THE RESPONSE OF PIPELINES IN HIGH PRESSURE, HIGH TEMPERATURE OFFSHORE ENVIRONMENT by ADELAJA, Adekunle Omolade (Matriculation Number: 069044075) B.Sc (Hons) OAU, M.Sc. Ibadan Thesis submitted to the School of Postgraduate Studies, University of Lagos in partial fulfilment of the requirements for the degree of Doctor of Philosophy in Mechanical Engineering Department of Mechanical Engineering Faculty of Engineering University of Lagos, Nigeria August, 2011 SCHOOL OF POSTGRADUATE STUDIES UNIVERSITY OF LAGOS, AKOKA, LAGOS, NIGERIA CERTIFICATION This is to certify that the thesis “THE RESPONSE OF PIPELINE IN HIGH PRESSURE, HIGH TEMPERATURE OFFSHORE ENVIRONMENT” Submitted to the School of Postgraduate Studies, University of Lagos for the award of the degree of Doctor of Philosophy (Mechanical Engineering) is a record of original research carried out by ADELAJA, Adekunle Omolade in the Department of Mechanical Engineering ___________________ _______________ _______________ Author’s Name Signature Date ___________________________ ________________ _______________ 1 st Supervisor’s Name Signature Date ___________________________ ________________ _______________ 2 nd Supervisor’s Name Signature Date ______________________________ ________________ _______________ 1 st Internal Examiner’s Name Signature Date _______________________________ _________________ _______________ 2 nd Internal Examiner’s Name Signature Date ____________________________ _________________ ________________ External Examiner’s Name Signature Date ______________________ _________________ ________________ SPGS Representative Signature Date ii DEDICATION This work is dedicated Firstly, to GOD, the only wise one, the Alpha and Omega, the Beginning and the End, Who Was, and Who is, and Who is to come, the Almighty and, to my late mother, Mrs. Felicia Modupe ADELAJA whom GOD used to raise my siblings and me after the death of my father twenty nine years ago iii ACKNOWLEDGEMENTS Foremost, I am eternally grateful to GOD, the source of life through whom all blessings flow for HIS abundant mercies and guidance through my academic endeavours till date. Special thanks to: my supervisors, Distinguished Professor V. O. S. Olunloyo (NNOM) and Professor O. Damisa for their tutelage, exemplary moral and intellectual commitment from the conception till the conclusion of this research work; my research mentors, Dr. A. A. Oyediran of AYT Corporation, USA and Dr. C. A Osheku of the Department of Systems Engineering for their intellectual commitment both in the problem formulation and methodology; Professor R. O. Fagbenle of the Department of Mechanical Engineering, Obafemi Awolowo University, Ile-Ife and Professor C. A. M. Sousa of the Department of Mechanical Engineering, University of New Brunswick, Canada for their suggestions and advice. I appreciate the tremendous supports of the Dean, Faculty of Engineering, Professor M. A. Salau, Professors S. O. Talabi, O. O. Mojola (visiting Professor), Dr. K. O. Aiyesimoju (alias Beckleylian scholar) for his advice and intellectual input, my Head of Department, Dr. J. S. Ajiboye, Drs. B. O. Okeke, S. J. Ojolo, K. T. Ajayi, O. O. Ajayi, M. O. Kamiyo, S. A. Oke, T. A. Fashanu, S. O. Adeosun, M. K. O. Ayomoh and all members of Distinguished Professor V. O. S. Olunloyo’s Saturday conference meeting. I thank Messrs L. O. Ighodaro (PhD student, Florida State University, USA), T. O. Olakoyejo (PhD student, University of Pretoria, South Africa), A. A. Oluwo and M. O. H. Amuda (Islamic University of Malaysia), Ismail Yakub (University of Princeton, USA) for their assistance in getting journal papers. Special thanks go to Dr. J. O. Akanmu for his kind advice and love. I am very, very grateful to my darling, God-given wife, Mrs. Ngozi Adelaja for her patience, prayers and understanding regarding my absence from home and my keeping late nights too often and too long during this programme. I cannot forget my brother Mr. Tope Adelaja for his concern, financial support and prayers; my uncle, Pastor K. T. iv Odesanya-George and my siblings; Mrs. Bisi Makinde, Dr. A. A. Adelaja and Mrs. O. A. Osoteku; mummy Olusanya and Dr.& Dr (Mrs) Osisanya. I am also thankful to Kehinde Orolu for teaching and guiding me in writing MATLAB programmes, Sanmi Dada and Adebola Ogunoiki for their assistance in drawings and animating some of my presentations. I will also like to show gratitude to my pastors; prominent among them are Pastors Edward Ayegbusi, K. O. Kalu, M. A. Adekoya, Jacob Odedina, Timothy Oyinlola, E. O. Olaniran and O. O. Ademola; to Brother Ropo Joseph for his ceaseless supports with internet facilities and generator services during my trying times and to Bro Femi Fagbami for his brotherly, God sent advice. Also, to my colleagues in the Department and beyond and friends too numerous to mention for their prayers and encouragements, I say thank you. I am likewise indebted to the earlier workers and experts in the engineering mechanics and analysis particularly, Professor M. P. Paidoussis, Emeritus Professor of Mechanical Engineering, McGill University, Canada, Professor C. D Mote Jr, President, University of Maryland, USA and Mr. M. Carr and co-workers at Boreas Consultants, UK, for sending some of their research papers to our research team. Above all, I am grateful to God the Father, the Son and Holy Ghost for this great thing He has done for me. v ABSTRACT This thesis explores the application of Euler-Bernoulli beam theory to derive the equations of motion of fluid conveying pipelines subjected to high pressure, high temperature offshore conditions. The vibration responses show that the higher the density of the transported fluid, the lower the critical velocity at which resonance will take place, hence heavier fluid should be pumped at lower velocity than lighter ones whereas heavier surrounding fluid (as in the case of salty and muddy swamp) behaves as a damper. In addition, the higher the inlet temperature, the higher the period of oscillation but the lower the critical velocity required to initiate resonance. The study also provides theoretical bases for the popular practice of burying of pipeline as a means of controlling buckling and further justifies the same means for the control of pipe walking where the geology permits it. Deformations such as pipeline buckling and pipe walking are found to be enhanced by increase in inlet temperature which is a function of the well condition. Furthermore, the study is shows that the hitherto central role attributed to transient response may not be the main driver for pipe walking since the magnitude of steady state is higher and may after all be responsible. Also, the results of the order of the contributing factors indicate that oscillatory strain, pressure, temperature, tension and friction among others play a significant role in the phenomenon of walking. vi TABLE OF CONTENTS CERTIFICATION ii DEDICATION iii ACKNOWLEDGEMENT iv ABSTRACT vi LIST OF FIGURES xi LIST OF TABLES xvii NOTATIONS xix CHAPTER 1: INTRODUCTION 1 1.1 Background to the Study 4 1.2 Statement of Problem 5 1.3 Aim and Objectives of the Study 5 1.4 Scope and Limitations of the Study 6 1.5 Significance of the Study 6 CHAPTER 2: LITERATURE REVIEW 7 2.1 Studies on the Dynamics of the Problem Alone 7 2.1.1 Linear theory of the dynamics of fluid conveying pipe 7 2.1.2 Nonlinear theory of the dynamics of fluid conveying pipe 12 2.2 Studies on the Heat Transfer Aspect of the Problem Alone 17 2.2.1 Fluid conveying heated pipe 17 2.2.2 Transverse deflection and longitudinal displacement 19 2.2.3 Energy analysis of the problem 20 2.2.4 Buried pipeline 23 2.3 Conceptual/ Theoretical Framework 24 2.4 Special Cases 29 2.4.1 Case A: General problem of unheated pipeline in onshore environment 29 2.4.2 Case B : Riser problem 30 2.4.3 Case C: HP/HT fluid conveying pipe lying on the horizontal seabed 31 vii 2.4.4 Case D: Unheated, unpressuried and untensioned horizontal pipeline 31 in offshore environment CHAPTER 3: ANALYSIS OF THE TRANSVERSE MOTION 33 3.1 Formulated Governing Differential Equations of Motion 33 3.2 Analysis of Transverse Motion Problem 34 3.3 Consideration of Actual Temperature Profile in the Pipe 36 3.3.1 Analysis of the energy transport problem 37 3.3.1.1 Analysis of the energy transport problem (fluid) 37 3.3.1.2 Analysis of the energy transport problem (pipe) 41 3.3.2 Polynomial approximation of the temperature distribution 45 3.3.3 Sensitivity analysis: error evaluation in temperature approximations 47 3.4 Pipe Temperature Polynomial Approximation and the Transverse Motion 49 3.4.1 Analytic solution for w via Integral Transform method 51 3.4.2 Transverse steady state and transient response 56 3.4.3 Sensitivity/ error analysis for the transverse motion 58 CHAPTER 4: ANALYSIS OF LONGITUDINAL MOTION AND THE PIPE 59 WALKING PROBLEM 4.1 Pipe Temperature Polynomial Approximation and the Longitudinal Motion 60 4.2 Analytic Solution of u Via Integral Transform Method 61 4.3 Longitudinal Steady State and the Transient Responses 71 4.4 Sensitivity/ Error Analysis for the Longitudinal Motion 74 CHAPTER 5: DISCUSSION OF RESULTS 76 5.1 Natural Frequencies 77 5.2 Transverse Displacement 88 5.3. Longitudinal Displacement 102 5.4 Model Validation 103 viii CHAPTER 6: CONCLUSION 118 6.1 Conclusion 118 6.2 Contributions to Knowledge 119 6.3 Recommendations 120 REFERENCES 121 APPENDIX A: MATLAB PROGRAM 137 A1 Matlab Program for Temperature Distribution in the Pipe & the 137 Polynomial Approximations A2 Matlab Program for Frequencies Response to Flow Velocity & the Effects 143 of Temperature Polynomial Approximation A2.1 Matlab Program for Transverse Frequency Variation with Inlet 158 Temperature A3 Matlab Program for Transverse Displacement 162 A3.1 Steady State Response to Mode 162 A3.2 Transient State Response to Mode 166 A3.3 Steady State Transverse Response to Inlet Temperature 168 A3.4 Transient State Transverse Response to Inlet Temperature 169 A3.5 Steady State Transverse Response to Mode: Sensitivity to Temperature 172 Polynomial Approximation order A3.6 Transient State Transverse Response to Mode: Sensitivity to Temperature 173 Polynomial Approximation order A3.7 Steady State Transverse Response to Sediment Coverage 175 A3.8 Transient State Transverse Response to Sediment Coverage 177 A4 Matlab Program Longitudinal Displacement 179 A4.1 Steady State Response to Mode and Sedimentation 179 A4.2 Transient State Response to Mode and Sedimentation 184 A4.3 Steady State Response: Sensitivity to Temperature Polynomial 187 Approximation ix A4.4 Transient State Response: Sensitivity to Temperature Polynomial 192 Approximation A4.5 Steady State Longitudinal Response to Inlet Temperature 197 A4.6 Transient State Longitudinal Response to Inlet Temperature 202 x LIST OF FIGURES Figure 1.1: Flowline/ Pipeline/ Riser System 4 Figure 2.1: The flow geometry of the dynamic interaction of pipeline on 26 Seabed when γ is zero Figure 2.2: The flow geometry of the dynamic interaction of pipeline on 26 Seabed when γ is not zero Figure 2.3: Control volume of fluid element and pipe element 27 Figure 2.4: Strain triangle 28 Figure 3.1: Fluid flow in a cylindrical pipe 37 Figure 3.2: Exact solution, linear and quadratic approximation 46 Figure 3.3: Exact solution and 7 th order polynomial approximation 47 Figure 3.4: Percentage maximum relative error versus position on the pipeline 49 Figure 5.1: Transverse natural frequency ω n(1) profile for the case 80 Ri = 0.4m, L = 2Km γ = 0.02, δ1 = 0.3721, δ 2 = 0.3111 Figure 5.2: Transverse natural frequency ω n(1) profile for the case 81 Ri = 0.4m, L = 2Km γ = 0.02, n = 1, δ 2 = 0.3111 Figure 5.3: Transverse natural frequency profile for the case 81 ω n(1) Ri = 0.4m, L = 2Km γ = 0.02, n = 1, δ1 = 0.3721 Figure 5.4: Transverse natural frequency ω n(1) profile for the case 82 Ri = 0.4m, L = 2Km γ = 0.02, n = 1, δ1 = 0.3721, δ 2 = 0.2 − 0.8 Figure 5.5: Transverse natural frequency profile for the case 82 ω n(2) Ri = 0.4m, L = 2Km γ = 0.02, δ1 = 0.3721, δ 2 = 0.3111 Figure 5.6: Transverse natural frequency ω n(2) profile for the case 83 Ri = 0.4m, L = 2Km γ = 0.02, n = 1, δ 2 = 0.3111 Figure 5.7: Transverse natural frequency profile for the case 83 ω n(2) Ri = 0.4m, L = 2Km γ = 0.02, n = 1, δ1 = 0.3721 Figure 5.8: Transverse natural frequency ω n(2) profile for the case 84 Ri = 0.4m, L = 2Km γ = 0.02, n = 1, δ1 = 0.3721, δ 2 = 0.2 − 0.8 xi Figure 5.9: Longitudinal natural frequency 1 profile for the case 84 Ω Ri = 0.4m, L = 2Km γ = 0.02, δ1 = 0.3721, δ 2 = 0.3111 Figure 5.10: Longitudinal natural frequency 1 profile for the case 85 Ω Ri = 0.4m, L = 2Km γ = 0.02, n = 1 Figure 5.11: Longitudinal natural frequency 2 profile for the case 85 Ω Ri = 0.4m, L = 2Km γ = 0.02, δ1 = 0.3721 Figure 5.12: Longitudinal natural frequency profile for the case 86 Ω Ri = 0.4m, L = 2Km γ = 0.02, n = 1 Figure 5.13: Transverse natural frequency ωn(1) profile for inlet temperature 86 the case Ri = 0.4m, L = 2Km γ = 0.02, n = 1, δ1 = 0.3721, δ 2 = 0.3111 Figure 5.14: Transverse natural frequency ω n(2) profile for inlet temperature 87 the case Ri = 0.4m, L = 2Km γ = 0.02, n = 1, δ1 = 0.3721, δ 2 = 0.3111 Figure 5.15: Transverse natural frequency ωn(1) profile for polynomial order for 87 the case Ri = 0.4m, L = 2Km γ = 0.02, n = 1, δ1 = 0.3721, δ 2 = 0.3111 Figure 5.16: Transverse natural frequency ω n(2) profile for polynomial order for 88 the case Ri = 0.4m, L = 2Km γ = 0.02, n = 1, δ1 = 0.3721, δ 2 = 0.3111 Figure 5.17: Steady-state transverse displacement profile for the case 90 ws = 2.5, δ1 = 0.3721, δ 2 = 0.3111 Ri = 0.4m L = 2Km , t = 600 γ = 0.02, U Figure 5.18: Transient-state transverse displacement wt profile for the case 90 = 2.5, δ1 = 0.3721, δ 2 = 0.3111 Ri = 0.4m, L = 2Km, n = 1 γ = 0.02, U Figure 5.19: Steady-state transverse displacement profile for the case 91 w s = 2.5, δ1 = 0.3721, δ 2 = 0.3111 Ri = 0.4m, L = 2km ,γ = 0.02, U Figure 5.20: Transient-state transverse displacement wt profile for the case 91 Ri = 0.4m, L = 2km, n = 1, = 2.5, δ1 = 0.3721, δ 2 = 0.3111 x = 0.5, γ = 0.02, U Figure 5.21a: Transient transverse displacement profile for the case Ri = 0.4m 92 wt x = 0.5, L = 2km, n = 1, γ = 0.02, U = 2.5, δ1 = 0.3721, δ 2 = 0.3111 , Θ = 150 0 C xii Figure 5.21b: Transient transverse displacement wt profile for the case Ri = 0.4m 92 = 0.5, L = 2km, n = 1, x γ = 0.02, U = 2.5, δ1 = 0.3721, δ 2 = 0.3111 , Θ=200 0 C Figure 5.21c: Transient transverse displacement wt profile for the case Ri = 0.4m 93 = 0.5, L = 2km, n = 1, x γ = 0.02, U = 2.5, δ1 = 0.3721, δ 2 = 0.3111 , Θ=250 0 C Figure 5.21d: Transient transverse displacement w t profile for the case Ri = 0.4m 93 = 0.5, L = 2km, n = 1, x γ = 0.02, U = 2.5, δ1 = 0.3721, δ 2 = 0.3111 , Θ=300 0 C Figure 5.21e: Transient transverse displacement wt profile for the case Ri = 0.4m 94 = 0.5, L = 2km, n = 1, x γ = 0.02, U = 2.5, δ1 = 0.3721, δ 2 = 0.3111 , Θ=350 0 C Figure 5.22: Steady-state transverse displacement ws for order of polynomial for 94 = 2.5, δ1 = 0.3721, δ 2 = 0.3111 the case: Ri = 0.4m, L = 2km ,γ = 0.02, U Figure 5.23: Transient transverse displacement wt for order of polynomial for the 95 case: Ri = 0.4m, x = 0.5, L = 2km, n = 1, γ = 0.02, U = 2.5, δ1 = 0.3721 δ 2 = 0.3111 Figure 5.24a: Transient transverse displacement w t profile for the case Ri = 0.4m 95 = 0.5, L = 2km, n = 1, x γ = 0.02, U = 2.5, δ1 = 0.3721, δ 2 = 0.3111 , m = 1 Figure 5.24b: Transient transverse displacement w t profile for the case Ri = 0.4m 96 = 0.5, L = 2km, n = 1, x γ = 0.02, U = 2.5, δ1 = 0.3721, δ 2 = 0.3111 , m = 2 Figure 5.24c: Transient transverse displacement profile for the case Ri = 0.4m 96 wt = 0.5, L = 2km, n = 1, x γ = 0.02, U = 2.5, δ1 = 0.3721, δ 2 = 0.3111 , m = 3 Figure 5.24d: Transient transverse displacement profile for the case Ri = 0.4m 97 wt = 0.5, L = 2km, n = 1, x γ = 0.02, U = 2.5, δ1 = 0.3721, δ 2 = 0.3111 , m = 4 Figure 5.24e: Transient transverse displacement profile for the case Ri = 0.4m 97 w t = 0.5, L = 2km, n = 1, x γ = 0.02, U = 2.5, δ1 = 0.3721, δ 2 = 0.3111 , m = 5 Figure 5.24f: Transient transverse displacement profile for the case Ri = 0.4m 98 wt = 0.5, L = 2km, n = 1, x γ = 0.02, U = 2.5, δ1 = 0.3721, δ 2 = 0.3111 , m = 6 Figure 5.24g: Transient transverse displacement profile for the case Ri = 0.4m 98 wt = 0.5, L = 2km, n = 1, x γ = 0.02, U = 2.5, δ1 = 0.3721, δ 2 = 0.3111 , m = 7 Figure 5.25: Steady-state transverse response ws to sedimentation for the case 99 Ri = 0.4m, L = 2km, U = 2.5, δ1 = 0.3721,δ 2 = 0.3111, γ = 0.02, m = 7 xiii Figure 5.26: Transient-state transverse response wt to sedimentation for the case 99 Ri = 0.4m , L = 2km, U = 2.5, x = 0.5, δ1 = 0.3721, δ 2 = 0.3111, γ = 0.02, m = 7 Figure 5.27a: Transient-state transverse response wt to sedimentation for the case 100 Ri = 0.4m , L = 2km, U = 2 . 5 , x = 0.5, δ1 = 0.3721, δ2 = 0.3111, γ = 0.02, m = 7, δ0 = 0 Figure 5.27b: Transient-state transverse response wt to sedimentation for the case 100 Ri = 0.4m , L = 2km, U = 2.5, x = 0.5, δ1 = 0.3721, δ 2 = 0.3111, γ = 0.02, m = 7 δ0 = 0.5r0 Figure 5.27c: Transient-state transverse response wt to sedimentation for the case 101 Ri = 0.4m , L = 2km, U = 2.5, x = 0.5, δ1 = 0.3721, δ 2 = 0.3111, γ = 0.02, m = 7 δ 0 = r0 Figure 5.27d: Transient-state transverse response wt to sedimentation for the case 101 Ri = 0.4m , L = 2km, U = 2.5, x = 0.5, δ1 = 0.3721, δ 2 = 0.3111, γ = 0.02, m = 7 δ 0 = 1.5r0 Figure 5.27e: Transient-state transverse response wt to sedimentation for the case 102 Ri = 0.4m , L = 2km, U = 2.5, x = 0.5, δ1 = 0.3721, δ 2 = 0.3111, γ = 0.02, m = 7 δ 0 = 2.0r0 Figure 5.28: Steady-state longitudinal response u profile for the case Ri = 0.4m 104 = 2.5, δ1 = 0.3721, δ 2 = 0.3111 L = 2km γ = 0.02, U Figure 5.29: Transient-state longitudinal response u profile for the case Ri = 0.4m 104 L = 2km x =1.0, γ = 0.02, U = 2.5, δ1 = 0.3721, δ 2 = 0.3111, n =1 Figure 5.30: Steady-state longitudinal response u profile for the case 105 R i = 2.5, δ1 = 0.3721,δ2 = 0.3111 n = 1 = 0.4m, L = 2km,γ = 0.02, U Figure 5.31: Transient-state longitudinal response u to inlet temperature for the 105 case Ri = 0.4m, L = 2km, =1.0, γ = 0.02, = 2.5, δ1 = 0.3721, δ 2 = 0.3111 x U Figure 5.32a: Transient-state longitudinal response u to inlet temperature for the 106 case Ri = 0.4m, L = 2km, =1.0, γ = 0.02, = 2.5, δ1 = 0.3721, δ 2 = 0.3111 x U Θ = 150 0 C xiv Figure 5.32b: Transient-state longitudinal response u to inlet temperature for the 106 case Ri = 0.4m, L = 2km, =1.0, γ = 0.02, = 2.5, δ1 = 0.3721, δ 2 = 0.3111 x U Θ = 200 0 C Figure 5.32c: Transient-state longitudinal response u to inlet temperature for the 107 case Ri = 0.4m, L = 2km, =1.0, γ = 0.02, = 2.5, δ1 = 0.3721, δ 2 = 0.3111 x U Θ = 250 0 C Figure 5.32d: Transient-state longitudinal response u to inlet temperature for the 107 case Ri = 0.4m, L = 2km, =1.0, γ = 0.02, = 2.5, δ1 = 0.3721, δ 2 = 0.3111 x U Θ = 300 0 C Figure 5.32e: Transient-state longitudinal response u to inlet temperature for the 108 case Ri = 0.4m, L = 2km, =1.0, γ = 0.02, = 2.5, δ1 = 0.3721, δ2 = 0.3111 x U Θ = 350 0 C Figure 5.33: Steady-state longitudinal response u to polynomial order for n = 1 108 = 2.5, δ1 = 0.3721,δ2 = 0.3111 case Ri = 0.4m, L = 2km γ = 0.02, U Figure 5.34: Transient-state longitudinal response u to polynomial order for case 109 Ri = 0.4m, L = 2km, = 2.5, δ1 = 0.3721, δ 2 = 0.3111 x =1.0,γ = 0.02, U Figure 5.35a: Transient-state longitudinal response u to polynomial order for case 109 Ri = 0.4m, L = 2km, =1.0, γ = 0.02, = 2.5, δ1 = 0.3721, δ 2 = 0.3111 x U n = 1, m = 1 Figure 5.35b: Transient-state longitudinal response u to polynomial order for case 110 Ri = 0.4m, L = 2km, = 2.5, δ1 = 0.3721, δ2 = 0.3111 x =1.0, γ = 0.02, U n = 1, m = 2 Figure 5.35c: Transient-state longitudinal response u to polynomial order for case 110 Ri = 0.4m, L = 2km, =1.0, γ = 0.02, = 2.5, δ1 = 0.3721, δ2 = 0.3111 x U n = 1, m = 3 Figure 5.35d: Transient-state longitudinal response u to polynomial order for case 111 Ri = 0.4m, L = 2km, =1.0, γ = 0.02, = 2.5, δ1 = 0.3721, δ2 = 0.3111 x U n = 1, m = 4 xv Figure 5.35e: Transient-state longitudinal response u to polynomial order for case 111 Ri = 0.4m, L = 2km, =1.0, γ = 0.02, = 2.5, δ1 = 0.3721, δ 2 = 0.3111 x U n = 1, m = 5 Figure 5.35f: Transient-state longitudinal response u to polynomial order for case 112 Ri = 0.4m, L = 2km, =1.0, γ = 0.02, = 2.5, δ1 = 0.3721, δ 2 = 0.3111 x U n = 1, m = 6 Figure 5.35g: Transient-state longitudinal response u to polynomial order for case 112 Ri = 0.4m, L = 2km, =1.0, γ = 0.02, = 2.5, δ1 = 0.3721, δ 2 = 0.3111 x U n = 1, m = 7 Figure 5.36: Steady-state longitudinal response u profile to sedimentation for the 113 case Ri = 0.4m, L = 2km, = 2.5, δ1 = 0.3721, δ2 = 0.3111 n = 1, m = 7 U Figure 5.37: Transient-state longitudinal response u profile to sedimentation 113 for the case Ri = 0.4m L = 2km = 1.0, = 2.5, δ1 = 0.3721, δ2 = 0.3111 x U n = 1, m = 7 Figure 5.38a: Transient-state longitudinal response u profile to sedimentation 114 for the case Ri = 0.4m, L = 2km, =1.0, = 2.5, δ1 = 0.3721, δ2 = 0.3111 x U n = 1, m = 7,δ0 = 0 Figure 5.38b: Transient-state longitudinal response u profile to sedimentation 114 for the case Ri = 0.4m, L = 2km, =1.0, = 2.5, δ1 = 0.3721, δ2 = 0.3111 x U n = 1, m = 7,δ0 = 0.5r0 Figure 5.38c: Transient-state longitudinal response u profile to sedimentation 115 for the case Ri = 0.4m, L = 2km, =1.0, = 2.5, δ1 = 0.3721, δ2 = 0.3111 x U n = 1, m = 7,δ0 = r0 Figure 5.38d: Transient-state longitudinal response u profile to sedimentation 115 for the case Ri = 0.4m, L = 2km, =1.0, = 2.5, δ1 = 0.3721, δ2 = 0.3111 x U n = 1, m = 7,δ0 = 1.5r0 Figure 5.38e: Transient-state longitudinal response u profile to sedimentation 116 for the case Ri = 0.4m, L = 2km, =1.0, = 2.5, δ1 = 0.3721, δ2 = 0.3111 x U xvi n = 1, m = 7,δ0 = 2.0r0 Figure 5.39: Contributions to Steady-state longitudinal response u profile for the 116 case Ri = 0.4m, L = 2km, = 2.5, δ1 = 0.3721, δ2 = 0.3111 U Figure 5.40: Comparison of three models ws profile for the case 117 Ri = 0.4m, L = 2km, = 2.5, δ1 = 0.3721, δ2 = 0.3111, n = 1, m = 7 U xvii LIST OF TABLES Table1.1: Examples of HP/HT wells and the operating conditions 2 Table 3.1: Coefficients of temperature polynomial approximation 46 Table 3.2: Polynomial orders and the errors compared with actual temperature profile 48 Table 3.3: Percentage relative error in transverse frequencies 58 Table 3.4: Percentage relative error in transverse steady state response at x = 0.5 58 Table 4.1: Percentage relative error in longitudinal steady state response at x =1.0 74 Table 5.1: Parametric values used for simulation of the study 76 xviii NOTATIONS Pipe Motion Analysis A Ao At A p A′ C1 C 2 CD E M Q f n f t δx D Dt F1 (t) F2 (t) g h I kb L m f m p mw Pipe internal cross sectional area Original cross sectional area of pipe at inlet Pipe cross sectional area Surface area of pipe Change in the surface area of the pipe Damping force per unit velocity in the transverse direction Damping force per unit velocity in the axial direction Hydrodynamic drag coefficient Young’s modulus of elasticity Bending moment Shearing force Normal force per unit length Tangential frictional force at fluid-pipe interface Elemental length Material derivative External force in the transverse direction External force in the longitudinal direction Acceleration due to gravity Depth of pipe below sea level Moment of inertia Stiffness of the sea bed Length of pipe Mass per unit length of the transported fluid inside the pipe Mass per unit length of the pipe Mass per unit length of sea water displaced by pipe during transverse motion xix m Sum of the masses per unit length of pipe and fluid Μ Sum of masses per unit length of pipe, fluid in pipe and external fluid displaced by pipe p Ph p To t U U ′ U & u u ′ u′′ u′′′ u IV u ~ u F u~ F w w′ w′′ w′′′ wIV wo (t) ~ w w F w~F x z Fluid pressure Hydrodynamic effect of the ocean Pressure drop Tension in pipe Time Velocity of fluid flowing inside pipe Differential of velocity with respect to x Differential of fluid velocity with respect to time Longitudinal displacement First order derivative of longitudinal displacement wrt x Second order derivative of longitudinal displacement wrt x Third order derivative of longitudinal displacement wrt x Fourth order derivative of longitudinal displacement wrt x Longitudinal response in Laplace plane Longitudinal response in Fourier plane Longitudinal response in Fourier-Laplace plane Transverse displacement First order derivative of transverse displacement wrt x Second order derivative of transverse displacement wrt x Third order derivative of transverse displacement wrt x Fourth order derivative of transverse displacement wrt x External excitation displacement Transverse response in Laplace plane Transverse response in Fourier plane Transverse response in Fourier-Laplace plane Axial displacement coordinate Transverse displacement coordinate xx ri Internal radius of pipe ro External radius of pipe [.] Dimensionless quantities Energy Transport Analysis D Dx ∂ ∂t ∂ ∂r ∂ ∂x r x R L Tf Tp k f k p c pf c pp Bi Nu Pe Fo hi ho Material derivative Partial derivative with respect to time Partial derivative with respect to radius Partial derivative with respect to axial position radius axial position Internal radius Length of pipe Fluid temperature Pipe temperature Thermal conductivity of fluid Thermal conductivity of pipe Specific heat at constant pressure of fluid Specific heat at constant pressure of pipe Biot number Nusselt number Peclet number Fourier number Heat transfer coefficient for the inner surface of the pipe Heat transfer coefficient for the external surface of the pipe xxi Ui Overall heat transfer coefficient qw Heat flow through the wall of the pipe pi Hankel roots/eigenvalues of the fluid equation bi Hankel roots/eigenvalues of the pipe equation mi Fourier transforms roots H n Hankel transform of order n J n Bessel function of the first kind of order n Kn Bessel function of the third kind Greek letters α γ ω n ω n(1 ) ω n( 2 ) Ω 1 Ω 2 θ φ ε Ω1 Ω 2 Ω 3 χ 1−χ7 ΔΘ p Θ Θ′ Coefficient of thermal expansivity Coefficient of area deformation Transverse natural frequency Transverse natural frequency Complimentary transverse natural frequency Longitudinal natural frequency Complimentary longitudinal natural frequency Angle between pipe element position and the x-axis Orientation of the system Axial strain Environmental fluid domain Fluid in pipe domain Soil domain Coefficients of the polynomial approximation Temperature change from inlet to outlet pressure change from inlet to outlet Temperature of the flowing fluid Temperature gradient xxii Θ f Θ F f Θ F f Θ p Θ F p Θ p F δ 1 δ 2 μ μ s ∇ 2 ∇ ρ w Φ ρ f ρ p α f α p Dimensionless temperature of fluid Dimensionless temperature of fluid in Fourier plane Dimensionless temperature of fluid Fourier-Hankel plane Dimensionless temperature of pipe Dimensionless temperature of pipe in Fourier plane Dimensionless temperature of pipe Fourier-Hankel plane m f m mw Μ Coefficient of sliding friction Sliding frictional coefficient of the interface of pipe sediment layer Laplacian operator Gradient operator Density of water Velocity potential Fluid density Pipe density Thermal diffusivity for fluid Thermal diffusivity for pipe xxiii