and fixed variables, generating realizationscorresponds to the random variables and then analysing each case usingdeterministic FE method. The results obtained from the multiple runs ofthe FE solver are used for computing the probability distribution and sta-tistical moments of the stochastic output. By post-processing the results,probability of failure of the system can also be calculated. Although MCS issimple to implement, the convergence rate of this method is extremely slow.Large numbers of FE runs are needed for this method to converge makingit infeasible for solving actual problems. As an improvement to the classi-cal MCS method, approaches, such as stratified sampling ([84] and [85]) andimportance sampling ([86], [87] and [88]) have been proposed by researchers.However, even these methods often suffer from the curse of dimensionality.An alternative to the simulation based approaches discussed above are thesurrogate based approaches. The primary idea here is to train a machinelearning model to act as a surrogate to the computationally expensive FEsolver. MCS is then performed by using the trained surrogate model. Pop-ular surrogate models available in the literature include polynomial chaosexpansion ([89] and [90]), neural networks ([73] and [91]), radial basis func-tions ([92] and [93]), analysis of variance decomposition ([94] and [95]) andKriging ([93] and [96]). However, even the most efficient surrogate modelsrequire hundreds of FE runs for generating the training data. This is prob-lematic for cases, such as the 3D pipe-soil interaction problem undergoingfault displacement as the nonlinear 3D FE solver is highly expensive.Fragility curves provide a relation between the probability of exceedinga certain state of damage for a structure with that of the intensity of loadacting on that structure. In regards to buried continuous pipelines thereare a number of studies that have dealt with producing fragility functionsof such systems, such as [97–105]. Most of these work however dealt withstudying the effect of transient seismic ground deformations on continuous132.4. MULTI-FIDELITY MODELSpipelines. Study by [102] involved probabilistic risk assessment of a buriedcontinuous pipeline under transverse permanent ground deformation due tolandslide and there by leading to the generation of fragility curves. Workby [104] covers damage due to both seismic wave propagation as well aspermanent ground deformation. A part of the current scope has been toutilize the developed multi-fidelity model to develop fragility curves for theconsidered pipeline-fault problem.Design of experiments are rational ways of investigating all possibilitiesin a test involving several factors and their respective ranges [106]. Designof experiments in the past have been used for a range of purposes, such asagriculture, chemical industries, pharmaceutical industries, manufacturingindustries, product development and various other fields [106]. The stan-dard orthogonal arrays by Taguchi [107] are extensions of the initial worksby [108] and [109]. The fundamental basis for fractional factorial arraysand orthogonal arrays is proposed by [108]. Methods to derive mixed-levelorthogonal arrays from difference matrices are proposed by [109].2.4 Multi-fidelity modelsUncertainty quantification exercises require a number of scenarios tobe analyzed. Advanced numerical models capable of predicting the truebehavior of a process require significant computational resources. In regardsto the current research, the experimentally evaluated geotechnical numericalmodels are computationally quite expensive. An uncertainty quantificationexercise which preserves the complexity and accuracy of a process requiresadvanced numerical models. Fitting a surrogate model with such models,also called HF models require large numbers of analyses to be completed. Onthe contrary, LF models or approximate models although computationallycheap are not able to capture the complexity of a process and certainly notaccurate.Multi-fidelity approaches leverage the accuracy and complexity of HFmodels with the efficiency of LF models to predict nearly accurate resultsin an efficient manner. Multi-fidelity surrogates are surrogate models, thatfit information between input variables and output parameters. [110] iden-tifies the primarily three methods to combine fidelities namely, adaptation,fusion and filtering. Adaptation works to improve the LF prediction basedon HF behaviour. Fusion on the other hand, simply integrates the LF andHF response. Filtering uses LF response to evaluate HF response at specificpoints. [111] categorizes multi-fidelity surrogate modelling under several142.4. MULTI-FIDELITY MODELStypes, such as multiplicative correction, additive correction and comprehen-sive correction. Multiplicative and additive correction methods modify thelower fidelity model with multiplicative or additive parameter. Comprehen-sive corrections introduce both multiplicative and additive corrections in thesame surrogate.A model g : x → y is a functional relationship between inputs x andoutputs y. Considering that the uncertainty in inputs is defined by stochas-tic variable X, uncertainty propagation characterizes the statistics of g(X),such as the model expectancy and variance. Variance reduction techniquesuse correlations between g(X) and a supplementary random parameter whosestatistical characteristics are obtained from the lower-fidelity models. [112]and [110] employed the control variate method as the multi-fidelity estima-tor. Given the limitation in computational resources, the objective hereis to minimize the mean square error of the multi-fidelity estimator. [110]assumed a HF model g1 and LF surrogate models g1, g2, g3, ....., gk and con-sidered the same input for all these models. The objective of the process isto evaluate the estimator:t = E[g1(Z))] (2.1)corresponding to the HF model g1 from input random variable Z. The kscenarios of the random variable Z are denoted as z1, z2, z3, ....zmk. Model giis evaluated for i = 1, 2, 3, ...., k at z1, z2, z3, ....zmi resulting in gi(z1), ........gi(zmi).mi ofm represents the number of assessments for model gi for i = 1, 2, 3, ....k.The Monte Carlo estimator is then expressed as:y−im = (1/m)m∑j=1gi(zj) (2.2)which estimates E[g1(Z))] by sampling z1 , z2 , z3 , .....zm of random vari-able Z where, i = 1, 2, 3, ....k. First the method estimates y−im from mimodel evaluations gi(z1), ........gi(zmi) for i = 1, 2, 3, ...., k. Thereafter, it es-timates Monte Carlo estimate y−im−1 from evaluations gi(z1), ........gi(zmi−1)with i = 2, 3, ...., k. The Multi-fidelity Monte-Carlo estimate is expressedas:tˆ = y−1m +k∑i=2ϵi(y−imi − y−imi−1) (2.3)The coefficients ϵ2 , ϵ3 , ....ϵk control the respective weights of (y−imi − y−imi−1).The framework for the multi-fidelity Monte Carlo estimator modifies the es-timated HF quantities with a sum of estimates of differences of lower-fidelity152.4. MULTI-FIDELITY MODELSquantities, there by fusing the outputs from both the high and LF modelsto predict the statistics of the HF model. The estimator is a function of thecoefficients ϵ2 , ϵ3 , ....ϵk and the number of model evaluations m0 ,m1 , ....mkand hence, these parameters are chosen in such a way to minimize the meansquare error of the estimator for a given computational budget.The work by [113] considers HF model Phigh(x, u) and LF model Plow(x, u)with the input vectors defined by the design variables x and model param-eters u. u is a random sample from random vector U(ω). The output fromthe HF model Phigh(x, U(ω)) is expressed as random variable A(x, ω). Thestatistic (mean and variance) of the HF model output is denoted as sA(x).The estimation of the mean for the HF model is considered as sA = E[A(ω)].The estimator corresponding to sA is sˆA . Considering constant values fordesign variables xk , the HF model is Phigh(xk , U(ω)) and the output isA(ω). Considering constant values for design variables xk , the LF modelis Plow(xk , U(ω)) and the output is B(ω). If independent and identicallydistributed random samples (ui , i = 1, 2, 3...) are drawn from U(ω), the HFand LF models Phigh(xk , U(ω)) and Plow(xk , U(ω)) are evaluated to deriveai and bi . The control variate multi-fidelity estimator is then given as:ŝA =1nn∑i=1ai + ϵ(sB − 1nn∑i=1bi) (2.4)ϵ is the control parameter and statistic sB = E[B(ω)]. The sB can beapproximated as: b¯m =1m∑mi=1 bi whenm > n. The multi-fidelity estimatorthus becomes:sˆA = a¯n + ϵ(b¯m − b¯n) (2.5)The basic idea here is to enhance the original estimator a¯n by (b¯m − b¯n)using a large number of evaluations m from the LF model. The variance ofthe multi-fidelity estimator is given as:V ar[ ˆsA,p ] = V ar[a¯n ]+ϵ2V ar[b¯m ]+ϵ2V ar[b¯n ]+2ϵCov[a¯n , b¯m ]−2ϵCov[a¯n , b¯n ]−2ϵ2Cov[b¯m , b¯n ](2.6)=ς2An+ϵ2ς2Bm+ϵ2ς2Bn+2ϵ1nmn∑i=1m∑j=1Cov[ai, bj ]−2ϵϖABςAςBn−2ϵ2 1nmm∑i=1n∑j=1Cov[bi, bj ](2.7)162.4. MULTI-FIDELITY MODELS=ς2An+ϵ2ς2Bm+ϵ2ς2Bn+2ϵ1nmn∑i=1Cov[ai, bi]−2ϵϖABςAςBn−2ϵ2 1nmn∑j=1Cov[bj , bj ](2.8)=1n(ς2A + ϵ2ς2B − 2ϵϖABςAςB)−1m(ϵ2ς2B − 2ϵϖABςAςB) (2.9)where, ς2B = V ar[B(ω)];ϖAB = Corr[A(ω), B(ω)]This variance can be minimized in terms of ϵ and r, where r = m/n toobtain the optimum value of V ar[ ˆsA,p ] which is expressed as:MSE[ ˆsA,p∗] = V ar[ ˆsA,p∗] = (1 +r∗w)[1− (1− 1r∗)ϖAB2]ςA2p(2.10)where, ϵ∗ = ϖAB ςAςB ; r∗ =√wϖ2AB1−ϖ2AB; ς2A = V ar[A(ω)]; ς2B = V ar[B(ω)];ϖAB =Corr(A(ω), B(ω)) being the variance correlation coefficient of A(ω) andB(ω).[114] provides an account of application of co-kriging in multi-fidelityoptimization. [115] and [116] discussed kriging from the fundamentals. Tomake a prediction at some point x in the domain, considering uncertainty,a random variable Y (x) is considered having a mean of µ and variance ς2.Considering two distinct points xi and xj and assuming that the associatedfunctional values are y(xi) and y(xj); these functional values are more likelyto be close if the distance |xi − xj | is small. The correlation between therandom variables is given as:Corr[Y (xi), Y (xj)] = exp(−d∑l=1θl |xil − xjl|pl) (2.11)If the distance |xi − xj | is infinite, the correlation tends to be zero and ifthe distance is zero, the correlation tends to be 1. The θl parameter controlsthe sensitivity of the correlation in the lth coordinate direction. The plparameter controls the smoothness of the correlation in the lth coordinatefunction. The uncertainty about the function’s value at the n points can beexpressed as:172.4. MULTI-FIDELITY MODELSy =Y (x1)Y (xn) (2.12)By equating the derivative of log-likelihood function with ς2 and µ tozero,µˆ =1′R−1y1′R−11(2.13)ςˆ2 =(y − 1µˆ)′R−1(y − 1µˆ)n(2.14)Replacing the above expression in the log-likelihood function becomes:− n2log(ςˆ2)− 12log |R| (2.15)The function may be maximized to estimate θˆl and pˆl (l = 1, 2, 3, ...d),as the expression depends on R. These estimated values may be used toderive µˆ and ςˆ2. To predict values at an unknown point x∗, a function y∗, isassumed. To assess its accuracy, the point (x∗, y∗) is added to the observeddata as the (n+1)th observation and the likelihood function is computed us-ing parameter values determined in maximum likelihood estimation. Usingthe determined fixed parameters, the log-likelihood is a function of y∗ andthe corresponding y∗ that maximizes the likelihood function is the krigingpredictor. The standard formula of kriging predictor is:ˆy(x∗) = µˆ+ r′R−1(y − 1µˆ) (2.16)[114] provides the fundamentals from co-kriging. Assuming there aretwo data sets, one with HF data ye at points Xe and LF data yc at pointsXc (Xe ⊂ Xc). The combined set can be expressed as:X =(XcXe)= (x1c , ....xncc , x1e, ....xnee )T (2.17)The value at point X is a sample from a Gaussian random variable.Y =(Yc(Xc)Ye(Xe))= (Yc(x1c), ....Yc(xncc ), Ye(x1e), ...Ye(xnee ))T (2.18)182.4. MULTI-FIDELITY MODELSIt is considered that, if the value of the HF model output is known at xi,no further information can be obtained from the LF model at that point.Gaussian processes Zc(•) and Ze(•) represent the local features of the LFand HF models. The HF model function can be expressed as:Ze(x) = ρZc(x) + Zd(x) (2.19)Where, ρ is a scaling function and represents a Gaussian process definingthe difference between Ze(x) and ρZc(x). The co-variance is defined as:cov {Yc(Xc), Yc(Xc)} = cov {Zc(Xc), Zc(Xc)} = ς2cψc(Xc, Xc) (2.20)cov {Ye(Xe), Yc(Xc)} = cov {ρZc(Xc) + Zd(Xc), Zc(Xe)} = ρς2cψc(Xc, Xe)(2.21)cov {Ye(Xe), Ye(Xe)} = cov {ρZc(Xe) + Zd(Xe), ρZc(Xe) + Zd(Xe)}(2.22)= ρ2cov {Zc(Xe), Zc(Xe)}+ cov {Zd(Xe), Zd(Xe)} (2.23)= ρ2ς2cψc (Xe, Xe) + ς2dψd (Xe, Xe) (2.24)C =(ς2cψc(Xc, Xc) ρς2cψc(Xc, Xe)ρς2cψc(Xe, Xc) ρ2ς2cψc(Xe, Xe) + ς2dψd(Xe, Xe))(2.25)where, ψc (Xe, Xc) for instance is a matrix of the form ψc for correlationsbetween Xe and Xc and is similar to Equation 2.20. Similar to kriging,the LF model data are evaluated independently and maximum likelihoodestimate of µc, ς2c , θc and pc are calculated by maximizing the log-likelihood.To estimate µd, ς2d , θd, pd and ρ from the difference Gaussian Process, thefollowing relation is considered:d = ye − ρyc(Xe) (2.26)where, yc(Xe) are the values of yc at locations common to Xe. By maxi-mizing the log-likelihood of d, the hyperparameters of µd, ς2d , θd, pd and ρ can192.4. MULTI-FIDELITY MODELSbe obtained. The co-kriging predictor based on the knows hyper-parametersis given as:yˆe(x(ne+1)) = µˆ+ uTC−1(y − 1µˆ) (2.27)u =(ρˆςˆ2cψc(Xc, xn+1)ρˆςˆ2cψc(Xe, xn+1) + ςˆ2dψd(Xe, xn+1))(2.28)Given the limitations with the simulation based and surrogate basedapproaches discussed above, we propose to use a special type of surrogatemodel, referred to as the multi-fidelity (MF) surrogate for uncertainty quan-tification of buried pipeline undergoing strike slip fault rupture in sand. InMF surrogate models, the training data is generated from two solvers - acomputationally expensive HF solver that yields highly accurate results anda computationally efficient LF solver that yields results of reasonable accu-racy. The idea is to have a small number of training samples from the HFsolver (in the order of tens) and significant number of training samples fromthe LF solver (in the order of hundreds). Since the LF solver is relativelyefficient, the data generating process in this case is relatively efficient. Asthe LF solver, the beam model previously discussed have been used. The3D model developed in the authors’ previous study is used as the HF solver.Among different MF surrogates available in the literature, we have used theMF Gaussian process (MF-GP) [82], also known as the co-Kriging [83] forthe current study. HF models yield highly accurate results but are computa-tionally expensive and time consuming in nature. HF models are generallymore detailed, complex leading to accurate results. On the other hand,LF models yield approximate results due to the simplified nature of themodels but are highly time efficient and require much less computationaleffort. LF models are generally more simplified in nature. Multi-fidelityapproaches aim to combine the accuracy of hi-fidelity models with the ef-ficiency of LF models to predict nearly accurate results in a reasonablyefficient manner. Multi-fidelity models generally involve the construction ofsurrogate models, which are approximations that fit to available data of aprocess and define a relationship between input variables and output quan-tities of interest. Construction of surrogate models lead to reduction in thecomputational efforts required for optimization or uncertainty quantificationproblems. Multi-fidelity models may however consider both surrogates aswell as non-surrogates.202.5. PROBABILISTIC GROUND DEFORMATION DUE TO FAULT RUPTURE AND THE LEECH RIVER FAULT ZONE2.5 Probabilistic ground deformation due to faultrupture and the Leech River Fault ZoneCharacterizing seismic risk to buried pipeline infrastructure and otherstructures due to surface rupture require due consideration to the uncer-tainties regarding surface displacement values as a result of fault rupture.Initial studies to probabilistically characterize fault displacement hazardscan be found in [117, 118]. Broadly, the work adopts the ‘Probabilisticseismic hazard analysis’ level 4 (PSHA) practice as provided in [119]. Theapproach to estimate the hazard involves several expert evaluations of seis-mic sources along with their potential for fault displacement. To considerfor uncertainties in their evaluations, the experts propose several weightedalternative evaluations. Fault displacements are classified as principal anddistributed to differentiate between the main plane of fault rupture and thesecondary ruptures occurring in the region adjacent to the main plane. Thebasic formulation by [117] estimates the fault displacement hazard as a prod-uct of the frequency of fault displacement occurrences at a point and theconditional probability representing the fault displacement above a certainmagnitude. In this regard, primarily two approaches namely the ‘displace-ment approach’ and ‘earthquake approach’ are used; the former uses pointspecific historical surface displacement data to define the frequency of faultdisplacement occurrences and the later uses seismic source evaluation to re-late the frequency of earthquake occurrences at the source to the frequencyof fault displacements. The work in [118] provides probabilistic distributionsfrom global data sets for normal faulting.Further improvement to [118] is [120] in which, data regarding strike-slip faulting ground deformations at faults, maps of fault trace prior to anearthquake and maps of surface rupture from various earthquakes are com-piled. These data are suitably processed to generate regression equationsfor principal fault displacement vs. distances along the fault, distributedfault displacement magnitudes vs. perpendicular distances from the princi-pal fault and off-fault probability of surface rupture. Data compiled includesfault displacement value and location data in addition to the correspondingmagnitude of earthquakes along the principal faults and distributed faultdisplacement data as a function of perpendicular distance from principalfault along with corresponding earthquake magnitudes. This approach [120]uses four probability distribution functions: firstly, to characterize the earth-quake magnitude and fault rupture location along the fault; secondly, todefine the perpendicular distance from the considered site to the fault trace;212.5. PROBABILISTIC GROUND DEFORMATION DUE TO FAULT RUPTURE AND THE LEECH RIVER FAULT ZONEthirdly, to characterize the fault displacement along the principal fault andlastly to characterize distributed fault displacements at locations perpendic-ular to the rupture. In addition, probabilities to define the ratios of cellsthat have ruptured in the principal fault region, distributed fault region andto define the probability of surface rupture for a given earthquake magnitudeare also used. The procedure considers uncertainty of both ”epistemic” and”aleatory” origins.An alternative approach to probabilistically characterize ground defor-mation due to fault rupture is proposed by [121] which uses stochastic sourcemodelling and analytical Okada solutions instead of regression models. Thismethod relies on analytical solution by [122]. In this method, firstly thesource parameters of the rupturing fault are generated by empirical methodsor other means. Uncertainties in fault rupture are considered in the analyses,both in terms of uncertainties in fault geometric parameters and fault slipdistributions. In order to do so, statistical scaling relationships from [123]are used, for modelling the source of the earthquakes probabilistically, whichdefine important features of a fault rupture. Other scaling relationships arealso available based on empirical equations, such as by [124–129]. Sourceparameters considered include geometry parameters of the fault, statisticsof the slip parameters for fault rupture and spatial distribution parametersfor fault rupture slip. These scaling relationships for the source parametersbased on a given rupture nature and earthquake magnitude are obtainedwith the use of publicly available extensive earthquake data and regressionanalyses. Thereafter, uncertainty is considered for the slip process [123] tostochastically generate earthquake slip. Ground displacements are finallycomputed probabilistically using analytical solutions based on Okada equa-tions [122, 130]. Okada equations generate the surface displacements forboth inclined shear as well as tensile faults. In Okada equations, the surfacedisplacements are deterministically calculated in a three-dimensional fieldallowing to calculate differential displacement between any two points for acertain earthquake. Surface displacements calculated at different locationshold physical consistency with one another.The Leech River Valley Fault Zone (LRVFZ) is about 60 km in lengthand is a potential threat for producing earthquakes having moment magni-tude (MW) more than 6 [131]. This fault zone is in the proximity of theCity of Victoria. [131] further highlights the destructive potential from sur-face ruptures arising due to earthquakes at shallow sources in the LRVFZ.[132] indicates that the LRVFZ can be divided into two parts towards theeast, one on the north probably connecting with the Devil’s Mountain Fault(DMF) and the other towards south probably connecting with the Southern222.6. PROBABILISTIC PIPELINE STRAIN RISK ANALYSISWhidbey Island Fault (SWIF). They further confirm the existence of crustalfault in the Victoria region and suggest the potential of micro seismicity dueto slow movement of the active crustal faults. [133] mapped the DMF andindicated that the DMF and LRVFZ may be parts of the same fault net-work. They estimate that this fault network has the potential of generatingearthquakes of MW 7.5 at very close vicinity of the City of Victoria.[134] conducted a seismic hazard study for the Victoria region by con-sidering the combined effects of the LRVFZ and DMF considering varyingfault lengths and slip lengths as well as inter fault interplay. This hazard isthereafter included in the 2020 – 6th Generation Seismic Hazard model ofCanada. Additional study, such as [135] also included LRVFZ as a poten-tial source of seismicity to evaluate probabilistic seismic hazard at Victoria.They predicted an increase of 1 to 23 percent from National Building Codeof Canada (NBCC) 2015 uniform hazard spectra ground motions due tothe inclusion of LRFZ as an earthquake source. Uncertainties with respectto LRVFZ source is suitably considered in this study. [136] conducted aprobabilistic seismic hazard analysis of the Victoria region due to seismicityfrom the LRFZ-DMF system using a probabilistic method based on the faultsource of this region. Their study concluded a 10 to 30 percent increase inseismic hazard in the Victoria region due to the inclusion of LRVFZ-DMFas a potential cause of earthquake.2.6 Probabilistic pipeline strain risk analysisMethods to convert seismic hazard to pipeline strain risk using pipelinestructural analysis can be found in works by [18] and [17]. In this approach,the seismic hazard is represented by a fault displacement vs mean annual rateof exceedance. The pipeline structural analysis provides a relation betweenfault displacement and generated strain. Pipeline strain risk is determinedby combining pipeline structural analysis and seismic hazard as a relationbetween mean annual rate of exceeding strain values vs the correspondingstrains. In this work, a comprehensive study on the pipeline response fora wide range of independent fault displacement components are conducted.However, it is highlighted that, an efficient way would be perform only a setof structural analysis for fault displacement components specific to the pipefault crossing system at hand.[19] performed a study to evaluate probabilistic pipeline risk based onprobabilistic permanent ground deformation hazard. The method computedfailure probability of buried continuous pipelines under different permanent232.6. PROBABILISTIC PIPELINE STRAIN RISK ANALYSISfault displacement levels as well as pipe cross-section properties. The studycombined probabilistic fault displacement hazard, pipeline mechanical re-sponse due to fault displacement loads and empirical fragility functions toderive annual exceedance rate of pipeline failure. [137] deals with simulationbased seismic risk assessment of gas distribution networks. [138] providesan assessment framework for seismic risk assessment of a gas pipeline infras-tructure at a regional scale. The method employs fragility functions fromliterature to estimate losses resulting from liquefaction and landslide hazarddue to earthquake. The basic framework consisted of hazard assessmentand risk assessment components. [97] conducted a comprehensive seismicrisk analysis using spatial coordinates of Azerbaijan natural gas pipeline inIran; the objective being to estimate corresponding losses.24Chapter 3Finite Element Modellingand Evaluation of BuriedPipeline Responseundergoing Fault RuptureDeformations3.1 IntroductionLiterature review suggested that no study until now has been attemptedto use non-linear soil strength constitutive relations to investigate pipelineresponse under fault movements. Employment of non-linear soil strengthinclude considerations to suitably evaluate soil non-linear mobilized shearstrength behavior in FE model with respect to triaxial test results for 3Dapplications. Further, the objective of this study is to develop an improved3D FE model for buried pipeline undergoing fault rupture in sand usingsuitable properties for the non-linear sand relevant for a large scale pipe-soiltest. The novelty of this work is two folds. First, we have incorporated anexperimentally evaluated non-linear soil constitutive model with pre-peakhardening and post-peak softening behavior. Secondly, the soil model iscalibrated suitably based on direct-shear soil test data corresponding to alarge-scale experimental test.To that end, the Mohr-Coulomb soil model has been modified basedon experimentally derived constitutive relations available in the literature.The updated Mohr-Coulomb model with pre-peak hardening and post-peaksoftening behavior has been imported into the FE package ABAQUS by de-veloping a user-defined subroutine, USDFLD. This developed model is thensuitably calibrated for a large-scale test based on direct shear soil test datafor that experiment. Thereafter, this calibrated soil model is used to simu-late large scale experiment of a pipe undergoing fault rupture. The accuracy253.2. PHYSICAL MODEL OF BURIED PIPELINEof the developed FE model is evaluated in two-stages. First, we evaluatethe material model with triaxial test results available in the literature. Themodel used in this study is found to yield good results as quantified lateron. Second, the detailed three-dimensional FE analysis results are comparedagainst other numerical results and experimental results of a full-scale testcarried out at Cornell University [1, 2, 39]. The developed FE model isfound to yield the best results.The rest of the paper is organized as follows. In section 3.2, details aboutthe physical system considered in this study has been provided. The detailsabout the developed FE model is discussed in section 3.3. Evaluation of thematerial model and the developed FE model are presented in section 3.4.Finally, section 3.5 provides the concluding remarks.3.2 Physical model of buried pipelineWe consider the large-scale experimental test carried out at Cornell Uni-versity. Two separate case studies are considered. The experiments consid-ered no internal pressure and no operating temperature on the pipeline andare carried out using split-boxes. The tests involved fixing one end of thepipe, its associated soil mass and applying a displacement on the other end ofthe pipe and its associated soil mass. Displacement applied is a combinationof displacement along the lateral direction of the pipe and a displacementalong the axial direction of the pipe. One of the tests involved a lateraldisplacement along with an axial compressive displacement and the othertest involved a lateral displacement along with an axial tensile displacement.The difference between the two experiments considered resides in the factthat one exerted compressive force in the pipe in addition to bending andthe other exerted tensile force in the pipe in addition to bending due to theexperimental displacement arrangement.Soil used for the large-scale tests are well graded glacio-fluvial sand. Theexperimental sand is tested using direct shear tests with normal stresses of15 kPa which is similar to the vertical stresses developed at a depth of 1.25m in sand situated above water table. This is the expected vertical stress ata depth to the centerline of the HDPE pipes used for these experiments. Thelength of the pipeline used in the experiments are 10.6 m in length. Theexperimental setup consisted of two soil boxes containing the pipe. Eachhalf of the soil box is 6.6 m in length by 3.2 m in width by 2.3 m in depth.Total length of the setup is however, 10.6 m. The boxes are aligned at 65◦with respect to the pipe. The pipe ran perfectly horizontally through the263.2. PHYSICAL MODEL OF BURIED PIPELINEboxes. The pipeline is reported to be fixed at each end of the box [139].Figure 3.2 shows the experimental configuration of the ‘tensile’ strike-sliptest as a FE mesh. The FE mesh is generated to closely represent the actualexperimental setup. Hence, the soil blocks are arranged to be staggeredinitially and gradually displaced along the fault plane. The pipe is modelledto be placed close to the right edge of the stationary soil mass similar to theexperimental arrangement. Soil strength are determined using direct sheartests. Consistent peak shear strength is maintained for the tests. The sandused for the large-scale test is termed “RMS graded sand”. The mean grainsize D50 of the soil is 0.67 mm [40]. It is reported that a consistent directshear peak friction angle of 39◦ − 40◦ and dry density of 15.5-15.8 kN/m3is maintained for these tests. A schematic representation of the describedmodel is shown in Figure 3.1. ‘α’ and ‘β’ in the schematic diagram are thefault angle in the vertical plane and fault angle in the horizontal plane. Asstated previously, the objective of this study is to develop a detailed three-dimensional FE model for simulating the physical model described below.αβ-β-αxyzBuried PipelineFault Movement DirectionFault Crossing Angle in the Horizontal PlaneFault Crossing Angle in the Vertical PlaneSoil DomainSoil Face and Pipeline End Axially FixedSoil Face and Pipeline End DisplacedSoil Face Laterally FixedSoil Face Vertically FixedSoil Face Laterally Fixed/DisplacedSoil Face Vertically Fixed/DisplacedFigure 3.1: Pipeline-fault schematic diagram273.3. NUMERICAL MODELLING3.3 Numerical modellingThe overall ABAQUS model is developed to closely represent the large-scale experimental pipe-soil specimen at Cornell University to simulate buriedpipeline response under fault movement loads. Pipeline segment is consid-ered to cross continuum soil partitioned into two blocks. One block beingstationary and the other being the moving block. The two blocks are sep-arated by a planar shear band. Since a finitely thick shear band is notmodelled here, the propagation of fault rupture through the soil is not sim-ulated here. Only the interaction of the moving and stationary soil blockswith that of the pipe and with each other is simulated. Interaction betweenthe soil blocks in the normal direction has been modelled using ”hard” con-tact and interaction between the soil blocks in the ”tangential” direction ismodelled using frictionless contact. The nature of loading on the pipe witha planar shear band is expected to be quite similar as compared to a finitethickness shear band. It is so expected as the normal stresses will get read-ily transferred from one block to the other due to ”hard” contact and theblocks can freely slide past one another due to ”frictionless” contact. Thefault rupture propagation nature will however, differ with a planar shearband when compared to a finite thickness shear band; however, it is notexpected to affect the pipeline performance significantly differently.Pipe end in the stationary block is axially restrained whereas the pipeend in the moving block is modelled to move in the intended direction andmagnitude along with the moving block. This arrangement of boundaryconditions correspond to a buried pipeline under restrained conditions attwo ends of the pipe which is expected to induce higher strains due tothe imposed fault displacements. Boundary surfaces of the stationary soilblock are restricted to displace in their normal directions but can move intheir tangential direction. Fault movement displacement is applied in theappropriate boundary surfaces in the moving block. Modelling ensured thatthe soil mass rupture is initiated from the bottom. Pipeline modelled isdevoid of any imperfection and soil surfaces and boundaries are modelled tobe perfectly horizontal and vertical surfaces.3.3.1 FE Modelling detailsThe finite element method (FEM) is used to study the performance ofburied pipeline undergoing fault rupture using ABAQUS [140]. The pipe ismodelled using 4-noded doubly curved general-purpose shell elements withreduced integration (ABAQUS element S4R). This element allow for trans-283.3. NUMERICAL MODELLINGverse shear deformation and based on the shell thickness, either use thickshell theory or discrete Kirchhoff thin shell theory. These properties makeit suitable to model pipes of various thicknesses. These elements possesssix degree of freedom at each node; three translations and three rotationsdefined in their global co-ordinate system. It consists of a single integra-tion point at the mid-surface to form the element internal force vector. Thenumber of integration points through the thickness of the element is chosento be 5. Soil continuum are modelled using general purpose 3D stress ele-ments (ABAQUS element C3D8R). These elements are 8-node linear brickelements with reduced integration. Reduced integration elements use a lowerorder integration to form the element stiffness reducing running times, es-pecially in three dimensions [140]. The integration point is located at thecenter of the element. Details on pipe-soil interaction are discussed below.2.3 m3.2 mDisplacementDisplaced Soil MassStationary Soil Mass6.6 mFigure 3.2: Pipeline-fault isometric meshThe optimal mesh configuration is determined based on a convergencestudy. Pipe mesh size of 25 mm and soil mesh size surrounding the pipeof 10 mm are used. A schematic representation of the model is shown inFigure 3.1. The FE model developed using ABAQUS is shown in Figure 3.2293.3. NUMERICAL MODELLINGFigure 3.3: Finite element model meshand Figure 3.3. The shear band width for sand can range in between 3 and25 D50 [65, 66]. [67] recommended a value of 16 D50 for sand shear bandwidth. In the current study, for a given D50 of 0.67 mm and a multiplicativefactor of 15, the shear band is calculated to be 10mm. [68] illustrates amethod to update the strain softening factor with a scaling factor to con-sider for scaling effects and reduce FE mesh related inaccuracies. For faultrupture propagation, an exponential scaling factor of 1 [67, 68] is used. Ad-ditionally, an element size of 10 mm is used, which is same as the shear bandwidth. Hence, no further modification to the strain softening factor is foundnecessary.3.3.2 Soil material modelFor sand, the Mohr-Column yield criterion is considered:τ = c− σtanϕ (3.1)where c is cohesion, ϕ is internal angle of friction, σ and τ are the normaland shear stresses acting on the plane where failure occurs. The abovecriterion assumes σ is negative in compression and implies, larger the normalforce, higher the shear the material can sustain. Primary features of stress-strain behavior of sand are its peak strength, initial elastic stiffness, pre-303.3. NUMERICAL MODELLINGpeak hardening and post-peak softening [56, 58]. Shear strength of sandsare influenced by its relative density, its confining pressure and are governedby its internal angle of friction [61, 68, 141–143]. Past test data show thatdense sands generate plastic strains before failure and undergo softening postpeak strength. Relatively softer sands exhibit lesser degrees of softening.A number of constitutive models are available in literature to define thisbehavior. The stress-strain behavior of dense sand can be grossly definedas elastic zone, pre-peak hardening zone [56], ranging from the initial yieldsurface up to the failure surface and the post peak softening zone, rangingfrom failure to higher deformations leading to the critical state conditions.The elastic modulus E, defining the behavior of sand in the elastic zone iscomputed according to [144] as:E = Kp′a(p′/p′a)n (3.2)where, the confining pressure is denoted as p′, the atmospheric pressure isdenoted as p′a and following [59, 60] K = 150 and n = 0.5.In the plastic zone, sand behaves as a nonlinear material where the fric-tion angle and the dilation angle vary with the plastic shear strain p. Notethat the variation of friction and dilation angle with the plastic shear strainsare different in the pre-peak plastic zone and in the post peak softening zone.In the pre-peak plastic zone, the variation of the friction and dilation anglesare represented based on [57] as:ϕ′= ϕ′in + sin−1[(2√γpγppγp + γpp)sin(ϕ′p − ϕ′in)] (3.3)ψ = sin−1[(2√γpγppγp + γpp)sin(ψp)] (3.4)where, ϕ′is the mobilized friction angle, ψ is the mobilized dilationangle and γpp is the plastic shear strain corresponding to the instant of peakinternal angle of friction and is calculated as [68]:γpp = γpc (p′p′a)m (3.5)where, γpc is a strain softening variable given as:γpc = C1 − C2ID (3.6)The constants C1 = 0.22, C2 = 0.11 and m = 0.25 have been used asrealistic values [68], which generally need to be calibrated from experimental313.3. NUMERICAL MODELLINGdata. ϕ′in is the internal friction angle at yielding. A ϕ′in of 29◦ has beenused. ϕ′p is the peak friction angle and obtained as [61]:ϕ′p = ϕ′c +AψIR (3.7)where, ϕ′c = 31◦ is the internal angle of friction at critical state condi-tions. Aψ for tri-axial test conditions assumes a value of 3. The relativedensity index is denoted as IR and defined as [61]:IR = ID(Q− lnp′)−R (3.8)where, Q = 10 and R = 1. ψp is the peak dilation angle and can becalculated according to [61] as:ψp =ϕ′p − ϕ′ckψ(3.9)k = 0.5 for triaxial conditions [61]. The post-peak nonlinear strainsoftening zone is defined according to [58] as:ϕ′= ϕ′c + (ϕ′p − ϕ′c)exp[−(γp − γppγpc)2] (3.10)ψ = ψpexp[−(γp − γppγpc)2] (3.11)Unfortunately, in ABAQUS, it is not possible to provide a varying di-lation and/or friction angle. Therefore, the above equations are introducedinto ABAQUS by using user-defined field (USDFLD).An elasto-plastic representation of the soil in combination with a Mohr-Coulomb yield surface is frequently applied for soil-structure interactionanalyses. The basic assumption in the Mohr criteria is that, the maximumshear stress controls failure. At yield, the critical value of shear stress isa function of the normal stress and corresponds to the point of tangencybetween the Mohr circle of stress and the yield surface. The Mohr-Coulombsoil model is an elastic perfectly plastic model which is widely used for thedesign applications in geotechnical engineering to simulate material responsedue to monotonic loading and is available in ABAQUS. The Mohr-Coulombcriterion assumes that yield occurs when the shear stress on any point in amaterial reaches a value that depends linearly on the normal stress in thesame plane. The Mohr-Coulomb material model however is not equipped toproduce the strain softening behavior observed in dense sand, as it employsa constant internal friction angle and dilation angle for the model.323.3. NUMERICAL MODELLINGFigure 3.4: Mobilized friction angle and dilation angleHence, the Mohr-Coulomb model is upgraded to include a non-linearbehavior using user subroutine. The modified Mohr-Coulomb model whichincludes pre-peak strain hardening and post-peak strain softening behavioris implemented into ABAQUS/Standard via. the user subroutine USDFLD.The subroutine allows for defining the variation of mobilized friction angleand mobilized dilation angle with the increase in deviatoric shear strain.The subroutine employs three solution dependent state variables and onefield variable. In order to implement the strain softening behavior, the max-imum and minimum principal plastic strain increment values are obtainedfor each increment using the current maximum and minimum principal plas-tic strains and the previous maximum and minimum principal plastic strainscalled from the last increment using subroutine GETVRM. These values arein-turn used to obtain the current plastic shear strain increment. The max-imum deviatoric shear strain is obtained as a field variable at the end ofeach increment integrating the deviatoric shear strain increments over theperiod of analysis, which is in turn used to define the corresponding mo-bilized friction angle and mobilized dilation angle in the ABAQUS inputfile. To ensure the strain softening models validity, single element tests arecarried out exhibiting softening behavior.333.3. NUMERICAL MODELLINGFigure 3.5: Mobilized friction angle and dilation angle for varying relativedensity3.3.3 Pipe material modelSelection of proper material models are an important aspect of the FEmodel. The current study involves two materials – high density polyethelene(HDPE) pipe and sand. The HDPE is modelled as an elasto-plastic mate-rial. von-Mises yield criterion is adopted. The adopted stress-strain curvebased on [145], considering a typical laboratory strain rate and ambienttemperature of 21◦C is shown in Figure 3.6.3.3.4 Soil-pipe interaction modellingContact modelling is crucial to simulate the interaction between the soiland the pipeline in an appropriate manner. Contact between pipeline andsoil introduce severe non-linearity in 3D soil-pipeline interaction problems.Pipeline outer surface and soil surface surrounding the pipe are defined asseparate surfaces namely the “master” and “slave” surface for “surface tosurface” contact definition. ABAQUS/Standard takes into considerationthe shape of both the surfaces while defining contact using “surface to sur-face” contact discretization by forming equations in an average sense overregions adjacent to “slave” surface nodes. A realistic tangential friction co-efficient of 0.3 is adopted for contact in the tangential direction between thesurfaces. Although in reality, friction can be varying, to keep the analy-343.3. NUMERICAL MODELLINGFigure 3.6: Poly ethylene stress strain curvesis simple, a constant friction has been assumed. A “hard” contact usingABAQUS/Standard non-linear “penalty” method is adopted for contact inthe normal direction between the surfaces. The non-linear “penalty” methodapproximates a “hard” normal contact with a nonlinear pressure-overclosurerelationship between the two surfaces.3.3.5 Solution procedureABAQUS/Standard with implicit formulation is used for all analyses.In ABAQUS/Standard, the solution for a non-linear problem is found byapplying the specified loads gradually and incrementally towards a final so-lution. The process involves discretizing the analysis into numerous smallload increments and obtaining equilibrium at the end of each load increment.Generally, it takes numerous iterations to determine acceptable solution toa given load increment. Hence, ABAQUS/Standard consumes significantamount of time, memory, storage and in some cases, has convergence prob-lems in reaching the ultimate solution [140]. The Newton-Raphson methodis used to solve the non-linear equilibrium equations. Displacement con-trolled solution strategy has been used. The desired displacement is appliedas a smooth step at the boundary surfaces of the moving soil block except-ing the top surface as well as the pipe end of the moving side. Axial strainsalong the pipe are noted for varying fault displacements at specific loca-tions similar to what has been reported from experiments and are in turn353.4. RESULTS AND DISCUSSIONcompared.3.4 Results and discussionIn this section, the performance of the adopted material model and thedeveloped FE model are presented to evaluate the accuracy of the proposedapproach. For both the cases, results obtained have been compared withexperimental results available in the literature. Moreover, comparison withother FE based work available in the literature have also been presented soas to evaluate the superiority of the proposed models.3.4.1 Evaluation of material modelFigure 3.7: Triaxial test single axis-symmetric elementTo simulate triaxial test, axisymmetric element (ABAQUS element CAX8R)is employed as shown in Figure 3.7. Based on sensitivity analysis, it is ob-served that the aspect ratio of the discretized element has almost no effecton the response. The optimal mesh size is found to be 100 mm. Figure 3.8and Figure 3.9 show the variation of deviatoric stress to mean effective stressratio with axial strains. Results obtained using triaxial test and other FE363.4. RESULTS AND DISCUSSIONFigure 3.8: Evaluation of material model considered in this studybased models are also reported. The test performed is for a confining pres-sure of 39 kPa with a relative density of 80 percent. It is observed thatthe material model used in this study closely matches with the triaxial testresults reported in [146]. To evaluate the accuracy of the current FE model,a comparison of root mean-square errors (RMSE) between FE results andexperimental results are performed. Root mean-square error is calculatedasRMSE =√1n∑(ai − bi)2 (3.12)where, ai is the FE analysis result data point and bi is the correspond-ing experimental result data point. n indicates the number of data-points.RMSE of FE4 model from [68] is found to be 0.0392. RMSE from the cur-rent FE model is found to be 0.0036. Finally, the effect of relative density(RD) on the stress-strain behavior is shown in Figure 3.9. RD of 80 percent,90 percent and 100 percent have been considered. An increase in RD resultsin a reduced peak strength.3.4.2 Evaluation of full-scale FE modelHaving evaluated the material model, we now focus on the evaluation ofthe full-scale FE model. A FE model to closely represent the Cornell Uni-versity large scale test specimen is created. The geometry of the pipeline373.4. RESULTS AND DISCUSSIONFigure 3.9: Effect of relative density on the stress strain behavior of sandcross-section considered for this test had an outside diameter of 400.5 mmwith a wall thickness of 24 mm. The displacement offset considered is 1.22m for this test. The objective is to ensure the capability of the developedFE model to accurately simulate soil-pipe interaction mechanism and thegenerated pipeline strains from the large-scale experimental test. For thepurpose of evaluation of the developed FE model two large scale tests fromCornell University are considered – HDPE pipe undergoing strike-slip faultrupture at a crossing angle of 65◦ with the fault plane under compression andtension. For sand, a ϕ′in of 29◦ is used for this analysis similar to the triaxialtest based on recommendations by [56] for typical contributions from differ-ent factors towards mobilized friction angle. A plane strain friction angleat peak direct shear strength of 43◦ and a plane strain cohesion of 1.2 kPais reported by [40] for the experimental tests. [139] reported a plane straincritical state friction angle for RMS graded sand of 41◦ ± 2◦. [139] presentsa detailed account of direct shear strength and dilation characteristics ofRMS graded sand.In this analysis, the plane strain peak friction angle ϕ′PSp and plane straincritical friction angle ϕ′PSc are converted to triaxial peak friction angle ϕ′TXpand triaxial critical friction angle ϕ′TXc using suitable correlations. Usingthe set of experimentally determined constitutive equations by [61], a rela-tive density index IR of 0.4 is obtained using Aψ of 5.0 [61] for plane strainconditions. Based on experimental study by [147], triaxial critical state fric-383.4. RESULTS AND DISCUSSIONFigure 3.10: Failure plane computed using the proposed FE modeltion angle ϕ′TXc is considered to be about 4◦ smaller than the plane straincritical state friction angle ϕ′PSc equaling 37◦. This observation is supportedby other studies, such as that by [148–150]. Using equation by [61], IRof 0.4 is used to obtain the peak triaxial friction angle ϕ′TXP of 38.2◦. Apeak dilation angle of 2.4◦ is estimated using expression in [61]. This valueis found to be consistent with peak dilation angle reported in [139] corre-sponding to a dry density of 15.5-15.8 kN/m3. Figure 3.10 to Figure 3.15show the results obtained from the proposed FE model. It is observed thatthe FE model is able to closely predict the failure plane in the soil as shownin Figure 3.10, when compared with experimental observations in [40]. Fig-ure 3.11 shows soil failure plane extending from pipe to the surface undera combination of axial and lateral displacement applied during the tensilefaulting test. Lateral interaction between the pipe and soil being primarilyconcentrated at the fault, failure planes are more prominent close to thefault. Peak plastic shear strain can be located in Figure 3.11 (c) due to393.4. RESULTS AND DISCUSSION(a) (b)(c) (d)Figure 3.11: Plastic shear strain (FV1) at different locations under 300 mmdisplacement – (a) 2 m from fault plane on the moving soil mass (b) 2 mfrom fault plane on the stationary soil mass (c) vicinity of the fault planeon moving soil mass, and (d) vicinity of the fault plane on static soil mass.the interaction between the deforming side of the pipe and the lateral soilboundary. Pipe cross-sectional east springline strain comparisons from theexperimental tensile test with that of the FE analysis are presented in Fig-ure 3.12. A close match is observed between experimental and FE analysisresults. To illustrate the superior performance of the proposed approach,the RMSE between the results obtained using the FE models and the exper-imental results is computed. Under 300 mm displacement, RMSE for theproposed FE model is 0.0017. RMSE for the ‘coupled’ model and ‘interface’model are found to be 0.0031 and 0.0035 respectively. Under 600 mm and900 mm displacements as well, the proposed approach is found to predict aclose match. Summary of the current FE analysis result error calculationsand error calculations from other FE analysis results from literature for theexperimental tensile test are shown in Table 3.1. Summary of mesh sensi-tivity on the peak tensile strains generated for the experimental tensile testFE analysis are presented in Table 3.2. A pipe mesh of 25 mm is selectedfor analyses.Pipe axial strain comparisons from the compressive test case with that403.4. RESULTS AND DISCUSSIONLength along pipeline (m)BendingstrainEastspringlineaxialstrainEastspringlineaxialstrainLength along pipeline (m)Eastspringlineaxial strain(a) (b)(c) (d)Figure 3.12: Strain profile (tensile) under a displacement of (a) 300 mm,(b) 600 mm and (c) 900 mm obtained using current study, interface model,coupled model and full-scale experimental test [1] (d) Bending strain profilefrom current FE analysis.of the FE analysis are presented in Figure 3.13. Axial strains are obtainedas the average of springline strains in accordance with [2]. A good match isobserved between experimental results and FEA. Comparison of strain plotsbetween current study, experimental data and beam element model by [2]are presented. Similar to the tensile case, RMSE has also been computed.Under 300 mm displacement, RMSE for current study and the beam elementmethod are found to be 0.3766 and 0.5528, respectively. Under 600 mmdisplacement, RMS error for the current study is 0.5018 and RMS error forbeam element model by [2] is 0.6537.It is observed that this analysis results are not significantly better thanTable 3.1: Performance of the proposed approach in predicting the eastspring line axial strain (under tension) under displacements 300 mm, 600mm and 900 mmMethodsInterface Model Coupled Model Current Study0.3 m 0.6 m 0.9 m 0.3 m 0.6 m 0.9 m 0.3 m 0.6 m 0.9 mRMSE (10E-3) 3.5 7.6 12 3.1 5.9 9.4 1.7 4.1 4.4413.4. RESULTS AND DISCUSSIONAxial strainLength along pipeline (m)(a) (b)(c)Length along pipeline (m)Axial strainLength along pipeline (m)Bending strainLength along pipeline (m)Figure 3.13: Strain profile (under compression) with a displacement of (a)300 mm and (b) 600 mm obtained using the proposed approach, beam-element model and full-scale experimental test [2] (c) Bending strain profilefrom current FE analysis.the beam on springs analyses by [2]. Detailed description of the compressivetest along with its actual experimental configurations are not available inliterature. It is suspected that the difference in the experimental test con-figuration with that of the numerical model may have contributed to not sogood match between the results, or a better prediction compared to beamon spring analyses by [2].Figure 3.14 compares the deformed shape of the pipe at varying displace-ments for both the compressive and tensile large-scale tests. Figure 3.15compares the axial strain of the pipe at varying displacements for both thecompressive and tensile large-scale tests. Pipe under compression failed dueto initiation of local buckle as evident from axial strain plots in Figure 3.13whereas pipe under tension failed due to tensile rupture. Initiation of lo-cal buckle resulted in an undulated axial strain profile at the center of thepipeline as can be seen in Figure 3.13.423.5. SUMMARYTable 3.2: Peak tensile strains for varying mesh sizes of the pipeline fordisplacements 300 mm, 600 mm and 900 mmMesh size 15 mm 25 mmFault displacement 0.3 m 0.6 m 0.9 m 0.3 m 0.6 m 0.9 mStrain (%) 3.8 6.2 7.8 3.8 6.2 7.8Mesh size 45 mm 75 mmFault displacement 0.3 m 0.6 m 0.9 m 0.3 m 0.6 m 0.9 mStrain (%) 3.7 6.0 7.8 3.5 5.9 -3.5 SummaryIn this study, we have developed a nonlinear three-dimensional FE modelfor studying structural performance of buried pipeline undergoing fault rup-ture displacements in sand. For modelling the constitutive relation of sand,a modified Mohr-Coulomb criterion is adopted which results in a nonlinearvariation of the sand shear strength with variation in plastic shear strain.The non-linear variation in shear strength is characterized by initial linearzone, pre-peak strain-hardening and post-peak strain-softening. A USDFLDbased user-subroutine is developed to include the same within ABAQUS.The adopted material model is evaluated with available experimental tri-axial test results. This material model is thereafter suitably calibrated fora large-scale test based on direct shear soil test data for that experiment.Thereafter, a detailed FE model with material and geometric nonlinearityfor a pipeline undergoing fault rupture displacements incorporating this newsand constitutive relation has been developed. The developed FE model isthen used to simulate the full-scale test carried out at Cornell University.Results obtained using the developed FE model has been evaluated againstexperimental results available in the literature. In order to evaluate theperformance of the developed FE model to predict pipeline strains in com-parison to those already available in the literature, comparative studies havebeen carried out. Proposed approach is found to yield superior result thanthe FE models which do not consider non-linear sand constitutive relation.This indicates that the analysis methodology presented here is relevant toperform detailed FE analyses of critical fault crossings involving major en-ergy transmission pipelines. The key findings of the study are as follows:(1) The nonlinear material model adopted for sand successfully provideda good match with triaxial test results for sand specimen. RMSE from thecurrent FE model is found to be 0.0036 in comparison to 0.0392 from [68].433.5. SUMMARY(2) The non-linear material model adopted for sand is found to have min-imal dependence on mesh size and aspect ratio of the elements. Non-linearmaterial model FE predictions show relatively better match with respect toavailable numerical results in the literature.(3) The non-linear FE model developed is found to provide good corre-lation with experimental results for large-scale tests. RMSE calculated forevaluation of Cornell University pipeline-fault rupture large scale “tensile”and “compressive” experiments with that of the developed FE model predic-tions showed errors are consistently lower than available numerical resultsin the literature for varying fault displacement magnitudes. At 300 mm dis-placement for the large-scale tensile test, an RMSE of 0.0017 is achieved incomparison to previous 0.0031 and 0.0035 for the ’coupled’ and ’interface’model by [RSO16]. At 600 mm displacement for the large-scale tensile test,an RMSE of 0.0041 is achieved in comparison to previous 0.0059 and 0.0076for the ’coupled’ and ’interface’ model by [RSO16]. At 900 mm displacementfor the large-scale tensile test, an RMSE of 0.0044 is achieved in compar-ison to previous 0.0094 and 0.012 for the ’coupled’ and ’interface’ modelby [RSO16]. At 300 mm displacement for the large-scale compressive test,RMSE for current study and the beam element method [2] are found to be0.3766 and 0.5528, respectively. At 600 mm displacement for the large-scalecompressive test, RMS error for the current study is 0.5018 and RMS errorfor beam element model by [2] is 0.6537. Additionally, failure patterns inthe FE analysis predictions are found to closely match experimental obser-vations.443.5. SUMMARY(a)(b)(c)(d)(e)Figure 3.14: Deformation of the pipe under compression and tension at adisplacement of (a) 300 mm (compression), (b) 600 mm (compression), (c)300 mm (tension), (d) 600 mm (tension), (e) 900 mm (tension).453.5. SUMMARY(a)(b)(c)(d)(e)Figure 3.15: Axial strain of the pipe under compression and tension at adisplacement of (a) 300 mm (compression), (b) 600 mm (compression), (c)300 mm (tension), (d) 600 mm (tension), (e) 900 mm (tension).46Chapter 4Multi-fidelity approach forUncertainty Quantification ofBuried Pipeline Responseundergoing Fault RuptureDisplacements in Sand4.1 IntroductionA new efficient approach for uncertainty quantification of pipeline struc-tural response undergoing reverse-slip fault rupture displacements in sandis presented using multi-fidelity Gaussian processes. Uncertainty quantifica-tion problems generally take large numbers of scenarios to be analyzed con-sidering variations in the influencing parameters. Analysing large numbers ofcomputationally expensive detailed geo-technical numerical models is prac-tically not feasible. Hence, a multi-fidelity approach employing Gaussianprocesses is proposed to tackle this problem which combines the accuracy ofa relatively few number of detailed geo-technical numerical models and theefficiency of large numbers of simplified numerical models to track the uncer-tainty response. The detailed model utilizes a previously evaluated pipe-soilFE model including a non-linear sand constitutive model implemented in FEsoftware ABAQUS. The simplified model utilizes beam element pipe and bi-linear soil springs. The multi-fidelity model is first trained using data fromHF and LF model analyses, thereafter cross-evaluated and subsequentlyused to quantify uncertainty in the peak compressive strains generated andto identify the most sensitive input variables. Finally, fragility curves arederived for a site specific pipe-soil fault rupture problem.474.2. PROBLEM STATEMENT4.2 Problem StatementThe objective of this research is to propose a solution to uncertaintyquantification of detailed and complex geo-technical problems, maintainingthe complexity of the numerical models capable of defining the true nature ofthose problems. As a demonstration, the case of a buried continuous pipelineundergoing fault rupture deformations is explored. A detailed geometricallyand materially nonlinear FE model, including a nonlinear sand constitutivemodel implemented into it is used for this purpose.A number of input factors are varied for the uncertainty quantificationanalysis. Input parameters that are varied for these analyses include outerdiameter of the pipe, wall thickness of the pipe, Young’s modulus of the pipesteel, yield strength of the steel, pipeline-fault crossing angle, peak frictionangle calculated from the confining pressure and relative density using theadopted constitutive relations, operating temperature in the pipeline andoperating pressure in the pipeline. A mean outer diameter of 609.6 mmis considered with lower and upper bounds of 600 mm and 620 mm with acoefficient of variation of 0.05. A mean pipe wall thickness of 9.52 mm is con-sidered with lower and upper bounds of 8.57 mm and 10.47 mm, respectivelywith a coefficient of variation of 0.05. A truncated Gaussian distribution isadopted for both these parameters [151]. The Young’s modulus mean isconsidered to be 207,000 MPa with lower and upper bounds to be 82,000MPa and 331,200 MPa respectively with a coefficient of variation of 0.15.The range considered for Young’s modulus is quite broad. However, sincea probability distribution is considered, bulk of the data is centered in theproximity of 207,000 MPa. The yield strength mean is considered to be 414MPa with lower and upper bounds to be 289.8 MPa and 540 MPa respec-tively with a coefficient of variation of 0.075. The ultimate strength meanis considered to be 517 MPa with lower and upper bounds to be 310 MPaand 725 MPa with a coefficient of variation of 0.1. A lognormal distributionis adopted for all three steel material parameters [152].Soil confining pressure is assumed to be at a mean of 39 kPa varyingfrom a minimum of 7.8 kPa to a maximum of 70 kPa with a coefficient ofvariation of 0.2 and a lognormal distribution. Relative density of the soilis assumed to be at a mean of 80 percent with a coefficient of variation of0.025 ranging from a minimum of 55 percent to a maximum of 95 percentwith a lognormal distribution. Operating temperature and pressure in thepipeline depend on the state of the pipe, if it is under operation or not.Hence, bi-modal distribution using a mixture of gaussian has been used[153]. Minimum and maximum temperature considered are 0◦C and 56.5◦C.484.2. PROBLEM STATEMENTMinimum and maximum pressure considered are 100 kPa and 10.35 MParespectively. Operating mean value has been assumed as 7 MPa.4.2.1 High-fidelity FE modelAs mentioned previously, [1, 34, 41, 43, 44, 48–50, 52–55] had employed3D soil continuum model to study the problem. [1, 53, 55] had consideredlinear variation of soil strength and dilatancy with varying plastic deviatoricstrain in their analyses. [68] employed non-linear variation of soil strengthas a relation to plastic shear strains to study 2D pipeline-soil interactionproblems. The authors [154] incorporated soil constitutive model with pre-peak hardening and post-peak softening behavior in the soil model to studythe structural performance of pipelines under fault movement loads usinga 3D continuum soil approach. The HF model adopted here is createdfollowing the experimentally evaluated pipeline-fault model developed in[154]. The FE model created here however, considers different geometricconfigurations and material properties for soil as well as pipe in view ofthe uncertainties associated with these parameters. The model length isconsidered to be 100 m as shown in Figure 4.1.6 m4 mFault PlaneIsometric ViewEnd ViewSide View100 mDisplacementMoving SideStationary SidePipeFigure 4.1: HF finite element model meshThe FE model is created using general purpose FE analysis software494.2. PROBLEM STATEMENT0 0.1 0.2 0.3 0.4 0.5Plastic shear strain (.p)05101520253035404550Angle (3)Friction angleDilation angle?'pAp.p pA?'inPre-peakhardeningPost-peaksoftening?'Figure 4.2: Non-linear soil constitutive relationABAQUS [140]. The pipeline steel is modelled using elastic-plastic behaviorwith yield criterion defined by von-Mises theory. The adopted stress-strainrelation is defined according to Ramberg-Osgood relation [22]. For sand,the yield criterion based on Mohr-Coulomb theory is considered. Figure 4.1shows the FE model mesh. Table 4.1 shows the effect of mesh size variationtowards the generated strains for the HF models.Salient characteristics of the mobilized soil strength for sand are its peakshear strength, stiffness in the initial elastic zone, strain hardening in the prepeak strength zone and strain softening in the post peak strength zone. Thisrelation of sand is defined by the relative density and confining pressure ofthe sand. A typical non-linear sand material constitutive relation is shownin Figure 4.2. ϕ′in is the initial friction angle and ϕ′p is the peak frictionangle. ψp is the peak dilation angle. γpp is the plastic shear strain at whichthe peak strength is reached. Figure 4.3, Figure 4.4 and Figure 4.5 presentsnonlinear mobilized friction angle curves and nonlinear mobilized dilationangle curves adopted for the 30 HF analyses. Literature review suggeststhe presence of a number of constitutive relations to define this behavior[57–67]. In terms of stress space, the mobilized strength behavior of sandcan be grossly classified as the elastic region at low strains within the initialyield surface, pre peak strength plastic zone defining the stress space from504.2. PROBLEM STATEMENTTable 4.1: Compressive strain percent generated for HF model analysis 1-3for varying fault displacement and mesh sizeSerial Number Mesh Size (mm) Analysis NumberFault Displacement (mm)25 100 200 3001501 0.0123 0.0923 0.2306 0.38552 2 0.0042 0.0636 0.1382 0.36403 3 0.0082 0.0723 0.1640 0.24434751 0.0123 0.0923 0.2306 0.38555 2 0.0042 0.0636 0.1382 0.36406 3 0.0082 0.0723 0.1640 0.244371001 0.0138 0.1061 0.2721 0.42408 2 0.0048 0.0776 0.1741 0.43319 3 0.0096 0.0846 0.1887 0.2858102001 0.0159 0.1144 0.2951 0.493411 2 0.0051 0.0788 0.1755 0.469512 3 0.0100 0.0919 0.2067 0.3004initial yield surface to the surface at failure and the post peak strength strainsoftening plastic zone defining the space from failure surface to critical state[56, 58].Soil material models with Mohr-Coulomb yield criterion are extensivelyused in geo-technical engineering problems [155–159]. The soil materialmodel in ABAQUS based on Mohr-Coulomb theory is upgraded to enableit to simulate the true nature of the sand behaviour in compliance with theconstitutive relations discussed above. An user subroutine USDFLD is usedto implement this behaviour.4.2.2 Low-Fidelity FE modelThe objective of utilizing a LF model is to arrive at approximate resultsusing limited computational resources. LF models use simplified materialmodels, formulations and geometry of the actual problem. However, the ma-terial properties, geometrical dimensions and loads in both the HF and LFmodels are same for a given case. A beam element FE model [17, 34, 45, 46]is adopted as the simplified model to simulate the pipeline response undergoing fault displacement using ABAQUS/Standard analysis software. Twotypes of elements are used to prepare the model. The pipe itself is mod-elled using PIPE32 elements which employ Timoshenko beam formulationallowing for both shear and bending deformation. These elements are 3node quadratic beam elements. The elements representing mutually per-pendicular soil springs supporting the pipelines are modelled using PSI34514.2. PROBLEM STATEMENT(three-dimensional pipe-soil interaction elements). These elements allow forbi-linear elastic perfectly plastic stiffness definition in the axial, lateral andvertical directions. One end of these springs are connected to the pipelineand the other end are connected to the ground. The intended fault displace-ments are applied in the moving side of the fault at the soil spring nodesrepresenting ground. Fault displacements are also applied on the pipe endon the moving side. The intended resultant fault displacement is convertedto axial, lateral and vertical components and applied to the axial, lateral andvertical springs respectively. Soil spring nodes representing ground on thestationary side and the pipe end on the stationary side are kept restrained.The complete model consisted of 400 elements with nominal element lengthof 0.5 m. Based on a mesh sensitivity study, the element length is found toyield satisfactory results where, no difference in results are noticed with fur-ther decrease in mesh size. The length of the pipeline section in the modelis considered to be 100 m. The steel pipeline material is modelled with abi-linear elastic-perfectly plastic stress-strain relation. The von-Mises yieldcriterion is used to define the steel material properties. Elastic perfectlyplastic relations are utilized for the steel. Various yield strength values areused based on the distributions as discussed in section 4.2. Static analy-ses are performed to analyze this problem. Figure 4.6 shows 1D schematicdiagram of the LF pipe-soil model.Soil springs are calculated using the fundamental soil properties basedon [4] guidelines. For all analyses, drained soil properties are assumed.Since, cohesionless soil, such as sand can drain water readily, drained soilproperties are used for this study. To derive the drained soil properties,undrained shear strength value is not used in the prescribed equations by[4]. Peak effective friction angle is used instead to obtain the peak forces.For axial soil springs, coefficient of pressure at rest is obtained based onthe internal friction angle. Maximum force values in the axial direction arecalculated based on pipe diameter, depth to pipe centerline, co-efficient ofpressure at rest and interface angle of friction for pipe and soil as shown in(Equation 4.1):TU = πDωc+ πDHγ¯(1 +K0)2tanδ (4.1)where D is pipe outside diameter, c soil cohesion representative of thesoil back fill, H is the depth to pipeline center line, γ¯ is effective unit weightof soil, K0 is coefficient of pressure at rest, ω is adhesion factor, δ is interfaceangle of friction for pipe-soil.For lateral springs, maximum force values are calculated based on inter-524.3. UNCERTAINTY QUANTIFICATION USING MULTI-FIDELITY SURROGATEnal friction angle of soil, pipe diameter, depth to pipe centerline and effectiveunit weight of soil as shown in (Equation 4.2):Pu = NchcD +Nqhγ¯HD (4.2)where Nch is horizontal bearing capacity factor for clay, Nqh is horizontalbearing capacity factor for sand.For vertical uplift springs, maximum force values are obtained using in-ternal friction angle of soil, depth to pipe centerline, pipe diameter, effectiveunit weight of soil as shown in (Equation 4.3):Qu = NcvcD +Nqvγ¯HD (4.3)where Ncv is vertical uplift factor for clay, Nqv is vertical uplift factorfor sand.For vertical downward springs, maximum force values are obtained basedon internal friction angle of soil, depth to pipe centerline, pipe diameter,total unit weight of soil as shown in (Equation 4.4):Qd = NccD +Nqγ¯HD +NγγD22(4.4)where Nc, Nq and Nγ are bearing capacity factors and γ is total unitweight of soil.4.3 Uncertainty quantification usingmulti-fidelity surrogateOne of the primary challenges in uncertainty quantification in buriedpipelines is scarcity of data. Simulating the buried pipeline response under-going fault rupture deformations using HF FE method is quite expensivefrom a computational resource point of view. As a result, one can only af-ford to run the simulator a limited number of times. An alternative is toresort to LF simulators however, this will result in a loss of accuracy. Inthis work, we propose to use multi-fidelity Gaussian process (MF-GP) foruncertainty quantification of buried pipeline undergoing reverse-slip faultrupture in sand. MF-GP combines data generated from a HF simulatorwith data generated from a LF simulator. This enables it to learn from veryfew (in the order of tens) HF data. In this section, we first provide a briefdescription of MF-GP followed by its use in uncertainty quantification.534.3. UNCERTAINTY QUANTIFICATION USING MULTI-FIDELITY SURROGATE4.3.1 Multi-fidelity Gaussian processMF-GP is a type of Gaussian process (GP) where one uses LF and HFdata by using a autoregressive scheme [160]:Ze (x) = ρZc (x) + Zd (x) , (4.5)where Ze is the MF-GP model also representing the HF model, Zc is a GPrepresenting the LF model, Zd is a GP representing the difference model Ze−ρZc and, in this setting, ρ is a scaling factor. The MF-GP model Ze can bedeveloped in two steps [161]. The first step involves training the GP modelZc with respect to the LF training data (Xc, Zc), which consists of tuning theZc hyper-parameters through maximization of the log-likelihood function.The second step involves training the Zd GP model with respect to thedifference training data (Xe − ρXc, Ze − ρZc), by simultaneously optimizingthe corresponding Zd model hyper-parameters and the ρ constant, againusing log-likelihood maximization (see [161] for details). Each one of the GPmodels Zc and Zd consists of a mean function and a correlation function. If itis assumed herein that the same form of mean functions f(x) are consideredfor models Zc and Zd. The MF-GP best linear unbiased estimator Ẑe isthen given as:Ẑe(x) = h(x)T b̂+ v(x)TV−1(Y −Hb̂), (4.6)where Y =[YcT , YeT]T, h(x) =[ρf(x)T , f(x)T]T, b̂ =(HTV−1H)−1HTV−1Y ,and v(x) is computed as:v(x) =[ρξ2c rc(x, xc)ρ2ξ2c rc(x, xe) + ξ2drd(x, xe)], (4.7)the H matrix is given as:H =f (xc1)T 0......f(xcnc)T0ρf (xe1)T f (xe1)T......ρf(xene)Tf(xene)T, (4.8)and the covariance matrix V takes the form,V =[ξ2cRc (Xc, Xc) ρξ2cRc (Xc, Xe)ρξ2cRc (Xe, Xc) ρ2ξ2cRc (Xe, Xe) + ξ2dRd (Xe, Xe)]. (4.9)544.3. UNCERTAINTY QUANTIFICATION USING MULTI-FIDELITY SURROGATECorrelation matrices Rc and Re in Equation 4.9 are defined in a similarway as for an ordinary GP model, with the difference that there are nowtwo of them, corresponding to models Zc and Zd, respectively. The samehold for correlation vectors rc and re in Equation 4.7. For further details onGP and how correlation matrices are defined in GP, interested readers mayrefer to [162, 163].The kernel function to define the correlation between the random vari-ables is given as Equation 4.10:cor[Y (xi), Y (xl)] = exp(k∑j=1θj |xij − xlj |pj ) (4.10)In this study constant terms are selected as mean functions of the twoGP models, denoted as µc and µd for models Zc and Zd, respectively. Withthis, the MF-GP estimator (Equation 4.6) is now written as:Ẑe(x) = b̂+ v(x)TV−1(Y − 1b̂), (4.11)where b̂ =(1TV−11)−11TV−1Y and H = 1, a vector of ones. Additionally,the prediction variance at x is [161]:ξ̂2e (x) = ρ̂2ξ̂2c + ξ̂2d − v(x)TV−1v(x). (4.12)One of the important challenges with ML algorithms is its extrapolationcapability. MF-GP being a Bayesian ML algorithm addresses extrapolationproblems based on priors. In this work, we have used GP priors.4.3.2 Uncertainty quantification using MF-GPThe uncertainty in the pipeline response is a function of all the variabil-ities in the input parameters, such as fault-pipe crossing angle, soil proper-ties etc. A total of 30 HF and 500 LF models are analyzed. The numberof HF and LF models are selected based on trial and error to achieve ahigh coefficient of determination with minimal computational effort. Ta-ble 4.3 presents the improvement in coefficient of determination of MF-GPmodel trained with increasing number of HF models. Input parameters arerandomly selected from uniform probability distributions of the upper andlower bound ranges for the numerical analyses inputs. Data (peak com-pressive strains) generated from the numerical analyses are fed to train themulti-fidelity surrogate model. Subsequently, the surrogate model is usedto predict peak compressive strains for a separate set of prediction data554.4. RESULTS AND DISCUSSIONwhere the input parameter values are randomly selected from correspondingpredefined probability distributions of the input parameters as discussed insection 4.2. Figure 4.7 shows the MF-GP uncertainty quantification processthrough a flowchart.The vulnerability of a structure can be expressed as a cumulative prob-ability of it reaching a failure limit state as a function of the load intensityit is subjected to. Generally, a fragility function is used for this purpose andassumes the shape of a log-normal function. The cumulative lognormal func-tion is adopted in accordance with [164] as shown below in (Equation 4.13):P (di|IM = x) = υ(ln(xθ )λ)(4.13)wherein, the probability that a structure will be at a failure state (di)caused by a load intensity IM = x, is P (di|IM = x); υ denotes the standardnormal cumulative distribution function; θ is the load intensity level denot-ing a failure probability of 50 percent; λ signifies the standard deviation forln(IM). Initially, realistic values for the parameters defining the fragilityequation are assumed. The parameters are finally obtained by maximizingthe logarithm of the likelihood function as shown below (Equation 4.14) :{θˆ, λˆ}= argmaxm∑j=1{ln(njzj)+ zjlnϕ(ln(xjθ )λ)+ (nj − zj) ln(1− ϕ(ln(xjθ )λ))}(4.14)where, m is the number of load intensity levels. At each load intensitylevel IM = xj , it is assumed that the structural analyses produce zj failuresout of nj analyses. To derive the fragility curves, the multi-fidelity model istrained at incremental fault displacements and strains are predicted for theprediction data set. Number of failure instances are calculated for each faultdisplacement using [22] compressive strain limit criterion. The load intensity(fault displacement) vs probability of failure data points are thereafter usedfor generation of the fragility curves. The same procedure is repeated formodels trained with only HF data points and only LF data points.4.4 Results and discussionResults presented include comparison of HF model vs LF model peakcompressive strains generated with increasing fault displacements. Com-pressive strains are studied since the pipeline fails under compression due564.4. RESULTS AND DISCUSSIONTable 4.2: Coefficient of determination for surrogate models trained withonly 30 HF models, only 500 LF models and combination of 30 HF and 500LF multi-fidelity modelsSerial Number Fault Displacement (mm) HF Model Results LF Model Results Multi-fidelity Model Results1. 25 0.699 0.620 0.8972. 100 0.604 0.576 0.9443. 300 0.476 0.571 0.924Table 4.3: Coefficient of determination for multi-fidelity models trained withvarying number of HF models and constant 500 LF modelsSerial Number Fault Displacement (mm) 20 HF 500 LF 25 HF 500 LF 30 HF 500 LF1. 25 0.759 0.885 0.8972. 100 0.800 0.907 0.9443. 300 0.696 0.911 0.924to buckling under reverse fault rupture displacements. Additionally, re-sults include cross-evaluations of the developed multi-fidelity model predic-tions with HF model predictions, sensitivity of the input parameters towardsthe generated peak compressive strains, probability distribution of the peakcompressive strains at various fault displacement levels and development ofpipeline fragility curves for this specific problem.4.4.1 Cross-evaluation testThe peak compressive strain outputs obtained from the HF and LF FEanalysis models at different fault displacements are used to train the multi-fidelity model. Figure 4.8 shows a representative stress plot of the HF FEmodel. Figure 4.9 to Figure 4.11 shows the comparison between HF andLF model peak generated compressive strains vs fault displacement values.Comparisons from a total of 30 analyses which are performed both for the HFand LF models are presented. The multi-fidelity models are cross-evaluatedusing the k-fold cross-evaluation technique. Using this technique, the HFdata set are split into ’k’ number of sections.In the first step, the first section is used to test the model and the re-maining sections are used to train the multi-fidelity model. In the secondstep, the second section is used to test the model and the remaining sec-tions are used to train the multi-fidelity model. Likewise, this process isrepeated for all the ’k’-sections. The multi-fidelity model test predictiondata from the k-fold are then compared with the original HF model ob-servations. The accuracy of the multi-fidelity model is then checked usingthe coefficient of determination (R-square). Figure 4.12, Figure 4.14 and574.4. RESULTS AND DISCUSSIONFigure 4.16 show excellent coefficient of determination of 0.897, 0.944 and0.924 for cross-evaluation tests at fault displacements of 25 mm, 100 mm and300 mm, respectively. Figure 4.13, Figure 4.15 and Figure 4.17 also showthe cross evaluation attempts on only HF data trained surrogate model andonly LF data trained surrogate model. Table 4.2 shows the comparison ofcoefficient of determination obtained from the multi-fidelity model, only HFdata trained surrogate model and only LF data trained surrogate model.Table 4.3 shows the coefficient of determination for the MF-GP model witha fixed number of 500 LF model data and varying number of HF modeldata. It is observed that the coefficient of determination increases with theincrease in number of HF model data. At 25 mm fault displacement, theMF-GP model bias error and variance are found to be 5.85E-6 and 2.15E-6respectively. At 100 mm fault displacement, the MF-GP model bias errorand variance are found to be 0.00036 and 2.42E-5 respectively. At 300 mmfault displacement, the MF-GP model bias error and variance are found tobe 0.312 and 4.095 respectively.As mentioned previously, the MF-GP model allowed the use LF modelswith its significantly less run time of 15 minutes in contrast to 48 hours ofthe HF models for 500 runs and still achieve high accuracy in comparisonto the HF models as demonstrated in Table 4.2 coefficient of determinationresults.4.4.2 Sensitivity analysis and generation of fragility curvesAfter the multi-fidelity model for pipeline crossing fault rupture to esti-mate peak compressive strains at certain fault displacement levels is evalu-ated, the model is used to predict peak compressive strains at those faultdisplacement levels. Suitable distributions of input parameters (10 parame-ters) as described previously are fed to the the multi-fidelity model to predictdistribution of peak compressive strains.The current study uses the Pearson’s correlation coefficient for sensitiv-ity analyses. Two of the most sensitive input parameters are identified as thepipe-fault angle and the sand confining pressure as shown in Figure 4.18 for25 mm and 100 mm fault displacements. For 300 mm displacement as shownin Figure 4.18, the most sensitive parameter is however pipe outer diameteras larger diameter pipes are more prone to local buckle. Other input pa-rameters which had their sensitivity within 10 percent include internal pres-sure, pipeline temperature, soil relative density, pipe steel ultimate strength,pipeline yield strength, steel yield strength, steel young’s modulus and pipewall thickness. Pipe outer diameter had a sensitivity of around 30 percent.584.4. RESULTS AND DISCUSSIONSince, the primary mode of failure in this problem is compressive buckling,the pipe-fault angle which governs the amount of compressive displacementis one of the most sensitive parameters. Similarly, the pipe outer diame-ter which greatly influences buckling is significantly sensitive here since, theprimary mode of failure for this problem is buckling. Confining pressure hassignificant sensitivity due to its influence towards soil strength with whichthe soil loads the pipe. Positive sensitivity indicates that the quantity ofinterest, peak compressive strain increases with the increase in the parame-ter value. Negative sensitivity indicates that the quantity of interest, peakcompressive strain decreases with the increase in the parameter value. Inthis regard, yield strength and young’s modulus can be seen having nega-tive sensitivity at lower displacements. However, their sensitivity changes topositive at higher displacements. At smaller displacements, higher young’smodulus and yield strength leads to lower peak compressive strains due tohigher resistance. However, at higher displacements as buckles starts toform, higher yield strength and young’s modulus will result in higher forcesfeeding into the buckle, resulting in higher peak compressive strains.Figure 4.19 - Figure 4.20 shows the variations of the predicted peakcompressive strains with pipe-fault angle and sand confining pressure fordifferent fault displacement values. At 25 mm fault displacement, fault rup-ture angle on the higher side showed no variation in compressive strains withthe variation in sand confining pressure. However, for lower fault ruptureangles, where the pipe is subjected to a higher degree of compressive load-ing, higher sand confining pressure resulted in higher compressive strainsdue to larger soil resistance. At 100 mm fault displacement, irrespectiveof the fault rupture angle, increase in sand confining pressure resulted inincreased peak compressive strains, quite reasonably due to higher soil re-sistance. Compressive strains are noted to be increasing with lesser faultrupture angle which will impart more compression loading to the pipe incomparison to a larger fault rupture angle. At 300 mm fault displacementhowever, no specific trend is observed with respect to the above two param-eters and is most likely due to the fact that at such high displacements, thepipe has undergone local buckling failure irrespective of the loading patternand soil resistance. It is observed that with the increase in fault displace-ment the mean peak compressive strains increased and also the range ofuncertainty increased. Figure 4.21 shows the probability distributions ofthe peak compressive strain predictions at 25 mm, 100 mm and 300 mmfault displacements.Figure 4.22 shows the fragility curves derived for this specific problemwhich plots probability of pipeline failure as a function of fault displacement.594.5. SUMMARY AND FUTURE WORKResults show that the fragility curve obtained using the MF-GP has a sig-nificantly different characteristic than the surrogate models which used onlythe HF model data and LF model data. The HF surrogate model showedhigher failure probabilities at lower fault displacements (less than 350mm)in comparison to both the MF-GP and LF surrogate model. The LF surro-gate model on the other hand showed significantly lower failure probabilitiesat lower fault displacements in comparison to the MF-GP and HF surrogatemodels. This is due to the fact that, the LF models in general predicted sig-nificantly lesser peak compressive strains than the corresponding HF ones.At higher fault displacements (350 mm and more) however the HF modelpredicted failure strains for lesser number of cases as a proportion of thetotal number of cases analyzed in comparison to the LF model, resultingin a smaller slope of the fragility curve post 350 mm fault displacement.Overall, it is observed that the multi-fidelity model reduces conservatismin the pipeline design in comparison to the HF surrogate model and at thesame time captures the structural vulnerability of the pipeline which is notcaptured by the LF surrogate model.4.5 Summary and future workIn this paper, a multi-fidelity approach employing Gaussian processesto study uncertainty in pipeline response subjected to fault displacementloads is developed. The multi-fidelity approach developed uses a previouslydeveloped and evaluated detailed pipe soil continuum model with a nonlinearsoil constitutive model implemented into it, as the HF model. The multi-fidelity model uses a beam element pipe supported by bi-linear soil springsas the LF model. Important inputs in the analysis that may contribute topeak compressive strain uncertainty are identified. Numerical analyses ofboth the HF and LF models are performed to generate the training data forthe multi-fidelity model. There after the evaluated multi-fidelity model isused to quantify uncertainty in the pipeline strains generated and developfragility curves. Results obtained have been summarized accordingly.The key findings of the study are as follows:(1) Uncertainty quantification in a complex geo-technical engineeringproblem, such as pipeline undergoing fault rupture deformations can bestudied efficiently employing multi-fidelity Gaussian processes.(2) Excellent coefficient of determinations are achieved between HF ob-served values and multi-fidelity model predicted values during cross-evaluationof results, there by establishing the validity of the multi-fidelity model. Co-604.5. SUMMARY AND FUTURE WORKefficient of determinations obtained for the multi-fidelity surrogate modelsfor fault displacements of 25 mm, 100 mm and 300 mm are 0.897, 0.944 and0.924 respectively.(3) Among the input parameters considered, the pipeline-fault crossingangle and the confining pressure of the soil are found to be the most sensi-tive parameters in influencing the peak compressive strains generated. Otherinput parameters which had their sensitivity within 10 percent include inter-nal pressure, pipeline temperature, soil relative density, pipe steel ultimatestrength, pipeline yield strength, steel yield strength, steel young’s modulusand pipe wall thickness. Pipe outer diameter had a sensitivity of around30 percent. Since, the primary mode of failure in this problem is compres-sive buckling, the pipe-fault angle which governs the amount of compressivedisplacement is one of the most sensitive parameters. Similarly, the pipeouter diameter which greatly influences buckling is significantly sensitivehere since, the primary mode of failure for this problem is buckling. Con-fining pressure has significant sensitivity due to its influence towards soilstrength with which the soil loads the pipe.The current case study entails uncertainty quantification of the struc-tural performance of a buried continuous pipeline undergoing fault ruptureloading in an efficient and accurate manner using MF-GP. The case studyundertaken here considers a site specific pipeline-fault movement case withdue regards to variations in the input parameters with respect to a specificsite. Further extension to this work would be to characterise the regionalseismic risk to pipelines due to fault rupture which involves integration ofpipeline uncertainty quantification at a specific site to probabilistic regionalground deformation hazard due to fault rupture.614.5. SUMMARY AND FUTURE WORK0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5Plastic shear strain253035404550Friction angle (Degree)Analysis 1Analysis 2Analysis 3Analysis 4Analysis 5Analysis 6Analysis 7Analysis 8Analysis 9Analysis 10(a)Pre-peakStrainHardeningPost-peakStrainSoftening0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5Plastic shear strain05101520253035Dilation angle (Degree)Analysis 1Analysis 2Analysis 3Analysis 4Analysis 5Analysis 6Analysis 7Analysis 8Analysis 9Analysis 10(b)Figure 4.3: Non-linear variation of (a) mobilized friction angle and (b) mobi-lized dilation angle with plastic shear strain used in HF soil material modelsfor analyses 1 to 10 among a total of 30 analyses624.5. SUMMARY AND FUTURE WORK0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5Plastic shear strain25303540455055Friction angle (Degree)Analysis 11Analysis 12Analysis 13Analysis 14Analysis 15Analysis 16Analysis 17Analysis 18Analysis 19Analysis 20(a)0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5Plastic shear strain0510152025303540Dilation angle (Degree)Analysis 11Analysis 12Analysis 13Analysis 14Analysis 15Analysis 16Analysis 17Analysis 18Analysis 19Analysis 20X: 0.01Y: 16.13(b)Figure 4.4: Non-linear variation of (a) mobilized friction angle and (b) mobi-lized dilation angle with plastic shear strain used in HF soil material modelsfor analyses 11 to 20 among a total of 30 analyses634.5. SUMMARY AND FUTURE WORK0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5Plastic shear strain253035404550Friction angle (Degree)Analysis 21Analysis 22Analysis 23Analysis 24Analysis 25Analysis 26Analysis 27Analysis 28Analysis 29Analysis 30(a)0 0.1 0.2 0.3 0.4 0.5 0.6Plastic shear strain05101520253035Dilation angle (Degree)Analysis 21Analysis 22Analysis 23Analysis 24Analysis 25Analysis 26Analysis 27Analysis 28Analysis 29Analysis 30(b)Figure 4.5: Non-linear variation of (a) mobilized friction angle and (b) mobi-lized dilation angle with plastic shear strain used in HF soil material modelsfor analyses 21 to 30 among a total of 30 analyses644.5. SUMMARY AND FUTURE WORKVertical Up SpringAxialSpringLateral SpringVertical Down SpringPipeDisplacementxyzFigure 4.6: Schematic diagram of LF pipe-soil modelHigh Fidelity ModelLow Fidelity ModelMF-GP ModelPrediction Input DataQuantification of Strain UncertaintiesBeam Element PipeSoil SpringsBi-linear Soil PropertiesShell Element PipeContinuum Element SoilNon-linear Soil Material ModelReliabilityTraining DataTraining DataPredictionFigure 4.7: MF-GP uncertainty quantification flowchart654.5. SUMMARY AND FUTURE WORK Figure 4.8: HF pipe-soil model: pipe von Mises stress plot and soil plasticshear strain plot664.5. SUMMARY AND FUTURE WORK0 100 200 300 400 500 600 700 800Fault displacement (mm)-5-4-3-2-101Peak compressive strain (%)Analysis 1 - HFAnalysis 1 - LFAnalysis 2 - HFAnalysis 2 - LFAnalysis 3 - HFAnalysis 3 - LFAnalysis 4 - HFAnalysis 4 - LFAnalysis 5 - HFAnalysis 5 - LF(a)0 100 200 300 400 500 600Fault displacement (mm)-5-4-3-2-101Peak compressive strain (%)Analysis 6 - HFAnalysis 6 - LFAnalysis 7 - HFAnalysis 7 - LFAnalysis 8 - HFAnalysis 8 - LFAnalysis 9 - HFAnalysis 9 - LFAnalysis 10 - HFAnalysis 10 - LF(b)Figure 4.9: Peak compressive strain percent generated in the pipe at variousinstants with increasing fault displacements: (a) analysis 1 - 5 (b) analysis6 - 10674.5. SUMMARY AND FUTURE WORK0 100 200 300 400 500 600 700 800Fault displacement (mm)-5-4-3-2-101Peak compressive strain (%)Analysis 11 - HFAnalysis 11 - LFAnalysis 12 - HFAnalysis 12 - LFAnalysis 13 - HFAnalysis 13 - LFAnalysis 14 - HFAnalysis 14 - LFAnalysis 15 - HFAnalysis 15 - LF(a)0 100 200 300 400 500 600 700 800Fault displacement (mm)-5-4-3-2-101Peak compressive strain (%)Analysis 16 - HFAnalysis 16 - LFAnalysis 17 - HFAnalysis 17 - LFAnalysis 18 - HFAnalysis 18 - LFAnalysis 19 - HFAnalysis 19 - LFAnalysis 20 - HFAnalysis 20 - LF (b)Figure 4.10: Peak compressive strain percent generated in the pipe at variousinstants with increasing fault displacements: (a) analysis 11 - 15 (b) analysis16 - 20684.5. SUMMARY AND FUTURE WORK0 100 200 300 400 500 600 700 800Fault displacement (mm)-5-4-3-2-101Peak compressive strain (%)Analysis 21 - HFAnalysis 21 - LFAnalysis 22 - HFAnalysis 22 - LFAnalysis 23 - HFAnalysis 23 - LFAnalysis 24 - HFAnalysis 24 - LFAnalysis 25 - HFAnalysis 25 - LF(a)0 100 200 300 400 500 600 700 800Fault displacement (mm)-5-4-3-2-101Peak compressive strain (%)Analysis 26 - HFAnalysis 26 - LFAnalysis 27 - HFAnalysis 27 - LFAnalysis 28 - HFAnalysis 28 - LFAnalysis 29 - HFAnalysis 29 - LFAnalysis 30 - HFAnalysis 30 - LF(b)Figure 4.11: Peak compressive strain percent generated in the pipe at variousinstants with increasing fault displacements: (a) analysis 21 - 25 (b) analysis26 - 30694.5. SUMMARY AND FUTURE WORK-0.005 0 0.005 0.01 0.015 0.02 0.025 0.03High-Fidelity Model Observations-0.00500.0050.010.0150.020.0250.03Model PredictionsMulti-Fidelity Model Trained DataFigure 4.12: Cross-evaluation data (peak compressive strain percent) at 25mm fault displacement for multi-fidelity model trained with both HF andLF data704.5. SUMMARY AND FUTURE WORK-0.005 0 0.005 0.01 0.015 0.02 0.025 0.03High-Fidelity Model Observations-0.00500.0050.010.0150.020.0250.03Model PredictionsOnly Low-Fidelity Model Trained DataOnly High-Fidelity Trained DataFigure 4.13: Cross-evaluation data (peak compressive strain percent) at 25mm fault displacement for multi-fidelity model trained with only HF or LFdata714.5. SUMMARY AND FUTURE WORK0 0.05 0.1 0.15High-Fidelity Model Observations00.020.040.060.080.10.120.14Model PredictionsMulti-Fidelity Model Trained DataFigure 4.14: Cross-evaluation data (peak compressive strain percent) at 100mm fault displacement for multi-fidelity model trained with both HF andLF data724.5. SUMMARY AND FUTURE WORK0 0.05 0.1 0.15High-Fidelity Model Observations00.020.040.060.080.10.120.14Model PredictionsOnly Low-Fidelity Model Trained DataOnly High-Fidelity Trained DataFigure 4.15: Cross-evaluation data (peak compressive strain percent) at 100mm fault displacement for multi-fidelity model trained with only HF or LFdata734.5. SUMMARY AND FUTURE WORK-10 -5 0 5 10 15 20 25 30 35 40High-Fidelity Model Observations-10-50510152025303540Model PredictionsMulti-Fidelity Model Trained DataFigure 4.16: Cross-evaluation data (peak compressive strain percent) at 300mm fault displacement for multi-fidelity model trained with both HF andLF data744.5. SUMMARY AND FUTURE WORK-10 -5 0 5 10 15 20 25 30 35 40High-Fidelity Model Observations-10-50510152025303540Model PredictionsOnly Low-Fidelity Model Trained DataOnly High-Fidelity Trained DataFigure 4.17: Cross-evaluation data (peak compressive strain percent) at 300mm fault displacement for multi-fidelity model trained with only HF or LFdata754.5. SUMMARY AND FUTURE WORK-60 -40 -20 0 20 40 60Sensitivity (%)Outer diameterWall thicknessYoungs modulusYield strengthUltimate strengthFault angleConfining pressureRelative densityTemperaturePressureInput parameters25mm displacement100mm displacement300mm displacementFigure 4.18: Sensitivity of input variables at 25 mm, 100 mm and 300 mmfault displacements764.5. SUMMARY AND FUTURE WORKConfining Pressure (kPa) Pipe-Fault Intersection Angle (Degree)0800.005900.01Peak Compressive Strain (%)600.015800.02407020 60(a)Pipe-Fault Intersection Angle (Degree)Peak Compressive Strain (%)Confining Pressure (kPa)0.04800.060.0890600.10.12800.1440 7020 60(b)Figure 4.19: Peak compressive strain predictions at (a) 25 mm fault dis-placement and (b) 100 mm fault displacement774.5. SUMMARY AND FUTURE WORKPeak Compressive Strain (%)Pipe-Fault Intersection Angle (Degree)Confining Pressure (kPa)080590106015802040 7020 60Figure 4.20: Peak compressive strain predictions at 300 mm fault displace-ment784.5. SUMMARY AND FUTURE WORK0 0.05 0.1 0.15Peak Compressive Strain (%)050100150200250Probability Density Function 25mm Displacement100mm Displacement-5 0 5 10 15 20Peak Compressive Strain (%)00.050.10.150.2Probability Density Function 300mm Displacement(b)Figure 4.21: Peak compressive strain predictions at (a) 25 mm fault dis-placement and (b) 100 mm fault displacement794.5. SUMMARY AND FUTURE WORK0 100 200 300 400 500 600Fault Displacement (mm)00.10.20.30.40.50.60.70.80.9Probability of FailureMultifidelity Model Data; - = 0.49; 3 = 321.81Only Highfidelity trained data; - = 0.98; 3 = 281.19Only Lowfidelity trained data; - = 0.21; 3 = 478.00Figure 4.22: Fragility curve for pipeline failure under compression80Chapter 5Regional ProbabilisticSeismic Risk Analysis ofBuried Pipelines due toPermanent GroundDeformation Hazard forVictoria, British Columbia,Canada5.1 IntroductionA novel method to estimate regional seismic risk to buried continu-ous pipelines probabilistically, due to earthquake rupture induced perma-nent ground deformations has been proposed. The seismic risk assessmentmethod has been illustrated for buried gas pipeline in the City of Victo-ria considering seismic risk from the Leech River Fault Zone (LRVFZ). Themethod considers uncertainty in a comprehensive manner on various aspectsand levels of the assessment, as well as utilizes experimentally evaluated com-plex geotechnical numerical models in an efficient manner. Major improve-ments of this method over existing studies include: (1) Stochastic sourcemodelling and analytical Okada solutions to regionally generate ground de-formations instead of regression equations to characterize ground deforma-tion along a fault (2) Complex experimentally evaluated geotechnical FEmodels employing non-linear soil constitutive relations instead of simple soilspring – beam element pipe models to evaluate pipeline response (3) Useof multi-fidelity Gaussian processes to ensure efficiency and limit requiredcomputational resource, given the high number of runs necessary for uncer-815.2. INTEGRATED RISK ASSESSMENT METHOD FOR BURIED PIPELINEStainty quantification exercises and the computationally expensive nature ofcomplex geotechnical FE models used. Probabilistic regional ground defor-mations generated has been suitably applied to the FE models, to transferthe uncertainty in ground deformation towards the pipeline response.The current study aims to quantify uncertainty in buried gas pipelineresponse in the City of Victoria due to permanent ground deformations re-sulting from LRVFZ fault rupture. The study probabilistically generatesthe ground deformations over a region to capture uncertainty and considersvariations in pipe geometry, soil conditions, operating conditions togetherusing a multi-fidelity Gaussian processes (MF-GP) analysis framework inassociation with FE analysis [154, 165]. The significance of this study incomparison to existing comparable studies [18, 19] is integration of proba-bilistic regional ground deformation with MF-GP surrogate model. Stochas-tic source modelling and analytical okada equations are used to characterizeprobabilistic ground deformation regionally. On the other hand, MF-GP sur-rogate model enabled experimentally evaluated complex pipe-soil numericalmodels are used to quantify uncertainty of pipeline response. The case studyperformed for the City of Victoria due to permanent ground deformationsresulting from LRVFZ fault rupture indicated potential for significant dif-ferential ground deformations at many instances and locations. A fragilitycurve is generated for pipe failure at a specific site based on [4] strain limit.5.2 Integrated risk assessment method for buriedpipelinesThe proposed methodology for integrated risk assessment of buried pipelinesfirst characterizes the permanent ground deformation over a region, proba-bilistically using the approach by [121]. Thereafter, the pipeline response isevaluated using FE analysis coupled with multi-fidelity Gaussian processes(MF-GP) surrogate model [165]. To consider uncertainties in the pipe re-sponse, variations in pipe geometry, soil conditions and operating conditionsare considered in this method. The integrated assessment of uncertain per-manent ground deformation and uncertain pipeline response result in theseismic risk assessment of pipelines due to permanent ground deformation.Figure 5.1 presents the method in a flow chart form.The probabilistic ground deformation approach computes ground defor-mations over a region using statistical models for earthquake source param-eters, simulated earthquake slip models and analytical Okada equations. Todo so, fault geometry is defined based on regional geological and seismo-825.2. INTEGRATED RISK ASSESSMENT METHOD FOR BURIED PIPELINESlogical studies and is discretized into sub-faults to allow heterogeneous slipdistribution over a fault plane. An earthquake scenario is defined based onregional seismicity and are typically represented by truncated exponentialand characteristic earthquake recurrence models. A logic tree approach isused to consider uncertainties in various parameters involved in the earth-quake occurrence modelling. Thereafter, based on an earthquake scenario,earthquake source parameters are sampled from statistical scaling relation-ships derived from the earthquake rupture model database. Subsequently,realistic fault slip fields are generated and ground deformations computedusing Okada equations, all constrained to the fault geometry set initially.The resultant probabilistic ground deformations are strictly of tectonic ori-gin and can be obtained at any locations within a region. More importantly,differential probabilistic ground deformation between any two points can alsobe estimated.To consider uncertainty in the pipeline response, large numbers of sce-narios considering variations in input parameters are required to be ana-lyzed. In addition to variation in pipe operating conditions, soil materialproperties, pipe geometry, variation of differential ground deformation be-tween two ends of the pipe section are also to be considered. This variabilityin differential ground deformation between the two ends of the pipe is ob-tained from the generated probabilistic ground deformation. From pipelineinfrastructure maps and fault alignments, potential sites for pipeline riskassessment can be identified. The probabilistic differential ground defor-mations between two surface points can be converted to local co-ordinatescorresponding to the pipe section under consideration and included in theFE analysis. In case of the HF type of FE models, these variations in dif-ferential ground deformations will translate to varying interface fault anglesbetween the two soil blocks both in the horizontal plane and vertical plane.Additionally, these will translate to variations in the axial, lateral and ver-tical differential displacements between the two sides of the pipeline, truefor both the HF and LF models. To derive training data for the MF-GPsurrogate model, the HF and LF models are to be fed with uniformly ran-domly sampled axial, lateral and vertical differential displacements. Thesedifferential displacements are to be obtained from the range of probabilisticdifferential ground deformations between two surface points of interest inlocal co-ordinates corresponding to the pipe section.Finally, to assess pipeline risk, the two primary variables of interestnamely peak axial tensile strain and peak axial compressive strain are ex-tracted from the pipeline at the end of both the HF and LF analyses. Thesedata are used to train the MF-GP model. Data from a small number of835.3. PROBABILISTIC GROUND DEFORMATION HAZARD ASSESSMENT FOR VICTORIA, BRITISH COLUMBIAHF models (tens) and a large number of LF models (hundreds) can be com-bined to predict the uncertainty in the pipeline response by leveraging theaccuracy and complexity of the HF models and the efficiency of the LFmodels in a MF-GP surrogate model framework. MF-GP surrogate mod-els use Gaussian processes to approximate relationship between the inputand output variables by fitting available data. To ensure reliability of theMF-GP model, it is first to be cross-evaluated and then used to study theuncertainty response of the buried pipelines.5.3 Probabilistic ground deformation hazardassessment for Victoria, British ColumbiaThe LRVFZ is located in the vicinity of the City of Victoria and is of im-portant consideration for seismic risk assessment in the region [131, 131, 133].To generate ground deformations probabilistically due to fault rupture, themethod proposed in [121] is adopted. Broadly, it involves the definition ofan earthquake scenario comprising of earthquake magnitude and faultingtype. Additionally, it requires the selection of a fault rupture source region,which reflect the seismological information of fault rupture in the region.The source region is subsequently divided into a number of sub-faults toenable random generation of candidate rupture models to consider uncer-tainty. To model the fault rupture source stochastically, first earthquakesource parameters comprising of geometry and slip distribution parametersare randomly sampled from [123]. Then, the geometry of the fault plane isselected and slip distribution is defined. Finally, Okada equations [122] areemployed to generate surface deformations. The final ground deformationscalculated are solely of tectonic origin and can be used to characterize prob-ability distributions of ground deformations in the North-South, East-West,and Up-Down directions.5.3.1 Leech River Valley Fault Zone (LRVFZ) SourceCharacterization and Rupture OccurrenceThe LRVFZ consists of a North-North East dipping zone [132]. TheNorth-East ward reverse faulting dipping region of the LRVFZ is close to70◦ steep and extends for about 60 km [131, 134, 135]. The fault geometryof the LRVFZ in the current study has been adopted from [134], with afault length of 67.8 km. A dipping angle of 70◦ is used [134], resultingin a width of 25 km. The fault geometry is, thereafter, divided into a845.3. PROBABILISTIC GROUND DEFORMATION HAZARD ASSESSMENT FOR VICTORIA, BRITISH COLUMBIAtotal of 525 sub-faults: 35 along strike and 15 along dip. Fault rupturesin the LRVFZ fault plane are simulated by rectangular finite-fault sources.The earthquake magnitudes are modelled based on [166, 167]. A truncatedexponential model and characteristic model are used to define the magnitudevs. annual frequency relationship. Details of the earthquake modellingapproach for the LRVFZ can be found in [165]. To this end, first the sliprate is obtained by considering the representative slip rate of 0.25 mm/year[3, 134, 168]. The fault zone area is calculated based on the adopted faultlength and width. A shear modulus of 35 GPa is adopted in the calculations[136]. A value of 0.796 is used for the slope parameter of the truncatedexponential model [169].A minimum magnitude of 6 is used [134]. The magnitude intervals todefine probability density value for the characteristic part and the magni-tude intervals for the characteristic part is set to 1.0 and 0.5 [134, 166].For each of the two earthquake magnitude models, variations of discretevalues of these parameters are considered. These parameters are randomlyselected using a logic-tree approach in association with a combination ofcharacteristic and truncated exponential magnitude-annual frequency rela-tionships to characterize a fault rupture and their occurrence. Once, a faultrupture occurs within a specified magnitude range of 0.1 bin in between6.0 and 7.7, earthquake magnitude is generated from a discrete distribu-tion and subsequently, stochastic source models are generated [121] for thatgiven earthquake magnitude using earthquake source scaling relationships[123], which consider uncertainties in fault rupture by means of variations inboth geometrical parameters and slip distributions. 1000 stochastic sourcesare generated for the selected magnitude range of 0.1 and one of them arerandomly selected for extraction of fault plane and fault slip information.5.3.2 Probabilistic Ground Deformation HazardAssessmentIn the ground deformation computation phase, the randomly selectedsource parameters based on magnitudes are utilized to define the fault plane.Additionally, an earthquake slip field is generated randomly based on thespatial slip distribution parameters which are also generated from the scalingrelationships [123]. An iterative approach is adopted to ensure consistencybetween the generated fault length, fault width and mean slip by matchingthe target and simulated seismic moment. Subsequently, Okada equationsare used to compute surface displacements of an elastic half space due to arupture in a rectangular fault. The inputs for the Okada computations are855.4. FINITE ELEMENT MODELS OF BURIED PIPELINESrake angle and slip value in addition to the rupture geometry, such as thestrike angle, dip angle, fault length, fault width and depth. Depth indicatesthe depth of the fault centroid, strike is the azimuth angle of fault tracewith respect to North, dip angle is the angle between the fault plane andhorizontal plane, fault length is the dimension of the fault along the strike,fault width is the dimension of the fault along the dip, rake angle is theangle between the hanging wall movement direction with that of the strikeand slip defines the value of hanging wall movement. To limit computationaltime, Okada equations are pre-computed at locations of interest in the East-West, North-South and Up-Down directions for every sub-fault by consid-ering an unit displacement (reverse faulting). A thorough review of the gaspipeline network for the City of Victoria is performed and a pipe locationsusceptible to differential permanent ground deformation (48◦25’59.49”N,123◦31’42.80”W) and (48◦26’27.67”N, 123◦29’6.64”W) is identified. Poststochastic synthesis of slip for all sub-faults, pre-computed deformation val-ues for unit slip can be scaled to derive the final ground deformations usingOkada equations [122]. Since the ground deformation computed using Okadaequations are physically consistent for a given rupture, differential deforma-tions in the East-West, North-South and Up-Down directions can also beevaluated at the two sites of interest and used for FE analyses.5.4 Finite Element Models of Buried PipelinesThe FE models used in the analyses involve both HF models and LFmodels. The FE models are created in ABAQUS [140]. [1, 34, 41, 43, 44, 48–50, 52–55] previously employed shell pipes with surrounding continuum soilto analyze pipeline response undergoing fault rupture deformations. The HFmodel in the current study employs shell pipeline and continuum soil withgeometric and material nonlinearity. The HF and LF models are created inaccordance with [165] which successfully performed a multi-fidelity uncer-tainty quantification analysis for buried pipeline undergoing fault rupturedisplacements. The HF model is based on [154]. The model employs tri-axialtest evaluated nonlinear soil constitutive models. It is based on a full-scalepipeline fault rupture experimental test evaluated FE model. [18, 34, 45, 46]employed beam elements to study this problem in a simplified FE approach.5.4.1 Variation in Model ParametersTo conduct an uncertainty quantification exercise, variation in pipelinegeometrical parameters, such as pipe diameter and wall thickness is con-865.4. FINITE ELEMENT MODELS OF BURIED PIPELINESsidered. The pipeline diameter at the location of study is 114 mm (emailcommunication with Fortis BC). Variations of pipe wall thickness from 104mm to 124 mm is considered. Pipe wall thickness varies from 6.02 mm to17.1 mm, which is the available range of wall thickness for the given pipediameter. Additionally, among operating conditions, internal pressure variesfrom no pressure up to a maximum of 550 kPa (email communication withFortis BC). The soil conditions at the concerned site is sandy in nature (BCSoil Information Finder Tool). Uncertainty in the sandy soil conditions isconsidered by varying confining pressure and relative density. Mean confin-ing pressure of 39 kPa is assumed with values ranging from 7.8 kPa to 70kPa and mean relative density of 80 percent is assumed with values rang-ing from 55 percent to 95 percent. The effect of uncertainty in permanentground deformation due to fault rupture is considered by varying the axial,lateral, and vertical displacement boundary conditions in the FE analyses.The uncertainty in permanent ground deformation is estimated from theprobabilistic regional ground deformations, as discussed in section 3. Thedifferential North-South, Up-Down and East-West probabilistic ground de-formations between the two ends of the pipeline section under considerationare converted into the pipe axial, lateral and vertical directions and appliedin the FE model. In this analysis, differential East-West, North-South andUp-Down displacements are varied as -0.185 m to 0.216 m, -0.458 m to 0.079m, and -1.238 m to 0.219 m respectively which is consistent with the valuesobtained from the probabilistic ground deformation at the site of interest.5.4.2 High-Fidelity FE ModelA total of 30 HF models are created. The pipeline segment is modelled byemploying 4-noded general reduced integration shell elements (S4R element).The soil continuum is modelled using three-dimensional reduced integrationstress elements (C3D8R element). A mesh convergence study is performedand an optimal mesh size is chosen accordingly. Pipe mesh size of 25 mmand soil mesh size surrounding the pipe of 10 mm are used. Figure 5.2 showshow the displacement boundary conditions for the FE analyses are obtainedfrom the regional ground deformations (section 3). The pipe material ispolyethelene PE80 and accordingly, a Young’s modulus of 800 MPa, poison’sratio of 0.4, yield strength of 21 MPa and ultimate strength of 30 MPa areconsidered. A yield criterion based on von-Mises criteria is adopted. The soilmaterial model is adopted based on the Mohr-Coulomb yield criterion withnon-linear constitutive relations. Sand shear strength response is a functionof the confining pressure and relative density and is directly dependent on875.5. UNCERTAINTY QUANTIFICATION USING MULTI-FIDELITY SURROGATE MODELLINGthe internal angle of friction [61, 68, 141–143]. The salient features of a sandconstitutive relation are its elastic stiffness, pre-peak hardening behavior,peak strength, and post-peak softening behavior [56, 58]. The default Mohr-Coulomb model in ABAQUS is not capable of modelling varying mobilizedfriction angle and dilation angle. The Mohr-Coulomb model in ABAQUS ishence modified using the user-defined field subroutine (USDFLD).5.4.3 Low-Fidelity FE ModelThe pipe is modelled using 3-noded quadratic PIPE32 elements, consti-tuting Timoshenko beam formulation allowing for both shear and bendingdeformation. The soil springs supporting the pipe in the mutually perpen-dicular axial, lateral, and vertical directions are modelled using 3D pipe-soilinteraction elements PSI34. These elements allow for bi-linear elastic per-fectly plastic force displacement response in simulating the soil resistance tothe pipeline. One end of these springs are connected to the pipe nodes andthe other end are connected to the ground. The ground ends are assignedto be fixed on the stationary side of the fault. The ground ends are assigneda certain movement displacement boundary condition on the moving sideof the fault. The LF FE model represents a 100m-long pipe segment andis made up of a total of 400 elements. Based on a mesh-sensitivity studythe element length is chosen to be 0.5 m. Out of the 400 elements, 200elements are PIPE32 elements and 200 elements are PSI34 elements. Thesoil resistances are calculated based on [4].5.5 Uncertainty Quantification usingMulti-Fidelity Surrogate ModellingUncertainty in pipeline response can be due to variations in the pipegeometry, soil conditions, operating conditions as well as variations in theground deformations resulting out of fault rupture. It is evident from [69–71, 73–76, 78–80] that variations in soil conditions can impact pipeline re-sponse significantly. Additionally, pipe geometry, such as diameter and wallthickness, can also influence pipeline response [73, 78, 80]. Uncertaintyquantification studies typically require large numbers of scenarios to be an-alyzed to gather information on structural response under varied conditions.Some of the more commonly used uncertainty quantification techniques in-clude simulation-based techniques, such as Monte-Carlo simulation [82, 83]or surrogate model-based techniques, such as polynomial chaos expansion885.6. RESULTS AND DISCUSSION[89, 90], neural networks [73, 91], radial basis function [92] and kriging[93, 96]. Monte-Carlo simulations require large numbers of scenarios torun in order to achieve desirable accuracy. Similarly, surrogate model-basedmethods involve training a surrogate model using data from several analysesand using it to predict data for a large number of cases using Monte-Carlosimulation. Both approaches practically require a large number of analy-ses to be performed. Traditionally, uncertainty quantification exercises forburied pipelines are performed using simplified numerical models or analyt-ical models [18, 19, 69–71, 73–76, 78–80] requiring minimal computationalefforts.An uncertainty quantification exercise allowing for maintaining the intri-cacy and reliability associated with detailed pipe-soil numerical models willprovide deeper and realistic insight of the behavior of the pipeline. Solvingdetailed pipe-soil numerical analysis problems, such as the current one athand involving continuum elements with non-linear soil constitutive rela-tions as well as geometric nonlinearity and contact nonlinearity requires sig-nificant amount of computational efforts. Hence, a multi-fidelity surrogate;multi-fidelity Gaussian processes (MF-GP) [170, 171] is used to quantifyuncertainty in buried pipeline response in Victoria, British Columbia dueto fault rupture induced ground deformations. [165] successfully appliedthe technique to study this problem for a hypothetical reverse faulting case.MF-GP learns from a small number of HF data (in the order of 10s) and alarge number of LF data (in the order of 100s) and predicts data as good asHF. This approach helps in reducing enormous amounts of computationalefforts as it allows for learning from a large number of computationally in-expensive LF data instead of an equal number of computationally expensiveHF data.5.6 Results and DiscussionThis section presents results regarding probabilistic ground deformationin the City of Victoria and its further application to quantify uncertaintyin structural response of buried pipelines. The results also include cross-evaluation of the MF-GP surrogate model for both the pipeline responseparameters of interest namely, peak axial compressive strains and peak ax-ial tensile strains. Uncertainty in pipeline response has been studied bygenerating probability density functions of pipeline peak axial compressiveand tensile strains as well as their relationship with the most sensitive inputvariables. A sensitivity analysis has also been included to identify the most895.6. RESULTS AND DISCUSSIONinfluential parameters towards the pipeline response.5.6.1 Probabilistic Ground Deformation Hazard Estimationfor LRFZFigure 5.4 and Figure 5.5 presents some of the critical rupture scenariosincluding the slip distribution along the fault as well as computed grounddeformations in the North-South, East-West and Up-Down directions. Itis to be noted that the location of interest in this study as identified by asquare and circle in Figure 5.4 and Figure 5.5 is seen to be having signifi-cant differential ground deformations. Since, ground deformations computedusing Okada equations regionally are physical consistent, the ground defor-mations computed for each scenario at point 1 and point 2 hold physical de-pendency. Thus, once the probabilistic ground deformations are computedat point 1 (48◦25’59.49”N, 123◦31’42.80”W) and point 2 (48◦26’27.67”N,123◦29’6.64”W), the differential probabilistic ground deformations are ob-tained by computing the difference in ground deformation between point1 and point 2 in the corresponding North-South, East-West and Up-Downdirections. Figure 5.8 and Figure 5.9 shows the differential probabilisticground deformation between point 1 and point 2.5.6.2 Cross-evaluation and uncertainty quantification ofpipeline responseThe peak axial compressive strains and peak axial tensile strains from theHF and LF FE models are utilized to train the MF-GP surrogate model.Thereafter, to ensure the validity of the models a k-fold cross-evaluationmethod is used. To do so, HF data are divided equally into 6 parts . There-after, the first part is used to test the surrogate model and the rest of thedata are used to train the surrogate model. Similarly, the second part isthen used to test the surrogate model and the rest of the data are used totrain the surrogate model. This process is repeated for all the parts. Thevalidity of the MF-GP surrogate model is measured with the coefficient ofdetermination (R-square). Figure 5.7 presents the cross-evaluation resultsfor both the axial tensile strains and axial compressive strains. High valuesof the coefficient of determination of 0.92 and 0.96 are obtained for the axialcompressive strains and axial tensile strains, respectively. For peak axialtensile strains, the most sensitive parameters are identified to be pipe outerdiameter, sand confining pressure and differential ground deformation in theup-down direction. For peak axial compressive strains, the most sensitive905.7. SUMMARYparameters are identified to be sand relative density, differential ground de-formation in the North-south direction and differential ground deformationin the Up-down direction.Figure 5.10 presents variations of the peak axial tensile strains and com-pressive strains with their corresponding most sensitive input variables. Thevariation of peak axial tensile strain with outer diameter is non-linear in na-ture. The variation of peak axial compressive strain with relative density islinear in nature and the strains increase with the increase in relative density.Figure 5.11 presents the generated histograms for the pipeline peak axialtensile and compressive strains for the given location and considered set ofinput parameters and their respective variations. The peak tensile strainhistogram gradually increases from 0 to about 0.025 and thereafter, dropsdrastically. The compressive strain histogram gradually increases from to0 to about 0.014 and thereafter, drops drastically. A cumulative lognormalfunction based on [164] is used to generate fragility curve for the pipelineat the site. To evaluate damage, strain limit based on [4] has been used.Figure 5.12 shows the probability of pipeline damage at the site of interestas a function of differential ground deformation. It is to be noted here that,based on ultimate strain capacity of PE 80 pipes, no significant failure isobserved.5.7 SummaryIn this paper, an integrated risk assessment technique for buried pipelinesunder fault rupture induced permanent ground deformation hazard is de-veloped. It can be applied both regionally and at specific sites. To doso, a previously available probabilistic ground deformation approach hasbeen employed. Additionally, a previously developed method to quantifyuncertainty in pipeline response using a MF-GP surrogate model has beenused. The primary contribution of this work has been to bridge these twoapproaches suitably to present an integrated risk assessment approach forburied pipelines. The method considers regional fault source characteristics,site specific soil conditions, pipeline geometry and operating conditions toquantify uncertainty in pipeline response. The method considers uncertaintyin a comprehensive manner on various aspects and levels of the assessmentas well as utilizes experimentally evaluated complex geotechnical numericalmodels in an efficient manner. The method uses MF-GP surrogate modelto ensure efficiency.The developed method is used to illustrate a case study for the city of915.7. SUMMARYVictoria buried gas pipeline. Pipeline risk assessment is conducted due torupture in the LRVFZ. The MF-GP surrogate model is successfully cross-evaluated with excellent coefficient of determinations of 0.92 and 0.96 be-tween HF observed values and MF predicted values for peak axial com-pressive strains and peak axial tensile strains, respectively. Finally, theuncertainty in peak axial tensile and compressive strains are presented ashistograms. Additionally, a fragility curve based on [4] strain limit is de-veloped. Its is also found that based on ultimate strain capacity of PE 80pipe no appreciable failure is identified at this site. Although the case studyperformed here is for a specific pipe location, the same approach can be usedfor multiple locations of concern over a region. In that case, either separateMF-GP surrogate models can be created for each site or, a global MF-GPsurrogate model can be created with wide variations in the input variablesconsidering every situation for all the sites and then separate prediction datacan be used to evaluate each site individually.925.7. SUMMARYProbabilistic Regional Ground Deformation (Goda 2017)Probability distribution of ground deformations in North-South (N-S), East-West (E-W) and Up-Down (U-D) Direction Probability distribution of differential ground deformations in N-S, E-W and U-D directions Probabilistic Regional Ground Deformation Probability distribution of differential axial, lateral and vertical displacements between two sides of the rupture in the pipe-soil FE modelPipeline orientationSoil informationProbability distributions of sand relative density, confining pressure, pipe diameter, wall thickness, internal pressureMulti-fidelity-Gaussian Process (MF-GP) model for uncertainty quantification of pipeline response (Dey et al., 2021)High-fidelity modelLow-fidelity modelUniformly random selected data within the probability distribution ranges to train MF-GP modelRandomly selected data from the probability distribution to predict response uncertaintyQuantification of pipeline response uncertaintyPipeline MF-GP ModelPipeline Finite Element Model InputsRisk AssessmentFigure 5.1: Integration of regional probabilistic ground deformation withpipeline response uncertainty quantification935.7. SUMMARYGlobal North-South, East-West, Up-Down DisplacementsLocal Axial, Lateral and Vertical DisplacementsNorth-South DirectionEast-West DirectionConverted toPipelineGrid – with minimal ground displacementGrid – with significant ground displacementFigure 5.2: Schematic diagram of FE model displacement boundary condi-tions derivations from probabilistic ground deformation analyses-124.4 -124.2 -124 -123.8 -123.6 -123.4Longitude (Degree)48.448.548.6Latitude (Degree)Figure 5.3: LRFZ finite fault model and sub-fault discretization along with[3] fault scarp information945.7. SUMMARYFigure 5.4: Sample critical rupture scenario slip distribution and regionalground deformation profiles in the East-west, North-south and Up-downdirection in the City of Victoria955.7. SUMMARYFigure 5.5: Sample critical rupture scenario slip distribution and regionalground deformation profiles in the East-west, North-south and Up-downdirection in the City of Victoria965.7. SUMMARYFigure 5.6: City of Victoria buried gas pipeline network under consideration975.7. SUMMARY-0.03 -0.025 -0.02 -0.015 -0.01 -0.005 0High-Fidelity Model Observations-0.03-0.025-0.02-0.015-0.01-0.0050Model Predictions(a)0 0.005 0.01 0.015 0.02 0.025 0.03 0.035 0.04High-Fidelity Model Observations00.0050.010.0150.020.0250.030.0350.04Model Predictions(b)Figure 5.7: Cross-evaluation of MF-GP surrogate model for (a) peak axialcompressive strains and (b) peak axial tensile strains985.7. SUMMARY-1.5 -1 -0.5 0 0.5Differential Up - Down Ground Deformation (m)050100150200250300350400Number of Data(a)-0.5 -0.4 -0.3 -0.2 -0.1 0 0.1Differential North - South Ground Deformation (m)050100150200250300350400Number of Data(b)Figure 5.8: Histograms of the simulated differential ground deformations inthe (a) Up-Down and (b) North-South direction at the location of interest995.7. SUMMARY-0.2 -0.15 -0.1 -0.05 0 0.05 0.1 0.15 0.2 0.25Differential East - West Ground Deformation (m)050100150200250300350400Number of DataFigure 5.9: Histograms of the simulated differential ground deformations inthe East-West direction at the location of interest1005.7. SUMMARY-0.03 -0.025 -0.02 -0.015 -0.01 -0.005 0Compressive strain65707580859095Relative density (%)(a)0 0.01 0.02 0.03 0.04 0.05 0.06 0.07Tensile strain104106108110112114116118120122124Outer diameter (mm)(b)Figure 5.10: Variation of predicted peak axial strains with the most sensi-tive parameters at the location of interest (a) compressive strain vs relativedensity (b) tensile strain vs outer diameter1015.7. SUMMARY0 0.01 0.02 0.03 0.04 0.05 0.06 0.07Peak axial tensile strain020406080100120140160180Number of data(a)-0.03 -0.025 -0.02 -0.015 -0.01 -0.005 0Peak axial compressive strain0102030405060708090Number of data(b)Figure 5.11: Histograms of the pipeline peak axial (a) tensile strains and(b) compressive strains generated at the location of interest1025.7. SUMMARY0 200 400 600 800 1000 1200 1400Differential Permanent Ground Deformation (mm)00.10.20.30.40.50.60.70.80.91Probability of Failure3=11.47,-=2.66Figure 5.12: Fragility curve for pipeline damage using [4] strain limits103Chapter 6Design of Buried Pipelineundergoing Fault RuptureDeformations in Sand usingTaguchi Design ofExperiments6.1 IntroductionNonlinear structural response of buried continuous steel pipeline under-going fault rupture deformation is studied in a systematic manner. A de-tailed and efficient analysis framework for design is proposed and explainedwith a case study. A three-dimensional nonlinear FE model previously de-veloped and evaluated by the authors is used for this study. Taguchi methodfor design of experiments is employed to evaluate the structural performanceof buried pipeline. It is also used to identify the influence of different pa-rameters, such as the fault crossing angle, faulting type (Reverse/Normal,Strike-slip), the operating conditions of the pipeline, geometry of the pipecross-section and material properties of the pipe and soil on the structuralbehavior of buried pipelines. The proposed method can be successfully em-ployed to derive peak strain demands as a function of fault displacementsfor a given set of input conditions in an efficient manner leading to an effi-cient and safe design solution to this problem. A case study involving NPS24 steel pipeline with a maximum operating internal pressure of 9.1 MPa isalso carried out. The method presented here is suitable for pipeline strainhazard analysis, applicable for major oil and gas transmission lines crossingseismically active faults.1046.2. PROBLEM STATEMENT6.2 Problem statementThe objective of the research includes establishing an appropriate frame-work to quantitatively evaluate the structural performance of buried contin-uous steel pipelines undergoing fault rupture deformations in a comprehen-sive manner. The aim is establishing a detailed structural analysis methodfor this problem suitable for design purposes which is highly efficient andreliable. To quantitatively evaluate the structural performance, engineer-ing demand parameters, such as tensile and compressive strain demand arerequired to be obtained from systematic FE analyses. The structural perfor-mance of the pipeline is affected by several parameters, such as fault type,fault-pipe crossing angle, pipe material grade, pipe diameter, pipe inter-nal pressure, pipe temperature and soil conditions. Considering the non-linearity of the system, the influence of the concerned parameters towardsthe peak pipeline strains generated is characterized by significant variability.To address this, the research employs a previously developed and exper-imentally evaluated FE model by the authors [154]. The pipe-soil FE modelemploys pipe material (steel) and soil (sand) non-linear material propertiesas well as geometric non-linearity. The steel is modeled using von-Misesyield criterion and elastic plastic Ramberg-Osgood non-linear stress strainrelation. Non-linear soil strength relations are used to model the soil. sec-tion 6.4 provide further specifics of the steel and soil material propertiesused. An experimentally evaluated Mohr-Coulomb yield criterion for sandwith pre-peak strain hardening and post-peak strain softening is adoptedusing suitable constitutive relationships. The adopted FE model [154] pro-vides a better match with experimental results in comparison to numericalstudies in literature. For the material model evaluation, root mean squareerror (RMSE) of the [154] model is found to be 3.6E-03 in comparison to3.9E-02 for the model by [68]. For large scale tensile experimental test [40]evaluations, RMSE of the [154] model at 300 mm, 600 mm and 900 mm faultdisplacements are found to be 1.7E-03, 4.1E-03 and 4.4E-03 respectively incomparison to 3.1E-03, 5.9E-03 and 9.4E-03 respectively for the [1] ’cou-pled’ model and 3.5E-03, 7.6E-03 and 12E-03 for the [1] ’interface’ model.For large scale compressive experimental test [40] evaluations, RMSE of the[154] model at 300 mm and 600 mm fault displacements are 0.3766 and0.5018 respectively in comparison to 0.5528 and 0.6537 respectively for the[2] model.A process employing Taguchi method to efficiently study several impor-tant parameters affecting the structural performance of pipeline undergoingfault-rupture deformations is presented. The Taguchi method is employed1056.3. TAGUCHI’S METHOD FOR SENSITIVITY ANALYSISto project the peak compressive and tensile strains generated for varyingfault displacement values as well as to study the influence of the variationof the factors towards the peak projected strains. These peak strain valuesprovide the strain demand. The research presents details of the methodadopted and the analysis technique employed to obtain pipeline strain de-mand for a chosen pipe diameter and maximum internal pressure in concertwith [22], as a case study. This analysis methodology is applicable to per-form pipeline strain hazard analysis for buried energy pipelines due to faultrupture.6.3 Taguchi’s Method for Sensitivity AnalysisBuried continuous energy pipeline undergoing fault rupture displacementis a complex phenomenon and the pipeline response is expected to be highlynon-linear. The effect of non-linear soil strength which includes pre-peakstrain hardening followed by post-peak strain softening, on the pipeline re-sponse has not been studied till now. The influence of soil type on thepipeline response is also expected to be non-linear. [46] reported that withthe increase in pipe diameter to wall thickness ratio, a reduction in peakcompressive strain is noted. Contrary to that, a pipeline with higher diame-ter to wall thickness ratio is expected to undergo local buckling more readily,resulting in higher strains. In another study, [46] reported a reduction ofcompressive strain with the increase in pipe wall thickness. Hence, the effectof diameter to wall thickness ratio towards the maximum pipeline strainsgenerated with increasing fault displacements can vary. [46] reported that,the faulting angle with respect to the pipe can have varying effects towardthe pipeline failure. [172] showed that the relation between maximum com-pressive strain generated with the fault crossing angle is highly non-linearand characterized by significant variability. [173] reported that the criticalfault displacement to initiate buckling increased with the increase in theinternal pressure up to an optimum level, beyond which, the critical faultdisplacement decreased with the increase in internal pressure. [48] showedthat, the influence of pipe internal pressure towards the maximum strainsmay vary depending on the value of other concerned parameters. Insteadof a parametric analysis approach as conducted by [1, 17, 31–38, 40–46, 48–50, 52–55], the method proposed here consider all the important factorsinfluencing the pipeline structural behavior in a comprehensive manner suit-able for design purposes.Figure 6.1 summarizes the proposed methodology. Firstly, the uncertain1066.3. TAGUCHI’S METHOD FOR SENSITIVITY ANALYSISSelection of parameters for analyses(Fault rupture – pipe crossing angle, Fault rupture type, Pipe diameter to pipe wall thickness ratio, Pipe steel grade, Soil type, Temperature, Pressure)Selection of number of levels(Low, Medium, High)Creation of Taguchi Design of Experiments (TDOE)Prediction of optimum performance (Peak strains) using TDOE- Post-processing of simulation results- Prediction of peak strain values as a function of fault displacementFigure 6.1: Analysis methodology using Taguchi design of experimentsfactors affecting the pipeline response are to be identified, such as pipe-faultcrossing angle, pipe geometric factors, soil and pipe material properties andpipe operating conditions. Range of steel grade, pipe wall thickness, operat-ing temperature can be decided based on concerned design codes and projectrequirements. Likewise, range of soil types can be decided based on avail-able data. Other important parameters, such as range of pipe-fault crossingangle and faulting types (Reverse/Normal, Strike-slip) can be decided basedon regional seismological information.Parametric analyses involving large numbers of factors and their respec-tive levels are not feasible, considering a full factorial design given the com-putationally expensive nature of the complex FE models used. To overcomethis, design of experiment employing Taguchi method is proposed. Basicsof Taguchi method can be found in [106]. Taguchi method includes a se-ries of orthogonal arrays to cover a wide range of experimental situationsas well as methods to analyze results. Design of experiments using orthog-onal arrays provide high experimental efficiency, however, are more suitablefor cases where there is minimal interaction between the factors. Details ofthe method and analysis technique adopted to generate strain envelopes areexplained here supported by a case study. After selecting concerned param-eters, a set of experiment is planned based on Taguchi approach. Pipeline1076.3. TAGUCHI’S METHOD FOR SENSITIVITY ANALYSISundergoing fault rupture deformation being a complex mechanism and thenon-linear pipeline response being characterized by significant variability asdiscussed, it’s difficult to predict the appropriate combination of level forvarious factors involved, that may lead to maximum strains for increasingfault displacements. The Taguchi method allows for predicting the maxi-mum strains possible at increasing values of fault displacement by projectingthe maximum value of the compressive and tensile strains. Subsequently,these strain values at discrete fault displacement values are combined to gen-erate maximum compressive and tensile strain envelope curves as a functionof fault displacement.The mixed level orthogonal array employed for this study consists of 6factors of 3 levels each and one factor of 6 level. The array A18(61by36) con-sidered for this study is a matrix of 18 rows and (1+6) = 7 columns in whichthe first 1 column has all the 6 elements and each of the next 6 columns have3 elements each. Additionally, another property of this array is that, in ev-ery 2 columns all the ordered pairs of levels appear a fixed number of times.The construction of the array involves difference matrix, Kronecker sums,saturated orthogonal arrays and column replacement. Taguchi’s A18(61by36)are based on [109] approach employing difference matrix D36. A differencematrix D36 is a 6 by 6 matrix whose columns have the property that the dif-ferences between any two columns is a column where each of the 3 elementsoccur equally often. Taguchi’s D36 matrix is given by Table 6.1 :Table 6.1: Matrix 10 0 0 0 0 00 0 1 1 2 20 1 0 2 1 20 2 2 1 1 00 1 2 0 2 10 2 1 2 0 1The Kronecker sum of a 6 by 6 difference matrix D6(3) and a 3 x 1 vectorb of 3 elements, column vector (0,1,2) is a matrix of 6 x 3 = 18 rows and6 columns created by adding the finite arithmetic from each element of thevector b to each element of the difference matrix D36. Hence, the Kroneckersum can be determined as Table 6.2:The matrix A3618 is further expressed as Table 6.3.To generate the final target orthogonal array, a matrix is constructed,comprising of 6 times 3 = 18 rows. 6 columns of the matrix are created fromthe Kronecker sum of the generated D36 difference matrix and column vector1086.3. TAGUCHI’S METHOD FOR SENSITIVITY ANALYSISTable 6.2: Matrix 20 + 0 0 + 0 0 + 0 0 + 0 0 + 0 0 + 00 + 1 0 + 1 0 + 1 0 + 1 0 + 1 0 + 10 + 2 0 + 2 0 + 2 0 + 2 0 + 2 0 + 20 + 0 0 + 0 1 + 0 1 + 0 2 + 0 2 + 00 + 1 0 + 1 1 + 1 1 + 1 2 + 1 2 + 10 + 2 0 + 2 1 + 2 1 + 2 2 + 2 2 + 20 + 0 1 + 0 0 + 0 2 + 0 1 + 0 2 + 00 + 1 1 + 1 0 + 1 2 + 1 1 + 1 2 + 10 + 2 1 + 2 0 + 2 2 + 2 1 + 2 2 + 20 + 0 2 + 0 2 + 0 1 + 0 1 + 0 0 + 00 + 1 2 + 1 2 + 1 1 + 1 1 + 1 0 + 10 + 2 2 + 2 2 + 2 1 + 2 1 + 2 0 + 20 + 0 1 + 0 2 + 0 0 + 0 2 + 0 1 + 00 + 1 1 + 1 2 + 1 0 + 1 2 + 1 1 + 10 + 2 1 + 2 2 + 2 0 + 2 2 + 2 1 + 20 + 0 2 + 0 1 + 0 2 + 0 0 + 0 1 + 00 + 1 2 + 1 1 + 1 2 + 1 0 + 1 1 + 10 + 2 2 + 2 1 + 2 2 + 2 0 + 2 1 + 2Table 6.3: Matrix 30 0 0 0 0 01 1 1 1 1 12 2 2 2 2 20 0 1 1 2 21 1 2 2 0 02 2 0 0 1 10 1 0 2 1 21 2 1 0 2 02 0 2 1 0 10 2 2 1 1 01 0 0 2 2 12 1 1 0 0 20 1 2 0 2 11 2 0 1 0 22 0 1 2 1 00 2 1 2 0 11 0 2 0 1 22 1 0 1 2 01096.3. TAGUCHI’S METHOD FOR SENSITIVITY ANALYSISTable 6.4: Matrix 40 0 0 0 0 0 00 1 1 1 1 1 10 2 2 2 2 2 21 0 0 1 1 2 21 1 1 2 2 0 01 2 2 0 0 1 12 0 1 0 2 1 22 1 2 1 0 2 02 2 0 2 1 0 13 0 2 2 1 1 03 1 0 0 2 2 13 2 1 1 0 0 24 0 1 2 0 2 14 1 2 0 1 0 24 2 0 1 2 1 05 0 2 1 2 0 15 1 0 2 0 1 25 2 1 0 1 2 0given by (0,1,2). This matrix forms the columns 2 to 7 of the target matrixand is an A3618 matrix. Now, the target matrix is generated by adding column1 to the previously constructed matrix. Column 1 is created consisting ofthree 1s, three 2s, three 3s, three 4s, three 5s and three 6s. The resultingsaturated orthogonal array A18(61by36) is given as Table 6.4.To project the maximum value of the desired performance output thefollowing technique is adopted. For this problem, a total of 7 factors canbe considered namely, ‘A’, ‘B’, ‘C’, ‘D’, ‘E’, ‘F’, ‘G’. ‘A’ is fault-pipe cross-ing angle, ‘B’ is fault type, ‘C’ is wall thickness, ‘D’ is steel grade, ‘E’ issoil type, ‘F’ is temperature and ‘G’ is internal pressure. ‘A’ can be con-sidered to comprise of 6 levels, ‘B’ to ‘G’ can be considered to consist of3 levels each. Possible input corresponding to a specific factor along withits level can be expressed as say, ‘A1’. ’A1’ being the input correspondingto factor ‘A’ and level ‘1’. Average of the performance output from all theset of experiments consisting a particular set of factor and level say, factor‘A’ and level ‘1’ can be expressed as ‘A1”. For a given fault displacement,this parameter needs to be evaluated for all possible combinations of factorsalong with their respective levels. Subsequently, maximum values of the1106.4. CASE STUDYparameter need to be stored for every factor and the corresponding levelsinfluencing the maximum value need to be identified. The combination ofthe factors along with their respective levels influencing the maximum val-ues define the optimum condition of input variables for a maximum desiredperformance output. Once the optimum condition of input variables is iden-tified as say, ‘A2’B3’C2’D3’E2’F1’G1”, projected peak performance outputcan be expressed according to [106] as Equation 6.1:Oprojected = X/M+(A′2−X/M)+(B′3−X/M)+(C′2−X/M)+...+(G′1−X/M)(6.1)where, ‘X’ is the sum of all the results and ‘M’ is the number of results.6.4 Case studyThe case study illustrates the proposed method using taguchi design ofexperiments to efficiently design buried pipelines faced with fault rupturehazard. To achieve this, the previously developed experimentally evaluatedFE modelling technique by [154] is used. To consider for variations in someof the most influential input factors in an efficient manner, taguchi design ofexperiments is employed to obtain the strain envelope curves and to studyinfluence of various factors towards the generated strains. NPS24 pipelineis considered for the case study. A maximum operational internal pressureof 9.1 MPa is selected and steel grades of Gr 290, Gr 386 and Gr 483 areselected. [22] is used to select a minimum wall thickness based on pressuredesign criteria considering the lowest grade from the selected steel grades.Subsequently, [22] is used to determine a maximum operating temperatureof 47◦C.6.4.1 Design of experiment using Taguchi methodA series of numerical experiments are first designed using taguchi methodand subsequently performed to investigate the effects of various factors to-wards the structural performance of the pipeline. Structural performancemeasures considered are tensile and compressive strains. Since, the numberof possible combinations for the input factors and their respective levels in afull-factorial experimental design is very large, a logical test plan involvingorthogonal arrays based on Taguchi approach is adopted. The numericalsimulations planned here are devoid of any repetitions for a specific testcombination. To study the nonlinear effect of the factors a minimum of1116.4. CASE STUDYthree levels are considered for all the factors. Six levels are considered forthe pipe-fault crossing angle factor to investigate a range of values. Hence,a mixed level array is employed. The design of experiment assumed thatthe factors are independent of one another. Hence, no interaction betweenthe factors are considered.The faulting mechanism type factor is assigned two levels namely, strikeslip type and normal/reverse faulting type. The primary difference betweenthese two categories is that, in the former, the rupture propagates from thelateral direction and the gravity load acts out of the bending plane of thepipe; where as in the later, the rupture propagates from the base and thegravity load acts in the bending plane of the pipe. The experimental de-sign adopted here considers six three-level factors and one six-level factor.Hence, to accommodate two levels for the faulting type factor in the ex-periment plan, a degraded column approach is adopted, wherein the thirdlevel is replaced with reverse/normal type faulting. The taguchi design ofexperiment adopted can be found in the Table 6.5. To summarize, analy-ses parametric factors consisted of fault-pipe crossing angle (-33.33◦, -20◦,-6.67◦, 6.67◦, 20◦ and 33.33◦); faulting type (Reverse/Normal and Strike-slip); pipe wall thickness (12.7 mm, 14.27 mm and 17.48 mm); steel grade(Gr 290, Gr 386 and Gr 483); soil type (soft sand, medium sand and densesand); temperature (0◦C, 23.5◦C and 47◦C) and pressure (0 MPa, 4.6 MPaand 9.1 MPa).6.4.2 Analysis of results using Taguchi method anddevelopment of envelope strain demand curvesFigure 6.2 to Figure 6.3 show plastic shear strain mobilized from theanalyses conducted at various instances and locations. Structural responseof the pipeline is evaluated in terms of the engineering demand parametersof interest namely, the maximum compressive strain and maximum tensilestrain. Projection of peak values of engineering demand parameters are con-ducted for varying fault displacements. For simplicity, fault displacementmagnitudes are discretized into a number of bins and peak strain valuesare projected for each of those bins. Peak strain envelopes are obtainedas functions of fault displacements. Figure 6.6 shows the maximum ten-sile strains obtained from the 18 numerical analyses with increasing faultdisplacements. 9 of the 18 analyses showed no accumulation of noticeablecompressive strains. Figure 6.6 also shows the maximum compressive strainsobtained from 9 of the 18 numerical analyses with increasing fault displace-ments. Figure 6.6 presents the developed compressive strain envelope and1126.4. CASE STUDYtensile strain envelope curves.Under reverse faulting or compressive strike slip faulting, the pipe failedin buckling; whereas, under normal faulting and tensile strike-slip faultingthe pipe failed in tensile rupture. Peak compressive strain is found to in-crease linearly with increase in fault displacement up to a certain level beforeassuming very large values. Its evident from the curve that with the onsetof buckling, strains tend to reach very high values. Peak tensile strains arefound to increase fairly linearly with increase in fault displacement beforereaching a plateau and increasing thereafter. The nature resembles some-what similarity to the steel stress-strain relation. Figure 6.4 and Figure 6.5show influence of all the factors towards the generated peak compressiveand tensile strains. Influence of a factor towards the generated strains arestudied by keeping all other factors at a constant level. Case 1 correspondsto results where all other factors are at their lowest level and case 2 corre-sponds to results where all other factors are at their highest level. Resultsare obtained at two discrete fault displacement values of 300 mm and 750mm.Results on the variance analysis (ANOVA) to evaluate the relative in-fluence of different input variables on the variance of the generated strainsare provided in Table 6.6 and Table 6.7. Peak strains are extracted at dis-crete fault displacement values and variance analysis is performed for bothcompressive and tensile strains. Tables presenting the results for tensilestrains and compressive strains are presented there. Pipe-fault crossing an-gle is identified as the most prominent factor influencing the variance ofpeak strains generated.Strain response as a function of variation of the input factors are foundto be non-linear in nature, as shown in Figure 6.4 and Figure 6.5. Figure 6.4shows that, compressive strains increased significantly with the decrease inpipe-fault crossing angle and vice versa for the tensile strains. This is rea-sonable since, decrease in pipe-fault angle results in more contraction in thepipe and increase in pipe-fault angle results in more extension of the pipe.For compressive strains, reverse faulting resulted in higher strains in com-parison to strike-slip faulting as shown in Figure 6.4. Higher strain levelsresulted due to thinner pipe walls as shown in Figure 6.4 due to increase inseverity of local failure with decrease in pipe wall thickness. Strain levelsare found to decrease with increase in steel grade as shown in Figure 6.5 pri-marily due to sooner yield in the lower grades. However, the strain responseis found to be varying mildly in relation to the steel grades. In general,higher peak strains are observed with the presence of internal pressure asshown in Figure 6.5 due to increase in severity of local failure zones in the1136.4. CASE STUDY(a) (b)(c) (d)(e) (f)(g) (h)Figure 6.2: Plastic shear strain in sand corresponding to analysis 7 (a) atfault side 1 in the vicinity of the fault (b) at fault side 1, 5 m away fromfault (c) at fault side 2 in the vicinity of the fault (d) at fault side 2, 5 maway from fault; analysis 8 (e) at fault side 1 in the vicinity of the fault (f)at fault side 1, 5 m away from fault (g) at fault side 2 in the vicinity of thefault (h) at fault side 2, 5 m away from fault1146.4. CASE STUDY(a) (b)(c) (d)(e) (f)Figure 6.3: Plastic shear strain in sand corresponding to analysis 12 (a) atfault side 1 in the vicinity of the fault (b) at fault side 1, 5 m away fromfault (c) at fault side 2 in the vicinity of the fault (d) at fault side 2, 5 maway from fault; analysis 11 (e) 5 m away from fault (f) at the vicinity offault1156.4. CASE STUDY 0246810121416-33.33 -20 -6.67Axial compressive strain (%)Fault-pipe crossing angle(ο)(b)012345-33.33 -20 -6.67 6.66 19.99 33.32Axial tensile strain (%)Fault-pipe crossing angle (ο)Case 1 - 300mmCase 2 - 300mmCase 1 - 750mmCase 2 - 750mm(a)00.511.522.531 2Axial compressive strain (%)Faulting type Level(d)012345671 2Axial tensile strain (%)Faulting type level(c)00.511.522.5312.7 14.27 15.84 17.41Axial compressive strain (%)Pipe wall thickness (mm)(f)012345678912.7 14.27 15.84 17.41Axial tensile strain (%)Pipe wall thickness (mm)(e)Figure 6.4: Influence of fault-pipe crossing angle on the peak axial strainsgenerated at 300 mm and 750 mm fault displacements: (a) tensile strain (b)compressive strain; Influence of faulting type: (c) tensile strain (d) compres-sive strain; Influence of pipe wall thickness: (e) tensile strain (f) compressivestrain1166.4. CASE STUDY 00.511.522.533.544.550 23.5 47Axial tensile strain (%)Temperature (οC)(e)00.511.522.533.544.551 2 3Axial tensile strain (%)Soil type level (c)00.511.522.53290 386 482Axial compressive strain (%)Steel grade (MPa)0123456290 386 482Axial tensile strain (%)Steel grade (MPa)00.511.522.531 2 3Axial compressive strain (%)Soil type level00.511.522.50 23.5 47Axial compression strain (%)Temperature (οC)(f)00.511.522.530 4.5 9Axial compressive strain (%)Pressure (MPa)(h)00.511.522.533.544.550 4.5 9Axial tensile strain (%)Pressure (MPa)(a) (b) (d) (g) Figure 6.5: Influence of steel grade on the peak axial strains generated at300 mm and 750 mm fault displacements: (a) tensile strain (b) compressivestrain; Influence of soil type level: (a) tensile strain (b) compressive strain;Influence of temperature: (a) tensile strain (b) compressive strain; Influenceof pressure: (a) tensile strain (b) compressive strain1176.4. CASE STUDY01234560 100 200 300 400 500 600 700Axial tensile strain (%)Fault displacement (mm)Strain Envelope CurveAnalyses 1 to 18(a)024681012141618200 100 200 300 400 500 600 700Axial compressive strain (%)Fault displacement (mm)Analyses 1 to 9Strain Envelope Curve(b)(b)Figure 6.6: Peak strain envelope as a function of fault displacement: (a)tensile (b) compressive1186.5. SUMMARYTable 6.5: Taguchi design of experiment for the problem under considerationSerial numberFault-pipecrossing angle (degree)Fault typeWall thickness(mm)Steel grade(MPa)Soil typeTemperature(degree C)Pressure(MPa)1 -33.33 Reverse/Normal 12.7 290 Soft 0 02 -33.33 Strike Slip 14.27 386 Medium 23.5 4.63 -33.33 Reverse/Normal 17.48 483 Dense 47 9.14 -20 Reverse/Normal 12.7 386 Medium 47 9.15 -20 Strike Slip 14.27 483 Dense 0 06 -20 Reverse/Normal 17.48 290 Soft 23.5 4.67 -6.67 Reverse/Normal 14.27 290 Dense 23.5 9.18 -6.67 Strike Slip 17.48 386 Soft 47 09 -6.67 Reverse/Normal 12.7 483 Medium 0 4.610 6.67 Reverse/Normal 17.48 483 Medium 23.5 011 6.67 Strike Slip 12.7 290 Dense 47 4.612 6.67 Reverse/Normal 14.27 386 Soft 0 9.113 20 Reverse/Normal 14.27 483 Soft 47 4.614 20 Strike Slip 17.48 290 Medium 0 9.115 20 Reverse/Normal 12.7 386 Dense 23.5 016 33.33 Reverse/Normal 17.48 386 Dense 0 4.617 33.33 Strike Slip 12.7 483 Soft 23.5 9.118 33.33 Reverse/Normal 14.27 290 Medium 47 0pipeline due to additional pressure load. Increase in temperature also ledto higher strains as shown in Figure 6.5 due to similar reasons. At lowerdisplacements of 300mm, pipe strains are found to increase with sand den-sity as shown in Figure 6.5 since its expected that the majority of the soil isstill in their pre-peak strain hardening phase. Higher sand density that is,soil strength results in higher pipe strains. Pipeline compressive strains arefound to increase for medium dense sands in comparison to loose sands andthen decrease for dense sands in comparison to medium dense sands for 750mm fault displacements. This is caused by the non-linearity in the soil mo-bilized shear strength comprising of strain-hardening and strain-softening.Similar non-linear response is observed for 750 mm tensile strains where thestrains first decrease and then increase with increase in soil strength.6.5 SummaryAn experimentally evaluated FE model for buried continuous pipelineundergoing fault rupture deformations is adopted. Thereafter, nonlinear 3DFE analyses of buried continuous energy pipelines under fault displacementloads considering variation in faulting mechanism, such as reverse faulting,normal faulting and strike slip faulting, variation in sand type, pipe wallthickness, pipe steel grade, internal pressure, temperature are performedto evaluate the structural performance of a typical oil and gas steel buriedpipeline. The analyses provided information on the tensile and compressivestrain demands on the buried pipeline. The findings of the study can be1196.5. SUMMARYTable 6.6: ANOVA analysis results for axial tensile strainsFault displacement (mm) FactorDegreesoffreedomSumofsquaresVarianceContribution(%)Pipe-fault crossing angle 5 0.559 0.111 49.14Fault type 1 0.069 0.069 6.12Wall thickness 2 0.121 0.060 10.65Steel grade 2 0.031 0.015 2.75Soil type 2 0.031 0.015 2.76Temperature 2 0.137 0.068 12.08Pressure 2 0.130 0.065 11.48100Error 1 0.056 0.056 4.99Pipe-fault crossing angle 5 5.005 1.001 53.56Fault type 1 0.572 0.572 6.13Wall thickness 2 0.198 0.099 2.12Steel grade 2 0.992 0.496 10.62Soil type 2 1.017 0.508 10.89Temperature 2 0.159 0.079 1.70Pressure 2 0.995 0.497 10.65200Error 1 0.402 0.402 4.30Pipe-fault crossing angle 5 18.904 3.780 70.50Fault type 1 0.054 0.054 0.20Wall thickness 2 1.947 0.973 7.26Steel grade 2 1.886 0.943 7.03Soil type 2 2.035 1.017 7.59Temperature 2 1.841 0.920 6.86Pressure 2 0.138 0.069 0.51300Error 1 0.005 0.005 0.02Pipe-fault crossing angle 5 19.219 3.843 66.42Fault type 1 0.085 0.085 0.29Wall thickness 2 0.953 0.476 3.29Steel grade 2 3.375 1.687 11.66Soil type 2 2.717 1.358 9.39Temperature 2 1.498 0.749 5.17Pressure 2 0.611 0.305 2.11400Error 1 0.473 0.473 1.63Pipe-fault crossing angle 5 23.841 4.768 72.28Fault type 1 0.476 0.476 1.44Wall thickness 2 0.934 0.467 2.83Steel grade 2 3.502 1.751 10.62Soil type 2 3.341 1.670 10.13Temperature 2 0.540 0.270 1.63Pressure 2 0.229 0.114 0.69500Error 1 0.114 0.114 0.34Pipe-fault crossing angle 5 28.352 5.670 74.24Fault type 1 1.102 1.102 2.88Wall thickness 2 0.985 0.492 2.58Steel grade 2 3.964 1.982 10.38Soil type 2 3.308 1.654 8.66Temperature 2 0.360 0.180 0.94Pressure 2 0.116 0.058 0.30600Error 1 0.00 0.00 0.00Pipe-fault crossing angle 5 28.408 5.681 74.34Fault type 1 1.295 1.295 3.39Wall thickness 2 0.891 0.445 2.33Steel grade 2 4.801 2.400 12.56Soil type 2 1.990 0.995 5.21Temperature 2 0.596 0.298 1.56Pressure 2 0.128 0.064 0.33750Error 1 0.099 0.099 0.251206.5. SUMMARYsummarized as follows:(1) Design of buried continuous pipelines faced with permanent grounddeformation hazard due to fault rupture can be conducted using a Taguchidesign of experiment approach considering variations in influential factorsaffecting pipeline response.(2) The projection of peak performance approach employed using Taguchidesign of experiments is an efficient way of estimating peak strain demandsfor tensile as well as compressive strains.(3) Only 18 analysis are carried out instead of a full factorial design oftotal 2916 possible combinations to arrive at the peak strain demands forincreasing fault displacement magnitudes.(4) Maximum compressive strain envelope generated from a case studyfor NPS 24 pipe from Taguchi design of experiment showed linearly increas-ing strains up to 1.33 percent for approximately 600 mm displacement andincreasing sharply thereafter. Maximum tensile strain envelope generatedfor the same case study showed a linear increase in strains up to 4.42 percentfor 300 mm fault displacement followed by a plateau and further increasefrom 5.2 percent at 750 mm fault displacement.(5) Analysis of variance showed that pipe-fault crossing angle is the mostinfluential factor affecting the variance in pipeline strains generated.1216.5. SUMMARYTable 6.7: ANOVA analysis results for axial compressive strainsFault displacement (mm) FactorDegreesoffreedomSumofsquaresVarianceContribution(%)Pipe-fault crossing angle 5 0.059 0.011 84.27Fault type 1 0.000 0.000 0.19Wall thickness 2 0.003 0.001 4.26Steel grade 2 0.001 0.000 2.73Soil type 2 0.002 0.001 2.82Temperature 2 0.002 0.001 3.53Pressure 2 0.000 0.000 0.86100Error 1 0.000 0.000 1.29Pipe-fault crossing angle 5 0.157 0.031 69.36Fault type 1 0.003 0.003 1.70Wall thickness 2 0.013 0.006 6.06Steel grade 2 0.008 0.004 3.88Soil type 2 0.012 0.006 5.44Temperature 2 0.009 0.004 4.38Pressure 2 0.012 0.006 5.36200Error 1 0.008 0.008 3.78Pipe-fault crossing angle 5 0.314 0.062 61.96Fault type 1 0.017 0.017 3.49Wall thickness 2 0.026 0.013 5.17Steel grade 2 0.024 0.012 4.88Soil type 2 0.023 0.011 4.67Temperature 2 0.023 0.011 4.54Pressure 2 0.045 0.022 8.95300Error 1 0.032 0.032 6.30Pipe-fault crossing angle 5 0.705 0.141 64.65Fault type 1 0.051 0.051 4.70Wall thickness 2 0.032 0.016 2.98Steel grade 2 0.053 0.026 4.92Soil type 2 0.027 0.013 2.51Temperature 2 0.049 0.024 4.54Pressure 2 0.106 0.053 9.76400Error 1 0.064 0.064 5.91Pipe-fault crossing angle 5 1.296 0.259 67.18Fault type 1 0.112 0.112 5.82Wall thickness 2 0.044 0.022 2.32Steel grade 2 0.050 0.025 2.63Soil type 2 0.053 0.026 2.76Temperature 2 0.057 0.028 2.96Pressure 2 0.220 0.110 11.44500Error 1 0.093 0.093 4.85Pipe-fault crossing angle 5 2.354 0.470 74.02Fault type 1 0.086 0.086 2.70Wall thickness 2 0.108 0.054 3.40Steel grade 2 0.052 0.026 1.64Soil type 2 0.149 0.074 4.70Temperature 2 0.080 0.040 2.52Pressure 2 0.252 0.126 7.94600Error 1 0.097 0.097 3.05Pipe-fault crossing angle 5 266.191 53.238 59.78Fault type 1 0.004 0.004 0.00Wall thickness 2 21.647 10.823 4.86Steel grade 2 28.260 14.130 6.34Soil type 2 33.094 16.547 7.43Temperature 2 34.026 17.013 7.64Pressure 2 39.602 19.801 8.89750Error 1 22.386 22.386 5.02122Chapter 7Summary and Conclusions7.1 SummaryThe overall purpose of this research is to develop an improved and robustmethod for seismic risk assessment of buried continuous energy pipelinessubjected to fault rupture permanent ground deformations, considering un-certainty at various levels. The specific objectives met for this research areas follows:1. Conducted a state-of-the art literature review of existing approachesto study pipeline structural behavior undergoing fault rupture deformations,as well as a literature review of existing easy to use soil (sand) constitutiverelations. This includes review of approaches to evaluate pipeline responseusing analytical approaches, simplified numerical approaches and complexnumerical approaches. Additional literature review are conducted on avail-able approaches to model nonlinear mobilized soil strength properties as wellas dilatancy properties.2. Developed an experimentally evaluated FE model for soil non-linearmaterial property and developed an experimentally evaluated detailed 3DFE model of pipeline undergoing fault rupture deformations. To do so,first a nonlinear Mohr-Coulomb soil model is implemented in FE softwareABAQUS. Thereafter nonlinear mobilized soil strength and dilation behav-ior is incorporated in ABAQUS. Axis-symmetric uni-element is used to eval-uate the model against triaxial test results for sand. Finally, this developedsoil model is suitably fed with soil material properties from a large scalepipeline-soil fault rupture test and used to evaluate a FE model of pipelineundergoing fault rupture against large scale experimental tests.3. Conducted a state-of-the art literature review of existing methods foruncertainty quantification and propagation as well as multi-fidelity-basedapproaches for uncertainty quantification. A detailed review on MonteCarlo simulation, surrogate based approaches and multi-fidelity surrogateapproaches for uncertainty quantification is carried out.4. Developed a multi-fidelity-based uncertainty quantification approachfor buried pipeline undergoing fault rupture deformations. This includes1237.1. SUMMARYidentifying suitable machine learning techniques applicable to this problemand implementing suitable methods for pipeline response uncertainty quan-tification. In this study a multi-fidelity Gaussian processes based method isused. Specifically, Gaussian processes regression has been used. Gaussianprocesses are probability distribution over functions and generalize multi-variate Gaussian distributions. To achieve this, small number of data fromHF models which are computationally expensive but yield highly accurateresults and data from large numbers of LF models which are computationallycheap but yield approximate results are utilized. The approach combinesthe accuracy of hi-fidelity models with the efficiency of LF models to predictnearly accurate results in a reasonably efficient manner.5. Conducted a state-of-the art literature review of existing approachesto probabilistically predict permanent ground deformation due to fault rup-ture. This includes review of regression equation based methods to quantifyfault rupture induced ground deformations as well as review of stochasticsource modelling based method to predict fault rupture induced ground de-formations. Additionally, literature review on existing methods to quantifyseismic risk to buried pipelines due to fault rupture induced permanentground deformations are also studied.6. Probabilistically estimated ground deformation hazard at a givenregion and integrate it with pipeline structural performance to determinepipeline strain hazard. The integrated pipeline seismic risk assessmentmethodology is first established and illustrated with a case study for thecity of Victoria gas pipelines.The integrated seismic risk assessment approach utilizes seismologicalinformation of the regional fault to stochastically model the earthquakesource and thereafter use Okada equations to analytically compute proba-bilistic regional ground deformations. Additionally, the method utilizes sitespecific soil information, pipeline orientation, pipeline structural and op-erating properties, computed probabilistic differential ground deformationin a multi-fidelity analysis framework to quantify uncertainty in pipelineresponse. The multi-fidelity Gaussian processes uncertainty quantificationincludes one of the variable input parameters as the site specific differen-tial ground deformation derived from the computed probabilistic differentialground deformation, thereby integrating the two studies.1247.2. CONCLUSIONS7.2 Conclusions1. Nonlinear material model for sand provided a match with experimen-tal results. An RMSE of 0.0036 is achieved with the current study insteadof existing 0.0392 [68]. Additionally, the pipe-soil FE model is found topredict large scale tensile experimental results with high accuracy. At 300mm displacement for the large scale test, an RMSE of 0.0017 is achieved incomparison to previous 0.0031 and 0.0035 for the ’coupled’ and ’interface’model by [1]. At 600 mm displacement for the large scale test, an RMSEof 0.0041 is achieved in comparison to previous 0.0059 and 0.0076 for the’coupled’ and ’interface’ model by [1]. At 900 mm displacement for the largescale test, an RMSE of 0.0044 is achieved in comparison to previous 0.0094and 0.012 for the ’coupled’ and ’interface’ model by [1].2. The nonlinear sand material model is found to have negligible de-pendence on mesh size and aspect ratio of the elements. Both the materialmodel as well as the pipe-soil model is found to provide better results com-pared to available numerical results.3. Coefficient of determinations of 0.897, 0.944 and 0.924 are achievedbetween HF observed values and multi-fidelity model predicted values duringcross-evaluation of results, there by establishing the validity of the multi-fidelity model. A significant savings of computational effort is achievedwhere 15 minutes run are used instead of 48 hours run for a total of 500analyses using 3.2GHz processor and 32 GB RAM.4. Probabilistic regional ground deformation approach is subsequentlylinked with multi-fidelity uncertainty quantification model to derive seis-mic risk of pipelines. The practical implications of this is significant. Themethod allows for quantifying uncertainty in pipeline response due to per-manent ground deformation hazard arising out of fault rupture over a wideregion. Uncertainty is considered in pipe response as well as fault rupture.5. A method employing Taguchi Design of Experiments is successfullyused to design pipelines faced with the risk of permanent ground defor-mations. The method predicted peak tensile strains and peak compressivestrains as a function of differential ground deformation between the two endsof the pipe for a varying set of input parameters. Additionally, the most in-fluential parameters are identified. The pipe-fault crossing angle is identifiedas the most influential factor affecting the variance in pipeline strains gener-ated. The method has significant computational benefits as only 18 analysesare carried out instead of a full factorial 2916 possible combinations to arriveat the peak strain demands with increasing differential ground deformations.125Bibliography[1] D. J. Robert, , K. Soga, T. D. O’Rourke, Pipelines subjected to faultmovement in dry and unsaturated soils, International Journal of Ge-omechanics 16 (5) (2016) 1–16. → pages xii, 10, 11, 12, 26, 41, 49, 86,105, 106, 125[2] X. Xie, Numerical analysis and evaluation of buried pipeline responseto earthquake-induced ground fault rupture, Rensselaer PolytechnicInstitute, 2008. → pages xiii, 26, 41, 42, 44, 105[3] K. Morell, C. Regalla, C. Amos, S. Bennett, L. Leonard, A. Graham,T. Reedy, V. Levson, A. 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Thesis/Dissertation
eng
Civil Engineering
Vancouver : University of British Columbia Library
Attribution-NonCommercial-NoDerivatives 4.0 International
Graduate
Tesfamariam, Solomon
Seismic risk assessment of energy pipelines under permanent ground deformation due to fault rupture
Text