Analytical Prediction of Pipeline Behaviorsin J-Lay on Plastic Seabed

Li-Zhong Wang1; Feng Yuan2; Zhen Guo3; and Ling-Ling Li4

Abstract: J-lay is widely accepted as a favorable method for deepwater pipeline installation. The configuration and internal force of thepipeline during the laying process are key factors for its safety, and the embedment induced during pipeline installation plays an importantrole for pipeline stability in service. Traditionally, the seabed in the analysis of J-lay is assumed to be elastic, whereas deepwater deposits areusually very soft, exhibiting low strength and obvious plasticity. This paper presents an analytical model for pipelaying on plastic seabed. Thismodel simplifies the pipeline as the combination of four segments: a natural catenary in water, a boundary-layer segment, a beam on theseabed (touchdown zone) where the soil deforms plastically, and a freely laid horizontal segment. The comparison with a traditional elastic-seabed model reveals that both models have similar pipeline configurations, although the elastic-seabed model slightly underestimates themaximum bending moment. In contrast, the pipeline embedment depths in the two models differ with deeper ultimate penetration predicted inthe plastic-seabed model. DOI: 10.1061/(ASCE)WW.1943-5460.0000109. © 2012 American Society of Civil Engineers.

CE Database subject headings: Underwater pipelines; Safety; Embedment.

Author keywords: Pipeline; J-lay; Analytical model; Plastic seabed; Embedment.

Introduction

The development of offshore oil and gas exploitation has drawnmuch attention to the application of pipelines and finding the mostsuitable and cost-effective methods for transporting crude oil andgas. During its life span, a pipeline may encounter the higheststresses during the installation process (Powers and Finn 1969;Brando and Sebastiani 1971), especially in deep water. Ensuringthe safety of pipelines during deepwater pipeline installation is ahuge challenge because of the hostile ocean environment. Amongall available pipeline installation methods, J-lay has been widelyaccepted as a favorable method for deepwater pipelaying. More-over, the benefits of J-lay are magnified as the water depthincreases (McDdonald et al. 1998).

To predict the pipeline behavior during J-lay process, formerresearchers have pioneered work through both analytical andnumerical methods. The numerical methods have become well de-veloped in recent decades. Some sophisticated numerical tools,such as OFFPIPE, SIMULA, and PIPELAY, are already availablefor both static and dynamic analyses in 2D or 3D spaces (Savik

et al. 2004; Callegari et al. 2003). With regard to the analyticalmethods, they are still important because they are simple and rig-orous theories that can help engineers understand the mechanisminvolving pipeline installation. Moreover analytical methods arecost-effective and can provide quick and reliable results comparedwith the numerical methods. The earliest analytical solution can betraced back to the work of Plunkett (1967), who adopted the per-turbation method to derive asymptotic expansion to consider thebehavior of stiffened catenary. The perturbation method has beenwell developed more in the following years (Dixon and Rultledge1968; Palmer et al. 1974; Guarracino and Mallardo 1999) and itproves simple and effective. However, it is not able to considerthe pipe embedment into the seabed. The importance of pipelineembedment has been recognized for a long time (Aubeny et al.2006; Chai and Varyani 2006). To consider the influence of pipe-line embedment, Lenci and Callegari (2005) created some analyti-cal models with linearly elastic seabed and obtained a consistent,analytical solution. Aubeny (2006) improved the linear elasticseabed by using nonlinearly elastic spring to model the pipeline-seabed interaction. Chai (2006) adopted a different elastic seabedby introducing an extra group of lateral seabed springs to considerthe influence of lateral force of the seabed. Actually, most state-of-the-art pipeline-laying analyse use either rigid or linear elasticcontact to simulate the seabed (Bridge and Laver 2004). Pipelineon elastic seabed can penetrate into the seabed by its self-weight. Infact, the penetration depth of the pipeline is always exceeded be-cause of its self-weight. Some suggest that such over-penetrationresults from the effect of pipeline dynamic movement during thelaying process. The pipeline dynamic movement is definitely animportant factor, yet this work demonstrates that the plastic defor-mation of the seabed also plays an important role when the pipelineinitially penetrates into the seabed. In deep water, the deposits at theseabed surface usually exhibit very low strength and obvious plas-ticity. Many researchers have investigated the interaction betweenthe pipeline and the plastic seabed (Randolph and White 2008;Cheuk et al. 2008; Hodder and Cassidy 2010) to reveal the staticvertical embedment of the pipeline. Simplifying the pipe–soil

1Professor, College of Civil Engineering and Architecture, ZhejiangUniv., Yuhangtang Road 388, Hangzhou 310058, Zhejiang, China (corre-sponding author). E-mail: wlzzju@163.com

2Ph.D. Candidate, College of Civil Engineering and Architecture,Zhejiang Univ., Yuhangtang Road 388, Hangzhou 310058, Zhejiang,China. E-mail: yuanfen5742@163.com

3Ph.D. Candidate, College of Civil Engineering and Architecture,Zhejiang Univ., Yuhangtang Road 388, Hangzhou 310058, Zhejiang,China. E-mail: nehzoug@163.com

4Doctor, College of Civil Engineering and Architecture, Zhejiang Univ.,Yuhangtang Road 388, Hangzhou 310058, Zhejiang, China. E-mail:lingzju@126.com

Note. This manuscript was submitted on July 2, 2010; approved onJune 16, 2011; published online on June 18, 2011. Discussion period openuntil August 1, 2012; separate discussions must be submitted for individualpapers. This paper is part of the Journal of Waterway, Port, Coastal, andOcean Engineering, Vol. 138, No. 2, March 1, 2012. ©ASCE, ISSN 0733-950X/2012/2-77–85/$25.00.

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interaction as a plain strain problem, most researchers focused on apipeline section on the seabed, so the effects on the embedmentalong the pipeline when laying pipeline when laying pipeline onthe plastic seabed have not yet been sufficiently considered. Palmer(2008) pointed out the importance of considering the plastic defor-mation of the seabed and suggested a pipe–soil interaction model ofshort pipeline segments on rigid plastic seabed. The model needsmore work, for two primary reasons. First, the model focuses onlyon the localized pipeline near the touchdown point, which makes itimpossible to investigate the overall pipeline behavior from the lay-ing vessel to the seabed. Second, the seabed reaction is assumed tobe uniformly distributed along the touch down zone of the pipeline,although most published works reveal that the reaction varies withthe penetration depth. Accurate prediction of the pipe–soil interac-tion is significant because it determines the embedment depth andinternal force distribution of the pipeline. The pipeline embedmentwill further influence the behavior of pipeline at work, especially incase of buckling and walking.

This paper presents an alternative, more reasonable analyticalmodel that takes into account the overall pipeline behavior andthe plastic deformation of the seabed. This model extends the local-ized model of Palmer (2008) and considers the overall pipelinebehavior from the water surface to the seabed, so the laying effect,which primarily refers to the load concentration of the pipeline nearthe touchdown zone, can be well considered. The seabed resistancecan be simplified to be either uniform or nonuniform according tothe seabed characteristics. Comparing the two characteristics with atraditional elastic-seabed model (Lenci S. and Callegari M. 2005)reveals the need for considering the plastic deformation. The twomodels agree on the internal loads distributions, except that theelastic-seabed model slightly underestimates the maximum bend-ing moment.

The pipeline embedment depths in the two models differ in thatthe plastic-seabed model predicts deeper ultimate penetration.

Governing Equations

As shown in Fig. 1, two coordinate systems are used in this model:one global coordinate system ðx; yÞwith its origin at (0, 0) of touch-down point (TDP), and another local coordinate system ðx1; y1Þwith its origin at P2. The model shown consists of four segments:1. Catenary segment: The segment suspended in water from P1 to

P2 is long and flexible, simplified as a natural catenary.2. Boundary-layer segment: The pipeline from P2 to the TDP is

considered an individual segment because it stays very close to

the seabed, exhibits boundary-layer phenomenon, and behaveslike a beam. The boundary-layer phenomenon has attracted theattention of many researchers. Croll (2000) pointed out that theboundary layer is a place where the suspended pipeline locallydeviates from the basic catenary suspended profile, and a zonewhere potential failures can occur, and the combination of hightension and local bending can produce critical design situa-tions. The boundary-layer segment is clearly defined by itshorizontal length l2 which is part of the final solution.

3. Touch down segment: This segment is a distinguishing featureof this model; it is laid on the seabed at the neighborhood of theTDP, behaving like a beam and reaching the maximum settle-ment at P3. Unlike the traditional models, the seabed support-ing the pipeline undergoes plastic deformation. The length ofthis segment is defined by its horizontal length l3, which is partof the final solution.

4. Horizontal segment: The fourth segment is laid horizontally onthe seabed at the maximum penetration depth at the left of P3.No more deformation of soil supporting this segment occursand the soil reaction at this segment balances only the pipelineself-weight.The four segments constituting this model are described by

different governing equations according to their different mechani-cal features, and they are properly connected through continuityconditions of the geometry and internal forces.

Catenary Segment

The well-known governing equation for catenary is used to modelthe segment suspended in water (Seyed 1992):

dðtanφÞds

¼ pH

ð1Þ

where φ is the inclination slope; s is the arc length of the pipeline;p is the submerged unit weight; f and H is the constant horizontalcomponent of the axial tension.

Considering that ds ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiðdx1Þ2 þ ðdy1Þ2

pand tanφ ¼ dy1

dx1,

Eq. (1) can be transformed in to

d2y1dx21

¼ pH

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1þ

�dy1dx1

�2

sð0 ≤ x1 ≤ l1Þ ð2Þ

where y1 is the deformed configuration in ðx1; y1Þ coordinate sys-tem with the origin located at the P2, l1 is the horizontal distance ofthe catenary segment, as shown in Fig. 1.

The solution of Eq. (2) can be derived by elementary algebra:

y1ðx1Þ ¼ c1 þHpcosh

�pHx1 þ c2

�ð0 ≤ x1 ≤ l1Þ ð3Þ

where c1 and c2 are unknown coefficients. The slope angle, curva-ture, and tension are then obtained as follows:

φðx1Þ ¼ arctan

�dy1dx1

�ð4Þ

κðx1Þ ¼dφds

¼ p3

H3

�d2y1dx21

��2ð5Þ

T1ðx1Þ ¼H2

pd2y1dx21

ð6Þ

The natural catenary is not able to sustain bending moment, butthe approximated bending moment can be expressed by the productof the bending stiffness EI and the curvature as follows:Fig. 1. Sketch of the model

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M1ðx1Þ ¼ EIκðx1Þ ¼EIp3

ðd2y1dx21Þ2H3

ð7Þ

Whereafter, the approximated shear can be expressed as thedifferentiation of the bending moment:

S1ðx1Þ ¼dM1

ds¼ dM1

dx1

dx1ds

¼ �2EI

�pH

�6 dy1dx1

�d2y1dx21

��4ð8Þ

wheredx1ds

¼ dx1ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiðdx1Þ2 þ ðdy1Þ2

p ¼ 1ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1þ ðdy1dx1

Þ2q ð9Þ

Boundary-Layer Segment

Considering that the inclination slope of the boundary-layersegment is small as it lies close to seabed, the governing equa-tion can be obtained by taking the force equilibrium in the verticaldirection:

EId4y2dx4

� Td2y2dx2

¼ p ð0 ≤ x ≤ l2Þ ð10Þ

where y2 as shown in Fig. 2, is the configuration of the boundarylayer; T is the unknown constant axial tension at the TDP; and thechange on the tension T can be neglected (Croll 2000). The solutioncan be written as follows:

y2ðxÞ ¼�p2T

x2 þ c3 þ c4xþ c5eðγxÞ þ c6eð�γxÞ ð0 ≤ x ≤ l2Þð11Þ

where c3, c4, c5, and c6 are the unknown coefficients and

γ ¼ffiffiffiffiffiTEI

rð12Þ

In contrast, Palmer (2008) obtained the governing equation bytaking the force equilibrium in the normal direction along the pipe-line length. Eq. (11) agrees well with Palmer (2008) except thatPalmer obtained it in the local coordinate system ðx; sÞ, where sis the distance measured along the pipe from the TDP.

The bending moment and shear can be obtained as:

M2ðxÞ ¼ �EId2y2dx2

ð13Þ

S2ðxÞ ¼ �EId3y2dx3

ð14Þ

Touch Down Segment

Traditional models with elastic seabed consider the whole seabed atthe left of the TDP to be a whole part where elastic deformationoccurs. In this work, seabed at the left of the TDP is divided intotwo segments, the touchdown segment and the horizontal segment,

as shown in Fig. 2. The soil beneath the touch down segment under-goes plastic deformation and a permanent trench is formed, whichis one of the distinguishing features of this model.

The pipe–soil interaction is very complicated and some pioneer-ing work has been done to reveal the interaction mechanismbetween the pipeline and the seabed (Murff et al. 1989; Merifieldet al. 2008; Steward and Randolph 1994). Most researchers con-cluded that the relationship between pipe embedment and the soilresistance is nonlinear, and some obtained the power law expres-sions. Aubeny et al. (2005) suggested the following:

RSud1

¼ a

�y3d1

�b

ð15Þ

where RðxÞ is the soil resistance to a pipe segment of a unit length;d1 is the outer diameter of the pipeline; y3ðxÞ is the pipeline embed-ment depth as shown in Fig. 2; coefficients a and b are constantsrelated with pipe roughness and the variation of shear strength ofsoil. The shear strength Su increases linearly with the pipelineembedment depth Su ¼ Su0 þ k1y3, where k1 is the increasing rateand Su0 is the shear strength in the mudline. Then, the soil resis-tance can be expressed as:

RðxÞ ¼ Su0ad1�b1 yb3 þ k1ad1�b

1 y1þb3 ð16Þ

where the coefficients a and b can be obtained from the work ofAubeny et al. (2005). The value of b is very small and locates inthe neighborhood of 0.2, so the relationship between soil resis-tance and the pipe embedment depth can be approximated asRðxÞ ¼ R0 þ k2y3, where R0 is the resistance at the seabed surfaceand k2 is the increasing rate of the soil resistance.

The inclination slope of the boundary layer is small, and thegoverning equation of the touch down segment can be derivedby taking the force equilibrium in the vertical direction:

EId4y3dx4

� Td2y3dx2

þ k2y3 ¼ p� R0 ð�l3 ≤ x ≤ 0Þ ð17Þ

where l3 is horizontal distance of the touchdown segment. ForT > 2

ffiffiffiffiffiffiffiffiffiEIk2

p, no real solution can be obtained, whereas for

T ≤ 2ffiffiffiffiffiffiffiffiffiEIk2

p, which should be checked at the end of calculation

and is always satisfied in this work, the general solution can beobtained as:

y3ðxÞ ¼p� R0

k2þ eαx½c7 cosðβxÞ þ c8 sinðβxÞ�

þ e�αx½c9 cosðβxÞ þ c10 sinðβxÞ�ð�l3 ≤ x ≤ 0Þ

ð18Þ

where c7, c8, c9, and c10 are unknown coefficients and

α ¼ 12

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi2

ffiffiffiffiffik2EI

rþ TEI

sð19a Þ

β ¼ 12

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi2

ffiffiffiffiffik2EI

r� TEI

sð19b Þ

where the change of the axial tension T is neglected (Lenci andCallegari 2005). Then the bending moment and shear can beobtained as:

M3ðxÞ ¼ �EId2y3dx2

ð20ÞFig. 2. Sketch of the segments near the touchdown zone

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S3ðxÞ ¼ �EId3y3dx3

ð21Þ

The governing equations of this segment are distinguishablefrom those of the pipeline on elastic seabed. Lenci and Callegari(2005) suggested the elastic seabed at the left of TDP is infiniteand pipeline laid on seabed behaves like an infinite beam, soEq. (18) is simplified with c9 and c10 vanishing because ofboundary conditions at x → �∞. In comparison, the seabed whereplastic deformation occurs is finite and limited within the horizontallength l3.

Meanwhile, the soil resistance can also be assumed to be uni-formly distributed as suggested by Palmer (2008). The soil resis-tance remains a constant R0 from seabed surface at the TDP to theend of the touch down segment at P3, which must be larger thanthe submerged weight p. The inclination slope of this segment issmall, so the governing equation of the touchdown segment can bederived as:

EId4y3dx4

� Td2y3dx2

¼ p� R0 ð�l3 ≤ x ≤ 0Þ ð22Þ

The general solution can be obtained as

y3ðxÞ ¼R0 � p2T

x2 þ c11 þ c12xþ c13 sinhðγxÞ þ c14 coshðγxÞð�l3 ≤ x ≤ 0Þ ð23Þ

where c11, c12, c13, and c14 are unknown coefficients. Then thebending moment and shear can be obtained by the same equationsas Eqs. (20) and (21).

In Eqs. (11) and (23), nine unknowns are included: c3, c4, c5, c6,c11, c12, c13, c14, and l3. At P3, the inclination slope, curvature, andshear force must all be zero. At the TDP, y2ð0Þ ¼ 0, y3ð0Þ ¼ 0, theinclination slope, curvature, and shear force should all be continu-ous. Therefore, we have nine unknowns and eight conditions in thetwo equations. Then the length l3 can be expressed as the functionof c5:

l3 ¼ ln

�R0

ð2c5γ2T þ R0 � pÞ� ffiffiffiffiffi

EIT

rð24Þ

Palmer (2008) suggested that c5 be zero for his anlytical solu-tion; however, in mathematical view, c5 is not zero and has influ-ence on the calculation results. If c5 ¼ 0, we have:

l3 ¼ ln

�R0

R0 � p

� ffiffiffiffiffiEIT

rð25Þ

which is similar to that obtained by Palmer (2008). The former isthe horizontal length in relation to the axial tension T , whereas thelatter is the arc length in relation to the horizontal component of theaxial force.

Eqs. (18) and (23) can be used according to different soil proper-ties. Eq. (18) is adopted in this work for seabed providing linearlyincreasing soil resistance.

Horizontal Segment

The horizontal segment is freely laid on the trench created by thepipe–soil interaction at the touchdown segment. The soil resistanceat this segment, which just balances the pipeline self-weight, is lessthan at the touchdown segment. Therefore, the soil beneath thepipeline will experience an unloading process, which is accompa-nied with soil rebound. Aubeny (2006) pointed out that the unload

stiffness of the soil is very large, which induces very small elasticrebound even when the seabed soil resistance decreases to zero dur-ing the unloading process. The results of You et al. (2008) indicatesthat the elastic rebound takes less than one-sixth of the pipe embed-ment when the seabed is unloaded to provide zero soil resistance.Randolph and Quiggin (2009) proposed a mathematical model ofthe seabed reaction force. The calculation results of the modelillustrate negligible soil rebound even when the seabed resistancedecreases to zero. The rebound during the unloading process is sosmall that it is ignored at this segment, so the pipeline at this seg-ment lies horizontally on the seabed with neither further penetrationnor rebound.

Algorithm

General Equations

For the seabed providing linearly increasing resistance, consideringthe continuity of displacement, inclination slope, bending moment,and shear at P1, TDP, P2, and P3, we face a system of 15 equations,which is highly nonlinear, with c1, c2, c3, c4, c5, c6, c7, c8, c9, c10,l1, l2, l3, T , and H as the unknowns:

P1: y1ðl1Þ ¼ D ð26a Þ

P1:dy1ðxÞdx1

jx1¼l1 ¼ tanðφ0Þ ð26b Þ

P2: � y2ðl2Þ ¼ y1ð0Þ ð27a Þ

P2: � dy2ðxÞdx

jx¼l2 ¼dy1ðx1Þdx1

jx1¼0 ð27b Þ

P2: M2ðl2Þ ¼ M1ð0Þ ð27c Þ

P2: S2ðl2Þ ¼ S1ð0Þ ð27d Þ

P2: T ¼ T1ð0Þ ð27e Þ

TDP: y2ð0Þ ¼ 0 ð28a Þ

TDP: y3ð0Þ ¼ 0 ð28b Þ

TDP:dy2ðxÞdx

jx¼0 ¼dy3ðxÞdx

jx¼0 ð28c Þ

TDP: M2ð0Þ ¼ M3ð0Þ ð28d Þ

TDP: S2ð0Þ ¼ S3ð0Þ ð28e Þ

P3:dy3ðxÞdx

jx¼�l3 ¼ 0 ð29a Þ

P3: M3ð�l3Þ ¼ 0 ð29b Þ

P3: S3ð�l3Þ ¼ 0 ð29c Þ

where D is the water depth; and φ0 is the inclination slope at P1.Compared with traditional models with elastic seabed, the maindifference of this model is the continuity conditions at P3.

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Equations at P1 give

c1 ¼ D� Hp cosðφ0Þ

ð30a Þ

tanh

�pHl1

�¼ � sinhðc2Þ þ coshðc2Þ sinðφ0Þ

coshðφ0Þ � sinhðc2Þ sinðφ0Þð30b Þ

Equations at P2 give

� y2ðl2Þ ¼ c1 þH coshðc2Þ

pð31a Þ

� dy2ðxÞdx

jx¼l2 ¼ sinhðc2Þ ð31b Þ

d2y2ðxÞdx2

jx¼l2 ¼�p

T coshðc2Þð31c Þ

d3y2ðxÞdx3

jx¼l2 ¼2p2 sinhðc2ÞH2cosh4ðc2Þ

ð31d Þ

H ¼ Tcoshðc2Þ

ð31e Þ

Equations at the TDP provide

pkþ c7 þ c9 ¼ 0 ð32a Þ

c3 þ c6 ¼ 0 ð32b Þ

f 4ðc7; c8; c9; c10;α; βÞ ¼ c4 þ γc5 ð32c Þ

f 5ðc7; c8; c9; c10;α;βÞ ¼�pT

þ γ2c6 ð32d Þ

f 6ðc7; c8; c9; c10;α; βÞ ¼ γ3c5 ð32e ÞE The equations at P3 yield

dy3ðxÞx

jx¼�l3 ¼ f 1ðc7; c8; c9; c10;T; l3Þ ¼ 0 ð33a Þ

d2y3ðxÞx2

jx¼�l3 ¼ f 2ðc7; c8; c9; c10;T ; l3Þ ¼ 0 ð33b Þ

d3y3ðxÞx3

jx¼�l3 ¼ f 3ðc7; c8; c9; c10;T; l3Þ ¼ 0 ð33c Þ

Simplified Equations

At a glance, the previous equations reveal that the unknown l1 ap-pears only in Eq. (30b), which indicates it has no influence on thesolution of other unknowns, so l1 is the last to be solved. Then, theprevious 15-equation system can be simplified into a 14-equationsystem excluding l1 and Eq. (30b).

Next, the combination of Eqs. (30a), (31a), and (31e) yield

coshðc2Þ ¼T

f½Dþ y2ðl2Þ�pþ Tg cosðφ0Þð34Þ

c1 ¼ � Tp� y2ðl2Þ ð35Þ

H ¼ f½Dþ y2ðl2Þ�pþ Tg cosðφ0Þ ð36Þ

The tension at P2 can then be obtained from Eq. (36):

T ¼ ½Dþ y2ðl2Þ�p cosðφ0ÞcosðφP2

Þ � cosðφ0Þð37Þ

where φP2is the inclination slope at the P2. If φP2

¼ 0, y2ðl2Þshould also be zero. Then Eq. (37) coincides with that of a naturalcatenary laid on rigid seabed as suggested by Lenci and Callegari(2005):

T ¼ Dp cosðφ0Þ1� cosðφ0Þ

ð38Þ

As the boundary-layer segment lies very close to the seabed,y2(l2) is negligible compared with the water depth D and φP2

isvery small. Thus, the tension T should be close to and slightlylarger than that obtained from Eq. (38).

As shown at the right of Eq. (31b), sinh(c2) represents the pipe-line inclination slope, so it is always positive. On the basis ofEq. (34) and the relationship of cosh2ðc2Þ ¼ sinh2ðc2Þ þ 1:

sinhðc2Þ ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi�

TðDpþ TÞ cosðφ0Þ

�2� 1

sð39Þ

To simplify the previous equation system, the unknowns c3, c4,c5, c6, c7, c8, c9, and c10 are expressed as the function of T , l2, and l3by combining Eqs. (32a)–(32e) and (33a)–(33c). Then, togetherwith Eqs. (34)–(36) and (39), they are inserted into Eqs. (31b)–(31d), which are then simplified to form a system of three equationswith only three unknowns (T , l2; and l3):

sinhðc2Þ þdy2x

jx¼l2 ¼ f 7ðT ; l2; l3Þ ¼ 0 ð40a Þ

d2y2x2

jx¼l2 þp

T coshðc2Þ¼ f 8ðT; l2; l3Þ ¼ 0 ð40b Þ

d3y2x3

jx¼l2 � 2p2 sinhðc2ÞH2cosh4ðc2Þ

¼ f 9ðT; l2; l3Þ ¼ 0 ð40c Þ

where the left of the three equations represent the discrepanciesbetween the inclination slopes, the pipeline curvatures, and thederivatives of the pipeline curvatures, respectively, of the catenarysegment and the boundary layer at P2.

Solution

The previous equation system is so complicated that no explicitsolution can be found. In this work, a numerical method is adopted.As shown in Fig. 3, the tension at P2 should also first be assumedbefore the calculation, and its lower bound obtained from Eq. (38)is suggested as the initial value. In addition, a two-dimensionalðl2; l3Þ space should be established, then a zone where the wantedðl2; l3Þ may exist is specified in this space: l2lower < l2 < l2upper,l3lower < l3 < l3upper, where the subscripts “lower” and “upper”represent the lower and upper bound respectively. Successively,the zone is meshed and the coordinates of the intersection pointsof the mesh grids are obtained. Next, the coordinates of the inter-section points, together with T , are substituted into the left ofEqs. (40a)–(40c). For an arbitrary intersection point ðl2i; l3iÞ, iff 6ðT; l2i; l3iÞ < δ1, f 7ðT; l2i; l3iÞ < δ2, and f 8ðT ; l2i; l3iÞ < δ3 areall satisfied, where δ1, δ2, and δ3 are specified small quantities,the point will be selected. Nevertheless, if one or more of the threeconditions are not satisfied, the tension T should be increased. Thenthe coordinates of intersection points and the adjusted T are again

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substituted into Eqs. (40a)–(40c) for calculation. The tensionshould be increased until all three inequalities are satisfied.

The whole calculation doesn't take much time for two primaryreasons. First, the tension at P2 is close to its lower bound. Inaddition, the magnitudes of lower bounds are larger than zero,and, as both the boundary layer and the touchdown segment aresmall components of the whole pipeline, the upper bounds of l2and l3 are not very large for a common pipeline in J-lay. The mag-nitudes of the upper bounds are related to the laying conditions andseabed properties, and they can be defined relatively largely forextreme conditions, whereas 200 m is sufficient for a commonpipeline in J-lay.

Proper δ1, δ2, and δ3 are important, for if high accuracy isrequired the mesh should be refined and the points to be checkedwill double or triple, resulting in more calculation time. For theinclination slope, δ1 ¼ 1 × 10�2 is enough, whereas for the pipe-line curvature and its derivative, higher accuracies are necessaryand this work adopts δ2 ¼ 1 × 10�5 and δ3 ¼ 1 × 10�5. Once T ,l2, and l3 are determined, the other unknowns can be obtainedreadily, and then the configuration and internal loads distributioncan be easily obtained.

In the cases in which the seabed provides uniformly distributedresistance, the solution process is generally the same.

Examples and Comparison

To illustrate the previous model with the seabed providing non-uniform resistance a typical pipeline is selected with its elasticmodulus E ¼ 2:1 × 1011 Pa, outer and inner diameters of the pipe-line d1 ¼ 0:6 m and d2 ¼ 0:55 m. The density of steel pipe isρs ¼ 7:85 × 103 Kg=m3, inclination slope at P1 is φ0 ¼ 80°, andthe density and depth of sea water are ρw ¼ 1:03 × 103 Kg=m3

and D ¼ 1; 000 m.The seabed with Su0 ¼ 0 kPa is adopted in this work as an

example, so the resistance at seabed surface R0 ¼ 0 kN=m. Inaddition, the pipeline is assumed to be rough. Deepwater pipelinesare usually laid on the seabed, penetrating by a fraction of about adiameter owing to self-weight and the effects of the laying process(Merifield et al. 2008). Aubeny et al. (2005) recommended, fory3ðxÞ=D < 0:5, the values of coefficients are a ¼ 5:95 andb ¼ 0:15, whereas for y3ðxÞ=D > 0:5, a ¼ 6:02 and b ¼ 0:20.Researchers widely accept that the soil strength increases with

the penetration depth and four different increasing rates k1 areselected: 1:5 kPa=m, 3 kPa=m, 6 kPa=m, and 12 kPa=m. Theembedment–resistance relationship of the power law expressionby Aubeny et al. (2005) is linearly fitted with k2 ¼ 5:636 kN=m2,11:272 kN=m2, 22:543 kN=m2, 45:086 kN=m2, respectively, forthe four seabeds, and the fitting lines agree with the predictedresistance–embedment curve (Fig. 4).

For such seabeds with linearly increasing soil resistance, tradi-tional models assume the deformation of the seabed to be elastic,and the increasing gradient of seabed reaction is considered thesubgrade coefficient. To illustrate the difference between the pre-sent model and the model with elastic seabed, an analytical modelcreated by Lenci and Callegari (2005) is used for comparison(called “elastic model” in this work to distinguish it from thepresent model). The major difference between the elastic modeland the present model is the seabed simplification; the former useselastic seabed, whereas the later model uses plastic seabed.

The overall pipeline configurations of the two models are sim-ilar for different seabeds, as shown in Figs. 5(a)–5(d). The pipelineconfigurations in the two different models share the same origin atthe TDP for each soil resistance. The intention is to provide a bettercomparison of the pipeline behaviors and seabed deformations atthe neighborhood of TDP. As observed in the figures and Table 1,

Fig. 3. Scheme of solution process

0.0 0.2 0.4 0.6 0.8 1.0

0

10

20

30

40

50

60

R=5.636y3

R=11.272y3

R=22.543y3

R=45.086y3

R (

kN/m

)

y3 (m)

k1=1.5 kPa/m by Aubeny

k1=3.0 kPa/m by Aubeny

k1= 6.0 kPa/m by Aubeny

k1= 12.0 kPa/m by Aubeny

fitting curve of k1=1.5 kPa/m

fitting curve of k1=3.0 kPa/m

fitting curve of k1=6.0 kPa/m

fitting curve of k1=12 kPa/m

Fig. 4. Soil resistances and the fitting curves

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the ultimate pipeline embedment depth y3ult in the elastic model isalways smaller than that in the present model, whereas the maxi-mum embedment depths y3max in the two models are very close.The maximum penetration depth in the elastic model results fromthe combined impact of both the pipeline self-weight and the layingeffects; however, the ultimate penetration of pipeline in the elasticmodel is uniquely determined by its self-weight. In contrast, in thepresent model the ultimate embedment depth is also the largestembedment depth induced by the combination of both self-weightand the laying effects.

Fig. 6(a) indicates that the vessel in the elastic model is obvi-ously further from the TDP position than that in the present modelwhen the increasing rate is as low as 1:5 kPa=m. As the soil be-comes stiffer, the vessel positions in the elastic segment modeland the present model get closer. The length of the boundary layerin the present model grows as the soil becomes stiffer, as it does in

the elastic model [Fig. 6(b)]. The boundary layer of the elasticmodel is much larger than that of the present model, with the largestgap between them 28.5 m for k1 ¼ 1:5 kPa=m and the smallest26 m for k1 ¼ 12 kPa=m. Meanwhile, the length of the touchdownsegment undergoes a sharply declining trend [Fig. 6(c)].

The embedment depth of the pipeline is critical for assessing itsstability at work. As shown in Fig. 6(d) and Table 1, the maximumand ultimate embedment depths of both models decrease with theincreasing soil stiffness.

The pipeline in the elastic model penetrates into the seabed andpeaks in the neighborhood of TDP, and then the pipeline graduallyclimbs up, as it gets further away from TDP, forming a concaveshape [Fig. 7(a)]. Finally, the penetration depth becomes stablewhen the seabed reaction balances the pipeline self-weight. Whenthe pipeline is continuously laid onto the elastic seabed, the ori-ginal concave seabed will bounce back and a new concave appears[Fig. 7(a)]. According to the characteristics of deepwater soft de-posits, the elastic deformation is negligible compared with plasticdeformation for deepwater soft deposits. In contrast, the deforma-tion of the plastic seabed is permanent and stable during the wholelaying process [Fig. 7(b)]. Because of no rebound phenomenon, theultimate penetration depth of pipeline on plastic seabed is far largerthan that on elastic seabed, [Figs. 5(a)–5(d) and 7(b)].

The variation of the shear strength of the seabed soil has littleinfluence on the overall distribution of bending moment along thepipeline. Therefore, the case of k1 ¼ 1:5 kPa=m is shown as anexample in Fig. 8, and the influence of the shear strength on the

-100 0 100 200 300 400 500

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x (m)

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x (m)

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x (m)

y (m

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x (m)

y (m

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-150 0-0.15

0.00

(a) (b)

(c) (d)

Fig. 5. Pipeline configurations on different seabeds: (a) k1 ¼ 1:5 kPa=m; (b) k1 ¼ 3 kPa=m; (c) k1 ¼ 6 kPa=m; (d) k1 ¼ 12 kPa=m

Table 1. Pipeline Embedment Depths

Soil strength increasingrate k1 (kPa=m) 1.5 3 6 12

Elastic model y3max (m) 0.618 0.337 0.186 0.104

y3ult (m) 0.536 0.268 0.134 0.067

Present model y3max (m) 0.583 0.326 0.186 0.108

y3ult (m) 0.583 0.326 0.186 0.108

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bending moment distribution is illustrated in Tables 2 and 3. Thecomparison of the two models reveals a generally good agreementamong the overall bending moment distributions. However, Table 2shows that the elastic model slightly underestimates the maximumbending moment by approximately 2–4%. The maximum bendingmoment appears at the boundary layer for both models, and Table 2reveals that the positions of the maximum bending moment ofthe two models get closer when the soil gets stiffer, with thegap between them decreasing from 4.84 m to a of just 0.17 m.Interestingly, the maximum bending moment position of the elasticmodel stays closer to the TDP than that of the present model whenthe seabed is very soft, although it surpasses that of the present

model when the shear strength increasing rate reaches 12 kPa=m.Another difference is the bending moments near the TDP. Pipelinein the elastic model will experience small, negative bending mo-ment when the seabed is bounces back from the concave, as shownin Fig. 8. This phenomenon does not occur to pipelines on plasticseabed. As Table 3 shows, the shear strength of the seabed soil

0 2 4 6 8 10 120.0

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increasing rate of shear strength (kPa/m)

y3max

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y3max

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y3ult

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pipe

line

embe

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t y3(m

)

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520

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535

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dist

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fro

m th

e T

DP

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e ve

ssel

(l 1+

l 2) (m

)

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75

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115

increasing rate of shear strength (kPa/m)

leng

th o

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ry-l

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men

t l2(m

)

present model elastic model

0 2 4 6 8 10 12

25

30

35

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60

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leng

th o

f to

uchd

own

segm

ent l

3(m)

increasing rate of shear strength (kPa/m)

present model

(a) (b)

(c) (d)

Fig. 6. Variation of segment lengths and embedment depth (a) distance from the TDP to the vessel; (b) length of the boundary-layer segment;(c) length of the touch down segment; (d) pipeline-embedment depth

Table 2. Maximum Bending Moments

Soil strength increasing ratek1 (kPa=m) 1.5 3 6 12

Elastic model Mmax (MN·m) 1.643 1.645 1.647 1.647

xM (m) 70 74 77 80

Present model Mmax (MN·m) 1.683 1.700 1.707 1.708

xM (m) 74.84 74.94 77.71 79.83

elastic seabed

new shapeoriginal shape

laying direction

concave

new shape

rigid plastic seabed

original shape

laying direction

(a) (b)

Fig. 7. Deformation of (a) elastic seabed and (b) plastic seabed duringthe laying process

Table 3. Bending Moments at P1, P2, and P3

Soil strength increasing rate k1 (kPa=m) 1.5 3 6 12

Bending moment at P1 (MN·m) 0.057 0.057 0.057 0.057

Bending moment at P2 (MN·m) 1.683 1.696 1.706 1.708

Bending moment at P3 (MN·m) 0 0 0 0

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has no influence on the bending moment at P1, which is very smallcompared with that at P2. The bending moment at P2 is large andclose to the maximum bending moment for seabeds with differentshear strengths. At the maximum pipeline penetration, the bendingmoment vanishes at P3.

Conclusions

A simple analytical model for predicting pipeline behavior duringinstallation has been presented, which allows for plastic defor-mation of the seabed to be considered. A system of 15 nonlinearequations were established, which transformed into a set of threenonlinear equations, and a numerical method was adopted to obtainthe final solution. This work has discussed the differences betweentwo kinds of seabed, the elastic seabed and the plastic seabed, onthe pipeline laying configuration, internal force, and embedment.

The relative distance from the TDP to the vessel, predicted bythe plastic seabed model, is larger than in the elastic model whenthe soil is soft, although the distances get closer when the seabedbecomes stiffer. Moreover, the length of the boundary layer in theplastic seabed model is considerably smaller than that of the elasticmodel. The maximum pipeline embedment depths in the plastic-seabed model and the elastic model are approximately the same;however, the ultimate penetration predicted by the present plastic-seabed model is always larger than that of the elastic model, whichis caused by the rebound phenomenon of the elastic seabed. Theultimate pipeline embedment in the elastic model is determinedby its self-weight, whereas in the present plastic-seabed model itis determined by the combined effect of both its self-weight andthe laying effect. With regard to the bending moment distribu-tion, the two models have close predictions, although the elasticmodel slightly underestimates the value of the maximum bendingmoment.

Acknowledgments

The authors would like to acknowledge the support of GrantNo. 51079128 from the National Natural Science Foundation ofChina and Grant No. R1100093 from the Excellent Youth Founda-tion of Zhejiang Scientific Commitee, China.

References

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2.0

P1

P2

TDP

P3

bend

ing

mom

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(M

N· m

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x (m)

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Fig. 8. Bending moment distributions of pipelines on seabed withk1 ¼ 1:5 kPa=m

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