STUDY OF HYDRATE DEPOSITION AND SLOUGHING OF GAS-DOMINATED PIPELINES USING NUMERICAL AND ANALYTICAL MODELS

Transcription

1 STUDY OF HYDRATE DEPOSITION AND SLOUGHING OF GAS-DOMINATED PIPELINES USING NUMERICAL AND ANALYTICAL MODELS by Zhijian Liu

2 c Copyright by Zhijian Liu, 2017 All Rights Reserved

3 A thesis submitted to the Faculty and the Board of Trustees of the Colorado School of Mines in partial fulfillment of the requirements for the degree of Doctor of Philosophy (Petroleum Engineering). Golden, Colorado Date Signed: Zhijian Liu Signed: Dr. Luis E. Zerpa Thesis Advisor Golden, Colorado Date Signed: Dr. Erdal Ozkan Professor and Head Department of Petroleum Engineering ii

4 ABSTRACT Nowadays, one of the most important technical issues facing the petroleum industry is flow assurance. Within flow assurance, gas hydrate formation poses the most severe threat to the transportation of hydrocarbons through long-distance pipelines. The prevention and remediation of hydrate problems cost billions of dollars in gas transmission pipelines. Transportation of oil and gas for deep-water reservoirs involves handling multiphase flows over long distances in cold sub-sea environments. The temperature of the flowing fluids decreases along the pipe due to the cold surrounding sea water. If the temperature in the subsea pipeline is low enough, hydrate formation may be induced. This doctoral thesis will study hydrate deposition and sloughing of gas-dominated pipelines using numerical and analytical models. This work proposed the following hypothesized plug evolution processes for gas-dominated system: (1) hydrate formation; (2) deposition; (3) sloughing;(4) jamming; and finally (5) plugging. Hydrate formation mechanism and prediction models are discussed first. Presence of water, high pressure and low temperature are required for gas hydrate formation. Fluid composition, pipeline surface roughness and wettability can significantly affect the deposition process. The heat transfer principle is also studied. An equation is developed to predict hydrate deposit thickness growth assuming that hydrate formation only occurs at interface area between water and gas (i.e., water film attached on the wall or water droplets entrained in the gas). The growth of hydrate deposits is mainly due to two parts: (1) water condensation on the pipeline wall; and (2) direct hydrate particle deposition from the gas phase. The model is verified using published experimental data. To investigate hydrate deposit sloughing risk, the mechanical properties of hydrate deposits are studied. Four different packing patterns are proposed: (1) simple cubic packing; iii

5 (2) rhombohedral packing; (3) tetrahedral packing and (4) orthorhombic packing. The packing patterns are based on the proposed assumption that all hydrate particles possess spherical shapes and uniform diameters. Different packing patterns will result in different porosity and mechanical properties. The reduced porosity of the hydrate deposits due to hydrate particle sintering is also studied. Two models are proposed to study the relationship between bridge diameter and porosity: (1) evenly coating model and (2) corner growth model. These two models contribute to the understanding of the annealing effect on the decrease of the hydrate deposit porosity. To investigate the sloughing phenomenon in pipelines, the finite element method (FEM) is applied to study the internal stress distribution inside the hydrate deposit. The compact force induced by the momentum of upstream gas molecules will result in collapse of hydrate deposit from the wall. This work also investigated the methods to characterize hydrate deposit: (1) pressure-temperature profile method, (2) back-pressure method, (3) average pressure method, and (4) pressure transient method. The combination of these methods can provide more detailed information about the deposit. The original models proposed by this work advance the understanding of how hydrate deposition and sloughing will eventually plug pipelines. This work made a good progress in the area of hydrate deposition and sloughing mechanism for gas dominated systems. iv

6 TABLE OF CONTENTS ABSTRACT iii LIST OF FIGURES viii LIST OF TABLES xi LIST OF SYMBOLS xii LIST OF ABBREVIATIONS xix ACKNOWLEDGMENTS xx DEDICATION xxi CHAPTER 1 INTRODUCTION What are gas hydrates? Methane hydrate as a potential energy source and challenges Challenges for artificial lifting of gas hydrate reservoirs Hydrate problems for flow assurance Heat transfer across the pipeline Danger of hydrate plug Flow assurance techniques for hydrate formation Four liquid flow models Oil-dominated systems Gas condensate systems Gas-dominated systems High-water-cut systems v

7 1.8 Hydrate plug evolution processes for gas dominated system Scope of work Overall project objectives CHAPTER 2 HYDRATE FORMATION AND DEPOSITION Hydrate formation mechanism Wettability of steel pipeline surface Pipeline surface roughness Hydrate formation prediction models Heat Conduction Heat convection Boundary and initial conditions Temperature profile of the cylinder Kinetic model for deposit thickness growth Hydrate deposition from water film condensed on the pipeline wall Model verification Hydrate deposition from water droplet in the carrier gas phase Summary CHAPTER 3 STUDY OF HYDRATE DEPOSIT PROPERTIES AND SLOUGHING Hydrate deposit properties Elastic wave velocities of hydrate deposits Packing patterns of hydrate particles Porosity of hydrate deposits Evenly coating model vi

8 3.2.2 Corner growth model Tensile strength Soughing mechanism study of hydrate deposits Sloughing study using finite element method Consequence of collapsed hydrate deposit: jamming Summary CHAPTER 4 HYDRATE DEPOSIT CHARACTERIZATION Pressure-temperature profile methods Hydrate formation in the wellbore Hydrate formation at the choke Hydrate formation in the pipeline Hydrate amount quantification Gas volume inside pipeline estimation during one-side-depressurization Use of back-pressure method to determine hydrate deposit thickness and length Average pressure method for locating hydrate deposit Pressure transient method Summary CHAPTER 5 SUMMARY AND CONCLUSIONS CHAPTER 6 SUGGESTIONS FOR FUTURE WORK REFERENCES CITED APPENDIX HYDRATE FORMATION ACROSS THE CHOKE vii

9 LIST OF FIGURES Figure 1.1 Common gas hydrate structures Figure 1.2 Hydrate PT curves for pure hydrocarbon gases (by PVTSIM R ) Figure 1.3 Hydrate PT curves for CH 4 and C 3 H 8 mixtures (by PVTSIM R ) Figure 1.4 Hydrate PT curves for CH 4, CO 2, H 2 S and N 2 (by PVTSIM R ) Figure 1.5 Hydrate PT curves for CH 4 and H 2 S mixtures (by PVTSIM R ) Figure 1.6 Gas-well loading flow regimes with change of the production rate, modified from Govier and Aziz Figure 1.7 Liquid loading of a gas hydrate well, showing hydrate wells will experience liquid loading soon after being put into production Figure 1.8 Mount Elbert Well Borehole Temperature Figure 1.9 Operating expenditure (OPEX) for a subsea production system indicating transportation of oil and gas costs over 50% of the total OPEX Figure 1.10 A conceptual model for the evolution processes of hydrate plugging in gas-dominated system Figure 2.1 A corrosion-pitted surface of gas pipeline Figure 2.2 Phase wetting map at different superficial velocities of oil and water in a horizontal two-phase pipe Figure 2.3 Algorithm for hydrate formation pressure calculation using K-factor methods Figure 2.4 Heat Transfer across pipeline wall with hydrate deposition (H: Hydrate deposit layer; C: Coating layer; P: Pipe wall) Figure 2.5 Hydrate deposition thickness growth on a cold pipe with a diameter 1/8 in. at different time viii

10 Figure 2.6 Schematic of hydrate deposition apparatus Figure 2.7 Model of hydrate deposit thickness growth compared with experimental data Figure 2.8 Hydrate deposition distance as a function of particle size Figure 3.1 Front view of simple cubic packing Figure 3.2 Front view of rhombohedral packing Figure 3.3 Different packing arrangement of spherical hydrate particles Figure 3.4 Conceptual of hydrate bridge growth evenly coating model Figure 3.5 Conceptual of hydrate corner growth model Figure 3.6 Mohr-Coulomb failure criterion (Hoek, Carranza and Corkum 2002) Figure 3.7 Global node and element numbering for mesh of 8 node rectangle (partial) Figure 3.8 One element with 8-node rectangle Figure 3.9 Meshing of the element of the hydrate deposit Figure 3.10 Displacement of the element of the hydrate deposit Figure 3.11 Normal stress distribution (σ x ) inside the hydrate deposit Figure 3.12 Normal stress distribution (σ y ) inside the hydrate deposit Figure 3.13 Shear stress distribution (τ xy ) inside the hydrate deposit Figure 3.14 Meshing of the element of the hydrate deposit Figure 3.15 Displacement of the element of the hydrate deposit Figure 4.1 Hydrate PT curves for annulus at different depths from wellhead (0 m) to bottom hole (4430 m), showing the systems fall within the hydrate formation region from wellhead to 70 m below the surface Figure 4.2 Hydrate PT curves for tubing at different depths from wellhead (0 m) to bottom hole (4430 m) ix

11 Figure 4.3 Hydrate PT curves for surface pipeline with difference choke sizes Figure 4.4 Hydrate PT curves across the compressor by OLGA R Figure 4.5 Hydrate-formation pressures and temperatures at different methanol concentration along the pipeline (Notz 1994) Figure 4.6 Schematic of hydrate deposit on the pipeline wall Figure 4.7 Schedule of valve at outlet Figure 4.8 Gas flowrate during four-point test Figure 4.9 Pressure profile after each depressurization Figure 4.10 Temperature profile after each depressurization Figure 4.11 Determination of hydrate deposit amount on the pipeline wall Figure 4.12 Simulation errors for deposit thickness (left) and length (right), the dotted lines represent the prescribed thickness (left) and length (right) by reducing the diameter of a section of pipeline (52 ft) by 1.5 inches Figure 4.13 Four-point test in the pipeline with length=1.79 miles Figure 4.14 Four-point test in the pipeline with length=17.8 miles Figure 4.15 Four-points test in the pipeline with length=178.2 miles Figure 4.16 Plot of pressure against different blockage positions in a gas flowline with different hydrate blockage location and length (Liu and Scott 2000). 93 Figure 4.17 Outlet type curve for the location of blockage (B = 10) (Liu and Scott 2000) x

12 LIST OF TABLES Table 2.1 Calculated Values of Parameters in the Reaction Rate Expression (Eqn. 2.57) Table 3.1 Porosities and numbers of contact points for different packing patterns Table 4.1 Fluid Composition Table A.1 Coefficients for the Dranchuk Abou Kassem s equation xi

13 LIST OF SYMBOLS a Radius of the base of the cap A Cross-sectional area of choke, in 2 a base The radius of the base of the cap, m A f g The interface area between pipeline wall and gas, m 2 A ij Surface area between phase i and j A p g The surface area between water droplet and gas, m 2 A plug Cross-sectional area of the fully evolved plug, ft 2 A s Surface area between the hydrocarbon phase and the water phase, m 2 A TP Area of the triangle base, m 2 b..... X-axis coordinate of the tangent point between hydrate particle and the bridge c X-axis coordinate of Center of the bridge curvature C Cohesion, psi C Backpressure coefficient C D Choke discharge coefficient C Friction Field fit friction coefficient, 507 lbm/s c p The specific heat capacity of pipeline wall, Btu/(lb F) or kj/(kg K) d Diameter of hydrate particle, µm D Droplet diameter,cm d Diameter of hydrate bridge at the narrowest point d Choke diameter, in xii

14 d B Thickness of hydrate blockage, inch dt/dx Temperature gradient in the x direction, F/ft or C/m E Young s modulus, psi f Fanning friction factor F B Blockage factor F VB Volume factor F Helmholtz free energy F Resultant down stream force, lbf f B Friction factor for partially blocked section of pipe G Gibbs free energy, J h Height of the cap H Enthalpy of system, J h SC Height of the cap, m h tp Height of the pyramid, m h v Convection heat transfer coefficient, W/m 2 K k Specific heat ratio, k = Cp C v K Bulk modulus, psi k Rate constants k Rate constants K i Equilibrium ratio of component i k v Thermal conductivity of a material, Btu/(ft-hr- F) or W/(m-K) L B Length of hydrate blockage, ft L DB Dimensionless blockage thickness,l DB = L B /L xiii

15 M g Molecular weight of the carrier gas inside the pipeline, g/mol m gas Amount of gas consumed during hydrate formation, kg M h Molecular weight of hydrate, g/mol n Backpressure exponent n Number of hydrate particles n(k) The number of surface excess moles of the component K P Pressure, psia P a v Average pipeline pressure, psia P i Pipeline pressure after i th depressurization, kpa P i Pipeline pressure after (i+1) th depressurization, kpa P in Pressure at the inlet p down Downstream pressure at the choke outlet P out Pressure at the outlet p up Upstream pressure Q g Production rate of gas, scf/day q Flow rate, bbl/day q Heat source per unit volume, Btu/(hr ft 3 ) or W/m 3 q Heat flux, Btu/(hr ft 2 ) or W/m 2 q r Heat flux per unit length of a cylinder pipeline wall, Btu/(hr ft 2 ) or W/m 2 r Radius of hydrate particle R Universal gas constant r gas Rate of gas consumption, kg/sec r hd Hydrate deposition rate, g/s xiv

16 r f The radius of flow path with hydrate deposit on the pipeline wall, m r p The radius of pipeline without hydrate deposit, m RF Roughness factor, ratio of the real and apparent surface area S The maximum velocity gradient within the gas flow field, sec 1 S Entropy of total system, kj/k s i Mole fraction of the i th component in the solid hydrate phase t Time, second T Pipeline temperature T The fluid temperature T eq Equilibrium temperature for hydrate formation/dissociation, R T i Pipeline temperature after i th depressurization, kpa T i Pipeline temperature after (i+1)th depressurization, kpa T K Temperature, K T s The pipe wall surface temperature, R T ses System temperature within the pipeline, R T up Upstream temperature, R v Plug velocity (ft/s) V Volume of the downstream/upstream pipeline V bridge Volume of the bridge V bulk Bulk volume of hydrate deposit, m 3 V deposition Volume of hydrate deposition, ft 3 V fl Velocity in saturating liquid, m/s V g Volume of gas blown out during depressurization, ft 3 xv

17 V gas Volume of gas inside the subsea pipeline, ft 3 v gc Critical gas velocity V hydrate Volume of solid hydrate particles in hydrate deposit, m 3 V liquid Volume of liquid inside the subsea pipeline, ft 3 V M Velocity measured, m/s V p Compressional-wave (i.e., P-wave) velocity, m/s V Pipe Volume of subsea pipeline, ft 3 V s Shear velocity, m/s V sc Volume of hydrate truncated by each boundary plane V SC Volume of a spherical cap, m 3 V TP Volume of triangular pyramid, m 3 V unitcell Volume of a unit cell V void Volume of void space in hydrate deposit, m 3 V R Velocity in solid hydrate deposit, m/s x Independent variable x DB Dimensionless blockage location,x DB = d B /d y y Dependent variable Dependent variable y i Mole fraction of component i in the gas phase z Compressibility factor of gas mixtures α α α Dependent variable Dependent variable Dependent variable xvi

18 α v Thermal diffusivity, m 2 /s m Mass of gas blown out during depressurization, kg p Pressure drop, psi T sub Sub-cooling temperature, R V Volume increase due to annealing hydrate particles γ s/g Interfacial energy at the interface between solid and gas phase γ s/l Interfacial energy at the interface between solid and liquid phase γ l/g Interfacial energy at the interface between liquid and gas phase γ ij Interfacial energy at the interface between the phase i and j γ sw Solid surface energy γ sl Solid-liquid surface energy µ Shear modulus, psi µ g Absolute viscosity of gas, g/(cm s) ν Poisson s ratio φ Porosity φ A Angle in the radius direction φ f Friction angle ρ Density, kg/m 3 ρ bulk Bulk density, kg/m 3 ρ f Fluid density, lbm/ft 3 ρ g Gas density ρ l Liquid phase density ρ p The density of pipeline wall, lb/ft 3 or kg/m 3 xvii

19 σ Interfacial tension σ ad The adhesion force between two hydrate particles, mn/m σ f The mechanical strength due to adhesion, N σ h Deposit thickness, m θ w Contact angle ς f The maximum shear stress the hydrate deposit can take without failure xviii

20 LIST OF ABBREVIATIONS BHP Bottom hole pressure CAPEX Capital expense CHP Casing Head Pressure FEM Finite element method G Gas GLR Gas liquid ratio H Hydrate NPV Net present value OPEX Operational expense PCP Progressive cavity pump si Hydrate with structure I sii Hydrate with structure II sh Hydrate with structure H THP Tubing head pressure W Water xix

21 ACKNOWLEDGMENTS I would like to express my sincere appreciation for my advisor, Dr. Luis E. Zerpa, without whose excellent advice and dedicated support, the completion of this work would be impossible. I would like to thank Dr. D. Vaughan Griffiths, my minor advisor, for his valuable guidance and assistance on the finite element modelling. I would like to thank the chair of my advisory committee, Dr. Carolyn Koh, for her for her interest and support for my research. I appreciate Dr. Rosmer Brito-Jurado and Dr. Mark Miller for serving in my advisory committee and for their helpful comments and suggestions on the manuscript of my thesis. I also owe my gratitude to my dear parents, my brother and Mengjiao Li for their support and encouragement. xx

22 This work is dedicated to my parents, who have such a high expectation on my education. xxi

23 CHAPTER 1 INTRODUCTION 1.1 What are gas hydrates? Gas hydrates are ice-like crystalline compounds formed by 3-dimensional hydrogen-bonded water lattices that trap small gas molecules. The water molecules are commonly known as host molecules, and the gas molecules are taken as guest molecules. No chemical bonds are established between the hydrocarbon and water molecules. The water molecules form a cage through hydrogen bonds, and the small gas molecules trapped inside are free to rotate within the void space. Water and gases can produce clathrate hydrates compounds at low temperatures and high pressures. Gas hydrates are classified into three main types based on their crystalline structure (Sloan et al. 2010) : structure I (si), structure II (sii) and structure H (sh) as shown in Figure 1.1. The usually encountered hydrates of interest in typical hydrocarbon productionsystemsaresiandsii.sihydratesaredescribedbytheformula8x*46h 2 O,where X could be CH 4, C 2 H 6, CO 2, or H 2 S. sii hydrates are described by the formula 8X 136H 2 O, where X could be N 2, C 3 H 8 or i-c 4 H 10. si hydrates are usually found in nature, while sii hydrates are usually found in pipelines. Hydrates usually form at the interface where both water and hydrocarbon gas are present (Zerpa et al. 2010). Hydrate formation involves two distinct processes: first nucleation and then crystal growth (Natarajan et al. 1994). The nucleation process is stochastic and currently unpredictable. The hydrate nucleation process involves small clusters of water and gas growing to a critical size (Sloan Jr and Koh 2007). Water molecules cluster around the dissolved gas molecules. The nucleus with critical radius is in equilibrium with the surrounding medium. For a smaller cluster than the critical size, it could either grow or shrink. Usually, the decomposition probability is very high. However, a larger cluster than the critical nucleus 1

24 can grow spontaneously. The combination of these clusters results in hydrate formation. The duration of nucleation process is referred to as induction time or lag time (Natarajan et al. 1994). The nucleation process can be classified into two categories: (1) homogeneous nucleation and (2) heterogeneous nucleation. Homogeneous nucleation, which is very rare, involves a solidification process in the absence of impurities. In the real world, heterogeneous nucleation is more common. This phenomenon occurs when a foreign body (dust or other particles) or surface (interface between two fluids or pipe wall) is present. Understanding the crystallization is the key to the kinetic inhibition. Figure 1.1: Common gas hydrate structures (Sloan et al. 2010). How to avoid operation in the hydrate formation zone is crucial for prevention of hydrate problems. Low temperature, high pressure and the presence of water and gas are required for hydrate formation in oil and gas production systems. Liquid water must be present to form hydrate. Dust or rust particles can facilitate hydrate formation by acting as nucleation sites. 2

25 The gas composition can affect the hydrate formation significantly. Different hydrocarbon gas molecules form hydrate at different temperatures and pressures. It is easier for large hydrocarbon gas molecules to initiate hydrate formation (see Figure 1.2). The presence of large hydrocarbon molecules in a gas mixture facilitates the formation of hydrates (see Figure 1.3). For example, a mixture of 40% methane and 60% propane forms hydrate at pressures as low as pure propane hydrate. N 2 forms hydrate at higher pressures. However, CO 2 and H 2 S can form hydrate more easily (see Figure 1.4). Sour gas can facilitate hydrate formation in the pipeline (see Figure 1.5). Larger molecules such as normal pentane cannot form hydrate, because its size cannot fit in any si or sii cages. However, cyclopentane does form hydrates at atmospheric pressure and low temperature. Figure 1.2: Hydrate PT curves for pure hydrocarbon gases (by PVTSIM R ). 3

26 Figure 1.3: Hydrate PT curves for CH 4 and C 3 H 8 mixtures (by PVTSIM R ). Figure 1.4: Hydrate PT curves for CH 4, CO 2, H 2 S and N 2 (by PVTSIM R ). 4

27 Figure 1.5: Hydrate PT curves for CH 4 and H 2 S mixtures (by PVTSIM R ). 1.2 Methane hydrate as a potential energy source and challenges The undiscovered methane hydrates represent one of the largest prospective energy resources on earth (Kvenvolden and Lorenson 2001). Large scale economical production from natural gas hydrate could reshape the worldwide energy supply. They are extremely abundant in nature and their vast amount has called worldwide attention from industry and academia. Theestimationofmethanegashydrate-in-placerangesfrom to m 3 for terrestrial gas hydrate; and to m 3 for oceanic gas hydrate (Wang and Economides 2011). The production of hydrates involves a complicated combination of physico-chemical mechanism (Zerpa and Koh 2015). There are also many challenging issues associated with how we can harness these potential resources. One of them is how to remove the produced water when gas hydrates dissociate. This is referred to as liquid loading, and leads to the inability of the produced gas to lift the accumulated liquid to the surface. Figure 1.6 illustrates how the overall production rate decreases with increasing liquid loading 5

28 as a function of production time. 1.3 Challenges for artificial lifting of gas hydrate reservoirs As the accumulated water column can exert a high backpressure on the well or even kill the well (i.e., prevent further production from the well), the water must be removed. However, there are many challenges associated with unloading hydrate wells. Insufficient reservoir energy and low GLR. Low pressure reservoirs are more prone to liquid loading problems. As hydrate reservoirs are located within relatively shallow depth, the hydrate reservoir pressure can be very low, which increases the probability of liquid loading. Using Nodal Analysis method, we found that the production rate of a hydrate well in North Slope of Alaska is very close to the critical velocity, below which water droplets begin to settle in the bottom of the well. (see Figure 1.7). Artificial lift methods have to be applied at the initial stage of production. The dissociated water can also hinder the flow of gas into the wellbore, because it reduces the gas relative permeability. Considering that in terms of mass, 86% of the hydrate is water, if the dissociated water is delivered to the surface, it can result is a GLR of ft 3 /bbl. Low GLRs can frustrate the application of artificial methods. A foam assisted lift, velocity strings or gas lift has to be excluded as an economical deliquification method. While for other pumping methods, the presence of gas interferes with pumping efficiency. Low surface and reservoir temperature. As most hydrate reservoirs are located in either permafrost or offshore environments, the surface and wellbore temperatures are very low (see Figure 1.8) compared to conventional gas reservoirs, where the reservoir temperatureismuchhigher(downholetemperaturecanbehigherthan200 Finconventional gas reservoirs). At higher temperatures, water is present as gaseous vapor, and would be easier to be carried to surface. However, the hydrate reservoir temperature is usually between F.Forthistemperaturerange, itwouldbemucheasierforthewatervaportocondense and fall down to the bottom. For permafrost hydrate reservoirs, the surface temperatures are usually at or below 32 F. The produced water can solidify into ice, which can possibly 6

29 Figure 1.6: Gas-well loading flow regimes with change of the production rate, modified from Govier and Aziz plug the choke. Sand production. Most hydrate reservoirs consist of unconsolidated sand formations (Dvorkin et al. 2003). Unconsolidated sand grains are usually held together by adhesive osmotic pressure forces (Oyeneyin 2008). The gas hydrate can also act as cementing material between grains (Waite et al. 2004). When the hydrate decomposes, the hydrate-bearing sediment structural strength is significantly degraded. As the water saturation increases, loss of capillary cohesion results in fine grain movement. The loose fine grains will be produced into the wellbore along with water and gas. The presence of sand is detrimental to some pumps because the sand will (1) accelerate equipment wear; (2) increase rod torque and power requirement; (3) create flow restriction at the pump intake and in the tubing (Lea et al. 2011). Serious abrasive effects on the tubing and equipment on surface can occur. The produced sand entrained by the fluids can also act as the nucleation site of secondary hydrate formation. The well should be gravel packed to prevent sand production (Saucier 1974). Formation of secondary hydrate. The hydrate reservoir temperature is usually between F, and pressures range from 1700 psi to 2610 psi (Wu et al. 2008). As the gas hydrate reservoirs are located either in permafrost or marine environments with water 7

30 Figure 1.7: Liquid loading of a gas hydrate well, showing hydrate wells will experience liquid loading soon after being put into production. depths below 300 m (Dickens et al. 1997), the wellhead temperature is very low. The low surface temperature may also trigger the formation of secondary hydrates or even ice. Due to the memory effect, hydrates form more readily from gas and water from melted hydrates than from fresh water (Makogon 1997). Therefore, the wellbore is at higher risk of being completely clogged with solid hydrates. Due to the presence of newly formed solids, some artificial methods cannot be applied to dewater the loaded hydrate wells. Once detected with liquid loading, the wells should be treated immediately. Otherwise, if liquid loading goes unnoticed, costly production loss and possible abandonment of wells can happen. When we determine the initial artificial lift method for hydrate wells, many factors such as well location, sand production, low GLR, low BHP, and low surface temperature, should be taken into consideration. Once a method is chosen, it usually stays in place and changing into another method can be costly (Clegg et al. 1993). The goal of artificial lift 8

31 Figure 1.8: Mount Elbert Well Borehole Temperature (Kneafsey et al. 2011). selection is to maximize the overall life efficiency of the operation. Therefore, when we choose an artificial lift for hydrate wells, we need to consider the water production during the wells overall life. When we design the artificial lift methods, their capabilities should match the wells productivity. Another challenge for production of gas hydrate as an economic future energy is the formation of secondary solid hydrate, which could plug the pipeline. As most hydrate reservoirs are located in either permafrost or offshore environments, the surface and wellbore temperatures are very low compared to conventional gas reservoirs, where the reservoir temperature is much higher. Downhole temperature of conventional gas reservoir can be higher than 300 F (Kabir et al. 1996). For permafrost hydrate reservoirs, the surface temperatures are usually at or below 32 F. The produced water can solidify into ice, which possibly plug the choke. The low ambient temperature may trigger the formation of secondary hydrates or even ice. Due to the memory effect, hydrates form more readily from gas and water from 9

32 melted hydrates than from fresh water (Makogon 1997). Therefore, the production from gas hydrate reservoirs is at high risk of hydrate formation in and plugging of the pipelines. 1.4 Hydrate problems for flow assurance A past study of about 110 oil companies, conducted by Welling and Associates in 1999, disclosed that the most important technical issue that the petroleum industry are now facing is flow assurance (Ziya 2000). Within flow assurance, gas hydrate problems pose the most severe threat to the transportation of oil and gas through pipelines. Hydrate accumulations that plug the pipelines can interrupt normal production schedules and cause economic loss or even generate a safety risk if not properly handled. Hydrate problem in the pipelines costs billions of dollars in the gas transmission pipelines. The operating expenditure (OPEX) for a subsea production system is shown in Figure 1.9. The transportation of oil and gas in the pipelines costs over 50% of the total OPEX. Transportation of hydrocarbon fluids is tightly related to flow assurance. Therefore, it is imperative to have a solid understanding on how hydrates can finally plug the production systems. Figure 1.9: Operating expenditure (OPEX) for a subsea production system indicating transportation of oil and gas costs over 50% of the total OPEX (Chandel 2015). 10

33 1.4.1 Heat transfer across the pipeline Prediction of temperature profile along the pipeline is very important in the prediction of hydrate formation (Dorstewitz and Mewes 1994; Shi et al. 2011). Hydrate problem management involves the thermal design of pipelines (Bai and Bai 2012). To prevent hydrate deposition on the wall of pipeline, insulation layers are usually added to the pipeline. It costs about $1, 000, 000 per mile for the insulation of subsea pipelines to prevent hydrate formation problems during offshore operations (Jassim et al. 2010). As fluids are transported from the reservoir to the platform, the fluid temperature decreases due to heat transfer, i.e. heat loss to the surrounding subsea environment. The design of multiphase transportation systems requires the flow assurance engineers to have a detailed knowledge about heat and mass transfer principles with hydrate formation. The purpose of this part of doctoral thesis is to discuss the system thermal behavior. The heat transfer occurring in nature falls into three categories: (1) conduction; (2) convection; and (3) radiation (Bergman and Incropera 2011; Kreith et al. 2012). Heat transfer across the pipeline is a combination of all of these three modes. Conduction occurs when there is a temperature difference across the pipeline wall. If there is a observed temperature gradient existing between the outer pipeline surface and the moving surrounding sea water (for subsea pipelines) or air (for onshore pipelines), convection will occur. The solid pipeline surface can emit electromagnetic energy, which process is called radiation. As the temperature conditions for hydrate formation inside the pipeline is usually below 10 C, heat loss due to radiation is negligible in comparison with the loss from conduction and convection. Thus, only conduction and convection will be considered in this part of the thesis. 1.5 Danger of hydrate plug It might take a few hours to form hydrate plugs in a pipeline. Once hydrates form, the situation deteriorates quickly. Once the plug forms, it will completely block the pipeline. It might take weeks (or even months) to remove a hydrate plug. Interruption of normal 11

34 production schedule can cause monetary losses (i.e., lower net present value, NPV). Additionally, hydrate could plug the inlet of flowmeter or pressure gauge, resulting in measuring and reading errors. According to DeepStar field studies in Wyoming, the plug lengths range between 25 and 200 ft (Hatton and Kruka 2002). There could be a huge pressure difference across the existing plug. The plug of interest usually dissociates itself radially, so the plug would detach at the pipe wall. Once the force difference across the plug exceeds the adhesive force at the wall, the plug travels like a projectile (Sloan et al. 2010). Downstream of the plug, gas will be compressed and the pressure increases. The plug slows down or even starts moving in the opposite direction. The downstream pressure could be much larger than upstream pressure due to the compression of gas, which could result in a rupture of the pipeline. Once the moving plug encounters any obstruction, such as an indentation, orifice, bend or even inclination of the pipeline, there could be a sudden impact which could also damage the pipeline. 1.6 Flow assurance techniques for hydrate formation There are different methods used in flow assurance to deal with hydrate formation in pipelines. Some methods are related to the prevention of hydrate formation, while a new trend is to allow the formation of hydrates, but preventing the formation of a hydrate plug. Here is a summary of these methods: Remove water from the system, as the presence of water can facilitate hydrate formation. The gas from the separator should be dehydrated first to reduce the water amount to a minimum level. The pipeline system is usually designed to prevent hydrate problem by reducing the water content with the dehydrator. Hydrate formation does not pose a threat during normal pipeline operations. Maintain reservoir fluids temperature above the hydrate-forming temperature by heating the fluids or insulating the pipeline. In offshore production facilities, the produced 12

35 liquids travel from the reservoir pores into wellbore bottom, and up to the wellhead, and then through the manifold into the flowline before they rise to the platform. The fluids retain the heat from the reservoir, but after about 30 to 100 miles of travelling in the cold subsea environment, their temperatures drop significantly. By applying an insulator on the surface pipelines, we can reduce the cooling of reservoir fluids and maintain them above hydrate-forming temperatures. Keep the operating pressure below the hydrate-forming pressure. Adding hydrate inhibitors into the system. Hydrate inhibition chemicals can be divided into two categories: (1) thermodynamic inhibitors (THI) and (2) low-dosage hydrate inhibitors (or LDHIs). The commonly used THIs are methanol, salts and MEG. These inhibitors can compete for the formation of hydrogen bonds with the water molecules, preventing water from participating in the hydrate structure. The injection volume of THIs is a function of the amount of water in the system, and could become economically infeasible at large water cuts, which motivated the development of LDHIs. The LDHIs can work at much lower concentrations (<= 1 wt%), since they are intended to operate at interfaces. LDHIs can be further classified into (a) kinetic hydrate inhibitors (KHIs) and (b) anti-agglomerants (AAs). The KHIs can effectively postpone the significant crystal nucleation and growth. The KHIs can absorb onto the crystal surface of hydrates and make it harder for the crystal to grow. AAs works to prevent the formed hydrate particles from aggregation. The AAs have one end dissolved in the oil phase with the other attached to the hydrate structure. The AAs can keep the hydrate particles suspended and flowable in the cold flow state. 1.7 Four liquid flow models To study the hydrate flow model, four different models are proposed(sloan et al. 2010): 13

36 1.7.1 Oil-dominated systems. These systems have a large amount of oil. Oil flows as a continuous phase with water emulsified as droplets due to shear or surfactants. Gas is either dissolved in the oil phase or scattered in the liquid water phase by turbulence. The interfaces between different phases are always frequently disturbed and very difficult to be characterized. Typically, the oil holdup is 50% or higher Gas condensate systems. Gas condensate are also called retrograde condensate gases or retrograde gas-condensates. The fluid is initially gas at the reservoir conditions. As reservoir pressure decreases, a portion of the gas is condensed into liquid. Condensates contains fewer heavy hydrocarbons than black oil. In gas condensate systems, the dispersion of water in the liquid hydrocarbon phase is limited to high shear rates. Instead, water will separate as a continuous phase flowing in the bottom of the pipe Gas-dominated systems These systems are the target for this doctoral thesis. They have small amounts of liquid water and oil. Liquid could be entrained as small spherical droplets in the gas. The liquid droplets can intermittently wet the inner surface pipeline wall. The droplets cannot be observed without magnification or special lighting due to their extraordinarily small size. The water can accumulate at the lower location of the pipeline system, providing an ideal location for hydrate formation and plugging. The accumulated water can form slugs, which have a destructive consequence on the hydrate deposit due to higher turbulent shear force High-water-cut systems At a later development stage of oil fields, huge amounts of water can be produced as a result of water coning and/or multilayer channeling. As more water is produced, the gas hydrate plugging risk will be increased (Zerpa et al. 2013). As water content increases, 14

37 emulsion phase inversion from oil-continuous to water-continuous can occur. When the water content is high (typically > 70%), water will flow as the continuous phase while oil will be scattered as droplets in the predominant water phase. As gas is dissolved in the oil droplet, hydrates form at the interface between oil droplets and continuous water phase (Zerpa et al. 2012a). A transient hydrate formation model, CSMhyk, can be adopted to predict hydrate formation for the gas-dominated and water-dominated pipelines (Zerpa et al. 2012b). 1.8 Hydrate plug evolution processes for gas dominated system The hypothesized plug evolution processes for gas-dominated system, shown in Figure 1.10, can be divided into five steps: (1) hydrate formation; (2) deposition; (3) sloughing; (4) jamming; and finally (5) plugging. Figure 1.10: A conceptual model for the evolution processes of hydrate plugging in gasdominated system. After the small water droplets entrained in the continuous gas phase are converted into hydrate particles, they can deposit on the pipe wall. Deposition along the pipeline grows layer by layer and forms a local restriction, which limits the pathway for hydrate particles. The shear stress acting on the interface between the hydrate deposit and flowing fluids increases. Sloughing of these deposits may occur due to increased drag forces exerted by the flowing fluids on the hydrate deposit. For the gas-dominated system, when gas flows across 15

38 the restriction, the flow path suffers a sudden decrease of flow area, resulting in higher gas velocity. Gas hydrates are inclined to form and accumulate at restrictions. Once the fluids flow across a choke, the temperature will decrease due to the Joule-Thompson effect, which could result in water condensation or even subject the fluids to hydrate formation conditions. Different choke flow models are available to predict pressure and temperature drops across the restriction (Guo et al. 2011). Restrictions with smaller diameters are more prone to hydrate formation. By decreasing the restriction diameter, downstream conditions enter the hydrate formation region. When collapsed hydrate debris are transported to this restricted location with limited passage, the debris could be trapped and jam the entire line. Hydrate jamming is the sudden arrest of the suspended hydrate debris or large particles dynamics in the pipeline. The jammed hydrate particles or debris are trapped locally, preventing them from flowing downstream and making them an obstacle for fluids upstream. Limited amount of fluids can still cross the local obstacle as the obstacle is still very porous. However, as more heat is lost into the external environment, the fluids contained in the pores of trapped solids are converted into solid ice or hydrate, resulting in a more rigid and less permeable local obstacle for the fluids. No significant amount of fluids can travel across the obstacle and plugging occurs. Motivated by the above observations, this doctoral thesis proposes to investigate how hydrate formation eventually evolves into a plug in the gas pipeline system. This doctoral thesis addresses the following basic issues: (1) How does the hydrate deposit growth on a pipe wall finally choke the lines; (2) How does sloughing of hydrate deposit occurs and the collapsed deposit debris blocks the pipelines at the restrictions; (3) How to predict the thickness, length, amount and location of hydrate deposit. 1.9 Scope of work Gas hydrate that forms in deep-water subsea pipelines can block the flow stream, becoming an important safety and economic concern. The flow assurance engineer is required 16

39 to deal with hydrate formation and plugs in subsea pipelines, which could be located a few miles away from the platform. Understanding the plugging formation process can help to successfully manage hydrate issues in pipelines. This doctoral research project proposes to address the following fundamental questions: (1) How does the growth of hydrate as a film on a pipe wall choke the lines? (2) How does sloughing of hydrate deposits occur and the collapsed deposit debris block the pipelines at flow restrictions? and (3) how to predict the thickness, length, amount and location of hydrate deposits? This doctoral thesis will concentrate on the study of gas-dominated systems. The following approaches will be followed: Propose four different packing patterns for the deposit structure; Investigate the mechanical properties of hydrate deposit based on the hydrate packing pattern and estimate how they affect sloughing using finite element methods; Analyze the parameters, such as production rates, fluid composition, temperature and pressure conditions inside the pipeline, which can affect sloughing and jamming and investigate how to prevent sloughing from happening. Make practical recommendations for hydrate plugging prevention in field application Overall project objectives This work represents an original contribution to the understanding of how hydrate plugs will form in gas-dominated pipelines from the sloughing of pipe wall hydrate deposits. The overall project objectives are as follows: to quantify gas hydrate deposit growth with time on a pipeline wall using theoretical models; to examine the mechanical properties of hydrate deposits on gas subsea pipelines based on proposed theoretical packing models; 17

40 to study the fundamental mechanism of gas hydrate sloughing from pipe wall based on internal stress distribution analysis using finite element method; to develop a combination of methods to quantify and locate the hydrate deposit on the subsea pipelines. A result of this work contributes to a better understanding of the mechanism of hydrate film growth on pipe wall that chokes the flow. The sloughing of hydrate deposits was modeled; along with the subsequent jamming of hydrate particles. The doctoral thesis makes practical suggestions for field flow assurance operation to prevent hydrate sloughing and plugging problem. The results provide operators more insight into how their field decisions could result in higher net present value (NPV). 18

41 CHAPTER 2 HYDRATE FORMATION AND DEPOSITION The fact that hydrate could block natural gas-transport pipelines was first found by Hammerschmidt (1934). To prevent hydrate problems during operations in gas-dominated systems, it is crucial to know how to predict the hydrate formation conditions, in terms of temperatures and pressures. It is also necessary to establish a model to predict the hydrate deposit growth during the design of facilities, operation and hydrate remediation procedures. This chapter discusses the current available thermodynamic models for hydrate-formation prediction and develops a model to understand how long it will take for the hydrate to achieve a certain thickness. 2.1 Hydrate formation mechanism The hydrate embryo formation rate is controlled by system temperature and surface energy. The nucleation process requires the cluster of gas and water molecules to overcome the Gibbs energy barrier (Fletcher 2009). The Gibbs Energy is defined by: G(p,T) U +pv T K S (2.1) or G H T K S (2.2) Here, G denotes the Gibbs free energy, and H stands for the enthalpy of system, which can be expressed as: H = U +pv (2.3) Enthalpy is a measurement of the system thermodynamic energy, and S represents Entropy of total system. 19

42 The Gibbs free energy is controlled by the surface energy, which is also related to wettability and contact angle. Surface roughness and coating can change the surface energy. Therefore, the surface energy of cold steel surface can affect the initial hydrate nucleation process, which could be altered by surface treatment (Na and Webb 2003) Wettability of steel pipeline surface For water-saturated gas flow, water can condense on the pipe wall. The condensation rate is smaller than that of hydrate formation by two orders of magnitude, meaning water condensation rate controls the rate of hydrate formation during the initial deposition (Rao et al. 2013a). When a water droplet contacts the cold pipeline surface, the eventual shape of the drop and whether the droplets wet the pipeline surface are heavily influenced by the molecular cohesion force magnitude (Weil 1981). Interfacial tension is the result of the competition of these forces. Interfacial energy can be defined by Young s equation (Young 1805) as: Contact Angle is defined by: ( ) δf γ ij = δa ij T,V,n(K) (2.4) γ s/g γ s/l = γ l/g cosθ (2.5) However, the wetting phenomenon on the pipe wall surface is not a static state as fluids are moving. The contact line between solid, liquid and gas is in motion (Yang et al. 1998). The liquid will move to expose its fresh surface and to wet the fresh steel surface. The dynamic contact angle arise from the surface roughness and/or heterogeneity. This phenomenon allows more steel surface to be wetted by water. Once the water wets the inner pipe wall, formation of hydrate occurs. Water chemistry, additives (drag reducing agent, scaling/corrosion inhibitor, etc.) and pipe wall surface state (coating, scale, etc.) can affect the wettability of the inner pipe wall. For example, iron carbonate film can make the surface more hydrophilic, i.e., water-wet (Tang et al. 2008). Rust is also found to improve 20

43 the wettability of steel surface by water (Lu and Chung 1998). Wetting can be classified into (1) physical wetting, (2) chemical wetting and (3) mechanical wetting depending on the nature of the attractive forces acting across a solid-liquid interface (Lawrence and Li 2000). For physical wetting, the attractive energy is achieved through the reversible van der Waals forces. As for the chemical wetting, adhesion is gained by the molecular reactions between the mating surfaces, resulting in chemical bonds. Most pipelines for hydrocarbon transportation in the field are made of carbon steel. The properties of the wetted surface, such as roughness and coating, can affect its wettability Pipeline surface roughness The building material for oil and gas transportation pipelines is commonly carbon steel (Li et al. 2008). The internal surface of pipeline are usually smooth and coated. However, as time goes on, the presence of water can result in internal corrosion, which increases the roughness of pipes (as seen in Figure 2.1, Abduh (2008)). The roughness of pipes has a great impact on the pressure loss and for fluids transported. Surface roughness is generally defined as the variations in height of the surface relative to a reference plane (Bharat and Bharat 2000). Absolute roughness measures the surface roughness of a material over which a fluid may flow. Absolute roughness plays a very important role in the calculation of pressure drop, especially for the turbulent flow regime. Dividing the absolute roughness by the pipe inner diameter results in the unitless relative roughness, which is usually used for the calculation friction factor. When abrasive materials, such as sands or high velocity fluids, are transported through the pipeline, they can cause damaging effects to the pipeline wall. Abrasive wear occurs when small hard particles are pressed against or sliding along the pipeline surface. This increases the roughness of the pipe wall. 21

44 The roughness of pipeline surface can affect the wetting contact angle, described by the Wenzels equation (Wolansky and Marmur 1999): RF (γ sw γ sl ) = γ lw cosθ w (2.6) Here, RF represent the roughness factor. Figure 2.1: A corrosion-pitted surface of gas pipeline (Abduh 2008). Pipeline with smoother surface will have smaller contact angle (Lawrence and Li 2000), which means it is easier for water to wet smoother pipeline. However, iron carbonate film turns the pipe surface more hydrophilic and makes water droplets wet the surface easier(tang et al. 2008). The roughness of pipe(carbon steel) in the petroleum industry is generally inch (25.4 µm). Besides, the wettability of pipeline wall, the superficial fluid velocities also control the phase wetting on the pipeline surface. Li et al. (2008) conducted extensive experimental studies on water wetting at different superficial velocities of oil and water. The results show flow pattern plays an important role in the phase wetting of the pipe surface, which is also 22

45 referred as phase inversion (Figure 2.2). Although no experimental data are available to demonstrate that phase inversion also occurs in a gas-dominated system, this experiment can provide more insight into the understanding of the water-wetting process on the surface of a pipeline wall in a gas dominated system. Figure 2.2: Phase wetting map at different superficial velocities of oil and water in a horizontal two-phase pipe (Li et al. 2008). 2.2 Hydrate formation prediction models For a gas dominated system, the most important phase is the gas phase. Water vapor, water liquid and hydrocarbon liquid could also be present. These phases can coexist in equilibrium in the pipeline. We need to have a quantitative understanding of the relationship among the pipeline temperature, pipeline pressure and the composition of gases to characterize the phase behavior of the fluid mixture. The following correlations are available to estimate the gas hydrate equilibrium locus (si and sii): (1) the K-value method (Ahmed 23

46 2013; Katz 1983; Kobayashi et al. 1987); (2) the gas gravity method (Katz 1959; McCain 1990); and some Computer Methods based on phase equilibrium. The K-value method: also know as Vapor/Solid Equilibrium Ratios Method. For this method, equilibrium ratios for the multi-components (gas, water and hydrate) are used for calculation of the hydrate frost point, which is similar to dew-point calculation. K vs i of a given component can be mathematically described by Eqn. 2.7: Also, n i=1 K vs i = y i s i (2.7) y i K vs i = 1.0 (2.8) One of the assumptions for this method is that there is sufficient water for hydrate formation. Thus, the water phase is excluded in the calculation. We need to use the K value charts for various gas to determine the hydrate formation conditions (GPSA 2004). The algorithm is shown in Figure 2.3. The application ranges for K factor method are: 0.7 < P < 7 MPa for 0 < T < 20 C; or 100 < P < 1000 psia for 32 < T < 68 F. It can also be very accurate for pure CH 4, C 2 H 6, H 2 S, CO 2. The gas gravity method: This method involves the use of a simple gas specific gravity chart. The composition of gas mixtures is not required (Katz 1959). Given the temperature, pressure and the specific gravity, we can locate the point on the chart. The condition falls into the hydrate-region if the point is located above and to the left of the gravity curves. Otherwise, it is located in the non-hydrate-region. The chart should be used with caution, as experimental data sometimes does not agree with the correlation on the chart. Makogon (1997) provided a simple equation for the specific gravity method. logp = β (T +kt 2 ) 1 (2.9) Here, T is in C and the unit of P is MPa. β and k can be expressed as follows: β = γ γ 2 (2.10) 24

47 k = γ γ 2 (2.11) Here, γ represents the gas specific gravity. Kobayashi and Katz (1949) proposed a relatively complicated equation to predict the hydrate formation temperature based on the gas gravity and temperature. 1 T = lnp lnγ lnp ln(p)ln(γ) ln(γ) ln(p) ln(p) 2 ln(γ) ln(p)ln(γ) ln(γ) ln(p) ln(p) 3 ln(γ) ln(p) 2 ln(γ) ln(p)ln(γ) ln(γ) 4 (2.12) Here, T is in R, P is in psia. Another simple equation was later developed to estimate the hydrate formation temperatures T (Towler and Mokhatab 2005): T = 13.47ln(P) ln(γ) 1.675ln(P)ln(γ) (2.13) Despite of the simplicity of these correlations, they can be very accurate for sweet gases with small amounts of propane. Baillie and Wichert (1987) proposed another chart method based on specific gravity to account for the presence of propane and H 2 S in gas mixtures. This method is popular for the calculation of hydrate formation conditions in sour gas. We can perform rapid hand calculations to predict the hydrate formation conditions using the above correlations. With the emergence and advance of computers, more software packages based on thermodynamic models are available for the hydrate calculation. The modelling hydrate formation using computer methods is based on phase equilibrium, which is beyond the scope of this doctoral thesis. 25

48 Figure 2.3: Algorithm for hydrate formation pressure calculation using K-factor methods (Carroll 2014). 26

49 2.2.1 Heat Conduction Heat transfer follows the direction of decreasing temperature. The rate of heat transfer q per unit area in the x direction can be expressed mathematically by the following equation for 1-D plane with a temperature distribution T(x). q = k dt(x) dx (2.14) This rate equation is the well known Fourier s Law (Zohuri and Fathi 2015). The constant k represents the thermal conductivity. If the pipe has one layer of uniform material, its thermal conductivity will be a constant. Then, the temperature distribution across the pipeline wall is linear and Eqn becomes, q = k v T2 T 1 x 2 x 1 (2.15) For 1-D radial steady state heat flow rate of a cylinder pipeline per unit length, the equation becomes, q r = 2πk v T 2 T 1 ln(r 2 /r 1 ) (2.16) For 1-D radial transient heat conduction without any disturbed heat source or sink, the heat flux is expressed by Eqn. 2.17: 1 r r ( k v r T ) T = ρ p c p r t (2.17) Here, the temperature change rate T t depends on the pipeline wall density ρ p, the thermal conductivity of pipeline wall k v and the specific heat capacity c p. The product ρc p is commonly accepted as the volumetric heat capacity, which measures the material s ability to store thermal energy. Another important material property in the analysis of heat transfer is the thermal diffusivity α v (m 2 /s), which measures the material s ability to conduct thermal energy compared with to store thermal energy (Touloukian 1970). It is expressed by: α v = k v ρc p (2.18) 27

50 The complete form of heat diffusion equation with a heating source or sink for a 3-D cylindrical system can be written as: ( 1 k v r T ) r r r + 1 r 2 φ A ( ) T k v + ( ) T T k v + q = ρ p c p φ A z z t (2.19) Heat convection Heat transfer by this mode is due to: (1) random molecular diffusion, and (2) macroscopic motion of the fluid (Atreya 2016). Heat convection occurs between a flowing fluid and a bounding surface when the two have temperature difference(bejan 2013). The surface-fluid interaction can result in a velocity boundary layer and a thermal boundary layer. A velocity/thermal boundary layer refers to the fluid region where the fluid velocity varies from zero at the surface to a finite value which is associated with the flow and not affected by the wall surface (Eckert 1956). Heat transfer by convection occurs within the boundary layer. More specifically, heat convection due to random molecular diffusion dominates near the solid wall surface. When the fluid velocity is zero, heat is transferred by random molecular diffusion only. Understanding the boundary layer phenomena play a vital role in the appreciation of heat convection mechanism (Tsou et al. 1967). Heat convection can be further classified into(1) forced convection, which occurs when the fluid flow is induced by external devices, such as a pump or fan; and (2) natural convection, which is caused only by the fluid density difference due to temperature variations (Yuge 1960). The general rate equation for heat convection is expressed by the Newton s law of cooling (Winterton 1999): q = h v (T s T ) (2.20) Here, the convective heat influx q has a linear relationship with the temperature difference between the pipe wall surface T s and the fluid T. h v is terms as convection heat transfer coefficient (W/m 2 K). It depends on the boundary layer conditions, which is affected by the 28

51 fluid flow pattern, surface geometry (Bejan 2013). In the heat transfer analysis, total thermal resistance R tot is used to measure a material s ability to resist a heat flow, which is described mathematically as: R tot = T,1 T,2 q x (2.21) q x is the heat loss, W; T,1 T,2 represents the overall temperature difference across the solid wall surfaces. For subsea pipelines, both the inner and outer pipeline surfaces are exposed to flowing fluids. Thus, it is necessary to discuss the internal and external convection which occur between the pipeline surface and the fluids. A composite wall model (Figure 2.4) is applied to study the heat transfer across the pipeline wall. The heat transfer equation for the series composite wall can be expressed mathematically as: q r = T,1 T,2 (2.22) Rt The heat transfer process in a insulated subsea pipeline without deposition happens as follows: Convection from the internal flowing fluids to the pipeline wall; Heat conduction through pipeline wall and internal coating to the outer surface of the pipeline; Heat convention from the pipe external surface to the surrounding fluid (sea water). However, if hydrate has already deposited on the pipe wall, the heat transfer process would be as seen in Figure 2.4: Convection from the internal flowing fluids to the surface of hydrate deposit. q r,1 = h v,1 (T,1 T s,1 )2πr p (2.23) 29

52 Figure 2.4: Heat Transfer across pipeline wall with hydrate deposition (H: Hydrate deposit layer; C: Coating layer; P: Pipe wall). Heat conduction from the surface of hydrate deposit to internal surface of pipeline wall or internal coating if any; The heat loss across the hydrate layer per unit length can be expressed as: q r,2 = 2πk v,2 T s,1 T 1 ln( rp+h h r p ) (2.24) The heat loss across the coating layer (if any) per unit length can be expressed as: T 1 T 2 q r,3 = 2πk v,3 ( ) (2.25) ln rp+hh +h c r p+h h Heat conduction from internal surface of pipeline wall or internal coating to the outer surface of the pipeline; T 2 T s,2 q r,4 = 2πk v,4 ( ) (2.26) ln rp+hh +h c+h p r p+h h +h c 30

53 Heat convention from the pipe external surface to the surrounding fluid (sea water). q r,5 = h v,5 (T,1 T s,1 ) 2π(r p +h h +h c +h p ) (2.27) The overall heat transfer rate for the composite pipeline wall with hydrate deposition can be described as: q r = T,1 T,2 ( ) ( ) ( ) 1 + ln rp+hh rp + ln rp+hh +hc rp+h h 2πr ph v,1 L 2πk v,2 L 2πk v,3 + ln rp+hh +hc+hp rp+h h +hc 1 L 2πk v,4 + L 2π(r p+h h +h c+h p)h v,5 L (2.28) Or in the form of: q r = T,1 T,2 R tot = UA(T,1 T,2 ) (2.29) Here, U, or U-value, denotes the overall heat transfer coefficient. If it is defined in terms of the inside pipe surface, then A = 2πr p L, and U can be expressed mathematically as: U = 1 + r ( ) p rp +h h ln + r ( ) p rp +h h +h c ln h v,1 k v,2 r p k v,3 r p +h h + r ( ) p rp +h h +h c +h p r p ln + k v,4 r p +h h +h c (r p +h h +h c +h p ) 1 h v,5 (2.30) Boundary and initial conditions To solve the heat equation to obtain the temperature profile, two boundary conditions must be known. Basically, there are three types of boundary conditions: (1) Dirichlet condition, where the surface temperature T s is maintained as constant; (2) Neumann condition (Pletcher et al. 2012; Reddy and Gartling 2010), which corresponds to a situation when the heat flux q at the surface is constant. It can be expressed mathematically by Eqn. 2.31; in the case of adiabatic or insulated surface, it can rewritten as Eqn. 2.32; and (3) convection surface condition, which is most encountered in heat transfer for the subsea pipeline system. It can be quantified by Eqn k T x x=0 = q s (2.31) 31

54 2.2.4 Temperature profile of the cylinder T x x=0= 0 (2.32) k T x x=0 = h[t T(0,t)] (2.33) For steady-state heat transfer in a cylinder pipeline without coating or deposition, Eqn can be rewritten as: 1 r r ( k v r T ) + = 0 (2.34) r k q For steady state heat transfer process, the temperature across the pipeline wall does not change with time and is only a function of location. Conversely, under transient state conditions, the temperature does vary with time. Applying boundary condition at the surface: dt dr r=0 = 0 and T r=od = T s,2 The first boundary condition is based on the fact that the temperature gradient at the center-line of the pipe is zero. Then, we can obtain the temperature profile equation as: T(r) = q T s,1 T ( s,2 r ) ln ( ) ln +T OD s,2 (2.35) OD ID From the above equation, we can see that the temperature distribution across the cylinder pipeline is logarithmic instead of linear. 2.3 Kinetic model for deposit thickness growth Hydrate deposition is undesirable due to its adverse effect on pressure loss and even pipeline plugging. Due to hydrate deposition onto the pipeline wall, the diameter of the flowing path decreases. The deposit thickness can be expressed as: δ h = r p r eff (2.36) 32

55 Here, r eff represents the effective diameter of the gas flow path, which is a function of time and the hydrate deposition rate Hydrate deposition from water film condensed on the pipeline wall Turner et al. (2005) investigated the formation/dissociation rate of gas hydrate, and proposed a model based on the the assumption that induction time of gas hydrate is very small and hydrate formation occurs instantaneously after sufficient sub-cooling. The rate of gas consumption g s cm 3 can be expressed as: r gas = dm gas dt where, dm gas dt = A s k 1 e k 2 Tses T sub (2.37) In the above equation, T sub is defined as the sucbooling, which represents the driving force facilitatinghydrateformation/dissociation. Thesub-coolingcanbearound6.5 F(Matthews et al. 2000). T sub = T eq T sys The hydrate formation/dissociation rate has a linear relationship with the temperature driving force. In Eqn. 2.37, k 1 and k 2 are rate constants, which are regressed from experimental data. The values given by Vysniauskas and Bishnoi (1983) and Englezos et al. (1987) are as follows: ln(k 1 ) = k 2 = 13,600 33

56 The units for k 1 are kg m 2 s K and k 2 is in K. The hydrate deposition rate r hd (g/s) can be expressed as: r hd = 10 6M hp sc RT sc r gas = 10 6M hp sc RT sc A fg φ w k 1 T sub e k 2/T ses (2.38) r hd is in g/s; p sc is in Kpa; T is in K; M h is in g/mol; R in J/(mol K) or KJ/(gmol K). Here, we only consider the hydrate formed by water film on the pipeline wall surface. A f g (cm 2 ) represents the interface area between water film on the pipeline wall and gas. φ w is the fraction of exposed surface area occupied by condensed water. The (1 φ w ) fraction represents the surface area occupied by hydrate particles deposited onto the wall from the bulk carrier gas phase. Therefore, the interface area between water film and carrier gas can be expressed mathematically by: A fg = 2πφ w r f L (2.39) Here, r f is in cm. The relationship between the mass of hydrate deposit and the shrinkage of flow path is: t 0 r hd dt+m hp = φπ ( r 2 p r 2 f) ρh L (2.40) The left term is calculated based on the hydrate deposition rate. The term m hp represents the mass of hydrate particles deposited from the carrier gas phase. To simplify our model, we first neglect this part. The right term represents hydrate mass based on layer growth. L is the length of a certain section of pipeline. This section of pipeline has uniform pressure, temperature and the same deposit thickness. Substituting r hd into Eqn. 2.40, we can obtain, t πr f L M hp sc φ w k 1 T sub e k 2/T ses dt = φπ ( ) rp 2 rf 2 ρh L (2.41) RT sc As the flow is under steady state flow, we can assume the temperature is constant. Note, every symbol in the below term does not change with time, except r fl πr fl L M hp sc RT sc φ w k 1 T sub e k 2/T ses 34

57 Thus we can move the constants outside of the integral sign, we obtain, Set and M hp sc RT sc ρ h φ w φ k 1 T sub e k 2/T ses t 0 r f Ldt = ( ) rp 2 rf 2 a = M hp sc RT sc ρ h φ w φ k 1 T sub e k 2/T ses (2.42) b = r 2 p Then Eqn becomes: a t 0 r f dt = b r 2 f (2.43) Make the substitution by writing y = t 0 r fdt, then r f = dy dt = y, Eqn becomes: Rearrange the above equation, we get: Thus, Rewrite the equation, we get the new form: By substituting dy = 1 d(b ay), then a ay = b (y ) 2 (2.44) y = b ay (2.45) dy dt = b ay (2.46) dy b ay = dt (2.47) 1 a (b ay) 1 2 d(b ay) = dt (2.48) Here, we can integrate the left side from y 0 to y and the right side from time 0 to t, we get 1 a y y 0 (b ay) 1/2 d(b ay) = t 0 dt (2.49) 35

58 So, we have Then 2 a 2 a (b ay)1 2 y y0 = t (2.50) [ ] (b ay) 1 2 (b ay0 ) 1 2 = t (2.51) As y = t r 0 fdt, when t = 0, y 0 = 0. Then Eqn becomes 2 [ ] (b ay) b 2 = t (2.52) a Thus, Then, Replace a and b by and y = b a 1 a [ at ] 2 2 +b1 2 (2.53) r f = dy dt = at 2 +b1 2 (2.54) a = M hp sc RT sc ρ h φ w φ k 1 T sub e k 2/T ses b = r 2 p then r f = r p 10 6 M hp sc RT sc ρ h φ w φ k 1 T sub e k 2/T ses t (2.55) Finally, the deposit thickness δh (mm) can be expressed as δh = 10 3 M hp sc RT sc ρ h φ w φ k 1 T sub e k 2/T ses t (2.56) Vysniauskas and Bishnoi(1983) had developed another gas consumption rate equation, which is expressed by: ( r gas = α A A s exp E ) ( a exp a ) P γ (2.57) RT T b 36

59 Applying the same methodology, we can obtain another deposit thickness growth equation: ( δh = 10 3 φ w M h p sc α A exp E ) ( a exp a ) P γ t (2.58) φ RT sc ρ h RT T b In Eqn. 2.58, δh is in mm; p sc is in Kpa; T is in K; M h is in g/mol; R in J/(mol K) or KJ/(gmol K); t is in min. All the other constants are shown in Table 2.1. Table 2.1: Calculated Values of Parameters in the Reaction Rate Expression (Eqn. 2.57) α A E a γ a b (cm 3 /cm 2 min bar γ ) (kj/gmol) ( K b ) Mean Value % Confidence Limits ± ±1.584 ±0.467 ± ±1.189 The above equation development is also based on the following assumption: Pressure, temperature is constant in the pipeline; Water condensed on the pipeline wall is all converted into hydrate Model verification Rao et al, (2013) show that the hydrate deposition consists of four stages (Figure 2.5): 1. Water film condensation: Water condensed on the steel pipe surface; 2. Initial growth: water condensation on the newly formed hydrate layer, some of the condensed water is soaked into the porous hydrate layer and the left liquid water is converted into hydrate; 3. Growth Period: a cyclic process of condensation and hydrate formation will results in hydrate growth layer by layer in a radial direction. The hydrate layer becomes like an homogeneous porous material made of solid hydrate matrix and pores filled with gas; 4. Annealing Period: once the outmost hydrate layer reaches thermodynamic equilibrium, hydrate deposit on the wall begin to anneal as a result of mass transfer of gas and heat loss into the ambient environment. 37

60 The process is very similar to that of frost formation (Le Gall et al. 1997). The deposition of hydrate can affect heat transfer to some extent due to the insulating effect of hydrate deposit. Thus, the deposition rate decreases as the hydrate deposit grows thicker and thicker. The growth rate and properties of hydrate deposit are determined by the gas consumption rate, which were further controlled by wall surface temperature, water and gas composition, and gas flow velocity. Figure 2.5: Hydrate deposition thickness growth on a cold pipe with a diameter 1/8 in. at different time (Rao et al. 2013b). 38

61 Rao et al. (2013b) conducted experimental studies on gas hydrate deposition on a cold surface in a water-saturated gas system. The experiment was conducted in a high pressure hydrate deposition apparatus, where the testing section consisted of a Jerguson cell. (as seen in Figure 2.6). The cross-section of the Jerguson cell is 1 inch 1 inch. The axial length is 9.5 inch. Before the Jerguson cell, a gas saturator filled with water was placed to saturate the gas. A flash drum is used to separate any liquid water droplets before entering the Jerguson cell. Gas was recirculated in the system by two series ISCO pumps. The pumps are operated at the flow rate of 190 ml/min, which resulted in laminar flow with Reynolds number about The entire system was kept inside a water bath. An 1/8 in. stainless steel pipe with low temperature was placed at the center of the Jerguson cell. Water starts to condense first on the surface of the steel pipe where the coldest spot is located in the cell. To initiate hydrate formation, the steel pipe temperature was temporarily dropped below ice freezing point (about 14 to 23 F). Once solid deposition were observed on the surface, the temperature was then increased above ice freezing point to ensure that no ice were present in the deposit. The deposit of hydrate grows outwardly. A video camera was placed to record the thickness of hydrate deposit growth, and the pressure and temperature of the entire system were controlled with Labview. The previous hydrate deposit growth model (Eqn. 2.56) was used to simulate this experiment. In Eqn. 2.56, φ and φ w are assumed to have the same value. The system pressure and temperature from the experiment data are 7.59 MPa and K, respectively. The hydrate deposit growth model result matches the experiment data to a satisfactory degree (Figure 2.7) and can be used for application Hydrate deposition from water droplet in the carrier gas phase The process of how a hydrate particle finally deposit on the surface of substrate is not clearly understood. Two conceptual models are proposed. 39

62 Figure 2.6: Schematic of hydrate deposition apparatus (Rao et al. 2013b). 1. Ballistic process of hydrate deposition. This conceptual deposition process assumes the hydrate particles maintain the spherical shape to simplify the dynamics of hydrate deposition. The principle of ballistic deposition is quite simple. It involves placing the particles on a layer of deposited particles or the steel substrate. The newly formed hydrate particles first touch the substrate or a deposited layer, then undergo a relaxation process along the steepest descent until they meet certain stability criterion to reach mechanical equilibrium. If the criterion is not met, the hydrate particles will continue rolling and the procedure is iterated until the equilibrium state is achieved. 2. Based on the general particle transport theory proposed by Yang et al.(1998), we can divide the deposition process of water droplets from the carrier gas conceptually into four stages: (a) When the distance from the solid boundary surface is far, the hydrate particle movement is controlled by the fluid convection, external body forces, such as drag force, gravity and particle collisions. (b) As the hydrate particles approach the wall surface to distances comparable to the particle size, additional forces start to act on the hydrate particles due to 40

63 Figure 2.7: Model of hydrate deposit thickness growth compared with experimental data (Rao et al. 2013a). the presence of the wall surface, which are known as particle-wall hydrodynamic interactions. These forces can significantly reduce the particle mobility. (c) When the distance becomes closer(1-100nm), the van der Waals and the electrical double-layer forces due to the interaction between the surface potentials of solid wall and hydrate particle start to affect the particles motion. These two colloidal forces determine the particle deposition behaviour in the near-wall region. The magnitude of these two colloidal forces depends on the particle size, Hamaker constant, surface potentials of the two interacting surfaces and ionic strength of the suspension medium (Elimelech et al. 2013). (d) As the distance becomes within 1nm, traditional continuum mechanisms may not be applied to investigate the particle behavior within such molecular dimension. 41

64 A CFD simulation conducted by Jassim et al. (2010) shows hydrate particles formed in the carrier gas phase can be transported downstream without adhering to the wall surface. Simulation results show hydrate particles with smaller diameter can travel longer distance (e.g. 6000m for particles with diameter 0.1 µm) and vice versa (Figure 2.8). The migration distance for hydrate particles with average diameter 40 µm is about 2 m before deposition. The flowing hydrate particle deposition and transport depend strongly on their particle size, chemical composition, and the nature of the carrier gas. The interaction of these mentioned parameters control, to a great extent, the nucleation, the particle size distribution and the deposition on the pipeline wall. Figure 2.8: Hydrate deposition distance as a function of particle size (Jassim et al. 2010). 42

65 First, we need to estimate the amount of gas in the pipeline by applying the real gas equation: At standard condition, we obtain: Combination of Equ and 2.60 will yields: P ses V ses = znrt ses (2.59) P sc V sc = nrt sc (2.60) V sc = T sc P sc P ses V ses zt ses (2.61) From the daily production data, the GLR can be estimated. The the volume of water droplet inside the pipeline can be expressed as: Here, the pipeline volume can be obtained by: V w = V sc GLR = T sc P sc GLR P sesv ses zt ses (2.62) V ses = 2πr p L The mean radius of water droplet particles is r w. As the hydrate particle is converted from water droplets, we assume the mean size of hydrate particle is the same as that of water droplet. Hinze (1955) showed that the size of liquid droplet entrained by the carrier gas is controlled by forces that act to shatter the droplet and surface tension force that tries to hold the droplet together. Two groups of numbers, a Weber group number N we and a viscosity group number N vi, are developed to analyze the splitting of water droplet. N we = µ csd σ (2.63) Where, S, sec 1, denotes the maximum velocity gradient in external fluid flow field; σ, dyne/cm, is the interfacial tension; µ c, phase; and D, cm, represents the diameter of droplet. g, is the absolute viscosity of the continuous cm sec 43

66 The viscosity group number can be expressed as N vi = µ d ρd σd (2.64) g Where, µ d,, denotes the absolute viscosity of the dispersed oil/water phase; and ρ cm sec d, g/cm 3, is the density of the dispersed phases. The viscosity group considers the gas viscosity effect. Droplet breakup happens when N we reaches a critical value (N we ) crit. The value of (N we ) crit depends on the fluid flow pattern and how the droplets deform. The mean diameter of water droplet can be determined by the following correlation (Pan and Hanratty 2002): d 32 = ( σr ) eff ρ g vg 2 The number of water droplet n w inside a section of pipeline is: n w = V w 4 3 πr3 w The surface area between water droplet and gas A p g can be calculated by (2.65) (2.66) Substitute Equ. (2.62) into the above equation, we can get: A p g = n w 4πr 2 w = 3V w r w (2.67) A p g = 3T sc P sc GLR P sesv ses zt ses r w (2.68) From Turner s gas consumption rate equation (Turner et al. 2005), we can obtain: r hd = r gas M h M g = M h M g A p g φ w k 1 T sub exp k2 Tses (2.69) Substitution of Eqn.(2.68) into Eqn.(2.69) will yield: r hd = 3T sc P sc GLR M h M g P ses V ses zt ses r w φ w k 1 T sub exp k2 Tses (2.70) 44

67 Assuming that all converted hydrate particles deposit on the pipe wall, the mass of hydrate deposition can be calculated by: i.e.: m hp = t 0 r hd dt = t 0 3T sc P sc GLR M h M g P ses V ses zt ses r w φ w k 1 T sub exp k2 Tses dt (2.71) m hp = 3T sc P sc GLR M h M g P ses V ses zt ses r w φ w k 1 T sub exp k2 Tses t (2.72) Substituting into Eqn and applying the same integration technique, we can obtain: δh = r 2 p T sc P sc GLR M h M g P ses V ses zt ses r w φ w k 1 T sub exp k2 Tses t r p 10 3 M hp sc RT sc ρ h φ w φ k 1 T sub e k 2/T ses t (2.73) Here, δh is in mm. Eqn is the hydrate deposit thickness growth model considering both water condensation and hydrate particle adhesion to the wall surface. Due to experimental data being unavailable, this model has not yet been validated. 2.4 Summary This chapter discusses the mechanism of hydrate formation and deposition. Hydrate formation is associated with nucleation and crystal growth. The wettability and roughness of pipeline wall surface determine the adsorption of water droplet on the wall, and thus have significant influence over the hydrate deposition process. Two hydrate formation prediction models are discussed in this chapter: (1) The K-value method; (2) The gas gravity Method. This chapter also discussed the heat transfer principles that occurs across the pipeline. The heat loss of fluids into the environment is mainly due to convection and conduction. The heat loss due to radiation is negligible. The most important contribution of this chapter is the development of the deposit thickness growth model. The water condensation onto the pipeline wall and the direct deposition of hydrate particles on the wall surface both contribute to the hydrate deposit. Eqn and 2.72 are developed for these two different processes. 45

68 The thickness growth equation (2.56) has been verified using data from the literature. The simulation results match the experiment data within satisfactory range. 46

69 CHAPTER 3 STUDY OF HYDRATE DEPOSIT PROPERTIES AND SLOUGHING Interruption of normal production schedule can cause monetary losses (i.e., lower net present value, NPV). It is believed that sloughing of hydrate deposits contributes to plugging of subsea pipelines. As hydrate deposit on the pipeline wall grows, the inner diameter of pipeline can be reduced, resulting in a local restriction. The increased shear stress exerted by thegasonthehydratedepositcanresultinthefailureofthedepositstructure, andsloughing occurs. The debris of collapsed hydrate deposit can clog the restriction. Once jammed, the situation can get worse quickly if not properly given prompt attention. Hydrate jamming is the sudden arrest of the suspended hydrate debris or large particles dynamics in the pipeline. The jammed hydrate particles or debris are trapped locally, preventing them from flowing downstream and making them obstacles for fluid upstream. Lack of experimental and field data led to a need for a conceptual model for simulating deposit sloughing. 3.1 Hydrate deposit properties Collapsed hydrate deposit debris can travel downstream and accumulate, jamming the pipeline. The mechanical properties of gas hydrate deposits, such as tensile strength, Poissons ratio, dynamic compressional-wave (M), bulk (K) and shear (G) moduli, should be known to assess the sloughing of hydrate deposits. However, these parameters of porous hydrate deposits are difficult to measure experimentally at typical high pressure and low temperature conditions. Therefore, we developed theoretical models to predict these parameters. 47

70 3.1.1 Elastic wave velocities of hydrate deposits To study the sloughing problem, we need to investigate how the hydrate deposits will respondtotheappliedforce. Theforceactingonthefinitebodyofdepositcanbetransmitted through the whole body due to the interactions of the particles making up the deposit. Internal forces arises to resist the externally applied forces (Langmuir 1916), and the solid deposit will deform under the influence of the force. Here, we consider the hydrate deposit as nonlinear elastic material, whose relationship between stresses and strains is nonlinear. When the force is removed, the deposit will return to its initial shape. If the force exceeds its failure criterion, the deposit sloughing will occur. Two shear strength parameters of interest are cohesion and angle of the internal friction. The Poisson s ratio and Young s Modulus of the sample can be determined given the density and the elastic wave velocities of hydrate deposits. Wyllie et al.(1956) conducted an experimental investigation to measure the elastic wave velocities (V M ) of porous media. The time-average formula can be described by: The elastic properties of hydrate deposit are defined as: The Young s Modulus: 1 V M = φ V fl + 1 φ V R (3.1) Poisson s ratio: E = ρ bulk V 2 p (1 2v)(1+v) ) (3.2) (1 v) v = V p 2 2Vs 2 2(V 2 p Vs 2 ) (3.3) We can use the above equations to estimate the elastic properties of hydrate deposits, which are the required input information for finite element analysis. The compressional velocities found in the literature are: V p = m/s for sii hydrate (Helgerud et al. 2009). The wave velocity for water is ft/s (Wyllie et al. 1956). Based on the above Eqn. 3.2 and Eqn. 3.3, we can obtain the compressional velocity of hydrate deposit with a porosity of 43% is 48

71 m/s. Eqn. 3.1 is not applied to obtain the shear velocity of hydrate deposit due to shear velocity of fluids being unavailable. Instead, we adopted the following equation to obtain the shear velocity: V s = V p φ (3.4) Eqn. 3.4 can be applied to ideal models for regular packing of spheres (Castagna et al. 1985). Given a porosity (43%), the calculated Poisson s ratio and Young s Modulus based on Eqn. 3.2 and Eqn. 3.3 is Gpa and Packing patterns of hydrate particles It is assumed that the mechanical properties of hydrate deposits on pipe walls are heavily controlled by the inter-particle contacts. A hydrate particle packing theory is used to predict the mechanical properties of porous hydrate deposits. For this packing theory, we assume the hydrate particles are spheres of uniform size. The packing configuration of aggregated hydrate particles plays a very important role in the study of the physical properties of hydrate deposit. Four possible arrangements for stable packing of uniform spherical particles are proposed: (1) simple cubic packing (as shown in Figure 3.1); (2) rhombohedral packing (as shown in Figure 3.2); (3) tetrahedral packing and (4) orthorhombic packing (as shown in Figure 3.3). Each packing pattern has its own unique mechanical properties in terms of porosity and tensile strength. The properties of hydrate deposit will be analyzed for all these four different packing patterns. In two dimensions, the configuration can be either packed into a square face or a rhombic face. For square faces, the centers of four neighboring spherical particles form a square shape. Each particle is in contact with four adjacent particles. However, for rhombic faces, the centers of four neighboring spherical particles form a rhombus shape. Each particle is in contact with six adjacent particles. In three dimensions, placing a second square array layer directly over a first square array layer can form a simple cubic packing. Simple cubic packing patterns have square faces in every dimension (see Figure 3.1). Orthorhombic packing 49

72 Figure 3.1: Front view of simple cubic packing. patterns have square faces on the horizontal plane and rhombic faces on the vertical plane. Rhombohedral packing patterns have rhombic faces in every dimension. Figure 3.2: Front view of rhombohedral packing. 3.2 Porosity of hydrate deposits The mechanical properties of hydrate deposits are heavily influenced by porosity. Porosity, φ, is a deposit property defined as the percentage of the gross solid hydrate volume occupied by pore fluids (gas or water). The space between hydrate particles is referred to as voids or pores. This is the space where water/gas can reside. 50

73 The porosity can be expressed as: φ = V void V bulk = 1 V hydrate V bulk (3.5) If the hydrate particles are of uniform size, the calculation of deposit porosity can be a simple exercise in solid geometry. Take the simple cubic packing pattern for example. One unit cell contains eight corner spheres. There is only 1 assembled sphere in each unit cell, because only 1/8 th of each sphere is actually located inside the unit cell. The rest 7/8 ths of each corner sphere is located in the other 7 adjacent unit cells. Figure 3.3: Different packing arrangement of spherical hydrate particles. The total volume of solids in a unit cell is: The volume of the unit cell is: V hydrate = π 3 r3 = 4π 3 r3 (3.6) V unitcell = (2r) 3 = 8r 3 (3.7) 51

74 Therefore, the porosity is: φ = V unitcell V hydrate V unitcell = 47.64% (3.8) For orthorhombic packing, For tetrahedral packing, φ = πr3 (2r) 2 ( 3r) = 39.54% (3.9) For rhombohedral packing, φ = 1 φ = πr3 (2r) 2 [2rsin(75.31 )] 4 5 2r 3r 2 3 πr3 ( r = 45.88% (3.10) ) = 11.14% (3.11) Thus, we obtain the porosities for these four different packing patterns (See Table 3.1). The calculation shows that cubic packing results in the most porous deposit, with a porosity of 47.64% in the ideal situation. Rhombohedral packing is the most compact arrangement of uniform spherical particles, with porosity of 11.14%, and will be more mechanically stable than the cubic packing. For a specific packing pattern, the particle diameter does not affect porosity. However, hydrate particles do not have the same size, nor can they be arranged in a cubic structure naturally, so hydrate deposits could have much less porosity than the 47% of the ideal cubic packing. And not all hydrate particles are spherical or round. The particles can exist in many shapes and pack in a variety of ways which could increase or decrease the deposit porosity and mechanical strength. Table 3.1: Porosities and numbers of contact points for different packing patterns Packing Simple Cubic Orthorhombic Tetrahedral Rhombohedral Pattern Packing Packing Packing Packing Porosity 47.64% 39.54% 45.88% 11.14% Numbers of contact points Poisson s Ratio Young s Modulus (MPa)

75 If the pipeline conditions are favorable for hydrate formation, the water inside pores can be converted into hydrate. Hydrate growth between the particles can reduce the deposit porosity. Two different models are proposed to study the porosity decrease due to hydrate annealing: (1) Evenly coating model and (2) Corner growth model. Here, we assume that the initial diameter of the hydrate particle is d, and the spherical particles grow evenly in all dimensions. The newly formed solid that connect two particles is called bridge and its diameter at the narrowest location is d, as shown in Figure 3.4. The diameter of the new hydrate particle is d 2 +d Evenly coating model Take a unit cell, with one spherical particle inscribe inside. The dimensions of the unit cell is d d d. Denote the volume of void space by V, then the original void volume can be calculated by: V = d π ( d 2) 3 = ( 1 π ) d 3 (3.12) 6 As the diameter of hydrate particles increase, the void volume shrinks. The radius of hydrate particles grows evenly on its surface(as shown in Figure 3.4). For each unit cubic, the increase volume of hydrate particle V is: V = π ) (d 3 d 3 6V sc (3.13) 6 V sc is the volume of hydrate truncated by each boundary plane, it is a spherical cap. V sc = π 6 h( 3a 2 +h 2) (3.14) [ ] [ V sc = π d2 +d 2 d {3d 2 d2 +d + 2 d 2 } (3.15) ] Therefore, we can obtain an expression to estimate the reduced porosity φ as a function of the bridge diameter d, φ = V V d 3 = [ ] [ π d 2 +d 2 d d {3d d 2} 2 d ] (3.16) d 3 53

76 Figure 3.4: Conceptual of hydrate bridge growth evenly coating model Corner growth model Due to the interfacial phenomenon, the interface between water and gas is often curved. The trapped water between particles is more likely to have the shape as Figure 3.5. Assuming the curvature of the bridge has a radius of R and then, Thus, ( ) d 2 2 +R + ( ) 2 ( ) 2 d d = 2 2 +R (3.17) Therefore, R = d 2 4(d d ) c = d +R = d + R R+ d 2 a = d 2 = c b c d 2 4(d d ) b = d d 2 + 4(d d ) 2 d 2 4(d d ) d d 2 4(d d ) +dd 4(d d ) (3.18) (3.19) (3.20) (3.21) 54

77 The variables a, b, and c are denoted in Figure 3.5. R = c 2R 2R+d c (3.22) Therefore, x 2 + ( y 1 d 2) 2 = ( ) 2 d (3.23) 2 y 1 = d 2 (d 2) 2 x 2 (3.24) y 2 2 +(x c) 2 = R 2 (3.25) y 2 = Thus the volume of the bridge V bridge : R 2 (x c) 2 (3.26) V bridge = = a 0 a 0 b + b 2πxy 1 dx+ 2πx(y 1 y 2 )dx a [ ] d ( d2 )2 x 2 2πx 2 dx (d 2πx{ d ) 2 2 x 2 2 a R 2 (x c) 2 }dx (3.27) = π 2 da π{[(d 2 )2 a 2 ] 3 2 d 3 + 2π 3 { [ (d 2 ) 2 b 2 ]3 2 8 }+ π 2 d( b 2 a 2) [ (d ]3 ) 2 2 b a 2 To calculate the term: b a 2πx R 2 (x c) 2 dx First, assume x c = Rsinα, 0 < α < π 2. When x = a 2 a 2πx R 2 (x c) 2 dx α = arcsin c a R 55

78 When x = b α = arcsin c a R Thus, the term b a 2πx R 2 (x c) 2 dx becomes = = arcsin c b R arcsin c a R arcsin c b R arcsin c a R arcsin c b R = arcsin c a R sin c b R arcsin c a R = [ 2πR3 3 2π(c Rsinα)Rcosαd(c Rsinα) 2π(Rsinα c)r 2 cos 2 αdα 2πR 3 sinαcos 2 αdα 2πR 3 cos 2 αd(cosα) cos 3 α = [ 2πR3 cos 3 α 3 ] arcsin c b R arcsin c a R ] arcsin c b R arcsin c a R πcr 2 ( arcsin c b R As α 1 = arcsin c a,cosα R 1 = 1 ( c a R arcsin c b R arcsin c a R arcsin c b R arcsin c a R arcsin c b R arcsin c a R ( πcr 2 sin2α 2 ) c a arcsin R ) 2 2πcR 2 cos 2 αdα πcr 2 (cos2α+1)dα πcr 2 cos2αdα ) arcsin c b R arcsin c a R arcsin c b R arcsin c a R πcr 2 dα (3.28) sin2α 1 = 2sinα 1 cosα 1 = 2 c a 1 R ( c a R ) 2 56

79 As α 2 = arcsin c b, cosα R 2 = 1 ( c b R ) 2 sin2α 2 = 2sinα 2 cosα 2 = 2 c b 1 R ( c b R ) 2 Therefore, equation 3.28 becomes, Figure 3.5: Conceptual of hydrate corner growth model. b a 2πx R 2 (x c) 2 dx 3 ( ) 3 2 ( ) 2 = 2πR3 3 { c a 1 c b 1 } R R ( πcr 2 arcsin c b ) c a arcsin R R πcr 2 c b ( ) 2 c b 1 c a ( ) 2 c a 1 R R R R (3.29) 57

80 Where, a = d b = 4dd (d d )+dd 2 2d 2 +4d(d d ) and c = d + d 2 4(d d ) Therefore, [ (d ]3 V bridge = π 2 da2 + 2π ) { a 2 d3 2 8 } [ (d ]3 + π 2 d( b 2 a 2) + 2π ) [ 2 2 (d ]3 ) { b 2 a 2 } ( ) 3 2 ( ) 2 2πR3 3 { c a 1 c b 1 } R R ( +πcr 2 arcsin c b ) c a arcsin R R +πcr 2 c b ( ) 2 c b 1 c a ( ) 2 c a 1 R R R R (3.30) Finally, an expression for the reduced porosity based on the corner growth model is obtained: φ = 1 πd V bridge d 3 (3.31) Tensile strength The tensile strength is the maximum amount of tensile stress that the material can support before failure. Its magnitude is the maximum force the material can withstand per unit area. The failure of a material can be explained by Mohr Coulum failure criterion, as shown in Figure

81 Figure 3.6: Mohr-Coulomb failure criterion (Hoek, Carranza and Corkum 2002). Assume an unit area (1 m 2 ) with uniform sphere hydrate particles. The number of particles is n, and their diameter is d in micrometers. Therefore, n = 1m = (3.32) µm2 πd 2 πd 2 4 The adhesion force f ad (mn/m) between two hydrate particles can be measured in the lab (Aspenes et al. 2010; Dieker et al. 2009; Taylor et al. 2007). Thus, the force f b (N) required to break them apart for simple cubic packing can be calculated: f b = ndf ad = f ad N (3.33) πd We can take f b as the mechanical strength due to adhesion. From the equation, we can see the smaller the particle diameter, the harder it is for the hydrate deposit to collapse from the wall. For rhombohedral packing, f b = n 3df ad = f ad N (3.34) πd The tensile strength of rhombohedral-packing hydrate deposit is larger than that of simple-cubic-packed one. Since the hydrate deposit is porous, it can actually hold small 59

82 amounts of gas or water. We can also use this model to estimate the shear strength of hydrate deposit, because they are equivalent. The fluids in the void spaces have insignificant resistance to shear, so the shear stress is taken entirely by the solid hydrate particles. 3.3 Soughing mechanism study of hydrate deposits Hydrate deposits on the pipe wall are subjected to shear forces due to the drag exerted by the gas stream. The shear stress at the interface between flowing fluids and hydrate wall deposit increases with further hydrate formation (i.e., increased deposit thickness). The decrease in flow path cross sectional area increases the fluid velocity. The sloughing of deposit on the wall can be explained by the Mohr- Coulomb failure criterion (Hoek et al. 2002). τ f = c+σtanφ (3.35) Here, τ f is the shear stress at failure, or the critical shear stress; c is the inherent cohesive strength, also known as cohesion; and φ is the angle of internal friction, tanφ represents the coefficient of internal friction. When sloughing happens, the shear stress along the failure plane reaches the shear strength of the hydrate deposit. One part of the hydrate deposit will slide relative to the other side of the failure plane, resulting in the collapse of the deposit structure. Generally, hydrate deposits will fail due to shear, instead of falling downward due to gravity. During hydrate deposit sloughing, the collapsed hydrate debris can travel downstream and jam the line. Hydrate particle jamming is the sudden arrest of the dynamics of suspended hydrate debris or large particles in the pipeline. The jammed hydrate particles or debris are trapped locally, preventing them from flowing downstream and making them an obstacle for fluids upstream Sloughing study using finite element method Stress analysis is used for the determination of the internal stress distribution in hydrate deposit. It is an essential tool for the study of deposit sloughing under prescribed loads. 60

83 Newton s laws of motion states that the external forces applied will be balanced by internal reaction forces. The reaction stress will propagate inside the deposit and create a continuous stress distribution throughout the body. The finite element method (FEM) can be used for structural analysis to compute the hydrate deposit structure deformations, internal stresses and stability (Dhatt et al. 2012; Zienkiewicz et al. 1977). By solving a system of algebraic equations (Eqn ), we can obtain the approximate values of the internal stress and deformations at discrete number of points over the domain. First, we need to divide the hydrate deposit into smaller, simpler parts, which are called finite elements. Then, assemble all these above-mentioned simple equations into the global system of equation. The mathematical solution of the analysis are used to verify the hydrate deposit s sloughing risk under various operating conditions. The solution usually involves the following process step by step (Huebner et al. 2008): Discretize the problem domain into smaller elements. Based on the hydrate deposit geometry, we need to divide the continuous medium into discrete elements for numerical calculation. However, spatial discretization can introduce errors (Lee and Cangellaris 1992), which is often referred as discretization errors. To avoid these errors, a simple 2-D plane geometry for the hydrate deposit is adopted. Assign nodes to each element and then select the interpolation function. To identify the nodes and element, a numbering system is needed, as shown in Figure 3.7. The elements are numbered 1, 2,, N e, depending on the geometry of the deposit. The nodes of an element are also numbered by 1, 2, 3 and 4 in the local node numbering system. All the local numbering should be related to the global numbering. Find the element properties. We need input information of the deposit, such as the poisson s ratio, Youngs Modulus and undrained cohesion. Assemble the individual element properties to find the system equations. Define the boundary conditions. 61

84 Figure 3.7: Global node and element numbering for mesh of 8 node rectangle (partial). Solve the system equations. We need to solve the equilibrium Eqn. 3.36, the strain displacement Eqn and the constitutive Eqn Stresses are related to deformation of the material which is governed by the constitutive equations. To simplify our model, the problem is modeled in two dimensions, with plane stress and plane strain. The stress distribution analysis of hydrate deposit on the pipeline wall falls into the category of plane-strain problems. We can study only unit-width slice of the hydrate deposit in the x-y plane to determine stresses and displacement. On the upstream side of the deposit, evenly distributed force is applied. The deposit will deform under the force applied. We can obtain the following equilibrium equations: σ x x + τ xy y +f x = 0 τ xy x + σ y y +f y = 0 (3.36) Where, σ x, σ y are normal stress, and τ xy are shear stress. f x and f y represents the body forces. 62

85 Or, in the matrix form, we can write as: [ ] 0 x ] x y σ σ 0 y fx = [ f y x τ y xy We can also write as: [A] T σ = f (3.37) We assume the hydrate deposit follows the linear elastic materials behavior. We also assume that the displacements of the hydrate deposit is much smaller than any dimension of the body. Thus, the deposit geometry and the constitutive properties at each element remains unchanged by the deformation. The 2-D elastic stress-strain relationships by the generalized Hooke s law can be expressed as: ǫ x = 1 E (σ x vσ y ) ǫ y = 1 E (σ y vσ x ) γ xy = 2(1+v) τ xy E ǫ z = v E (σ x +σ y ) (3.38) Where, E is the Young s Modulus and v is the Poisson s ratio. These values can be obtained from the Eqn. 3.1, 3.2 and 3.3. The porosity used here in Eqn. 3.1 is by assuming simple cubic packing pattern. We can write the strain-stress relationship in matrix form: σ x 1 v v 0 σ y E = v 1 v 0 (1+v)(1 2v) 1 τ xy 0 0 v 2 We can also write that as: ǫ x ǫ y γ xy σ = [D]ǫ (3.39) 63

86 Assuming small strain, then ǫ x = u x ǫ y = v y γ xy = u y + v x (3.40) Where, ǫ x, and ǫ y are normal strains and γ xy is the shear strains. We can write in the matrix form, ǫ = [A]e (3.41) where, e = [ ] u v Here, u, v represents the displacement at the x and y direction, respectively. Combining Eqn , the Eqn becomes: [A] T [D][A]e = f (3.42) This is displacement finite element formulation. Expanding this equation, we can obtain: [ E(1 v) 2 u (1+v)(1 2v) x + 1 2v 2 u 2 2(1 v) y + v 2 v 2 1 v x y + 1 2v 2(1 v) [ E(1 v) 2 v (1+v)(1 2v) x + 1 2v 2 2(1 v) 2 v y + v 2 1 v 2 u x y + 1 2v 2(1 v) Considering the following trial solution across the element: ] 2 v +f x = 0 (3.43) x y ] 2 u +f y = 0 (3.44) x y ũ = N 1 u 1 +N 2 u 2 +N 3 u 3 +N 4 u 4 +N 5 u 5 +N 6 u 6 +N 7 u 7 +N 8 u 8 ṽ = N 1 v 1 +N 2 v 2 +N 3 v 3 +N 4 v 4 +N 5 v 5 +N 6 v 6 +N 7 v 7 +N 8 v 8 (3.45) The undetermined parameters N i represent the shape functions. To simplify the problem, we use two-dimensional elements with rectangle shape(as shown in Figure 3.8).Using the Lagrangian interpolation concepts, we can develop the interpolation 64

87 functions for rectangle elements. Figure 3.8: One element with 8-node rectangle. For 8-node rectangle, the shape equation would be: N i = c 1 +c 2 x+c 3 y +c 4 x 2 +c 5 xy +c 6 y 2 +c 7 x 2 y +c 8 xy 2 (3.46) From element boundary condition, we can set up 8 equations. Thus, we can obtain: N 1 = 1 (1 x)(1 y)( x y 1) 4 N 2 = 1 2 (1 x)(1 y2 ) N 3 = 1 (1 x)(1+y)( x+y 1) 4 N 4 = 1 2 (1 x2 )(1+y) N 5 = 1 (1+x)(1+y)(+x+y 1) 4 (3.47) N 6 = 1 2 (1+x)(1 y2 ) N 7 = 1 (1+x)(1 y)( x y 1) 4 N 8 = 1 2 (1 x2 )(1 y) Thus, we derived the shape equations for 8-node rectangle. 65

88 ǫ x ǫ y = γ xy N 1 N 0 2 N 0 3 x x N 0 1 N y y N 1 N 1 N 2 N 2 N 3 y x y x y x 0... N 3 y... 0 x... N 3 N 8 0 x N 8 y N 8 N 8 y x To approximate the geometric domain of hydrate deposit, mesh generation is usually required for finite element analysis (Hole 1988). The method of mesh generation can be classified into two major modes in industry: mapped meshing and free meshing. Free mapping, also known as automatic meshing, allows the individual element to obtain any arbitrary shape. Mapped meshing require the subdivision of the problem into standard shapes, such as triangle, quadrilaterals or pentahedra. In this work, we adopt the mapped meshing technique by subdividing the hydrate deposit domain into rectangle elements (Figure 3.9). This technique can also provide great control over mesh density. For this problem, the length of the deposit is 1 m and the width is 0.1 m. The dimension of the element is 0.02 m 0.02 m. Thus, the total number of the element is 5 50 = 250. The connectivity information of each element need to be prepared as input information. Every individual element is attached to 8 nodes clock-wisely. The coordinates of each node also need to be provided as input information for finite element analysis. u 1 v 1 u 2 v 2... u 8 v 8 Figure 3.9: Meshing of the element of the hydrate deposit. The hydrate deposit has 5 layers with different properties. The layers close to the pipeline wall are exposed to low temperature for longer time, and thus, water residing inside the pores of deposit can be converted into ice or hydrate, reducing the porosity and increasing 66

89 the tensile strength. The freedom of the nodes located on the wall surface is zero. These can not move in the vertical and horizontal direction. Since each node has two degree of fixed freedom and there are 101 fixed nodes, the number of fixed freedom is 202. All the nodes attached to the wall surface have zero freedom. All the other nodes can move in either vertical or horizontal direction. Forces are applied to the nodes located on the left side and down side of the hydrate deposit. The number of loaded nodes is 111. Using the software package for finite element programming (Griffiths and Lane 1999; Smith et al. 2013), we can obtain the following results (Figure Figure 3.13). The displacement of each node are shown in Figure All the elements located on the upstream side of the deposit are subjected to more severe deformation than the rest. The interior elements, which are not directly exposed to the horizontal impact from the upstream gas, do not experience significant displacement from their original locations. The normal stress (σ x ) distribution inside the hydrate deposit (Figure 3.11) indicates the elements are all under compression in the horizontal direction. The normal stress (σ y ) distribution (Figure 3.12) indicates the elements are all under tension in the vertical direction. Figure 3.10: Displacement of the element of the hydrate deposit. Figure 3.11: Normal stress distribution (σ x ) inside the hydrate deposit. 67

90 Figure 3.12: Normal stress distribution (σ y ) inside the hydrate deposit. The shear stress distribution (Figure 3.13) represents the magnitude and direction of the shear force acting across all the elements of the hydrate deposit. The shear force is the resultant of upstream pressure and friction applied forces on the deposit. The simulation results also show the magnitude of these stresses follows the orders: σ x < τ < σ y. When the stress exceeds the tensile strength of hydrate deposit, sloughing will happen. Unfortunately, as the the angle of internal friction is not available from published data, no quantitative analysis about failure criteria is conducted in this work. Ambient pressure and contact forces with gas create internal stresses that are concentrated on deposit surfaces which are exposed to the collision impact from upstream gas. The exterior elements on the left side of deposit are subjected to more severe deformation. Sloughing of the deposit will occur at this region first. Interior elements inside the hydrate deposit do not experience significant deformation. Figure 3.13: Shear stress distribution (τ xy ) inside the hydrate deposit. Another simulation was performed by varying the length of each layer of the hydrate deposit (Figure 3.14 and Figure 3.15). The layers have a decreasing length from the wall surface: 1 m, 0.8 m, 0.6 m, 0.4 m and 0.2 m, respectively. The number of loaded nodes is 111 and the number of fixed freedoms is 202. Figure 3.15 shows the deformation of these 68

91 elements. The results show all the exterior elements will experience more sever deformation. Sloughing will occur here first and prevent hydrate deposit growth on the wall. Figure 3.14: Meshing of the element of the hydrate deposit. Figure 3.15: Displacement of the element of the hydrate deposit. 3.4 Consequence of collapsed hydrate deposit: jamming The debris of collapsed hydrate deposit can clog the restriction. Once jammed, the situation can get worse quickly if not properly given prompt attention. Hydrate jamming is the sudden arrest of the suspended hydrate debris or large particles dynamics in the pipeline. The jammed hydrate particles or debris are trapped locally, preventing them from flowing downstream and making them an obstacles for fluid upstream. The hydrate debris are driven into a jammed state by externally applied pressure from upstream. The whole bulk is in stress equilibrium with no possibility of escape for any fragment (Bi et al. 2011). If the jam has not anneal into a solid plug, the jam can easily breakupwheneverthestressdirectionchangesevenbyasmallamount. Thejamcansupport a very large applied load if it is in the same direction as the jamming (Liu and Nagel 1998). However, if a force is applied in a different direction the chain force can easily fall apart. The direction change of applied force can change the chain force structure. As the hydrate debris is porous, it can responds elastically if the radially applied force is below a threshold 69

92 values. If the hydrate debris is hardened to a certain extent that it cannot deform, the jam becomes fragile and responds plastically to the radially applied force. 3.5 Summary This chapter first investigates the mechanical properties of hydrate deposit. Four different packing pattern are proposed: (1) simple cubic packing; (2) rhombohedral packing; (3) tetrahedral packing and (4) orthorhombic packing. The hydrate particles are assumed to be in spherical shape and have the uniform diameter. The porosity for these four different packing pattern have also been calculated (see Table 3.1). Cubic packing results in the most porous deposit, with aporosity of 47.64%. Rhombohedral packing is the most compact arrangement, with porosity only 11.14%, and is most me-chanically stable than the cubic packing. As more water in the pore is converted into hydrate, the porosity will decrease. Two models are proposed to study the relationship between bridge diameter and porosity: (1) evenly coating model and (2) Corner growth model. Equations 3.16 and 3.31 are developed based the geometry of simple cubic packing pattern. The hydrate deposit experiences combined stress resulting from ambient pressure and friction with gas. Finite element method is applied to study the internal stress distribution inside the hydrate deposit. Simulation results show the region subjected to the lateral force of gas stream is more likely to collapse first. The collapsed hydrate debris can travel down and jam the entire flow-line. 70

93 CHAPTER 4 HYDRATE DEPOSIT CHARACTERIZATION The current trend in offshore oil and gas production is advancing into deeper waters, making it increasingly necessary for the petroleum industry to develop cost effective solutions for developing oil and gas fields in deep waters. Hydrate deposition on the inner surface of pipe wall is a very costly problem. It can affect the performance of subsea pipeline. Hydrate deposition is induced due to the reduced fluid temperature in the deep-water environment. If favourable hydrate formation conditions exist, hydrate deposition or even blockage can occur within hours. Hydrate buildup along the pipeline can affect the fluid flowing behavior inside the pipeline by decreasing the effective flow path and increasing the friction between the fluids and pipeline wall surface. Early detection and quantification of hydrate deposition can prevent production disruptions. It is very important to obtain the pressure and temperature information along the pipeline. Commercial software can be used to determine if a section of pipeline is located within hydrate formation conditions. Hydrates are unlikely to form in the well-bore, because the fluids from a reservoir still maintain high temperature. However, for reservoirs located in arctic areas, the wellbore could be at risk of hydrate formation. Due to JouleThomson effect, rapid change in temperature of a gas can induce hydrate formation in valves, nozzles or orifices. The backpressure method can be used to obtain hydrate deposit thickness and length during depressurization. The average pressure method and pressure transient method can be used to locate the hydrate deposition. The current available mathematic methods to determine hydrate deposit location are: (1) pressure-temperature profile method; (2) average pressure method (Liu and Scott 2001); (3) pressure transient method (Liu and Scott 2000); (4) pressure wave propagation technique (Adewumi et al. 2000; Chen et al. 2007); (5) kinetic model (Jassim et al. 2010; Turner et al. 2005). However, none of these methods can be applied in all types of conditions and 71

94 situations. The combination of these techniques can improve our ability to determine the location of hydrate deposits. They should not be used alone. 4.1 Pressure-temperature profile methods Before applying the pressure-temperature profile methods, the geometry of the pipeline should be investigated first. The pipeline layout usually follows the terrain. Water tends to accumulate in low spots of the pipeline due to gravity. Hydrate can only form when the system pressure and temperature conditions are within the hydrate stability region. The pressure-temperature profile method consists of estimating the temperature and pressure profile along the pipeline, and comparing it to the hydrate equilibrium conditions to determine if the system would be within the hydrate formation region. Multiphase flow modelling can be used to estimate the system pressure and temperature conditions. Understanding the flow behavior of fluids inside the pipeline requires a sound knowledge of fluid properties such as density, viscosity, surface tension, and phase behavior. The reservoir hydrocarbon fluids enter the well-bore first. Then, the fluids travel to the surface through the well tubing or annulus, whose geometry is typically vertical, but it could include an initial horizontal section. Then, the fluids flow through the well head into a flowline, which could have an irregular geometry depending on the terrain, to the separation/processing facility. Generally, hydrate cannot form inside the wellbore due to high temperature of the reservoir fluids and the geothermal gradient. However, for arctic reservoirs, the low surface temperature may induce hydrate formation in the wellbore (Figure 4.1) Hydrate formation in the wellbore During arctic and offshore operations, hydrate formation can be encountered in the tubing strings, which may lead to blockage of the pipe or choke. If production is shut-in for a few days, in the case of arctic regions, the tubing string cools off because of low temperature in the permafrost. In the case of offshore facilities, the subsea pipelines cools down during shut- 72

95 in because of low temperatures at the sea floor. Once the well starts flowing, the presence of free water can form hydrates and potentially plug the well. Hydrate risk assessment during restart procedure has been studied using a transient hydrate prediction model (Zerpa et al. 2011). Figure 4.1: Hydrate PT curves for annulus at different depths from wellhead (0 m) to bottom hole (4430 m), showing the systems fall within the hydrate formation region from wellhead to 70 m below the surface. A transient multiphase flow simulator (OLGA R ) was used to study the most likely locations where hydrates form in arctic wellbores. The results show that there is hydrate formation in the annulus, while the whole tubing condition is located within the non-hydrate zone (as shown in Figure 4.2) Hydrate formation at the choke The choke is used to directly regulate the amount of gas entering the well or the production rate. Zerpa et al. (2011) has studied the risk of a hydrate plug across the wellhead 73

96 valves with different openings. Figure 4.2: Hydrate PT curves for tubing at different depths from wellhead (0 m) to bottom hole (4430 m). The flow through the choke may be either critical (or sonic) flow or sub-critical (or subsonic) flow. For critical flow, gas travels through the choke at the speed of sound. There exists pressure discontinuity at the choke, as the downstream pressure changes do not affect the upstream pressure, and vice versa. The surface chokes are usually sized for critical flow, which helps to stabilize well production rates and separation operation conditions (Ikoku 1992). For sub-critical flow, gas travels through the choke at a speed lower than the speed of sound. The flow rate depends both on the upstream and downstream pressure. Sub-critical flow happens in subsurface control equipment, such as subsurface tubing safety valves, bottom-hole chokes/regulators, and check valves. A critical downstream-to-upstream 74

97 pressure ratio is used as the criterion to define the flow regime across the choke, ( pdown p up ) c = ( ) k 2 k 1 k +1 (4.1) Here, the value of k is usually 1.28 for natural gas. Thus, the critical pressure ratio is usually about 0.55 (Brown 1977). When gas flows across the choke, the gas suffers a sudden decrease of flow area, resulting in higher velocity. Once the fluids flow across the choke, the gas expands and the temperature will decrease due to the Joule-Thompson effect, which could cause water condensation or bring the fluids to hydrate formation conditions. In extreme cases, this could completely block the flow. For critical flows, the temperature at the choke downstream can be predicted using the following equation: T down = T up z up z outlet ( poutlet p up )k 1 k (4.2) Most current models just assume z up z outlet = 1. Here we use an iteration method to predict the downstream temperature, which is described in detail in the Appendix. When fluid mixtures flow through restrictions, they also suffer a significant pressure drop. Different choke flow models are available to predict pressure drops (Guo et al. 2011). During gas lift operations, the injection pressure determines the upstream pressure of the choke, and the downstream pressure is determined by the casing head pressure (CHP). The choke size determines the downstream temperature, which could affect the formation of hydrates. The size of gas injection choke has a great impact on well operations because it controls the gas injection rate. A choke with an overly large diameter is indicated by a CHP above the design pressure and can cause reopening of upper unloading valves and excessive lift gas usage. Smaller than designed chokes can limit gas injection rates and can reduce the liquid production. Unloading of the well may not be complete because of a small choke sizes. Chokes with smaller diameters are more prone to hydrate formation (See Figure 4.3). By decreasing the choke diameter, downstream conditions enter the hydrate formation region. In 75

98 thecased = 1.5in., theentirepipelinehavenoriskofhydrateformation. WhenD = 1.0in., the downstream condition enters hydrate formation region, while choke upstream condition is outside of the hydrate formation region. However, for a smaller diameter of 0.5 in., some portion of the upstream pipeline also enters the hydrate formation region. Figure 4.3: Hydrate PT curves for surface pipeline with difference choke sizes. During gas lift operations in arctic wells, hydrate formation is a serious concern. When the injection choke at the wellhead is plugged or frozen, there will be lower casing head pressure (CHP) and higher injection pressure. The CHP will go below the closing pressure of the operating gas lift valve. No gas can be injected into the tubing. As the well does not have sufficient energy to lift the produced liquid to the surface, production rates will decrease rapidly, and liquid will start to accumulate at the bottom of the wellbore. The production of liquid at the wellhead almost goes to zero and the well ceases to produce after some time. Meanwhile, the tubing head pressure (THP) drops to that of separator. Improper gas injection rates can also be used as symptoms to detect hydrate plug in the 76

99 injection pipeline. Lower than design gas injection rates may indicate partly closed valves or a plugged, frozen injection choke. As the size of the gas injection choke controls the injection rate, the choke inside diameter should be carefully designed. Two-pen pressure recorders can be installed to evaluate the performance of gas lift wells. Casing and tubing head pressures vs. time are measured and recorded by two-pen pressure recorders on circular preprinted charts. A plugging/freezing injection choke at the wellhead can slowly decrease the CHP with the THP also dropping, which can be detected from the two-pen pressure recorder. Once symptoms of hydrate formation in the pipeline have been detected, we can increase the opening of injection choke to inject more gas through the choke to minimize the Joule-Thompson effect. Also, larger amount of injection gas will cool more slowly. The temperature will not drop significantly before entering the well. Injection of gas into the tubing requires high pressure. It is impossible to lower injection pressure much to avoid hydrate formation. However, we can control the temperature of injected gas. For offshore operation in winter, the windy environment can cause a lot of heat loss due to convection and conduction. As compression of gas in the compressor is an exothermic process, the gases will be heated after compressor (See Figure 4.4). The temperature in the compressor can reach over 100 C (Demma 2005). However, when the gas reaches the wellhead, the temperature is only 20 C, which could be lower during colder weather. The presence of water can facilitate hydrate formation. The lift gas from the separator should be dehydrated first before injection to reduce the water amount to a minimum level. Once hydrate formed, the situation deteriorates quickly, and eventually cause injection failure. Freezing can be eliminated by heating or injection of chemical inhibitors into the gas lift Hydrate formation in the pipeline To determine the flow conditions for hydrate formation, we need to obtain the pressure and temperature along the pipeline by examining fluid dynamics and heat mass transfer. 77

100 Estimation of pressure drop in the pipeline plays an important role in the design of pipeline systems to guarantee uninterrupted transport of fluids (Zerpa 2013). Figure 4.4: Hydrate PT curves across the compressor by OLGA R. General energy equations are used to solve multiphase flow problems in the pipelines. Eqn. 4.3 is the general form for fluid flow in the pipeline. U 1 + mv2 1 2g c + mgz 1 g c +p 1 V 1 +q W = U 2 + mv2 2 2g c + mgz 2 g c +p 2 V 2 (4.3) Where, q = heat leaving or entering the pipeline system; if q has a negative sign, it means heat loss to the environment. U is the internal energy stored in the fluids. It cannot be measured but can be established by a relative value. The term mv2 2g c represents the kinetic energy of fluids due to its velocity. For pipelines with high gas velocity this term can be significant. The term mgz g c represents the potential energy. It is a predominant term for vertical flow or highly deviated flow. But for horizontal flow, it is negligible. The pressure volume term PV considers the fluid energy of compression or expansion. The term W 78

101 represents the energy added by the pump or compressor. The entropy change can be defined by: S 2 S 1 = 2 1 dq T = 2 1 mc p dt T (4.4) S is the entropy of the system. The entropy change depends entirely on the initial and final states. It is also related to internal energy by Eqn The entropy is also affected by lost energy of the fluids due to friction. U = S2 S 1 Tds+ V2 V 1 p( dv)+ CHEM EFFECT SURFACE + + TENSION EFFECT Here, S 2 S 1 Tds =heat effect; and V 2 V 1 p( dv) = compression effect. etc (4.5) The general Eqn. 4.3 can solve many flow problems and is the basis upon which many multiphase flow correlations are developed (Barnea 1987; Brill and Mukherjee 1999; Govier and Aziz 1977; Petalas and Aziz 2000). Bendiksen et al. (1991) developed a dynamic two-fluid model, OLGA R, to simulate transient flows in pipelines. Advanced methods are also developed to better describe multiphase flow in pipeline with hydrate formation. Turner et al. (2005) developed a hydrate kinetic model, which was incorporated into OLGA R 2000, to predict the hydrate formation and fluid rheology of gas hydrates in the oil pipeline. Sinquin et al. (2004) also investigated the rheological and flow properties of fluids with gas hydrate suspensions. Commercial software is also available to predict the pressure-temperature profile along the pipeline. We can use these temperature and pressure data to roughly estimate the potential blockage location (see Figure 4.5). The pipeline enters the hydrate formation zone at9milesfromthewellheadasshowninfigure4.5by(notz1994). Thesectionfrom20miles to 33 miles are at higher risk of hydrate blockage. Other mathematical methods, discussed as the following section, can be used to further verify if the blockage is located within the mile section. Pipeline geometry and subsea terrain information should be gathered to validate the blockage location. Hydrates are more likely to form at the bend, fittings, valves or other restrictions, where flow area or/and geometry are changed. The pipeline dip is also 79

102 an ideal location where hydrate can form due to the easy accumulation of free water. Figure 4.5: Hydrate-formation pressures and temperatures at different methanol concentration along the pipeline (Notz 1994). It takes only hours to form hydrate plugs in a pipeline. It might take from days to months to remove the plug. It is very important to recognize the formation of hydrate plugs before the situation gets worse. Typically, field data indicates there is pressure fluctuation in the pipeline before formation of a hydrate plug. The spikes in the pressure vs. time curve warn of severe hydrate deposition on the wall of the pipeline (Hatton and Kruka 2002). 4.2 Hydrate amount quantification The deposition of hydrate can be easily detected using rate-time monitoring techniques (Thrasher 1995). At present, it is very difficult to precisely detect and characterize hydrate deposition. The present solutions require use of very expensive alternative methods, such as gamma ray absorption pipe scanner (Hu et al. 2005; Bruvik et al. 2010; Hammer and Johansen 1997; Abdul Majid 2013), fibre optics technology (Clayton and Milanovic 80

103 2003; Tamachkiarow and Flemming 2003; Yan and Chyan 2010) and diameter expansion variation measurement technique (Devegowda 2005). These alternative modern methods can provide means for detecting the presence of solid deposition. But they are still in development stage. Besides, the high pressure environment may cause premature failure of devices. The lack of continuous monitoring of solid deposition can result in unexpected and undesirable shutdowns. Hence, it is desired to provide a mathematical model to detect and characterize hydrate deposit on the wall of pipeline, based on easily measured parameters such as pressures, temperatures and production data. This work address the techniques to quantify the hydrate deposit amount on the pipe wall. The detection of hydrate deposition does not impede the flow of fluid through the pipeline. Only daily production data are needed. Quantification of solid deposition amount, thickness and length could help with: (1) monitoring chemical injection program; (2) designing pigging schedules and assess pigging risk; (3) planning intervention procedures; (4) predicting the possibility of hydrate sloughing with information of deposit thickness. Wax, resin or even sand can also deposit on the pipe wall. The flow-line can be kept clear by pigging at a certain frequency, e.g. once per month. However, if equipment malfunction occurs, the line will become clogged and possible shut down because the previously established pigging frequency is now insufficient. To estimate the amount of hydrate deposition, we also need to know the volume of gas and water inside the pipelines (see Eqn. 4.6). The volume of water inside the pipeline can be estimated from the daily production data. The volume of hydrate will be calculated during one-side-depressurization. V pipe = V liquid +V deposition +V gas (4.6) Characterization of hydrate deposition requires detailed knowledge of amount of gas, water trapped inside the pipeline. The schematic of pipeline with hydrate deposit is shown in Figure

104 Figure 4.6: Schematic of hydrate deposit on the pipeline wall Gas volume inside pipeline estimation during one-side-depressurization During one-side depressurization, multipoint test of gas pipeline is conducted (Yi and Scott 1999). The average pressure inside the pipeline after the pipeline is shut-in is recorded. Then, the real gas equation is used to predict the volume of hydrate deposit inside the pipeline. The processes are as follows: (1) Simultaneous shut-in procedures are conducted at both ends of the pipeline. Measure and record the average pressure and temperature at the outlet of the pipeline. (2) The valve at the outlet is fully open, and then closed when there is a significant pressure drop observed inside the pipeline. The amount of gas blown out can be calculated by the real gas equation. (3) The valve is partially open again with opening = 0.1. When opening =1, the valve is fully open; when opening = 0, the valve is fully closed; when opening =0.1, 10% of the valve area is allowed to open. Pressure and temperature is measured and recorded. The valve is closed when significant pressure drop observed again. (4) Step 3 is repeated for at least two times to get better accuracy. The valve schedule is shown in Figure 4.7. The composition of gas is shown in Table 4.1. To model the one-side-depressurization process, we are using the transient multiphase simulator: OLGA R to obtain the required data for calculation. The gas flowrate is also recorded during these operations (see Figure 4.8). We can use these data to obtain the 82

105 amount of gas blown out during each depressurization process. Detailed amount of gas blown out can be calculated. The temperature and pressure data can be found in Figure 4.9 and Figure Table 4.1: Fluid Composition Component Moles Component Moles C C 19 -C C C 27 -C C C 33 -C ic C 40 -C nc C 47 -C ic C 54 -C nc C 62 -C C C 71 -C C H 2 O C N C CO C 10 -C Consider the pipeline as a container with fixed volume. The volume of pipeline can be easily obtained. Then, apply EOS for real gases: The amount of gas blown out can be calculated by: m = P iv gas M z i RT i PV = m zrt (4.7) M P i+1v gas M z i+1 RT i+1 = P scv bg M RT sc (4.8) The volume of gas blown out, V bg, can be represented by the following expression: P sc T sc V bg = P i z i T i P i+1 z i+1 T i+1 V gas (4.9) The time allowing the fluids in the pipeline flow under steady-state is 1 hour. Close the valves at inlet and outlet at the same time. After 20 mins, open the outlet for 2 min and then close the valve for 20 mins. Plotting Psc T sc V bg vs P i+1 z i+1 T i+1, we can obtain a straight line, as shown in Figure From Eqn 4.9, the slope of the straight line is the volume of gas trapped inside the pipeline after 83

106 Figure 4.7: Schedule of valve at outlet. initial shut-in. However, real data would deviate from the line due to measurement errors. To improve the accuracy, more test points are recommended. Practically, at least four points are needed to obtain satisfactory results. Applying this technique with OLGA simulation, we can get a straight line with slope = m 3, which represents the volume of gas inside the pipeline Use of back-pressure method to determine hydrate deposit thickness and length Back-pressure technique has been well recognized as an effective way to monitor well performance. Back-pressure methods have been first used to effectively monitor the production performance of gas wells and later adopted to oil wells (Cullender 1955, Fetkovich 1975). This technique is also very effective in the analysis of pipeline performance. According to the severity of blockage, two categories of blockage are proposed: (1) partially restricted blockage and (2) full blockage (commonly referred as plug). For partial blockage restriction, there is still good communication between the upstream and down- 84

107 Figure 4.8: Gas flowrate during four-point test. stream of blockage. For full blockage, due to the limited porous and permeable properties of hydrate deposit, there is still limited to insignificant communication across the hydrate deposit. A blockage factor F B is used to determine the severity of how the pipeline is blocked (Scott and Satterwhite 1998). When the pipeline is totally plugged, F B = 0; when the pipeline has no flow assurance problem, F B = 1; in most scenarios, 0 < F B < 1. The value of F B can provide a lot of information regarding flow assurance practice such as hydrate formation inhibitor injection. Rate-pressure equation with partial blockage (Blockage Factor F B ) can be expressed mathematically as: ( Q g = CF B p 2 in pout) 2 n (4.10) Taking a log on both sides of Eqn and 4.10 and rearranging the equations, we can obtain: Log 10 ( p 2 in p 2 out ) = Log 10 Q g n Log 10C n (4.11) 85

108 Figure 4.9: Pressure profile after each depressurization. Log 10 ( p 2 in p 2 out ) = Log 10 Q g n Log 10CF B n (4.12) Plotting Log 10 (p 2 in p 2 out) vs LogQ g, a straight line is obtained with slope 1/n and intercept log 10 (CF B )/n. The data obtained from a multipoint test before formation of hydrate deposit is used to develop a baseline curve (Eqn. 4.11). It can be used to be compared with data obtained during the multipoint test after the blockage restriction is formed. Any deviation from the baseline indicates the presence of a hydrate blockage restriction on the pipeline. The deviation can be characterized by the blockage factor F B. F B is related to the deposit thickness and length. Subtracting Eqn from Eqn yields: F B = 10 nδl (4.13) 86

109 Figure 4.10: Temperature profile after each depressurization. Here, δl is the vertical distance obtained from the two straight lines from Eqn and Scott and Satterwhite (1998) have developed the following equation for blockage factor for rough pipeline: F B = ( f B f 1 )( ) 5 d L P d B +1 L B (4.14) L P )( L B The volume of hydrate deposit can be written as: V B = π(d 2 d 2 B)L B (4.15) Combining of Eqn. 4.13, 4.14 and 4.15, we can develop two equations with two unknowns: the deposit thickness and length. To solve these nonlinear equation system, the Newton- Raphson method is applied. 87

110 Figure 4.11: Determination of hydrate deposit amount on the pipeline wall. The rate-pressure relationship of gas pipelines follows the general form of backpressure equation, which is a widely used empirical equation. Q g = C ( p 2 in p 2 out) n (4.16) Using Modified Newton-Raphson Method to arrange the equations, we obtain: [ ] f1 (d B,L B ) f 1 (d B,L B ) [ ] [ ] d B L db f1 B (d = B,L B ) L B f 2 (d B,L B ) Then, f 2 (d B,L B ) d B f 2 (d B,L B ) L B [ db ] i+1 = [ db ] i [ ] db + L b L B L B The process is repeated until successive solutions vectors from the equation are hardly changing. To model this problem, we carry out simulations to verify the validity of this method using PIPESIM R. The ID of a pipeline section is decreased to simulate the presence of hydrate deposit. The difference between the IDs of the pipeline is the thickness of hydrate 88

111 deposit. In this case, the deposit thickness is 1.5 inches and can be used to measure the accuracy of simulation results. The friction coefficient of the hydrate deposit is assumed to be the same as that of the pipeline. Deviations from this established base line indicate how severe the pipeline is plugged. The solid deposition shift the back-pressure line to the left (Figure 4.13). Significant deviations from the established back-pressure curve can be employed to estimate the degree of blockage. Figure 4.12: Simulation errors for deposit thickness (left) and length (right), the dotted lines represent the prescribed thickness (left) and length (right) by reducing the diameter of a section of pipeline (52 ft) by 1.5 inches. These more detailed information about the deposition can help the field engineers in: (1) designing pigging schedules and assess pigging risk; (2) planning intervention procedures; and (3) predicting the possibility of hydrate sloughing. More simulation cases are run by varying the location of hydrate deposit. The hydrate deposit is located at 52 ft, 2352 ft, 4704 ft, 7056 ft, 9408 ft from the pipeline inlet separately. Simulation results show this method tends to overestimate the amount of hydrate deposit. The errors for hydrate deposit thickness and length are %, %, separately, as shown in Figure

112 Figure 4.13: Four-point test in the pipeline with length=1.79 miles. To better verify the applicability of this method, a sensitivity analysis is conducted on the pipeline length. Three more cases are conducted by varying the total length of the pipeline: 1.79 miles, 17.8 miles and miles, separately. The deposit is located at the inlet for each case. The simulation results show that the longer the pipelines, the larger error will be encountered, as shown in Figure 4.14 and Figure Application of this method to a long pipeline must be used with caution. For the pipeline with length larger than 20 miles, this method is not recommended. Unfortunately, the longer the tieback is, the more likely the pipeline can get plugged due to a larger extent of heat loss to environment. Hydrate depositing on the inner walls of the pipeline could partially, and in some cases completely, block the flow (Liu et al. 2016). Blockage often occurs at the flow restrictions such as valves, chokes, fittings, and other locations where the pipeline geometry may change suddenly (Liu and Zerpa 2016). The investigation of this topics are beyond the scope of this work. This section provides a mathematical method to detect, characterize and determine 90

113 the extent of deposition and thus enable remedial procedures. Figure 4.14: Four-point test in the pipeline with length=17.8 miles Average pressure method for locating hydrate deposit Liu and Scott (2000) developed a type curve for locating partial gas pipeline blockages. Figure 4.16 shows how the position of hydrate blockage can affect the pressure distribution along the flow line. Line P 2 in p 2 out represents the pressure profile along the pipeline without any hydrate deposition. The squared pressure is linearly distributed along the pipeline during steady state flow. Line P 2 in p 2 out and Line P 2 in p 2 out represents a sudden pressure drop at x = x B1 and x = x B2, respectively, with a very small blockage length. Line P 2 in p 2 out indicates a case when a long partial blockage exists. Line P 2 in p 2 out denotes the existence of two blockages in the pipeline system. The general average 91

114 pressure equation for gas flowline is given (Liu and Scott 2001): P av = 2 1 {P 3P0F 2 in 3 Pout+ 3 ( VB ) B +1 FVB ([P ] 2 in (F B +x DB +L DB )P [ ] ) 0 P 2 B +L in x DB P } DB (4.17) Here, P 2 0 is an intermediate variable, which can be expressed mathematically by: P 2 0 = P2 in P 2 out F B +1 (4.18) F VB represents the volume factor: Figure 4.15: Four-points test in the pipeline with length=178.2 miles. F VB = 1 (1 d 2 DB)L DB (4.19) F B is the blockage factor, which can be obtained using backpressure method. This method is based on: (1) only one blockage deposit is present in the flowline; (2) single gas phase flow in the pipeline; (3) the inlet and outlet pressure difference is small. 92

115 If the hydrate blockage is short, then L D B goes approximately 0, and F B is close to 1. Eqn can be reduced to: P av = P 2 0 {Pin 3 Pout 3 + [ ] Pin 2 (F B +x DB )P [ P 2 1.5} in x DB P0] 2 (4.20) If the flow-line is fully blocked by the hydrate, Eqn is reduced to: P av = 2 3 Pin 3 Pout 3 Pin 2 P2 out = P2 in +P in P out +P 2 out P in +P out (4.21) In Eqn the variables F B and F VB can be obtained from the back-pressure method. Therefore, the value of x DB can be found by solving Eqn Once x DB obtained, the location of hydrate deposit x B can be obtained. Figure 4.16: Plot of pressure against different blockage positions in a gas flowline with different hydrate blockage location and length (Liu and Scott 2000). 93