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Dynamic Response of Deepwater LazyWave Catenary Riser

Songcheng Li, 2H Offshore Inc. & Chau Nguyen, 2H Offshore Inc.

1 Abstract

Lazy-wave catenary risers have gained popularity as a viable solution to improve fatigue and strength performance at the touchdown zone of a simple catenary riser. With the objective to provide technical reference for a lazy-wave shaped riser, this paper focuses on the study of lazy-wave configurations and dynamic responses when the riser is supported from high motion floating production platforms such as semi-submersibles and FPSOs. The study explores the behavior of lazy-wave risers with respect to a variety of input parameters, such as critical curvature radii, hang-off angle, top tension and buoyancy distribution. A systematic approach to pinpoint driving factors and critical locations is discussed. Equations are also presented to provide an analytic and deterministic approach to a desired lazy-wave shape for further numerical assessment of strength and fatigue responses.

2 Introduction

The application of lazy-wave catenary riser (LWR) has been popular in deep water application due to its adjustable payload on vessel and its options to control dynamic strength and fatigue response along the riser [1]. With help of commercial software or self-developed optimization tools [2], current practices rely on numerical approaches using trial and error or iterative procedures to explore a lazy-wave riser configuration [3], which demands a huge effort for optimization. It is time consuming and not cost effective, especially during preliminary screening stage or riser study for a project. The challenge encountered in the optimization of lazy-wave configuration is associated with the lack of parameterized equations for the configurations.

1

In this work, configuration of lazy-wave riser is derived analytically with two design input options. The key factors for dynamic strength and fatigue responses are explored and expressed as equations of the lazy-wave configuration parameters, which provide insight and perspective in lazy-wave riser behavior. This work simplifies the riser optimization process and can be used as a framework for lazy-wave configuration design.

3 Static Configuration of SCR and LWR

A simple steel catenary riser (SCR) is considered a cord of uniform density and cross-section area hanging on two ends under gravity and buoyancy force in water [4], although slight variation in density or cross-section may exist due to strakes, marine growth or special joints such as taper stress joint. A typical SCR is characterized by downward wet weight along its length. A lazy-wave catenary riser (LWR) is a special SCR with a segment of its length equipped with external buoyancy modules, where its upward buoyancy force in water is greater than its downward gravity force and thus an equivalent negative “gravity” force. A typical LWR consists of three segments, each segment a catenary, namely the hang-off catenary (hanging and jumper sections), the buoyancy catenary (lift and drag sections) and the touchdown catenary, as illustrated in Figure 3.1. The buoyancy catenary lies between the hang-off catenary and the touchdown catenary. A typical LWR has a sag bend and an arch bend. The elevation difference between the top of the arch bend and the bottom of the sag bend is termed arch height. As a general rule of thumb, the buoyancy force provided by the buoyancy modules is around twice the self-weight of the steel pipe with internal fluid. The variation of the net buoyancy force from the buoyancy modules produces high-arch, mid-arch or low-arch LWR configurations, or a shaped SCR[5]. A shaped SCR is defined as a degenerated LWR with no sag bend or arch bend. That is, its lowest elevation along the hang-off catenary coincides with the highest elevation along the buoyancy catenary at their connection point. A sag bend along the hang-off catenary and an arch bend along the buoyancy catenary distinguish a LWR from a shaped SCR. 2

The equations in the subsequent sections apply to SCR, LWR and shaped SCR.

8000 7000 6000 5000 4000 3000 Drag Point 2000 1000 0 0 1000 2000 3000 4000 5000 6000 7000 Horizontal Distance from Hangoff (ft) Jumper Section Sag Bend Touchdown Point Hangoff Catenary Buoyancy Catenary Touchdown Catenary

Elevation from Seabed (ft)

Hangoff Location Hanging Section Lift Section Lift Point Arch Bend Drag Section

Figure 3.1 – Example Configuration of Lazy-Wave Catenary Riser

3.1

General Characteristics of SCR

As a special SCR, the LWR shares some common characteristics of SCR such as correlations between curvature, riser mass density, arc length and hang-off angle. A catenary is governed by a hyperbolic cosine function [6] in Equation (1) as illustrated in Figure 3.2:

? x a y ? a (cosh ? 1) ? (e a ? e a ? 2) , a 2 x x

(1)

where a is the curvature radius of a catenary at its origin, since curvature k or

d2y | 2 | 1 dx , k? 3 ? dy 2 ? 2 a cosh 2 x ? ?1 ? ( ) ? a dx ? ? where

(2)

dx ? ds cos ? , dy ? ds sin ? ,

(3)

at origin x ? 0 gives

3

k?

1 . a

(4)

Y

Y X ds ds dx dy T Qds=mgds T+dT β

X O

Figure 3.2 – Coordinate System of Catenary

Curvature Equation (2) can be rewritten as

k? 1 a cosh 2 x a ? a , (a ? y ) 2

which dontes the maximum curvature or the lowest curvature radius along a catenary occurs at the origin y=0. The catenary curve is symmetric about the Y-axis in Figure 3.2. Without loss of generality, the absolute values of X and Y are used for the equations in this paper. The arc length of a catenary from its origin can be obtained from

S ? ? ds ? ? The inclination angle β suffices

x ? dy ? 1 ? ? ? dx ?a sinh . a ? dx ?

2

(5)

tan ? ? From horizontal equilibrium

dy x S ? sinh ? . dx a a

(6)

? T cos ? ? (T ? dT ) cos ? ? 0 ,

one obtains

d (T cos ? ) ? 0 ,

4

which implies a constant horizontal force along a catenary. By defining horizontal force N and wet weight per unit length Q: N ? T cos ? ? const , Q ? mg , (7)

where m is the wet mass per unit length and g the acceleration of gravity, the relationship between the horizontal force N and wet weight per unit length Q can be derived using vertical equilibrium equation: ? T sin ? ? Qds ? (T ? dT ) sin ? ? 0 , or or

d (T sin ? ) ? Qds , d (T cos ? tan ? ) ? Qds .

(8)

Substituting (3) and (7) into (8) gives

d (N dy dy ) ? Qdx 1 ? ( ) 2 , dx dx

d2y dy ? Q 1 ? ( )2 . 2 dx dx

or

N

(9)

Substituting Equation (1) to (9), the relationship between the horizontal force N and wet weight per unit length Q is obtained:

N ? aQ ? const , or a ? N . Q

(10)

This implies that the curvature radius at origin can be adjusted by varying the horizontal force N or the wet weight per unit length Q.

3.2

General Characteristics of Lazy-Wave Riser

The static equilibrium of the LWR illustrated in Figure 3.3 (a) and (b) indicates that, the resultant forces at the sag bend B and the arch bend D are horizontal and both equal to the horizontal force N at the touchdown point. In the vertical direction, there are no shear forces at points B and D, which requires static equilibrium of the weight of the jumper section BC and net buoyancy force of the lift section CD, as well as the drag section DE and the touchdown catenary EF. In other words, the net buoyancy force of the lift section CD “lifts” the wet weight of the jumper section, and that of the drag section “drags” the wet weight of the touchdown section. 5

Lift Section D C N B Jumper Section

N

N D E Drag Section

Touchdown Catenary (a) (b) F

N

Figure 3.3 – Static Equilibrium Analysis of the LWR

In local coordinate systems x-B-y, u-D-v and p-F-q, as shown in Figure 3.4, the hang-off catenary ABC, the buoyancy catenary CDE and the touchdown catenary EF can be expressed as the following equations:

y ? ai (cosh x p u ? 1) , v ? a j (cosh ? 1) , q ? ak (cosh ? 1) . ai ak aj (11)

where ai , a j and ak are the curvature radii at their corresponding origins B, D, and F.

Hangoff Catenary (i) S1 A β θ S2 Buoyancy Catenary (j) S3 S4 Touchdown Catenary (k) S5

y1

y u D v x

V

y3 y2

B C

y4

E q

ya y5

p

ys

F x1 x2 x3 H x4 x5

Figure 3.4 – Local Coordinate Systems of Catenary Equations

6

Assuming mi , m j and mk the wet masses of steel pipes with internal fluid at unit length of the hang-off catenary, the buoyancy catenary and the touchdown catenaries, the wet weights of riser per unit length are

Qi ? mi g , Q j ? F ? m j g , Qk ? mk g ,

(12)

where F is the net buoyancy force from the buoyancy modules pointing upward. Application of the dimensions in Figure 3.4 to Equation (5), the arc lengths of the jumper section and the lift section become

S 2 ? ai sinh

x2 x , S 3 ? a j sinh 3 . ai aj

(13)

The vertical equilibrium in Figure 3.3 (a) requires S 2Qi ? S3Q j , or S2 Q j ? S3 Qi (14)

Equation (14) implies that the lengths of the jumper section and the lift section are proportional to their vertical weight at unit length. Further derivations using Equation (10) give

S 2 x2 y2 Q j ai ? ? ? ? . S3 x3 y3 Qi a j

(15)

Similarly, for the drag section and the touchdown catenary, the following is true:

S 4 x4 y4 Qk a j ? ? ? ? . S5 x5 y5 Q j ak

(16)

Usually same materials and same geometric properties are utilized for both the hang-off catenary and the touchdown catenary, that is

Qi ? Qk or ai ? ak .

(17)

Combination of Equations (15) through (18) obtains

S ? S5 Q j S 2 S5 Q j ? ? ? , or 2 . S3 S 4 Qi S3 ? S 4 Qi

3.3

(18)

Configurations of Lazy-Wave Riser

For a lazy-wave configuration, there are usually two design input options. One option give the lengths of the three catenaries, and the other the elevations of the sag bend and the arch bend.

7

For design input option 1, given the lengths of the hang-off catenary Si , the buoyancy catenary S j , either the water depth V at the hang-off location A or the touchdown catenary length S k , the configuration of the LWR is determined. Without loss of generality, the touchdown catenary length S k is assumed provided herein, as presented in Equations (19) through (21).

Si ? S1 ? S 2 S j ? S3 ? S 4 Sk ? S5

(19) (20)

The total length of the LWR is therefore

S ? Si ? S j ? S k

(21)

Combination of Equations (19) through (21) provides the arc length of the hanging section:

S1 ? S ? (1 ? Qj Qi

)S j .

(22)

On the other hand, application of Equation (6) to the hang-off location gives

S1 ? ai tan ? ? ai cot ? ,

(23)

where ? is the given top hang-off angle, and ? ? 900 ? ? is the inclination angle at the hang-off location, as shown in Figure 3.4. Substituting Equation (22) into Equation (23) obtains the curvature radius of the sag bend or the touchdown point:

Qj ? ? ai ? ? S ? (1 ? ) S j ? tan ? . Qi ? ?

(24)

Substituting Equations (23) and (24) into Equations (5), (10) and (11), one obtains the spread and height of the hanging section:

x1 ? ai arcsin h(cot ? ) , y1 ? ai (cosh x1 ? 1) . ai

Equations (11), (13) and (19) give the spread and height of the jumper section:

x2 ? ai arcsin h( Si ? S1 x ) , y2 ? ai [cosh( 2 ) ? 1] . ai ai

Equation (15) gives the spread and the height of the lift section: 8

x3 ?

x2Qi yQ , y3 ? 2 i . Qj Qj

Equations (16) and (21) provide the length of the drag section:

S4 ? Qi ( S ? Si ? S j ) . Qj

Subsequently, the curvature radius at arch bend is

aj ? Qi ai , Qj

the spread and height of the drag section are

x4 ? a j arcsin h( S4 x ) , y4 ? a j [cosh( 4 ) ? 1] , aj aj

and the spread and height of the touchdown catenary are

x5 ? ai arcsin h( S ? Si ? S j ai

) , y5 ?

y4Q j Qi

.

With all the dimensions calculated above, the sag bend height is determined by:

y s ? y 4 ? y5 ? y 2 ? y3 ,

and the arch bend height becomes

y a ? y 4 ? y5 .

The water depth V at the hang-off location can be checked by

V ? y1 ? y 4 ? y 5 ? y 2 ? y3 .

For design input option 2, the LWR configuration can also be uniquely determined if the sag bend height ys , the arch bend height ya and the water depth V at the hang-off location A are provided. Starting with the height of the hanging section

y1 ? V ? ys ,

Equations (6) and (11) give the curvature radius at the sag bend as

ai ? y1 . cosh[arcsin h(cot ? )] ? 1

(25)

The curvature radius at the arch bend is given by Equations (15) and (25) as

aj ? Qi y1 . Q j cosh[arcsin h(cot ? )] ? 1

(26)

9

The spread of the hanging section is calculated from Equation (25) as

x1 ? ai arcsin h(cot ? ) .

Given the arch height, or the vertical distance between the arch bend and the sag bend,

y 2 ? y3 ? y a ? y s . Q j ( ya ? y s ) Qi ? Q j

(27)

Equations (15) and (27) produce the heights of the jumper section and the lift section:

y2 ?

, y3 ?

Qi ( ya ? ys ) . Qi ? Q j

Subsequently, the spreads of the jumper section and the lift section are thus

x2 ? ai arccos h(

y2 y ? 1) , x3 ? a j arccos h( 3 ? 1) . ai aj

The same derivations apply to the drag section and the touchdown catenary. Given the arch bend elevation

y 4 ? y5 ? y a ,

(28)

Equations (16) and (28) yield the heights of the drag section and the touchdown catenary:

y4 ?

Q j ya Qi ya , y5 ? . Qi ? Q j Qi ? Q j

Hence their spreads can be expressed as

x4 ? a j arccos h(

y y4 ? 1) , x5 ? ai arccos h( 5 ? 1) . ai aj

As a result, the arc length of each catenary can be calculated using Equation (5) as follows:

Si ? ai (sinh

x1 x x x x ? sinh 2 ) , S j ? a j (sinh 3 ? sinh 4 ) , S k ? S5 ? ai sinh 5 . ai ai ai aj aj

For both design input options, the total horizontal distance from the hang-off location to the touchdown point is

H ? x1 ? x2 ? x3 ? x4 ? x5 ,

the horizontal force at any point along the LWR and at touchdown point is

N ? aiQi ? a j Q j ? ak Qk ,

and the top tension at hang-off location is

T ? N / sin ? .

(29)

(30)

10

Combination of Equations (29) and (30) gives

ai ?

T sin ? T sin ? T sin ? , aj ? , or ak ? . Qi Qj Qk

(31)

4 Dynamic Response of Lazy-Wave Riser

4.1

Wave and Drift Motion Response of Lazy-Wave Riser

The first order motions in heave, surge and sway directions at the hang-off point A are mainly translated into heave motions at the sag bend B due to the constraint of the length of the hangoff catenary and the drag of the buoyancy modules at lift point C, as shown in Figure 4.1. For example, the horizontal surface motion ?x as a function of time at point A becomes a heave dominated motion ?y at sag bend B due to the motion difference in amplitude and phase between A and C.

Y P ?x A

R Q ?y B

C X

Figure 4.1 – Heave Motion at Sag Bend due to Horizontal Motion at Hang-Off Point

As opposed to the first order motion, the relatively long period of the second order motion gives the more time for the LWR to respond globally, especially along the buoyancy catenary. The second order motions at the hang-off point A turn into both horizontal motion and heave motion at the sag bend, as shown in Figure 4.2. For example, a horizontal second order motion at the hang-off point A is followed by dominant horizontal motions at the lift point C and the drag point E. This generates open and close movements of the sag bend or the arch bend, fluctuating curvature radii along LWR, especially at such critical locations as the sag bend B, the arch bend D and touchdown point F. 11

Hangoff Catenary ?xA A β θ

Buoyancy Catenary

Touchdown Catenary

D

?uD

V

B

?uC C ?uB E ?uE

F x1 x2 x3 x4 x5

?uF

Figure 4.2 – Illustration of Second Order Motion along LWR

Techically, the critical locations for curvature radius refer to critical zones. The local top point D may travel along the buoyancy catenary, such that the points near the arch bend D on the riser take turns to become the local top elevation to account for tension variations and section length adjustment during dynamic motions. In this sense, the arch bend is actually a zone in length instead of a fixed point, so are the sag bend and the touchdown point. The length of these critical zones varies with dynamic motions as well as vessel offsets, as shown in Figure 4.3. At a far offset of 10% water depth, a LWR shape with a decent arch height at near offset may degenerate into a low arch LWR, or even a shaped SCR if failed to optimize the buoyancy catenary for large offsets. Extreme far and near offset positions should be checked for a LWR configuration to avoid undesired buckling problem of a low-arch configuration, and to avoid local high stress at the arch bend at near offset. Critical curvatures along LWR are associated with extreme vessel positions under driving load conditions and are functions of the horizontal force N or top tension T and hang-off angle θ as indicated in Equation (31). A configuration with higher top tension T or greater hang-off angle 12

θ gives larger critical curvature radii and generally provides a better dynamic strength response. Restricted by vessel payload, sag bend can also be adjusted for a given top tension cap. The density, length and thickness of the buoyancy modules are among other key parameters for optimization of LWR configuration.

Figure 4.3 – Variation of LWR Configuration with Large Offsets

4.2

Strength Response of Lazy-Wave Riser

Taper stress joint or flex joint is widely used for stress relief at the hang-off location, while other critical locations along LWR have to rely on the optimization of the buoyancy catenary. The horizontal motion and the heave motion at the hang-off location lead to an open and close movement of the sag bend, the arch bend and the touchdown zone as illustrated in Figure 4.2. The lowest curvature radii at these critical locations governs their stress response since stress and bending moment are proportional to curvature, as shown in Figure 4.4. The highest stress frequently occurs at the arch bend where the most critical curvature is more likely when the lift point and the drag point move out of phase with the vessel at near offset position. On the other hand, the preference of high net buoyancy force for arch height optimization also results in lower curvature at the arch bend. Other critical curvatures occur at the touchdown point and 13

the sag bend. For an optimized LWR, the stress at the touchdown point and the sag bend is normally not driving in strength response, which is another feature of the LWR.

0.8 0.7 Von Mises / Yield Stress Ratio 0.6 0.5 4000 0.4 3000 0.3 0.2 0.1 0.0 0 1000 2000 3000 4000 5000 6000 Horizontal Distance from Hangoff (ft) 2000 7000

6000

5000

1000

0 7000

Figure 4.4 – Strength Response of High Arch Lazy-Wave Riser

Riser mass density is another factor for LWR optimization. A lighter riser or smaller Qi including internal fluid improves the strength response at the sag bend and the touchdown point as suggested in Equation (12). As equivalent negative mass along the buoyancy catenary where Q j ? F ? m j g , lower net buoyancy force F favors a greater curvature radius, but may result in an undesirable low-arch configuration at vessel far offset. For instance, the stress response of a low-arch lazy-wave configuration with sag bend and arch bend elevations of 2900 ft and 3100 ft, respectively, is shown in Figure 4.5. Heave motion at the sag bend becomes whipping and buckling wave motion between sag bend and the arch bend, which results in local curvature and stress much higher than those otherwise at the sag bend or the arch bend. The structural and hydrodynamic damping effect is significantly compromised in this case.

14

Elevation from Seabed (ft)

0.8

7000

Von Mises Stress / Yield Strength Ratio

0.7

6000

0.5

4000

0.4

3000

0.3

2000

0.2

1000

0.1 0 1000 2000 3000 4000 5000

0 6000

Horizontal Distance from Hangoff (ft)

Figure 4.5 – Strength Response of Low Arch Lazy-Wave Riser

4.3

Motion Fatigue Response of Lazy-Wave Riser

Strength response is driven by extreme curvature at critical locations, while fatigue response is controlled by curvature fluctuation range associated with higher stress range. The highest curvature fluctuation occurs at the sag bend, the arch bend or the touchdown point. The low fatigue lives occur at these high curvature fluctuation zones, as shown in Figure 4.6. Critical fatigue damage is observed at the touchdown point and at the riser hang-off location. The fatigue damage at the touchdown zone is driven by soil-structure interaction under more frequent occurrence of small stress cycles from low seastates, while at the top hang-off section it is driven by combination of tension and bending moment from less frequent but high stress range from high seastates. Three case studies are presented in this section for sensitivity of the fatigue response of LWR attached to an internally turret-moored FPSO. The first case explores the first order fatigue sensitivity to arch heights. Three configurations with different arch heights – namely high arch, mid arch and low arch – are developed by 15

Elevation from Seabed (ft)

0.6

5000

varying the length of the buoyancy catenary. The vessel payload remains the same in all configurations with a constant hang-off angle and sag bend elevation. It is observed that fatigue life at the touchdown point (TDP) improves with arch height. With the same buoyancy thickness, a higher arch configuration requires a longer buoyancy catenary that provides more damping to motions. The buoyancy lengths of the three arch heights are compared in Figure 4.7, and their first order motion fatigue lives are presented in Table 4.1. The fatigue life of the mid-arch configuration at the TDP improves slightly from the low-arch configurations, but that of the high-arch configuration doubled. A higher arch diverts and damps cable wave motion in more favorable directions. This also indicates nonlinear redistribution of structural damping and hydrodynamic damping between different configurations.

1.E+12 1.E+11 1.E+10 Minimum Fatigue Life (years) 1.E+09 1.E+08 1.E+07 1.E+06 1.E+05 1.E+04 1.E+03 1.E+02 1.E+01 1.E+00 0 1000 2000 3000 4000 5000 6000 Horizontal Distance From Hangoff (ft) Fatigue Life Slick Section Buoyancy Section 0 7000 2000 3000 4000 6000 Elevation above Seabed (ft) 7000

5000

1000

Figure 4.6 – Example of Fatigue Response along Lazy-Wave Riser

16

7000

6000 Elevation from Seabed (ft)

5000

Archbend

Sag Bend Height: 2400ft Arch Bend Heights: Low Arch 2600ft Mid Arch 3100ft High Arch 3600ft

4000

3000

2000

Sagbend

1000

0 0 1000 2000 3000 4000 5000 6000 7000 Horizontal Distance From Hangoff (ft) Low Arch Mid Arch High Arch Buoyancy 1742 ft Buoyancy 2261 ft Buoyancy 2722 ft

Figure 4.7 – Buoyancy Lengths Variation with Arch Heights Riser Configuration Arch Type Arch Height (ft) High Arch 1200 Mid Arch 700 Low Arch 200 Minimum Fatigue Life at TDP (years) 1,500 750 670

Table 4.1 – First Order Motion Fatigue Life Variation with Arch Configurations

The second case compares the first order fatigue responses to sag bend elevations. Two configurations with different sag bend elevations are developed as shown in Figure 4.8. The hang-off angle and arch height are kept constant. The lower sag bend configuration improves the TDP fatigue life by 80%, however decreases the fatigue response at top of the riser by 13%, as compared in Table 4.2. The fatigue life increase at the TDP can be justified from Equation (31). The lower sag bend configuration has longer hang-off catenary length hence greater hang-off tension and smaller TDP curvature and bending stress range. The higher dynamic tension fluctuation near the hang-off location adversely contributes to the fatigue performance at top of the riser.

17

7000 6000 Elevation from Seabed (ft) 5000 Buoyancy 2722 ft 4000 3000 2000 Low Sagbend 1000 0 0 1000 2000 3000 4000 5000 6000 7000 8000 Horizontal Distance From Hangoff (ft) Lazy Wave Sagbend 2400ft Lazy Wave Sagbend 600ft Buoyancy 10inch Thickness Buoyancy 10inch Thickness High Sagbend Buoyancy 2189 ft

Figure 4.8 – Lazy-Wave Configuration with Varying Sag Bend Elevations Location Minimum Motion Fatigue Life (years) Sag Bend 2400 ft Sag Bend 600 ft 1,500 2,700 2,400 2,100

Touchdown Point Top Taper Stress Joint

Table 4.2 – Motion Fatigue Life Sensitivity to Sag Bend Elevations

The third case study is about fatigue response to current loading. Background current interferences riser motion and helps dissipating cable wave energy in an addition to hydrodynamic damping. As shown in Figure 4.9, the TDP fatigue life improves with increased background current speed. Background current profile should be used with caution to reduce conservativeness. Generally, application of background profile with 50% occurrence increases TDP fatigue life 2~3 times from without background current depending on variation of current directions and speeds.

18

14,000

12,000 Minimum Fatigue Life (years)

10,000

8,000

6,000

4,000

2,000 Base Case (no background current) 0 0 0.1 0.2 0.3 0.4 0.5

Surface Speed of Background Current Profile (knots)

Figure 4.9 – Effect of Background Current on Motion Fatigue Life at Touchdown Point

5 Summary

Consisting of three catenaries, namely the hang-off catenary, the buoyancy catenary and the touchdown catenary, a lazy-wave riser has better strength and fatigue responses than SCR. The buoyancy catenary produces effective hydrodynamic and structural damping to attenuate cables waves from vessel motions propagating along lazy-wave riser. The critical locations for strength and fatigue responses are at the top hang-off location, the sag bend, the arch bend and the touchdown point. This work provides parameterized equations for configuration optimization and strategic analysis. The damping efficiency of the buoyancy catenary and the variation of the curvature radii at the critical locations in conjunction with the top tension and hang-off angle are the driving factors for lazy-wave riser strength and motion fatigue responses. The horizontal force along the riser is a constant and a function of net riser wet weight at the hanging section and the top hang-off angle. The curvature at the critical locations is a function of horizontal force and submerged weight of riser section, which drives dynamic response. Dynamic response is sensitive to arch height, sag bend elevation, top hang-off angle and background current loading. 19

6 References

[1] [2] Torres, A.L.F.L. et al (2002), “Lazy-wave steel rigid risers for turret-moored FPSO”, OMAE’02/OFT-28124 Jaco, B.P. et al (2008), “Synthesis and optimization of steel catenary risers configurtations through evolutionary computation,” 5th Report for Petrobras, COPPE/UFRJ Edmundo Queiroz de Andrade et al (2010) “Optimization Procedure of Steel Lazy Wave Riser Configuration for Spread Moored FPSOs in Deepwater Offshore Brazil”, OTC 20777 Hugh Howells, (1995). “Advances in Steel Catenary Riser Design: Advances in Steel Catenary Riser design”, DEEPTEC '95, Aberdeen, February 1995 Bin Yue et al, (2010) “Improved SCR Design for Dynamic Vessel Applications”, OMAE2010-20406, Beijing, June 2010 Lockwood, E.H. (1961) “A Book of Curves”, Cambridge University Press.

[3]

[4] [5] [6]

20

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