GUIDANCE NOTE LECTURE 12 THERMAL EXPANSION ANALYSIS OVERVIEW The calculation procedure for determining the longitudinal pipeline response can be formulated on the basis of strain. The longitudinal strain component can be determined through linear superposition of Poisson effect due to internal pressure ( p ), thermal expansion effects ( T ), soil restraint ( s ), and residual lay tension ( H ). The corresponding pipeline axial or longitudinal displacement can be obtained by integrating the strain expression over the pipeline length. For the present analysis, the hoop stress can be defined by the expression, h p i p e (12.1) The longitudinal stress component may or may not be statically determinate and is dependent on the imposed boundary condition. The boundary conditions can include the effects of soil reaction loads, anchor restraints, linepipe bend resistance and residual pipelay tension forces. Using Hooke s law, the longitudinal stress strain relationships can be defined as: (12.2) where E Elastic or Young s modulus (typical: E 207GPa) Poisson s ratio (typical: 0.3) Coefficient of thermal expansion (typical: 1.1710-5 m/m/ C) Temperature differential The operating temperature profile for pipeline transportation systems is not constant along the length. The temperature gradient is dependent on a number of factors that include product type (i.e. oil or gas), internal pressure profile, physical thermal properties (e.g. coating, pipeline and soil), metocean conditions (e.g. temperature, current speed) and pipeline cover (e.g. entrenched, or non-entrenched). One objective of a flow assurance study is to determine the pipeline pressure and temperature profile. The thermal gradient can be idealised by an exponential decay function, (12.3) where o Initial temperature differential at the origin (x 0) e Exponential function Guidance Note Lecture 12 Thermal Expansion Analysis Page 1 of 6
x Distance along pipeline from origin Temperature decay length factor, Product mass flow rate c Product specific heat q Rate of heat transfer The analysis must evaluate the pipeline response on a systematic basis and consider the loading history for the respective loading conditions such as: Prior to installation As-laid Flooded Hydrotest Operation Change in operational parameters Shut-in Shutdown and restart The fundamental expressions to determine the pipeline behaviour in terms of the circumferential and longitudinal stress-strain response, longitudinal reaction loads and longitudinal pipeline displacement are presented. FULL AXIAL RESTRAINT BOUNDARY CONDITION A fully end constrained boundary condition can occur at an anchor block or pig trap, and Pipeline End Manifold (PLEM) or Pipeline End Termination (PLET) sled. For a fully end constrained pipeline, the longitudinal strain ( l 0) and deflection ( 0) components are zero and the longitudinal stress response can be determined from Equation (12.1) and Equation (12.2) assuming a constant, uniform temperature field, l h E o p i p e E o (12.4) Equation (12.4) represents the true through wall thickness stress in the pipeline and the true through wall thickness pipeline force can be determined by multiplying Equation (12.4) by the linepipe cross-sectional are12. END FREE BOUNDARY CONDITION An end free boundary condition can occur at locations where no physical longitudinal restraint exists; for example at a riser bend extending from the seabed to the production platform. Since the pipeline is not restrained axially, the Poisson effect and thermal expansion component does not produce stress in the linepipe. The pipeline longitudinal stress is only due to the end cap effect (Equation 2.5), Guidance Note Lecture 12 Thermal Expansion Analysis Page 2 of 6
l p i p e 4t (12.5) Combining Equation (12.1) and Equation (12.5) into the longitudinal stress strain relationship (Equation 12.2), and assuming a constant, uniform temperature field the longitudinal strain component is, l p i p e 4tE ( 1 2 )+ o (12.6) Assuming a free free end boundary condition, the longitudinal deflection at the pipeline end is, l dx L p i p e ( 1 2 )+ o 2 4 te L (12.7) The expression for deflection (Equation 12.7) assumes that the pipeline at midspan does not move. PARTIAL RESTRAINT BOUNDARY CONDITION An intermediate situation occurs where the longitudinal pipeline response is partially restrained by frictional resistance. A partially restrained boundary condition can occur with pipelines resting on the seabed near the tie-in location with risers or expansion spools, pipelines fully buried within a trench or pipelines resting on frictional guides and supports. For soil/pipeline interaction events, typically a virtual anchor point exists at some distance away from the riser tie-in location with the riser or expansion spool whether or not the pipeline is fully buried. As shown in Figure 12.1, the anchor point represents a position on the pipeline such that axial displacement is fully restrained. Frictional resistance has integrated sufficient force along the anchor length to restrain axial movement. The distance required to develop the anchor point is dependent on the extent of contact, frictional coefficients and normal stress at the soil/pipeline interface. In general, frictional guides or supports do not develop a virtual anchor point but can influence the pipeline stress strain response. In the transition zone, the pipeline is in static equilibrium between the longitudinal pipeline wall force, longitudinal pressure force on the product or contents and the seabed frictional force. The horizontal shear force in the vertical riser segment is typically not considered in the expansion stress analysis. The sum of longitudinal forces can be defined as, F x fx+ l ( A e ) ( p i p e A e ) 0 (12.8) Guidance Note Lecture 12 Thermal Expansion Analysis Page 3 of 6
where f Frictional resistance force per unit length x Pipeline length Thus the equilibrium equation can be expressed in terms of the longitudinal stress, l ( p i A p e A e ) fx ( A e ) (12.9) Figure 12.1 Isometric of Subsea Riser Tie-in Expansion Spool. Guidance Note Lecture 12 Thermal Expansion Analysis Page 4 of 6
Based on compatibility and stress boundary requirements at anchor point, the longitudinal stress can also be defined by Equation (12.4). Equating the expression in Equation (12.4) with Equation (12.9) yields, (12.10) Thus, the distance to the anchor point from the free end of the pipeline, which is typically the riser or expansion spool tie-in location, can be defined as, x x z 1 f ( p i p e A e ) ( A e )( h E o ) (12.11) The maximum pipeline axial strain can be determined by substituting Equation (12.9) into Equation (12.2), which yields, l 1 E ( p i p e A e ) A e p i p e f + o E A e x (12.12) The maximum deflection can be determined by integrating the expression (12.12) over the pipeline length required to mobilize the anchor point (x z), 1 E ( p i p e A e ) A e p i p e f x + o x E A e x2 (12.13) The expressions can be simplified through consideration of the idealised geometric pipeline properties for a thin wall pipeline, where the mean radius is defined as R 1 ( 2 t). The hoop stress equation yields, (12.14) The longitudinal stress component is, (12.15) Guidance Note Lecture 12 Thermal Expansion Analysis Page 5 of 6
Substituting (12.14) in Equation (12.4) yields, (12.16) Equating (12.15) and (12.16) yields the distance to the anchor point from the free end of the pipeline, (12.17) Substituting Equation (12.14) and Equation (12.15) into Equation (12.2), the longitudinal strain expression can be expressed as, (12.18) and the axial displacement can be expressed as, (12.19) Guidance Note Lecture 12 Thermal Expansion Analysis Page 6 of 6