Coupled Dynamic Analysis for the Riser-Conductor of Deepwater Surface BOP Drilling Systemdownloads.hindawi.com/journals/sv/2018/6568537.pdf · 2019-07-30 · method for the riser-conductor

Research ArticleCoupled Dynamic Analysis for the Riser-Conductor ofDeepwater Surface BOP Drilling System

Kanhua Su 12 Stephen Butt2 Jianming Yang 2 and Hongyuan Qiu 2

1School of Petroleum Engineering Chongqing University of Science and Technology Chongqing China2Faculty of Engineering and Applied Science Memorial University of Newfoundland St Johnrsquos NL Canada

Correspondence should be addressed to Kanhua Su sukanhua126com

Received 31 May 2017 Accepted 21 November 2017 Published 11 January 2018

Academic Editor Yuri S Karinski

Copyright copy 2018 Kanhua Su et al This is an open access article distributed under the Creative Commons Attribution Licensewhich permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

Deepwater surface BOP (surface blowout prevention SBOP) drilling differs from conventional riser drilling system To analyzethe dynamic response of this system the riser-conductor was considered as a beam with varied cross-sections subjected to loadsthroughout its length then an equation ofmotion and free vibration of the riser-conductor string for SBOPwas developedThefinitedifference method was used to solve the equation of motion in time domain and a semianalytical approach based on the concept ofsection division and continuation was proposed to analyze free vibration Case simulation results show that the method establishedfor SBOP system natural frequency analysis is reasonable The mode shapes of the riser-conductor are different between coupledand decoupled methods The soil types surrounding the conductor under mudline have tiny effect on the natural frequency Giventhat some papers have discussed the response of the SBOP riser this work focused on the comparison of the dynamic responses onthe wellhead and conductor with variable conditionsThe dynamic lateral displacement the bending moment and the parametersrsquosensitivity of the wellhead and the conductor were analyzed

1 Introduction

Several operators have developed surface BOP (surfaceblowout prevention SBOP) drilling technology for deepwaterdrilling SBOP drilling differs from a conventional riserdrilling system the BOP stack is located at the surfacebelow the drill floor of the platform not at the seabedAnother key difference is that the riser of the SBOP drillingsystem is designed to contain wellbore pressure whereasa conventional drilling riser does not contain pressureThe SBOP drilling system can use smaller 2nd- or 3rd-generation semisubmersible rigs for operation Moreoverit has illustrated a considerable amount of day rate savingover traditional drilling methods using subsea BOP [1] Aspresented in the IADC (International Association of DrillingContractors) guidelines for SBOP from floating MODUs [2]the key components of the SBOP drilling system includethe surface BOP stack upper transition joint casing riserlower transition joint seabed isolation device (SID) subseawellhead conductor and casing strings

Surface BOP is not a new concept but it was extended intodeepwater only a few years ago Based on successful drillingcampaigns in Asia Shell extended its SBOP technology tothe more demanding offshore operations in Brazil with theimplementation of a SID Unocal and Transocean pioneeredthe application of the SBOP from floating drilling unitswhich began in early 1996 in the relatively benign environ-ment of Southeast Asia (Kozicz 2006) [3] The deepwaterwell 1-SHEL-14-RJS (block BM-C-10 in the Campos Basinoffshore Brazil) was successfully drilled in water depth of2887m by Shell in 2003 The 3397mm size P110 grade and35Mpa casing riser was adopted in this well (Brander etal 2004) [4] Total used the SBOP technique in 2000mwater depth on its Donggala block in Indonesia For thisSBOP drilling system a 3397mm casing riser complete withspecially designed fatigue enhanced connections was used(Simondin et al 2004) [5] In 2008 SBOP technology wasimplemented for both drilling and completion operations forthe Parque das Conchas deepwater development in block BC-10 offshore of Brazil A dedicated 4064mm X80 grade and

HindawiShock and VibrationVolume 2018 Article ID 6568537 15 pageshttpsdoiorg10115520186568537

2 Shock and Vibration

41Mpa high pressure riser replaced the marine riser in theSBOP system (Tarr et al 2009) [6] A compact deepwaterdrilling ship for SBOP operations was built (Claassen et al2010) [7]

Typical configurations of the casing riser include diam-eters of 2731mm 3397mm and 4064mm At the top andbottom of the casing riser heavy walled transition joints arerequired to distribute stresses [2] Due to the small size ofthe casing riser a 7620mm or smaller conductor is adoptedto be jetted into the seabed as a support for the wellheadand other casing strings Similar to a conventional riser thehigh pressure casing riser is under complex forces imposed byvesselmotions waves and current and soil interaction whichare transferred to the subsea wellhead and conductor

Specialized computer programs are generally used topredict conventional riserrsquos behavior under the designedconditions [8] The motion equation of the riser is usuallyconverted to a system of finite length elements using eithera finite difference or a finite element technique Botke(1975) used a derivation of the riser equations and finitedifferencemethod (FDM) of solution [9] Gardner and Kotch(1976) described finite element method (FEM) applied to theriser [10] Subsequently many researchers have studied thedynamics of the deepwater drilling riser

For the SBOP drilling system the design loads includebending loads coming from the riser stress joint and well-head above themudline and soil reaction below themudlineIADC identified that the casing riser and conductor analysisshould be conducted in a coupled manner [2] As the loadson the bottom of the riser will be transferred to the wellheadand conductor the coupled concept of riser and conductor isconcerned [11] King et al (1993) developed a new approachto analyze the behavior of a drilling riser and conductor asa complete entity and the FEM was used in his research[12] Although FEM is able to represent more details of theriser including the connection points the mesh should becarefully implemented with good modeling practices Su etal (2008) studied the stability of the subsea wellhead andthe conductor bearing capacity in deepwater drilling andproposed a wellhead stability analysis method with FDM[13] Yan et al (2015) analyzed the wellhead stability problemin deepwater drilling using the pile element and nonlinearspring element of ANSYS [14] Results show that the coupledmethod to analyze the conventional riser-conductor systemis reasonable

However the casing riser analysis for the SBOP drillingsystem is conducted in accordance with API RP 16Q for theconventional riser Some literatures discussed the dynamicbehavior of the SBOP drilling riser Morooka et al (2008)presented a numerical simulation to estimate the riser behav-ior for a drilling system with surface BOP but the researchdid not consider the coupling effect between the wellheadand conductor [15] Dib et al (2009) analyzed the fatiguelifetime of the SBOP riser in various modes of operation andalso ignored the effect of the conductor [16] For the SBOPriser-conductor system as the transition joints replace theflexball joint this allows for the consideration of the riser-conductor as a beam with varied cross-sections subjected toloads throughout its length Therefore it is feasible to model

the system considering the coupling between the riser and theconductor for the SBOP system and to solve the equationswith FDM and other semianalytical methods

In this paper a coupled time-domain dynamic FDMmethod for the riser-conductor of the SBOP system isderived and a semianalytic method is developed for thefree vibration of riser-conductor These methods are moreconvenient for the analysis coupled with the riser SIDwellhead and conductor of the deepwater SBOP drillingsystem

2 Equation of Motion of Riser-Conductor inTransverse Vibration for SBOP Drilling

As the transition joints connect to the surface BOP andSID there is no rotary joint on the riser-conductor for theSBOP system Thus the riser-conductor can be consideredas a beam with varied cross-sections subjected to loadsThe riser-conductor of the SBOP system is modeled as avariable section Euler-Bernoulli beam undergoing transversevibration under axial force as is shown in Figure 1(a) Theforces acting on an element of the riser-conductor system oflength dx are shown in Figures 1(b) and 1(c)

21 Equation ofMotion of Riser-Conductor System By assum-ing (1) the riser-conductor is a Euler-Bernoulli beam (2) theriser and casing string are both linear elastic (3) the drillingstring has no effect on its bending rigidity and (4) the vesselwave current and riser all move in a plane then the riser canbemodeled as a beam subjected to loads throughout its lengthwith boundary conditions at the top and bottom ends

According to the Euler-Bernoulli theory [17 18] theequation ofmotion for the forced lateral vibration of the riser-conductor system can be obtained as in

12059721205971199092 [119864 (119909) 119868 (119909) 1205972119910 (119909 119905)1205971199092 ] + 120597120597119909 [119875 (119909) 120597119910 (119909 119905)120597119909 ]

+ 119898 (119909) 1205972119910 (119909 119905)1205971199052 + 119888119904 (119909) 120597119910 (119909 119905)120597119905+ 119896119891 (119909) 119910 (119909 119905) = 119865 (119909 119905)

(1)

where 119910 is the lateral displacement of the riser-conductorsystem m 119909 is the depth coordinate as shown m 119898 is theriser-conductor systemmass per unit length kg 119888119904 is the riser-conductor system damping coefficient Nsdotsm 119896119891 is the riser-conductor system stiffness Ns 119864 is Youngrsquos modulus of theriser-conductor system Nm2 119868 is the areamoment of inertiaof cross-section m4 119875 is the effective tension or compressiveforce on the riser-conductor N and119865 is the force on the riser-conductor system N

The external forces on the riser SID and wellhead can becomputed using the Morison equation (American PetroleumInstitute 2001) [8] this equation has been widely used inriser dynamic analysis As the riser transfers its forces tothe conductor and casing strings the lateral force 119902 on theconductor and casing strings is zero

Shock and Vibration 3

ySea level

Casing riser

Mudline

Upper transition joint

Lower transition joint

Soil reactiondx

Wellheadamp SID

Conductorq

Casing strings

Drilling platform or ship

Surface BOP

dx

o

Wave

Current

Fw(x t)

L1

L2

(a)

M

P

Q

Fwdx

mdx2y

dt2

P + dP

Q+Q

xdx

M+M

xdx

(b)

M

P

Q

mdx2y

dt2

M +M

xdx

P + dP

qdx kfydx

Q+Q

xdx

(c)

Figure 1 Forces diagram of SBOP system

The bending stiffness of the riser can be calculatedeasily however the strings under the mudline are muchmore complicated containing the conductor cement andsurface casing Su et al (2008) described a method to obtainthe equivalent bending rigidity [13] the equivalent bendingrigidity of the casing string can be determined by

119864119868 (119909)|119909=119871uj 119909=119871 lj = 12058764119864stl (1198634rj minus 1198894rj) 119864119868 (119909)|119909=119871119903 = 12058764119864stl (1198634119903 minus 1198894119903) 119864119868 (119909)|

119909=119871sid= 12058764119864stl (1198634rs minus 1198894rs)


119864119868 (119909)|1198711lt119909le1198711+1198712 = 12058764119864stl [(1198634119903 minus 1198894119903) + (1198634119904 minus 1198894119904)]+ 0612058764 119864ce (1198894119888 minus 1198634119904)

1198711 = 119871uj + 119871119903 + 119871 lj + 119871 sid + 119871ml

1198712 = 119871119888 + 119871 sc(2)

where 119864stl is the elastic modulus of steel Pa 119864ce is the elasticmodulus of the cement sheath between the conductor and thesurface casing Pa 119863119903 119889119903 are the outer and inside diametersrespectively of the riser m 119863rj 119889rj are the outer and insidediameters of the transition joint m 119863rs and 119889rs are theequivalent outer and inside diameters of the SID m 119863119888 119889119888are the outer and inside diameters of the conductor m 119863119904119889119904 are the outer and inside diameters of the surface casingstring m 119871uj 119871119903 119871 lj 119871 sid 119871ml are the lengths of the uppertransition joint riser lower transition joint SID and theconductor above the mudline m and 119871119888 119871 sc are the lengthsof the conductor under themudline and surface casing stringrespectively m

The riser mass per unit length should include the massof the riser itself and the internal drilling mud [8] Theconductor and casing string mass per unit length include themass of the conductor the surface casing and the cementsheath between them The axial force of the riser-conductorcan be obtained as in (3) For the riser the axial force is itseffective tension [8 13] while in general the axial force onthe SID may be tensile and the force on the conductor andcasing string is compressive

1198751 (119909)min = 1198790 minus 1205874sdot 119892 [(1205881199031198632rj minus 1205881199031198892rj + 1205881198981198892rj minus 1205881199081198632rj) (119871uj + 119871 lj)+ (1205881199031198632119903 minus 1205881199031198892119903 + 1205881198981198892119903 minus 1205881199081198632119903) 119871119903]

1198752 (119909)min = 1198751 (119909)min minus 1205874sdot 119892 [(1205881199031198632rs minus 1205881199031198892rs + 1205881198981198892rs minus 1205881199081198632rs) 119871 sid+ 120588119888 (1198632119888 minus 1198892119888) (119871ml + 119871119888) + 120588119904 (1198632119904 minus 1198892119904) 119871 sc+ 120588ce (1198892119888 minus 1198632119904) 119871 sc] + 119865119891 (119909)

(3)

where 1198751 is the vertical force along the riser N 1198752 is thevertical force along the SID conductor and surface casingstring N 1198790 is the top tension of riser N 120588119903 is the riserdensity kgm3 120588119898 is the drillingmud density kgm3 120588119908 is theseawater density kgm3 120588119888 is the conductor density kgm3 120588119904is the surface casing string density kgm3 120588ce is the cementsheath density kgm3 119892 is the acceleration of gravity ms2119882119888 is the weight per length of the casing string Nm and 119865119891is the outside friction of the conductor and casing string N

The stiffness of the conductor and casing string can bederived from p-y curves under period loads according tothe actual soil considerations [19] The damping constant ofthe conductor and casing string can be expressed with thematerial damping and the radiation damping (Gazetas andDobry 1984) [20]

22 Boundary and Initial Conditions As (1) is a fourth-order equation four boundary conditions are needed For theriser the lateral displacement and bending moment of theupper transition joint are taken as two boundary conditions[21] For the conductor and casing string the shear forceand the bending moment at their bottom are two lowerboundary conditions and can be assumed to be zero Thesefour boundary conditions can be represented by

119910 (0 119905) = 119878 (119905)119872 (0 119905) = 1198641198681205972119910 (119909 119905)1205971199092

100381610038161003816100381610038161003816100381610038161003816119909=0

= 119870ru120579ru

119872(1198711 + 1198712 119905) = 119864119868 (119909) 1205972119910 (119909 119905)1205971199092100381610038161003816100381610038161003816100381610038161003816119909=1198711+1198712

= 0

119876 (1198711 + 1198712 119905) = 119864119868 (119909) 1205973119910 (119909 119905)1205971199093100381610038161003816100381610038161003816100381610038161003816119909=1198711+1198712 = 0

(4)

The initial condition of the equation of motion is

119910 (119909 0) = 119910static (119909) (5)

where 119872 is the bending moment Nsdotm2 119876 is the shearforce N 119910static is the lateral displacement of the static riser-conductor system m 119870ru is the upper rotational stiffness ofthe transition joint Nsdotmrad and 120579ru is the upper rotationangle of the transition joint rad119878(119905) is the horizontal deviation of drilling platformmotion from its initial location and it sums the mean offsetand the platform drift responding to random waves (Sextonand Agbezuge 1976) [22] It can be expressed as follows

119878 (119905) = 1198780 + 119878119871 sin(2120587119905119879119871 )

+ 119899sum119894=1

119878119899 cos (119896119899119878 (119905) minus 120596119899119905 + 120601119899 + 119886119899) (6)

where 1198780 is the mean offset of the platform m 119878119871 is the driftamplitude of the platform m 119879119871 is the drift period of theplatform s and 119878119899 119896119899 120596119899 120601119899 120572119899 are the wave amplitudewave number circular frequency initial phase and phasedifference of the wave n respectively These parameters canbe obtained by wave theory and response curves in [21 22]

23 Equation Solution Using FDM It is difficult to solve theequations analytically therefore numerical simulation withthe finite difference method was adopted in this paper


The riser-conductor string is divided into 119899 equal portionsand the length of each section is ℎ By using the three-point difference format to replace the first- and second-orderderivative schemes the five-point difference format takesplace of the fourth-order derivative scheme subsequentlyin (1) Then the finite differential equations of the riser-conductor string can be obtained which can be shown asfollows

119886119894119910119895+1119894 = ℎ119894 minus 119887119894119910119895119894+2 minus 119888119894119910119895119894+1 minus 119889119894119910119895119894 minus 119890119894119910119895119894minus1119894minus1 minus 119891119894119910119895119894minus2minus 119892119894119910119895minus1119894

119886119894 = 119898119894Δ1199052 + 1198881199041198942Δ119905119887119894 = (119864119868)119894+1Δℎ2119888119894 = minus2 (119864119868)119894+1 minus 2 (119864119868)119894 + 119875119894Δℎ2119889119894 = (119864119868)119894+1 + 4 (119864119868)119894 + (119864119868)119894minus1 minus 2119875119894Δℎ2 minus 2119898119894Δ1199052 + 119896119894119890119894 = minus2 (119864119868)119894 minus 2 (119864119868)119894minus1 + 119875119894Δℎ2119891119894 = (119864119868)119894minus1Δℎ2119892119894 = 119898119894Δ1199052 minus 1198881199041198942Δ119905ℎ119894 = 119865119895119894

(7)

According to the difference scheme the differential equa-tions of the upper boundary condition are expressed as in (8)and the lower boundary condition are shown as in (9)

1199101198950 = 119904 (119895Δ119905)119910119895minus1 = (2 minus 119870119903Δℎ11986411198681)(minus2 minus 119870119903ℎ11986411198681) 119910

1198951 + 4(2 + 119870rΔℎ11986401198680)119910

1198950

(8)

119910119895119873+1 = 2119910119895119873 minus 119910119895119873minus1119887119873119910119895119873+2 = (2119887119873 minus 119873119873) 119910119895119873+1 minus (119887119873 minus 119891119873) 119910119895119873

minus (2119891119873 minus 119873119873) 119910119895119873minus1 + 119891119873119910119895119873minus2(9)

When 119905 = 0 removing the time items in the equationsabove the initial conditions can be obtained by the differ-ence equations The static lateral deformation of the riser-conductor system can be easily solved using a matrix or theGlesser method

1199100119894 = 119910static (119894) 119894 = minus1 0 1 119873 + 2 (10)

Starting from the initial conditions the responses at aseries of discrete time instants can be obtained through directintegration MATLAB was employed to solve the model bytime step Through iterative calculation the displacementoffset angle bending moment shear force and soil reactionforce at each node and any time were calculated

3 Free Vibration Equations and Solution forSBOP Riser-Conductor System

For the free vibration of the riser-conductor system (1)reduces to

12059721205971199092 [119864 (119909) 119868 (119909) 1205972119910 (119909 119905)1205971199092 ] + 120597120597119909 [119875 (119909) 120597119910 (119909 119905)120597119909 ]

+ 119898 (119909) 1205972119910 (119909 119905)1205971199052 + 119896 (119909) 119910 (119909 119905) = 0(11)

Assuming that the system is a uniform beam (11) reducesto

119864 (119909) 119868 (119909) 1205974119910 (119909 119905)1205971199094 + 119875 (119909) 1205972119910 (119909 119905)1205971199092+ 119898 (119909) 1205972119910 (119909 119905)1205971199052 + 119896 (119909) 119910 (119909 119905) = 0

(12)

The solution of (12) can be calculated according to thebeam theory then the natural frequency and the correspond-ing natural mode shape of the riser-conductor system can beexpressed as follows

119884 (119909) = 119860 cosh (120573119909) + 119861 sinh (120573119909) + 119862 cos (120574119909)+ 119863 sin (120574119909)

120573 = ( 1198752119864119868 + ( 1198752

411986421198682 + 1198981205962

119864119868 minus119896119891119864119868)12)12

120574 = ( minus1198752119864119868 + ( 1198752

411986421198682 + 1198981205962

119864119868 minus119896119891119864119868)12)12

(13)

where 119860 119861 119862 and 119863 are constants that can be found fromthe initial conditions 120596 is the natural frequency and 119884(119909) isthe corresponding natural mode shape of the system

The riser-conductor of the deepwater SBOP system con-sists of several sections of different diameters shown inFigure 2 therefore (11) cannot be directly used to solvethe problem Based on the concept of section division andcontinuation (Cui et al 2012) [23] a semianalytical approachfor analyzing free vibration of the SBOP riser-conductor withvariable cross-section is proposed However each section ofthe conductor system is with constant cross-section and canbe treated as a uniform beam So the natural frequency andthe mode shape of each segment can be solved with (13)


Node 0

Node 1

Node 2

Segment 1

Segment 2

Segment

Segment i + 1

i minus 1

Segment i Li

Segment Nminus 1

Segment N

Node N

Node Nminus 1

Node i + 1

Node i minus 1

Node i

x

y

Figure 2 The segmental diagram of SBOP riser-conductor

For the segment 119894 the 119894th natural mode shape is

119884119894 (119909) = 119860 119894 cosh (120573119894 (119909 minus 119909119894minus1))+ 119861119894 sinh (120573119894 (119909 minus 119909119894minus1))+ 119862119894 cos (120574119894 (119909 minus 119909119894minus1))+ 119863119894 sin (120574119894 (119909 minus 119909119894minus1))

(14)

Let 119883119894(119909) = 120573119894(119909 minus 119909119894minus1) 119883119883119894(119909) = 120574119894(119909 minus 119909119894minus1) (119894 =1 2119873 + 1 1199090 = 0)Then (14) becomes

119884119894 (119909) = 119860 119894 cosh (119883119894) + 119861119894 sinh (119883119894) + 119862119894 cos (119883119883119894)+ 119863119894 sin (119883119883119894) (15)

Therefore the (119894+1)th natural mode shape of the segmentis expressed as follows

119884119894+1 (119909) = 119860 119894+1 cosh (119883119894+1) + 119861119894+1 sinh (119883119894+1)+ 119862119894+1 cos (119883119883119894+1) + 119863119894+1 sin (119883119883119894+1) (16)

Since the deflection slope moment and shear force ofthe 119894th segment and the (119894 + 1)th segment at node 119894 are equalassuming 119909 = 119909119894

119884119894+1 (119909119894) = 119884119894 (119909119894)1198841015840119894+1 (119909119894) = 1198841015840119894 (119909119894)(119864119868)119894+1 11988410158401015840119894+1 (119909119894) = (119864119868)119894 11988410158401015840119894 (119909119894)(119864119868)119894+1 119884101584010158401015840119894+1 (119909119894) minus 119875119894+11198841015840119894+1 (119909119894)= (119864119868)119894 119884101584010158401015840119894 (119909119894) minus 1198751198941198841015840119894 (119909119894)

(17)

By substituting (15) and (16) into (17) the following isobtained

[[[[[[

119860 119894+1119861119894+1119862119894+1119863119894+1

]]]]]]=[[[[[[[[[[[

11989941198981 11989931198981 minus11989921198982 minus11989911198982120573119894120573119894+1 1198993 (1198981 + V1)

120573119894120573119894+1 1198994 (1198981 + V1)120574119894120573119894+1 1198991 (1198982 + V2)

minus120574119894120573119894+1 1198992 (1198982 + V2)minus11989941198983 minus11989931198983 11989921198984 11989911198984minus120573119894120574119894+1 1198993 (1198983 + V1)

minus120573119894120574119894+1 1198994 (1198983 + V1)minus120574119894120574119894+1 1198991 (1198984 + V2)

120574119894120574119894+1 1198992 (1198984 + V2)

]]]]]]]]]]]

[[[[[[[

119860 119894119861119894119862119894119863119894

]]]]]]] (18)

where

1198991 = sin (120574119894119897119894) 1198992 = cos (120574119894119897119894) 1198993 = sinh (120573119894119897119894)

1198994 = cosh (120573119894119897119894)1198981 = (119864119868)119894 1205732119894 + (119864119868)119894+1 1205742119894+1(119864119868)119894+1 (1205732119894+1 + 1205742119894+1) 1198982 = (119864119868)119894 1205742119894 minus (119864119868)119894+1 1205742119894+1(119864119868)119894+1 (1205732119894+1 + 1205742119894+1)


200 400 600 800 1000 1200 1400

Mode 1Mode 2Mode 3

Mode 4Mode 5Mode 6

Mode 1Mode 2Mode 3

Mode 4Mode 5Mode 6

x (m)x (m)

minus11minus09minus07minus05minus03minus01 0

010305070911

Nor

mal

ized

ampl

itude

minus003minus002minus001 13

51

0001002003

Nor

mal

ized

ampl

itude

1381

1379

1377

1375

1373

1371

1369

1367

1365

1363

1361

1359

1357

1355

1353

Figure 3 First five natural mode shapes for SBOP riser-conductor

200 400 600 800 1000 1200 1400x (m)

x (m)

x (m)

x (m)

Nor

mal

ized

ampl

itude

minus11minus09minus07minus05minus03minus01

010305070911

0 200 400 600 800 1000 1200 1400N

orm

aliz

ed a

mpl

itude


010305070911

0 200 400 600 800 1000 1200 1400

Nor

mal

ized

ampl

itude


0 200 400 600 800 1000 1200 1400

Nor

mal

ized

ampl

itude


010305070911

TTR = 15 coupledTTR = 12 decoupled

TTR = 12 coupledP = 0 coupled TTR = 15 coupled

TTR = 12 decoupledTTR = 12 coupledP = 0 coupled


TTR = 12 coupledP = 0 coupled



Figure 4 First four natural mode shapes for 4 situations

000501

01502

02503

035

1 2 3 4 5 6Mode

Nat

ural

Fre

quen

cy (H

z)

0

00501

015

02025

03

1 2 3 4 5 6Mode

Nat

ural

Fre

quen

cy (H

z)

Clay 1Clay 2Clay 3



Figure 5 Mode shapes and natural frequency with variable TTR and soil type


200 300 400 500 600 700 800 900 1000

Time (s)minus20minus10

0102030405060

Plat

form

mot

ion

ampl

itude

(m)

Offset + drift + waveOffsetdirft + wave

DriftWave

Figure 6 Dynamic response of the platform

Bottom of USJ

Top of LSJ

00280029

003003100320033003400350036

Wellhead

0010011001200130014001500160017

minus0001minus00009minus00008minus00007minus00006minus00005minus00004minus00003

202530354045505560

Late

ral d

ispla

cem

ent

(m)

45505560657075

Late

ral d

ispla

cem

ent

(m)

1700

1900

1500

2300

2500

2100

2900

270040

0

600

800

200

1200

1400

1000

Time (s)

6062646668707274767880

Late

ral d

ispla

cem

ent

(m)

3400

3200

3000

3800

4000

4200

4400

3600

Riser (minus300 m)

2527293133353739

Late

ral d

ispla

cem

ent

(m)

Late

ral d

ispla

cem

ent

(m)

Late

ral d

ispla

cem

ent

(m)

5900

5700

5500

5300

5100

4900

4700

4500 40

0

600

800

1000

1200

140020

0

1700

1900

2100

2300

2500

2700

2900

1500

000001000020000300004000050000600007

Late

ral d

ispla

cem

ent

(m)

Late

ral d

ispla

cem

ent

(m)

3000

3400

3600

3800

4000

4200

4400

3200

5900

5700

5500

5300

5100

4900

4700

4500

Under mudline minus10 mUnder mudline minus5m

Mudline

Riser (minus600 m)

Time (s) Time (s)

Time (s)Time (s)Time (s)

Time (s)

Time (s)

Figure 7 Lateral displacement at different positions

1198983 = (119864119868)119894 1205732119894 minus (119864119868)119894+1 1205732119894+1(119864119868)119894+1 (1205732119894+1 + 1205742119894+1)

1198984 = (119864119868)119894 1205742119894 + (119864119868)119894+1 1205732119894+1(119864119868)119894+1 (1205732119894+1 + 1205742119894+1)

V1 = 119875119894+1 minus 119875119894(119864119868)119894+1 (1205732119894+1 + 1205742119894+1) V2 = 119875119894 minus 119875119894+1(119864119868)119894+1 (1205732119894+1 + 1205742119894+1)

(19)


Wellhead Mudline

23235

24245

25255

26

662646668

77274

889092949698

100102104106

Top of LSJ

780800820840860880900920940960

800820840860880900920940

0102030405060

Bottom of USJ

400

600

800

200

1200

1400

1000

Time (s)

Riser (minus300 m) Riser (minus600 m)


Bend

ing

mom

ent

(KNmiddotm

)

4547495153555759

Bend

ing

mom

ent

(KNmiddotm

)

Bend

ing

mom

ent

(KNmiddotm

)

Bend

ing

mom

ent

(KNmiddotm

)

Bend

ing

mom

ent

(KNmiddotm

)

Bend

ing

mom

ent

(KNmiddotm

)

Bend

ing

mom

ent

(KNmiddotm

)

Bend

ing

mom

ent

(KNmiddotm

)

3200

3400

3600

3800

4000

4200

4400

3000

Time (s)

4700

4900

5100

5300

5500

5700

5900

4500

Time (s)

1700

1900

2100

2300

2500

2700

2900

1500

Time (s)40

0

600

200

800

1200

1400

1000

Time (s)

4700

4900

5100

5300

5500

5700

5900

4500

Time (s)

3200

3400

3600

3800

4000

4200

4400

3000

Time (s)

400420440460480500520540560580600

1700

1900

1500

2300

2500

2700

2900

2100

Time (s)

Figure 8 Dynamic bending moment on the riser wellhead and conductor

Let

119860 (119894+1) = [119860 119894+1 119861119894+1 119862119894+1 119863119894+1]119879 119860 (119894) = [119860 119894 119861119894 119862119894 119863119894]119879

119885(119894) =[[[[[[[[[[

11989941198981 11989931198981 minus11989921198982 minus11989911198982120573119894120573119894+1 1198993 (1198981 + V1)120573119894120573119894+1 1198994 (1198981 + V1)

120574119894120573119894+1 1198991 (1198982 + V2)minus120574119894120573119894+1 1198992 (1198982 + V2)minus11989941198983 minus11989931198983 11989921198984 11989911198984minus120573119894120574119894+1 1198993 (1198983 + V1)

minus120573119894120574119894+1 1198994 (1198983 + V1)minus120574119894120574119894+1 1198991 (1198984 + V2)

120574119894120574119894+1 1198992 (1198984 + V2)

]]]]]]]]]]

(20)

Then (18) becomes

119860 (119894+1) = 119885(119894)119860 (119894) (21)

From (21)

119860 (119873) = 119885119860 (1) (22)

where

119885 = 119885(119873minus1)119885(119873minus2) sdot sdot sdot 119885(2)119885(1) (23)

As (23) is the function of the natural frequency 120596 ofthe riser-conductor the relationship of the undetermined


0

15

30

45

60

75

90

105

120

Bottom of USJTop of LSJ

5

55

6

65

7

75

45 55 65 75 85Lateral displacement (m)

780

800

820

840

860

880

900

920

940

960

001 0015 002 0025 003 0035 004Lateral displacement (m)

WellheadMudline

0

100

200

300

400

500

600

minus0001 minus00005 0 00005 0001Lateral displacement (m)

5 10 15 20 25 30 35 40 45 50 550Lateral displacement (m)

Riser (minus300 m)Riser (minus600 m)

Bend

ing

mom

ent (

kNmiddotm

)Be

ndin

g m

omen

t (kN

middotm)

Bend

ing

mom

ent (

kNmiddotm

)Be

ndin

g m

omen

t (kN

middotm)


Figure 9 Moment-displacement curves

coefficient between the 119873th segment and the 1st segment isestablished

Assuming 119878(0) = 0 follows (4) from the boundaryconditions we get (24)

1198601 = 1198621 = 0

[ (119864119868)119873 1205732119873 cosh (120573119873119897119873) (119864119868)119873 1205732119873 sinh (120573119873119897119873) minus (119864119868)119873 1205742119873cos (120574119873119897119873) minus (119864119868)119873 1205742119873 sin (120574119873119897119873)((119864119868)119873 1205733119873 minus 119875119873120573119873) sinh (120573119873119897119873) ((119864119868)119873 1205733119873 minus 119875119873120573119873) cosh (120573119873119897119873) ((119864119868)119873 1205743119873 + 119875119873120574119873) sin (120574119873119897119873) minus ((119864119868)119873 1205743119873 + 119875119873120574119873) cos (120574119873119897119873)]

[[[[[[

119860119873119861119873119862119873119863119873

]]]]]]= 0

(24)

Solving (24) using the secant iteration method by MAT-LAB to obtain the natural frequencies 120596i 119894 = 1 2 theiteration formation is shown as in

120596119896+1 = 120596119896 minus 119891 (120596119896) 120596119896 minus 120596119896minus1119891 (120596119896) minus 119891 (120596119896minus1) (25)

Then (22) is used to get the constants119860 (119894) 119894 = 1 2 119873and the corresponding natural mode shapes of the riser-conductor

4 Case Study and Discussion

A case study with the parameters given in Table 1 is carriedout

Seabed soil conditions vary substantially around theworld However to simplify the calculation process andcompare the results the soil type below the mudline 0ndash60mis assumed as all clay or sand layerThe six soil type propertiesare listed in Table 2


Table 1 Basic parameters

Parameters ValueWater depthm 13610Length of casing riserm 13500OD of casing riserm 03397ID of casing riserm 03112Platform offset water depth 30Tension ratio of riser (TTR) 12Velocity of surface windmsdotsminus1 10Tide velocitymsdotsminus1 05Current velocitymsdotsminus1 10Significant wave heightm 12Significant wave periods 8Platform slow drift amplitudem 93Platform slow drift periods 2598Elastic modulus of steelGPa 2100Density of steelkgsdotmminus3 78500Density of seawaterkgsdotmminus3 10300Density of drilling mudkgsdotmminus3 12000Wellhead above the mudlinem 30Height of SIDm 80Equivalent outer diameter of SIDm 08Length of transition jointm 100Equivalent OD of transition jointm 03683Equivalent WT of transition jointm 00508Length of conductorm 600OD of conductormm 0762WT of conductormm 00254Elastic modulus of cement sheathGPa 180Length of surface casingm 5000

41 Natural Frequencies and Mode Shapes of the SBOP Riser-Conductor System The natural frequencies with the datagiven in Tables 1 and 2 with the proposed method are listedin Table 3 And Figure 3 shows the first six mode shapesof the riser-conductor of the SBOP drilling system As seenespecially in Figure 3 the amplitudes of the mode shapes onthe SID wellhead and conductor are very small because thesoil reaction causes stiffness in these sections

The mode shape comparison results for 4 situations (119875 =0 coupled TTR = 12 coupled TTR = 15 coupled TTR =12 decoupled) are shown in Figure 4 From these figures itcan be observed that the mode shapes have some differencebetween the coupled and decoupled method however thereis no obvious difference when the mode number is greaterthan 3 In addition axial force has great effect on the modeshape and natural frequency because axial force directlyaffects the stiffness of bending

Although the TTR has little effect on the mode shape itseffect on the natural frequencies for the modes is obvious asshown in Figure 5 It can be seen that with the increase of TTRthe natural frequency of the riser-conductor increases as wellHowever it would not have an effect on the natural frequencymuch for the SBOP riser-conductor It was also observed that

the soil types surrounding the conductor under the mudlinehave very tiny effect on the natural frequency for the riser-conductor for SBOP drilling system

42 The Dynamic Response of the SBOP Riser-Conductor Toanalyze the dynamic response of the riser-conductor for theSBOPdrilling system lateral displacement bendingmomentand soil reactions at the different positions of the riser-conductor string are compared Given that some papers havediscussed the response of the SBOP riser this work focuseson the comparison of the dynamic responses on the wellheadand the conductor with variable conditions

421 Dynamic Response of the Platform The P-M spectrumhas been employed to calculate the motion of the platformand the simulation results are shown in Figure 6 From thisfigure themotion amplitude of the platform is relatively smallwithout considering offset and driftThismotion consideringmore conditions will obviously cause the riser response

422 Lateral Displacement at Different Positions of the Riser-Conductor String For the SBOP drilling system some keyjoint points are critical to the drilling operation This workfocuses on 4 positions on the riser namely the bottomof the upper transition joint (USJ) the elevation 300munder the mean water level (MWL) the elevation 800munder the mean water level (MWL) and the top of thelower transition (stress) joint (LSJ) and 4 positions on theconductor including the subsea wellhead (WH) themudline(ML) minus5m under the ML and minus10m under the ML

The first four pictures in Figure 7 show the lateraldisplacements at different locations on the riser and thelast four pictures show the displacements on the wellheador the conductor The lateral displacement amplitude at theelevation of 800m of the riser under the MWL is the highestin the first four pictures the displacement amplitude at thewellhead is more than other places on the conductor underthe ML Their periods are the same in the time domainfollowing the platform motion

423 Dynamic Bending Moment on the Riser the Wellheadand the Conductor In Figure 8 the bending momentrsquosvariations at different positions on the riser the wellhead andthe conductor are presented The largest bending moment ofthe riser focuses on the LSJ and themoment becomes smallerat the place of the conductor under the ML minus5m

424 Moment-Displacement Curves Comparing the bend-ingmoment and the lateral displacement in the same positionsimultaneously in Figure 9 the change of displacement isfound to have a certain delay compared with the bendingmoment on the conductor and there is no delay on the riserThe reason is that the nonlinear soil reaction acted on theconductor under the mudline

It is also found that the displacement varies more atthe bottom of the USJ than at the top of the LSJ Thedisplacement-moment curves change on the negative 119909-axisunder the mudline minus10m because the displacement of the


Table 2 Soil parameters

Clay 1 Clay 2 Clay 3 Sand 1 Sand 2 Sand 3Undrained shear strengthkPa 200 600 600 Angle of internal frictiondegree 300 395 395Submerged unit weight(kNm3) 70 70 80 Submerged unit weight(kNm3) 85 85 100

Table 3 First six-order natural frequencies

Mode 1 Mode 2 Mode 3 Mode 4 Mode 5 Mode 6Natural frequenciesHz 004895 008687 014998 018547 021846 027866

minus0005 0005 0015 0025 0035Lateral displacement (m)

minus200 0 200 400 600 800 10000

10

20

30

40

50

60

minus30 minus20 minus10 0 10 20 30Soil reaction (kPa)

005

115

225

335

445

5

0005 0015 0025 0035

0123456789

10

400 600 800 1000 3

4

5

6

7

8

9

12 16 20 24 28

60

50

40

30

20

10

0

Dep

th (m

)

60

50

40

30

20

10

0

Dep

th (m

)

60

50

40

30

20

10

0D

epth

(m)

Dep

th (m

)

t = 0 st = 2325 st = 2605 st = 2974 s

t = 3300 st = 3624 st = 3949 st = 4273 s

t = 200 st = 2325 st = 2605 st = 2974 s

t = 3300 st = 3624 st = 3949 st = 4273 s

t = 200 st = 2325 st = 2605 st = 2974 s

t = 3300 st = 3624 st = 3949 st = 4273 s

Bending moment (kNmiddotm)

Figure 10 Deformation and stresses of the conductor in a drift period

conductor under a certain depth will be negative or equalzero

425 Deformation and Stresses of the Conductor in a DriftPeriod The deformation and stress of the casing stringbelow the mudline and the soil reaction force around thecasing string are analyzed in a slow-drift period of a drillingplatform When the vibration reaches the steady state 8time points (t = 200 s 2325 s 2605 s 2974 s 3300 s 3624 s3949 s and 4273 s) are taken in one period The lateraldisplacement bending moment and soil reaction along theconductor are shown in Figure 10

It can be seen from Figure 10 that the deformation stressand surrounding soil reaction of the casing string changewith time in the slow-drift period In a period the maximum

lateral displacement is along the column the relative changesin bending moment and maximum soil reaction are 7766 and 32 respectively

426 Sensitivity Analysis of Parameter Changes to Wellheadand Conductor

vspace10pt(1) Platform Offset and Conductor Size The dif-ferent platform offsets of 0 1 2 3 4 5 67 and 8 water depth and different conductor sizes (OD= 7620mm 6604mm and 5080mm no conductor) arechosen to compare the results as shown in Figure 11 Thedisplacement and the bending moment are raised withthe offset increase and those have a rapid increase in thecondition when there is no conductor


0005

01015

02025

03035

04045

05

0 1 2 3 4 5 6 7 8Offset ( WD)

Late

ral d

ispla

cem

ent a

t WH

(m)

400500600700800900

10001100120013001400

0 1 2 3 4 5 6 7 8Offset ( WD)

0

002

004

006

008

01

012

0 1 2 3 4 5 6 7 8Offset ( WD)

400500600700800900

1000110012001300

0 1 2 3 4 5 6 7 8Offset ( WD)

No conductor ( minus2m)OD = 5080 mm ( minus2m)OD = 6604 mm ( minus2m)OD = 7620 mm ( minus2m)


No conductor (WH)OD = 5080 mm (WH)OD = 6604 mm (WH)OD = 7620 mm (

Bend

ing

mom

ent a

t WH

(kNmiddotm

)

Late

ral d

ispla

cem

ent a

tminus5

m (m

)

Bend

ing

mom

ent a

tminus5

m (k

Nmiddotm

)

WH)

No conductor (WH)OD = 5080 mm (WH)OD = 6604 mm (WH)OD = 7620 mm (WH)

Figure 11 Effect of the platform offset and conductor size on wellhead and conductor

0 5 10 15 20 25Offset (m)

400

600

800

1000

1200

1400

1600

0 5 10 15 20 25Offset (m)

Bend

ing

mom

ent (

kNmiddotm

)

0001002003004005006007

Late

ral d

ispla

cem

ent (

m)

Riser OD = 4064 mm (WH)Riser OD = 3397 mm (WH)Riser OD = 2731mm (WH)

Riser OD = 2731 mm ( minus5m)Riser OD = 3397 mm ( minus5m)Riser OD = 4064 mm ( minus5m)

Riser OD = 4064 mm (WH)Riser OD = 3397 mm (WH)Riser OD = 2731 mm (WH)


Figure 12 Effect of the platform surge amplitude and riser size on wellhead and conductor

(2) Platform SurgeAmplitude andRiser Size Figure 12 displaysthe sensitivity of the lateral displacement and the bendingmoment to the variable surge amplitudes of platform and the

riserrsquos outer diameters In Figure 12 the lateral displacementand the bending moment of the wellhead increase with thegreater surge amplitude The 4064mm diameter riser will


00005

0010015

0020025

0030035

004

Clay 1 Clay 2 Clay 3 Sand 1 Sand 2 Sand 3

Late

ral d

ispla

cem

ent (

m)

WHML(minus5m)

WHML(minus5m)

0100200300400500600700800900

1000


Bend

ing

mom

ent (

kNmiddotm

)

Figure 13 Effect of the soil properties at wellhead mudline and minus5m on conductor

transfer more force to the wellhead and the conductor sothe 3397mm diameter riser is the best for the SBOP drillingsystem

(3) Soil Properties As the seabed soil supports the conductorthe soil property is very critical for the transverse stress anddeformation of the conductor and the wellhead Figure 13compares the displacement and the bending moment of thewellhead and the position of the conductor at the mudlineand 5m below the mudline using 6 soil types (see Table 2)The parameter of undrained shear strength (USS) is moresensitive to clay than the submerged unit weight A lowerUSSof clay will cause weak conductor support The parameter ofthe angle of internal friction is also more sensitive to sandthan the submerged unit weight Also the bending momentdoes not change more than the lateral displacement of thewellhead and the conductor

5 Conclusion

For the SBOP drilling system the riser-conductor is con-sidered as combination sections with different cross-sectionssubjected to loads throughout its length and a FDM solutionis derived in time domain Results show that the displacementamplitude at the wellhead is more than in other places ofthe conductor under mudline The largest bending momentof the riser focuses on the LSJ and the moment becomessmaller at the place of the conductor under the ML minus5mThe deformation stress and surrounding soil reaction of thecasing string change with time in the slow-drift period

Based on the concept of section division and continua-tion a semianalytical approach for analyzing free vibrationof the SBOP riser-conductor with variable cross-section isproposed which can actually be applied to any variablecross-section And this method established for SBOP systemnatural frequency analysis is reasonable Results show thatthe mode shapes have some difference between the coupledand decoupled method The natural frequencies at diversemodes have little variation with variable TTR The soil typessurrounding the conductor under mudline have very tinyeffect on the natural frequency of the riser-conductor system

Conflicts of Interest

The authors declare that there are no conflicts of interestregarding the publication of this article

Acknowledgments

This work was done at the Drilling Technology Laboratory(DTL) at theMemorial University ofNewfoundland CanadaThis work was supported by the National Natural ScienceFoundation of China (Grant no 51004119) the ChongqingResearchProgramof Basic Research andFrontier Technology(Grant no CSTC2015jcyjA90021) and the Academician LedSpecial Project of Chongqing Science and Technology Com-mission (Grant no cstc2017zdcy-yszxX0009)

References

[1] P Azancot EMagne and J Zhang Surface BOPmdashManagementSystem Design Guidelines Society of Petroleum Engineers 2002

[2] ldquoGuidelines for Surface BOP Drilling from Floating MODUsrdquoin Proceedings of the International Association of Drilling Con-tractors (IADC) 2015

[3] J R Kozicz ldquoSurface BOPmdashRecent Experience and FutureOpportunitiesrdquo in Proceedings of Society of Petroleum EngineersMumbai India 2006

[4] G Brander E Magne and T Newman ldquoSurface BOP technol-ogy steps into deeper water with DP vesselsrdquo Oil Gas Journalvol 102 no 17 p 65 2004

[5] A Simondin D MacPherson N Touboul and G Ragnes ldquoAdeepwater well construction alternative surface BOP drillingconcept using environmental safe guardrdquo in Proceedings ofthe IADCSPE Drilling Conference pp 153ndash160 Society ofPetroleum Engineers TX USA March 2004

[6] B Tarr T Taklo L A Olijnik et al ldquoSurface BOP systemoperational experience offshore Brazil in 1900 m of waterrdquo inProceedings of the SPEIADCDrillingConference andExhibitionAmsterdam The Netherlands 2009

[7] L C Claassen S M Hendriks and G H T Zijderveld ldquoNew-build compact deepwater drillship designed for surface BOPsystemrdquo in Proceedings of the Offshore Technology Conference2010 OTC 2010 pp 1391ndash1400 USA May 2010


[8] American Petroleum Institute API RP 16Q- RecommendedPractice for Design Selecfion Operafion and Maintenanceof Marine Drilling Riser Systems Washington DC USAAmerican Petroleum Institute 2001

[9] J C Botke ldquoAn Analysis of the Dynamics of Marine RisersrdquoTech Rep Delco Electronics 1975

[10] T N Gardner and M A Kotch ldquoDynamic Analysis of Risersand Caissons by the Element Methodrdquo in Proceedings of theOffshore Technology Conference Houston TX USA

[11] J M Shaughnessy W T Daugherty R L Graff and T DurkeeldquoMore Ultradeepwater Drilling Problemsrdquo in Proceedings of theSPEIADC Drilling Conference Feburary 2007

[12] G W King K Burton and T Hodgson ldquoA coupled analysisapproach to the assessment of marine drilling systemsrdquo SPEDrilling amp Completion vol 8 no 2 pp 131ndash137 1993

[13] K Su Z Guan and Y Su ldquoMechanical Stability Analysis of Sub-sea Wellhead for Deepwater Drillingrdquo Oil Drilling ProductionTechnology vol 30 no 6 pp 1ndash4 2008

[14] W Yan Z-J Chen J-G Deng H-Y Zhu F-C Deng and Z-LLiu ldquoNumericalmethod for subseawellhead stability analysis indeepwater drillingrdquoOcean Engineering vol 98 pp 50ndash56 2015

[15] C K Morooka R I Tsukada and D M Brandt ldquoNumericalsimulations of ocean drilling system behavior with a surfaceor a subsea BOP in waves and currentrdquo in Proceedings of the27th International Conference on Offshore Mechanics and ArcticEngineering OMAE pp 469ndash476 Germany June 2008

[16] M W Dib J Lou L Zhu and M Bassey ldquoSBOP DrillingEnables EfficientDrilling in ExtremeWaterDepthsrdquo inProceed-ings of the Offshore Technology Conference Houston TX USA2009

[17] S S RaoVibration of Continuous Systems JohnWiley Sons IncHoboken NJ USA 2007

[18] Tsinghua University Mechanical Vibration China MechinePress1980

[19] M Novak ldquoDynamic stiffness and damping of pilesrdquo CanadianGeotechnical Journal vol 11 no 4 pp 574ndash598 1974

[20] G Gazetas and R Dobry ldquoHorizontal response of piles inlayered soilsrdquo Journal of Geotechnical Engineering vol 110 no1 pp 20ndash40 1984

[21] T Wang X Zhang and W Zhu ldquoVessel Motion Effects onNonlinear Dynamics of Deepwater Drilling Riserrdquo Journal ofShip Mechanics vol 14 no 6 pp 606ndash617 2010

[22] R M Sexton and L k Agbezuge ldquoRandom Wave and VesselMotion Effects on Drilling Riser Dynamicsrdquo in Proceedings ofthe Offshore Technology Conference Houston Texas 1976

[23] C Cui H Jiang and Y-H Li ldquoSemi-analytical method forcalculating vibration characteristics of variable cross-sectionbeamrdquo Journal of Vibration and Shock vol 31 no 14 pp 85ndash882012

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41Mpa high pressure riser replaced the marine riser in theSBOP system (Tarr et al 2009) [6] A compact deepwaterdrilling ship for SBOP operations was built (Claassen et al2010) [7]

Typical configurations of the casing riser include diam-eters of 2731mm 3397mm and 4064mm At the top andbottom of the casing riser heavy walled transition joints arerequired to distribute stresses [2] Due to the small size ofthe casing riser a 7620mm or smaller conductor is adoptedto be jetted into the seabed as a support for the wellheadand other casing strings Similar to a conventional riser thehigh pressure casing riser is under complex forces imposed byvesselmotions waves and current and soil interaction whichare transferred to the subsea wellhead and conductor

Specialized computer programs are generally used topredict conventional riserrsquos behavior under the designedconditions [8] The motion equation of the riser is usuallyconverted to a system of finite length elements using eithera finite difference or a finite element technique Botke(1975) used a derivation of the riser equations and finitedifferencemethod (FDM) of solution [9] Gardner and Kotch(1976) described finite element method (FEM) applied to theriser [10] Subsequently many researchers have studied thedynamics of the deepwater drilling riser

For the SBOP drilling system the design loads includebending loads coming from the riser stress joint and well-head above themudline and soil reaction below themudlineIADC identified that the casing riser and conductor analysisshould be conducted in a coupled manner [2] As the loadson the bottom of the riser will be transferred to the wellheadand conductor the coupled concept of riser and conductor isconcerned [11] King et al (1993) developed a new approachto analyze the behavior of a drilling riser and conductor asa complete entity and the FEM was used in his research[12] Although FEM is able to represent more details of theriser including the connection points the mesh should becarefully implemented with good modeling practices Su etal (2008) studied the stability of the subsea wellhead andthe conductor bearing capacity in deepwater drilling andproposed a wellhead stability analysis method with FDM[13] Yan et al (2015) analyzed the wellhead stability problemin deepwater drilling using the pile element and nonlinearspring element of ANSYS [14] Results show that the coupledmethod to analyze the conventional riser-conductor systemis reasonable

However the casing riser analysis for the SBOP drillingsystem is conducted in accordance with API RP 16Q for theconventional riser Some literatures discussed the dynamicbehavior of the SBOP drilling riser Morooka et al (2008)presented a numerical simulation to estimate the riser behav-ior for a drilling system with surface BOP but the researchdid not consider the coupling effect between the wellheadand conductor [15] Dib et al (2009) analyzed the fatiguelifetime of the SBOP riser in various modes of operation andalso ignored the effect of the conductor [16] For the SBOPriser-conductor system as the transition joints replace theflexball joint this allows for the consideration of the riser-conductor as a beam with varied cross-sections subjected toloads throughout its length Therefore it is feasible to model

the system considering the coupling between the riser and theconductor for the SBOP system and to solve the equationswith FDM and other semianalytical methods

In this paper a coupled time-domain dynamic FDMmethod for the riser-conductor of the SBOP system isderived and a semianalytic method is developed for thefree vibration of riser-conductor These methods are moreconvenient for the analysis coupled with the riser SIDwellhead and conductor of the deepwater SBOP drillingsystem

2 Equation of Motion of Riser-Conductor inTransverse Vibration for SBOP Drilling

As the transition joints connect to the surface BOP andSID there is no rotary joint on the riser-conductor for theSBOP system Thus the riser-conductor can be consideredas a beam with varied cross-sections subjected to loadsThe riser-conductor of the SBOP system is modeled as avariable section Euler-Bernoulli beam undergoing transversevibration under axial force as is shown in Figure 1(a) Theforces acting on an element of the riser-conductor system oflength dx are shown in Figures 1(b) and 1(c)

21 Equation ofMotion of Riser-Conductor System By assum-ing (1) the riser-conductor is a Euler-Bernoulli beam (2) theriser and casing string are both linear elastic (3) the drillingstring has no effect on its bending rigidity and (4) the vesselwave current and riser all move in a plane then the riser canbemodeled as a beam subjected to loads throughout its lengthwith boundary conditions at the top and bottom ends

According to the Euler-Bernoulli theory [17 18] theequation ofmotion for the forced lateral vibration of the riser-conductor system can be obtained as in

12059721205971199092 [119864 (119909) 119868 (119909) 1205972119910 (119909 119905)1205971199092 ] + 120597120597119909 [119875 (119909) 120597119910 (119909 119905)120597119909 ]

+ 119898 (119909) 1205972119910 (119909 119905)1205971199052 + 119888119904 (119909) 120597119910 (119909 119905)120597119905+ 119896119891 (119909) 119910 (119909 119905) = 119865 (119909 119905)

(1)

where 119910 is the lateral displacement of the riser-conductorsystem m 119909 is the depth coordinate as shown m 119898 is theriser-conductor systemmass per unit length kg 119888119904 is the riser-conductor system damping coefficient Nsdotsm 119896119891 is the riser-conductor system stiffness Ns 119864 is Youngrsquos modulus of theriser-conductor system Nm2 119868 is the areamoment of inertiaof cross-section m4 119875 is the effective tension or compressiveforce on the riser-conductor N and119865 is the force on the riser-conductor system N

The external forces on the riser SID and wellhead can becomputed using the Morison equation (American PetroleumInstitute 2001) [8] this equation has been widely used inriser dynamic analysis As the riser transfers its forces tothe conductor and casing strings the lateral force 119902 on theconductor and casing strings is zero


ySea level

Casing riser

Mudline



Soil reactiondx

Wellheadamp SID

Conductorq

Casing strings


Surface BOP

dx

o

Wave

Current

Fw(x t)

L1

L2

(a)

M

P

Q

Fwdx

mdx2y

dt2

P + dP

Q+Q

xdx

M+M

xdx

(b)

M

P

Q

mdx2y

dt2

M +M

xdx

P + dP

qdx kfydx

Q+Q

xdx

(c)







1198711 = 119871uj + 119871119903 + 119871 lj + 119871 sid + 119871ml

1198712 = 119871119888 + 119871 sc(2)





(3)




119910 (0 119905) = 119878 (119905)119872 (0 119905) = 1198641198681205972119910 (119909 119905)1205971199092

100381610038161003816100381610038161003816100381610038161003816119909=0

= 119870ru120579ru

119872(1198711 + 1198712 119905) = 119864119868 (119909) 1205972119910 (119909 119905)1205971199092100381610038161003816100381610038161003816100381610038161003816119909=1198711+1198712

= 0

119876 (1198711 + 1198712 119905) = 119864119868 (119909) 1205973119910 (119909 119905)1205971199093100381610038161003816100381610038161003816100381610038161003816119909=1198711+1198712 = 0

(4)


119910 (119909 0) = 119910static (119909) (5)


119878 (119905) = 1198780 + 119878119871 sin(2120587119905119879119871 )

+ 119899sum119894=1

119878119899 cos (119896119899119878 (119905) minus 120596119899119905 + 120601119899 + 119886119899) (6)







(7)



1198951 + 4(2 + 119870rΔℎ11986401198680)119910

1198950

(8)




1199100119894 = 119910static (119894) 119894 = minus1 0 1 119873 + 2 (10)




12059721205971199092 [119864 (119909) 119868 (119909) 1205972119910 (119909 119905)1205971199092 ] + 120597120597119909 [119875 (119909) 120597119910 (119909 119905)120597119909 ]

+ 119898 (119909) 1205972119910 (119909 119905)1205971199052 + 119896 (119909) 119910 (119909 119905) = 0(11)


119864 (119909) 119868 (119909) 1205974119910 (119909 119905)1205971199094 + 119875 (119909) 1205972119910 (119909 119905)1205971199092+ 119898 (119909) 1205972119910 (119909 119905)1205971199052 + 119896 (119909) 119910 (119909 119905) = 0

(12)


119884 (119909) = 119860 cosh (120573119909) + 119861 sinh (120573119909) + 119862 cos (120574119909)+ 119863 sin (120574119909)

120573 = ( 1198752119864119868 + ( 1198752

411986421198682 + 1198981205962

119864119868 minus119896119891119864119868)12)12

120574 = ( minus1198752119864119868 + ( 1198752

411986421198682 + 1198981205962

119864119868 minus119896119891119864119868)12)12

(13)




Node 0

Node 1

Node 2

Segment 1

Segment 2

Segment

Segment i + 1

i minus 1

Segment i Li

Segment Nminus 1

Segment N

Node N

Node Nminus 1

Node i + 1

Node i minus 1

Node i

x

y




(14)


119884119894 (119909) = 119860 119894 cosh (119883119894) + 119861119894 sinh (119883119894) + 119862119894 cos (119883119883119894)+ 119863119894 sin (119883119883119894) (15)


119884119894+1 (119909) = 119860 119894+1 cosh (119883119894+1) + 119861119894+1 sinh (119883119894+1)+ 119862119894+1 cos (119883119883119894+1) + 119863119894+1 sin (119883119883119894+1) (16)


119884119894+1 (119909119894) = 119884119894 (119909119894)1198841015840119894+1 (119909119894) = 1198841015840119894 (119909119894)(119864119868)119894+1 11988410158401015840119894+1 (119909119894) = (119864119868)119894 11988410158401015840119894 (119909119894)(119864119868)119894+1 119884101584010158401015840119894+1 (119909119894) minus 119875119894+11198841015840119894+1 (119909119894)= (119864119868)119894 119884101584010158401015840119894 (119909119894) minus 1198751198941198841015840119894 (119909119894)

(17)


[[[[[[

119860 119894+1119861119894+1119862119894+1119863119894+1

]]]]]]=[[[[[[[[[[[

11989941198981 11989931198981 minus11989921198982 minus11989911198982120573119894120573119894+1 1198993 (1198981 + V1)

120573119894120573119894+1 1198994 (1198981 + V1)120574119894120573119894+1 1198991 (1198982 + V2)


minus120573119894120574119894+1 1198994 (1198983 + V1)minus120574119894120574119894+1 1198991 (1198984 + V2)

120574119894120574119894+1 1198992 (1198984 + V2)

]]]]]]]]]]]

[[[[[[[

119860 119894119861119894119862119894119863119894

]]]]]]] (18)

where

1198991 = sin (120574119894119897119894) 1198992 = cos (120574119894119897119894) 1198993 = sinh (120573119894119897119894)

1198994 = cosh (120573119894119897119894)1198981 = (119864119868)119894 1205732119894 + (119864119868)119894+1 1205742119894+1(119864119868)119894+1 (1205732119894+1 + 1205742119894+1) 1198982 = (119864119868)119894 1205742119894 minus (119864119868)119894+1 1205742119894+1(119864119868)119894+1 (1205732119894+1 + 1205742119894+1)


200 400 600 800 1000 1200 1400

Mode 1Mode 2Mode 3

Mode 4Mode 5Mode 6

Mode 1Mode 2Mode 3

Mode 4Mode 5Mode 6

x (m)x (m)


010305070911

Nor

mal

ized

ampl

itude


51

0001002003

Nor

mal

ized

ampl

itude

1381

1379

1377

1375

1373

1371

1369

1367

1365

1363

1361

1359

1357

1355

1353


200 400 600 800 1000 1200 1400x (m)

x (m)

x (m)

x (m)

Nor

mal

ized

ampl

itude


010305070911

0 200 400 600 800 1000 1200 1400N

orm

aliz

ed a

mpl

itude


010305070911

0 200 400 600 800 1000 1200 1400

Nor

mal

ized

ampl

itude


0 200 400 600 800 1000 1200 1400

Nor

mal

ized

ampl

itude


010305070911









000501

01502

02503

035

1 2 3 4 5 6Mode

Nat

ural

Fre

quen

cy (H

z)

0

00501

015

02025

03

1 2 3 4 5 6Mode

Nat

ural

Fre

quen

cy (H

z)

Clay 1Clay 2Clay 3





200 300 400 500 600 700 800 900 1000


0102030405060

Plat

form

mot

ion

ampl

itude

(m)


DriftWave


Bottom of USJ

Top of LSJ

00280029

003003100320033003400350036

Wellhead

0010011001200130014001500160017


202530354045505560

Late

ral d

ispla

cem

ent

(m)

45505560657075

Late

ral d

ispla

cem

ent

(m)

1700

1900

1500

2300

2500

2100

2900

270040

0

600

800

200

1200

1400

1000

Time (s)

6062646668707274767880

Late

ral d

ispla

cem

ent

(m)

3400

3200

3000

3800

4000

4200

4400

3600

Riser (minus300 m)

2527293133353739

Late

ral d

ispla

cem

ent

(m)

Late

ral d

ispla

cem

ent

(m)

Late

ral d

ispla

cem

ent

(m)

5900

5700

5500

5300

5100

4900

4700

4500 40

0

600

800

1000

1200

140020

0

1700

1900

2100

2300

2500

2700

2900

1500

000001000020000300004000050000600007

Late

ral d

ispla

cem

ent

(m)

Late

ral d

ispla

cem

ent

(m)

3000

3400

3600

3800

4000

4200

4400

3200

5900

5700

5500

5300

5100

4900

4700

4500


Mudline

Riser (minus600 m)

Time (s) Time (s)


Time (s)

Time (s)


1198983 = (119864119868)119894 1205732119894 minus (119864119868)119894+1 1205732119894+1(119864119868)119894+1 (1205732119894+1 + 1205742119894+1)

1198984 = (119864119868)119894 1205742119894 + (119864119868)119894+1 1205732119894+1(119864119868)119894+1 (1205732119894+1 + 1205742119894+1)

V1 = 119875119894+1 minus 119875119894(119864119868)119894+1 (1205732119894+1 + 1205742119894+1) V2 = 119875119894 minus 119875119894+1(119864119868)119894+1 (1205732119894+1 + 1205742119894+1)

(19)


Wellhead Mudline

23235

24245

25255

26

662646668

77274

889092949698

100102104106

Top of LSJ

780800820840860880900920940960

800820840860880900920940

0102030405060

Bottom of USJ

400

600

800

200

1200

1400

1000

Time (s)



Bend

ing

mom

ent

(KNmiddotm

)

4547495153555759

Bend

ing

mom

ent

(KNmiddotm

)

Bend

ing

mom

ent

(KNmiddotm

)

Bend

ing

mom

ent

(KNmiddotm

)

Bend

ing

mom

ent

(KNmiddotm

)

Bend

ing

mom

ent

(KNmiddotm

)

Bend

ing

mom

ent

(KNmiddotm

)

Bend

ing

mom

ent

(KNmiddotm

)

3200

3400

3600

3800

4000

4200

4400

3000

Time (s)

4700

4900

5100

5300

5500

5700

5900

4500

Time (s)

1700

1900

2100

2300

2500

2700

2900

1500

Time (s)40

0

600

200

800

1200

1400

1000

Time (s)

4700

4900

5100

5300

5500

5700

5900

4500

Time (s)

3200

3400

3600

3800

4000

4200

4400

3000

Time (s)

400420440460480500520540560580600

1700

1900

1500

2300

2500

2700

2900

2100

Time (s)


Let

119860 (119894+1) = [119860 119894+1 119861119894+1 119862119894+1 119863119894+1]119879 119860 (119894) = [119860 119894 119861119894 119862119894 119863119894]119879

119885(119894) =[[[[[[[[[[

11989941198981 11989931198981 minus11989921198982 minus11989911198982120573119894120573119894+1 1198993 (1198981 + V1)120573119894120573119894+1 1198994 (1198981 + V1)


minus120573119894120574119894+1 1198994 (1198983 + V1)minus120574119894120574119894+1 1198991 (1198984 + V2)

120574119894120574119894+1 1198992 (1198984 + V2)

]]]]]]]]]]

(20)

Then (18) becomes

119860 (119894+1) = 119885(119894)119860 (119894) (21)

From (21)

119860 (119873) = 119885119860 (1) (22)

where




0

15

30

45

60

75

90

105

120


5

55

6

65

7

75


780

800

820

840

860

880

900

920

940

960


WellheadMudline

0

100

200

300

400

500

600




Bend

ing

mom

ent (

kNmiddotm

)Be

ndin

g m

omen

t (kN

middotm)

Bend

ing

mom

ent (

kNmiddotm

)Be

ndin

g m

omen

t (kN

middotm)





1198601 = 1198621 = 0


[[[[[[

119860119873119861119873119862119873119863119873

]]]]]]= 0

(24)



























minus200 0 200 400 600 800 10000

10

20

30

40

50

60


005

115

225

335

445

5

0005 0015 0025 0035

0123456789

10

400 600 800 1000 3

4

5

6

7

8

9

12 16 20 24 28

60

50

40

30

20

10

0

Dep

th (m

)

60

50

40

30

20

10

0

Dep

th (m

)

60

50

40

30

20

10

0D

epth

(m)

Dep

th (m

)

t = 0 st = 2325 st = 2605 st = 2974 s

t = 3300 st = 3624 st = 3949 st = 4273 s

t = 200 st = 2325 st = 2605 st = 2974 s

t = 3300 st = 3624 st = 3949 st = 4273 s

t = 200 st = 2325 st = 2605 st = 2974 s

t = 3300 st = 3624 st = 3949 st = 4273 s










0005

01015

02025

03035

04045

05

0 1 2 3 4 5 6 7 8Offset ( WD)

Late

ral d

ispla

cem

ent a

t WH

(m)

400500600700800900

10001100120013001400

0 1 2 3 4 5 6 7 8Offset ( WD)

0

002

004

006

008

01

012

0 1 2 3 4 5 6 7 8Offset ( WD)

400500600700800900

1000110012001300

0 1 2 3 4 5 6 7 8Offset ( WD)




Bend

ing

mom

ent a

t WH

(kNmiddotm

)

Late

ral d

ispla

cem

ent a

tminus5

m (m

)

Bend

ing

mom

ent a

tminus5

m (k

Nmiddotm

)

WH)



0 5 10 15 20 25Offset (m)

400

600

800

1000

1200

1400

1600

0 5 10 15 20 25Offset (m)

Bend

ing

mom

ent (

kNmiddotm

)

0001002003004005006007

Late

ral d

ispla

cem

ent (

m)









00005

0010015

0020025

0030035

004


Late

ral d

ispla

cem

ent (

m)

WHML(minus5m)

WHML(minus5m)

0100200300400500600700800900

1000


Bend

ing

mom

ent (

kNmiddotm

)




5 Conclusion





Acknowledgments


References



























RoboticsJournal of




VLSI Design



Shock and Vibration







Journal of



Volume 2018



Volume 2018


Journal of




SensorsJournal of



RotatingMachinery





Propagation






Hindawi


Advances in

Multimedia



ySea level

Casing riser

Mudline



Soil reactiondx

Wellheadamp SID

Conductorq

Casing strings


Surface BOP

dx

o

Wave

Current

Fw(x t)

L1

L2

(a)

M

P

Q

Fwdx

mdx2y

dt2

P + dP

Q+Q

xdx

M+M

xdx

(b)

M

P

Q

mdx2y

dt2

M +M

xdx

P + dP

qdx kfydx

Q+Q

xdx

(c)







1198711 = 119871uj + 119871119903 + 119871 lj + 119871 sid + 119871ml

1198712 = 119871119888 + 119871 sc(2)





(3)




119910 (0 119905) = 119878 (119905)119872 (0 119905) = 1198641198681205972119910 (119909 119905)1205971199092

100381610038161003816100381610038161003816100381610038161003816119909=0

= 119870ru120579ru

119872(1198711 + 1198712 119905) = 119864119868 (119909) 1205972119910 (119909 119905)1205971199092100381610038161003816100381610038161003816100381610038161003816119909=1198711+1198712

= 0

119876 (1198711 + 1198712 119905) = 119864119868 (119909) 1205973119910 (119909 119905)1205971199093100381610038161003816100381610038161003816100381610038161003816119909=1198711+1198712 = 0

(4)


119910 (119909 0) = 119910static (119909) (5)


119878 (119905) = 1198780 + 119878119871 sin(2120587119905119879119871 )

+ 119899sum119894=1

119878119899 cos (119896119899119878 (119905) minus 120596119899119905 + 120601119899 + 119886119899) (6)







(7)



1198951 + 4(2 + 119870rΔℎ11986401198680)119910

1198950

(8)




1199100119894 = 119910static (119894) 119894 = minus1 0 1 119873 + 2 (10)




12059721205971199092 [119864 (119909) 119868 (119909) 1205972119910 (119909 119905)1205971199092 ] + 120597120597119909 [119875 (119909) 120597119910 (119909 119905)120597119909 ]

+ 119898 (119909) 1205972119910 (119909 119905)1205971199052 + 119896 (119909) 119910 (119909 119905) = 0(11)


119864 (119909) 119868 (119909) 1205974119910 (119909 119905)1205971199094 + 119875 (119909) 1205972119910 (119909 119905)1205971199092+ 119898 (119909) 1205972119910 (119909 119905)1205971199052 + 119896 (119909) 119910 (119909 119905) = 0

(12)


119884 (119909) = 119860 cosh (120573119909) + 119861 sinh (120573119909) + 119862 cos (120574119909)+ 119863 sin (120574119909)

120573 = ( 1198752119864119868 + ( 1198752

411986421198682 + 1198981205962

119864119868 minus119896119891119864119868)12)12

120574 = ( minus1198752119864119868 + ( 1198752

411986421198682 + 1198981205962

119864119868 minus119896119891119864119868)12)12

(13)




Node 0

Node 1

Node 2

Segment 1

Segment 2

Segment

Segment i + 1

i minus 1

Segment i Li

Segment Nminus 1

Segment N

Node N

Node Nminus 1

Node i + 1

Node i minus 1

Node i

x

y




(14)


119884119894 (119909) = 119860 119894 cosh (119883119894) + 119861119894 sinh (119883119894) + 119862119894 cos (119883119883119894)+ 119863119894 sin (119883119883119894) (15)


119884119894+1 (119909) = 119860 119894+1 cosh (119883119894+1) + 119861119894+1 sinh (119883119894+1)+ 119862119894+1 cos (119883119883119894+1) + 119863119894+1 sin (119883119883119894+1) (16)


119884119894+1 (119909119894) = 119884119894 (119909119894)1198841015840119894+1 (119909119894) = 1198841015840119894 (119909119894)(119864119868)119894+1 11988410158401015840119894+1 (119909119894) = (119864119868)119894 11988410158401015840119894 (119909119894)(119864119868)119894+1 119884101584010158401015840119894+1 (119909119894) minus 119875119894+11198841015840119894+1 (119909119894)= (119864119868)119894 119884101584010158401015840119894 (119909119894) minus 1198751198941198841015840119894 (119909119894)

(17)


[[[[[[

119860 119894+1119861119894+1119862119894+1119863119894+1

]]]]]]=[[[[[[[[[[[

11989941198981 11989931198981 minus11989921198982 minus11989911198982120573119894120573119894+1 1198993 (1198981 + V1)

120573119894120573119894+1 1198994 (1198981 + V1)120574119894120573119894+1 1198991 (1198982 + V2)


minus120573119894120574119894+1 1198994 (1198983 + V1)minus120574119894120574119894+1 1198991 (1198984 + V2)

120574119894120574119894+1 1198992 (1198984 + V2)

]]]]]]]]]]]

[[[[[[[

119860 119894119861119894119862119894119863119894

]]]]]]] (18)

where

1198991 = sin (120574119894119897119894) 1198992 = cos (120574119894119897119894) 1198993 = sinh (120573119894119897119894)

1198994 = cosh (120573119894119897119894)1198981 = (119864119868)119894 1205732119894 + (119864119868)119894+1 1205742119894+1(119864119868)119894+1 (1205732119894+1 + 1205742119894+1) 1198982 = (119864119868)119894 1205742119894 minus (119864119868)119894+1 1205742119894+1(119864119868)119894+1 (1205732119894+1 + 1205742119894+1)


200 400 600 800 1000 1200 1400

Mode 1Mode 2Mode 3

Mode 4Mode 5Mode 6

Mode 1Mode 2Mode 3

Mode 4Mode 5Mode 6

x (m)x (m)


010305070911

Nor

mal

ized

ampl

itude


51

0001002003

Nor

mal

ized

ampl

itude

1381

1379

1377

1375

1373

1371

1369

1367

1365

1363

1361

1359

1357

1355

1353


200 400 600 800 1000 1200 1400x (m)

x (m)

x (m)

x (m)

Nor

mal

ized

ampl

itude


010305070911

0 200 400 600 800 1000 1200 1400N

orm

aliz

ed a

mpl

itude


010305070911

0 200 400 600 800 1000 1200 1400

Nor

mal

ized

ampl

itude


0 200 400 600 800 1000 1200 1400

Nor

mal

ized

ampl

itude


010305070911









000501

01502

02503

035

1 2 3 4 5 6Mode

Nat

ural

Fre

quen

cy (H

z)

0

00501

015

02025

03

1 2 3 4 5 6Mode

Nat

ural

Fre

quen

cy (H

z)

Clay 1Clay 2Clay 3





200 300 400 500 600 700 800 900 1000


0102030405060

Plat

form

mot

ion

ampl

itude

(m)


DriftWave


Bottom of USJ

Top of LSJ

00280029

003003100320033003400350036

Wellhead

0010011001200130014001500160017


202530354045505560

Late

ral d

ispla

cem

ent

(m)

45505560657075

Late

ral d

ispla

cem

ent

(m)

1700

1900

1500

2300

2500

2100

2900

270040

0

600

800

200

1200

1400

1000

Time (s)

6062646668707274767880

Late

ral d

ispla

cem

ent

(m)

3400

3200

3000

3800

4000

4200

4400

3600

Riser (minus300 m)

2527293133353739

Late

ral d

ispla

cem

ent

(m)

Late

ral d

ispla

cem

ent

(m)

Late

ral d

ispla

cem

ent

(m)

5900

5700

5500

5300

5100

4900

4700

4500 40

0

600

800

1000

1200

140020

0

1700

1900

2100

2300

2500

2700

2900

1500

000001000020000300004000050000600007

Late

ral d

ispla

cem

ent

(m)

Late

ral d

ispla

cem

ent

(m)

3000

3400

3600

3800

4000

4200

4400

3200

5900

5700

5500

5300

5100

4900

4700

4500


Mudline

Riser (minus600 m)

Time (s) Time (s)


Time (s)

Time (s)


1198983 = (119864119868)119894 1205732119894 minus (119864119868)119894+1 1205732119894+1(119864119868)119894+1 (1205732119894+1 + 1205742119894+1)

1198984 = (119864119868)119894 1205742119894 + (119864119868)119894+1 1205732119894+1(119864119868)119894+1 (1205732119894+1 + 1205742119894+1)

V1 = 119875119894+1 minus 119875119894(119864119868)119894+1 (1205732119894+1 + 1205742119894+1) V2 = 119875119894 minus 119875119894+1(119864119868)119894+1 (1205732119894+1 + 1205742119894+1)

(19)


Wellhead Mudline

23235

24245

25255

26

662646668

77274

889092949698

100102104106

Top of LSJ

780800820840860880900920940960

800820840860880900920940

0102030405060

Bottom of USJ

400

600

800

200

1200

1400

1000

Time (s)



Bend

ing

mom

ent

(KNmiddotm

)

4547495153555759

Bend

ing

mom

ent

(KNmiddotm

)

Bend

ing

mom

ent

(KNmiddotm

)

Bend

ing

mom

ent

(KNmiddotm

)

Bend

ing

mom

ent

(KNmiddotm

)

Bend

ing

mom

ent

(KNmiddotm

)

Bend

ing

mom

ent

(KNmiddotm

)

Bend

ing

mom

ent

(KNmiddotm

)

3200

3400

3600

3800

4000

4200

4400

3000

Time (s)

4700

4900

5100

5300

5500

5700

5900

4500

Time (s)

1700

1900

2100

2300

2500

2700

2900

1500

Time (s)40

0

600

200

800

1200

1400

1000

Time (s)

4700

4900

5100

5300

5500

5700

5900

4500

Time (s)

3200

3400

3600

3800

4000

4200

4400

3000

Time (s)

400420440460480500520540560580600

1700

1900

1500

2300

2500

2700

2900

2100

Time (s)


Let

119860 (119894+1) = [119860 119894+1 119861119894+1 119862119894+1 119863119894+1]119879 119860 (119894) = [119860 119894 119861119894 119862119894 119863119894]119879

119885(119894) =[[[[[[[[[[

11989941198981 11989931198981 minus11989921198982 minus11989911198982120573119894120573119894+1 1198993 (1198981 + V1)120573119894120573119894+1 1198994 (1198981 + V1)


minus120573119894120574119894+1 1198994 (1198983 + V1)minus120574119894120574119894+1 1198991 (1198984 + V2)

120574119894120574119894+1 1198992 (1198984 + V2)

]]]]]]]]]]

(20)

Then (18) becomes

119860 (119894+1) = 119885(119894)119860 (119894) (21)

From (21)

119860 (119873) = 119885119860 (1) (22)

where




0

15

30

45

60

75

90

105

120


5

55

6

65

7

75


780

800

820

840

860

880

900

920

940

960


WellheadMudline

0

100

200

300

400

500

600




Bend

ing

mom

ent (

kNmiddotm

)Be

ndin

g m

omen

t (kN

middotm)

Bend

ing

mom

ent (

kNmiddotm

)Be

ndin

g m

omen

t (kN

middotm)





1198601 = 1198621 = 0


[[[[[[

119860119873119861119873119862119873119863119873

]]]]]]= 0

(24)



























minus200 0 200 400 600 800 10000

10

20

30

40

50

60


005

115

225

335

445

5

0005 0015 0025 0035

0123456789

10

400 600 800 1000 3

4

5

6

7

8

9

12 16 20 24 28

60

50

40

30

20

10

0

Dep

th (m

)

60

50

40

30

20

10

0

Dep

th (m

)

60

50

40

30

20

10

0D

epth

(m)

Dep

th (m

)

t = 0 st = 2325 st = 2605 st = 2974 s

t = 3300 st = 3624 st = 3949 st = 4273 s

t = 200 st = 2325 st = 2605 st = 2974 s

t = 3300 st = 3624 st = 3949 st = 4273 s

t = 200 st = 2325 st = 2605 st = 2974 s

t = 3300 st = 3624 st = 3949 st = 4273 s










0005

01015

02025

03035

04045

05

0 1 2 3 4 5 6 7 8Offset ( WD)

Late

ral d

ispla

cem

ent a

t WH

(m)

400500600700800900

10001100120013001400

0 1 2 3 4 5 6 7 8Offset ( WD)

0

002

004

006

008

01

012

0 1 2 3 4 5 6 7 8Offset ( WD)

400500600700800900

1000110012001300

0 1 2 3 4 5 6 7 8Offset ( WD)




Bend

ing

mom

ent a

t WH

(kNmiddotm

)

Late

ral d

ispla

cem

ent a

tminus5

m (m

)

Bend

ing

mom

ent a

tminus5

m (k

Nmiddotm

)

WH)



0 5 10 15 20 25Offset (m)

400

600

800

1000

1200

1400

1600

0 5 10 15 20 25Offset (m)

Bend

ing

mom

ent (

kNmiddotm

)

0001002003004005006007

Late

ral d

ispla

cem

ent (

m)









00005

0010015

0020025

0030035

004


Late

ral d

ispla

cem

ent (

m)

WHML(minus5m)

WHML(minus5m)

0100200300400500600700800900

1000


Bend

ing

mom

ent (

kNmiddotm

)




5 Conclusion





Acknowledgments


References



























RoboticsJournal of




VLSI Design



Shock and Vibration







Journal of



Volume 2018



Volume 2018


Journal of




SensorsJournal of



RotatingMachinery





Propagation






Hindawi


Advances in

Multimedia




1198711 = 119871uj + 119871119903 + 119871 lj + 119871 sid + 119871ml

1198712 = 119871119888 + 119871 sc(2)





(3)




119910 (0 119905) = 119878 (119905)119872 (0 119905) = 1198641198681205972119910 (119909 119905)1205971199092

100381610038161003816100381610038161003816100381610038161003816119909=0

= 119870ru120579ru

119872(1198711 + 1198712 119905) = 119864119868 (119909) 1205972119910 (119909 119905)1205971199092100381610038161003816100381610038161003816100381610038161003816119909=1198711+1198712

= 0

119876 (1198711 + 1198712 119905) = 119864119868 (119909) 1205973119910 (119909 119905)1205971199093100381610038161003816100381610038161003816100381610038161003816119909=1198711+1198712 = 0

(4)


119910 (119909 0) = 119910static (119909) (5)


119878 (119905) = 1198780 + 119878119871 sin(2120587119905119879119871 )

+ 119899sum119894=1

119878119899 cos (119896119899119878 (119905) minus 120596119899119905 + 120601119899 + 119886119899) (6)







(7)



1198951 + 4(2 + 119870rΔℎ11986401198680)119910

1198950

(8)




1199100119894 = 119910static (119894) 119894 = minus1 0 1 119873 + 2 (10)




12059721205971199092 [119864 (119909) 119868 (119909) 1205972119910 (119909 119905)1205971199092 ] + 120597120597119909 [119875 (119909) 120597119910 (119909 119905)120597119909 ]

+ 119898 (119909) 1205972119910 (119909 119905)1205971199052 + 119896 (119909) 119910 (119909 119905) = 0(11)


119864 (119909) 119868 (119909) 1205974119910 (119909 119905)1205971199094 + 119875 (119909) 1205972119910 (119909 119905)1205971199092+ 119898 (119909) 1205972119910 (119909 119905)1205971199052 + 119896 (119909) 119910 (119909 119905) = 0

(12)


119884 (119909) = 119860 cosh (120573119909) + 119861 sinh (120573119909) + 119862 cos (120574119909)+ 119863 sin (120574119909)

120573 = ( 1198752119864119868 + ( 1198752

411986421198682 + 1198981205962

119864119868 minus119896119891119864119868)12)12

120574 = ( minus1198752119864119868 + ( 1198752

411986421198682 + 1198981205962

119864119868 minus119896119891119864119868)12)12

(13)




Node 0

Node 1

Node 2

Segment 1

Segment 2

Segment

Segment i + 1

i minus 1

Segment i Li

Segment Nminus 1

Segment N

Node N

Node Nminus 1

Node i + 1

Node i minus 1

Node i

x

y




(14)


119884119894 (119909) = 119860 119894 cosh (119883119894) + 119861119894 sinh (119883119894) + 119862119894 cos (119883119883119894)+ 119863119894 sin (119883119883119894) (15)


119884119894+1 (119909) = 119860 119894+1 cosh (119883119894+1) + 119861119894+1 sinh (119883119894+1)+ 119862119894+1 cos (119883119883119894+1) + 119863119894+1 sin (119883119883119894+1) (16)


119884119894+1 (119909119894) = 119884119894 (119909119894)1198841015840119894+1 (119909119894) = 1198841015840119894 (119909119894)(119864119868)119894+1 11988410158401015840119894+1 (119909119894) = (119864119868)119894 11988410158401015840119894 (119909119894)(119864119868)119894+1 119884101584010158401015840119894+1 (119909119894) minus 119875119894+11198841015840119894+1 (119909119894)= (119864119868)119894 119884101584010158401015840119894 (119909119894) minus 1198751198941198841015840119894 (119909119894)

(17)


[[[[[[

119860 119894+1119861119894+1119862119894+1119863119894+1

]]]]]]=[[[[[[[[[[[

11989941198981 11989931198981 minus11989921198982 minus11989911198982120573119894120573119894+1 1198993 (1198981 + V1)

120573119894120573119894+1 1198994 (1198981 + V1)120574119894120573119894+1 1198991 (1198982 + V2)


minus120573119894120574119894+1 1198994 (1198983 + V1)minus120574119894120574119894+1 1198991 (1198984 + V2)

120574119894120574119894+1 1198992 (1198984 + V2)

]]]]]]]]]]]

[[[[[[[

119860 119894119861119894119862119894119863119894

]]]]]]] (18)

where

1198991 = sin (120574119894119897119894) 1198992 = cos (120574119894119897119894) 1198993 = sinh (120573119894119897119894)

1198994 = cosh (120573119894119897119894)1198981 = (119864119868)119894 1205732119894 + (119864119868)119894+1 1205742119894+1(119864119868)119894+1 (1205732119894+1 + 1205742119894+1) 1198982 = (119864119868)119894 1205742119894 minus (119864119868)119894+1 1205742119894+1(119864119868)119894+1 (1205732119894+1 + 1205742119894+1)


200 400 600 800 1000 1200 1400

Mode 1Mode 2Mode 3

Mode 4Mode 5Mode 6

Mode 1Mode 2Mode 3

Mode 4Mode 5Mode 6

x (m)x (m)


010305070911

Nor

mal

ized

ampl

itude


51

0001002003

Nor

mal

ized

ampl

itude

1381

1379

1377

1375

1373

1371

1369

1367

1365

1363

1361

1359

1357

1355

1353


200 400 600 800 1000 1200 1400x (m)

x (m)

x (m)

x (m)

Nor

mal

ized

ampl

itude


010305070911

0 200 400 600 800 1000 1200 1400N

orm

aliz

ed a

mpl

itude


010305070911

0 200 400 600 800 1000 1200 1400

Nor

mal

ized

ampl

itude


0 200 400 600 800 1000 1200 1400

Nor

mal

ized

ampl

itude


010305070911









000501

01502

02503

035

1 2 3 4 5 6Mode

Nat

ural

Fre

quen

cy (H

z)

0

00501

015

02025

03

1 2 3 4 5 6Mode

Nat

ural

Fre

quen

cy (H

z)

Clay 1Clay 2Clay 3





200 300 400 500 600 700 800 900 1000


0102030405060

Plat

form

mot

ion

ampl

itude

(m)


DriftWave


Bottom of USJ

Top of LSJ

00280029

003003100320033003400350036

Wellhead

0010011001200130014001500160017


202530354045505560

Late

ral d

ispla

cem

ent

(m)

45505560657075

Late

ral d

ispla

cem

ent

(m)

1700

1900

1500

2300

2500

2100

2900

270040

0

600

800

200

1200

1400

1000

Time (s)

6062646668707274767880

Late

ral d

ispla

cem

ent

(m)

3400

3200

3000

3800

4000

4200

4400

3600

Riser (minus300 m)

2527293133353739

Late

ral d

ispla

cem

ent

(m)

Late

ral d

ispla

cem

ent

(m)

Late

ral d

ispla

cem

ent

(m)

5900

5700

5500

5300

5100

4900

4700

4500 40

0

600

800

1000

1200

140020

0

1700

1900

2100

2300

2500

2700

2900

1500

000001000020000300004000050000600007

Late

ral d

ispla

cem

ent

(m)

Late

ral d

ispla

cem

ent

(m)

3000

3400

3600

3800

4000

4200

4400

3200

5900

5700

5500

5300

5100

4900

4700

4500


Mudline

Riser (minus600 m)

Time (s) Time (s)


Time (s)

Time (s)


1198983 = (119864119868)119894 1205732119894 minus (119864119868)119894+1 1205732119894+1(119864119868)119894+1 (1205732119894+1 + 1205742119894+1)

1198984 = (119864119868)119894 1205742119894 + (119864119868)119894+1 1205732119894+1(119864119868)119894+1 (1205732119894+1 + 1205742119894+1)

V1 = 119875119894+1 minus 119875119894(119864119868)119894+1 (1205732119894+1 + 1205742119894+1) V2 = 119875119894 minus 119875119894+1(119864119868)119894+1 (1205732119894+1 + 1205742119894+1)

(19)


Wellhead Mudline

23235

24245

25255

26

662646668

77274

889092949698

100102104106

Top of LSJ

780800820840860880900920940960

800820840860880900920940

0102030405060

Bottom of USJ

400

600

800

200

1200

1400

1000

Time (s)



Bend

ing

mom

ent

(KNmiddotm

)

4547495153555759

Bend

ing

mom

ent

(KNmiddotm

)

Bend

ing

mom

ent

(KNmiddotm

)

Bend

ing

mom

ent

(KNmiddotm

)

Bend

ing

mom

ent

(KNmiddotm

)

Bend

ing

mom

ent

(KNmiddotm

)

Bend

ing

mom

ent

(KNmiddotm

)

Bend

ing

mom

ent

(KNmiddotm

)

3200

3400

3600

3800

4000

4200

4400

3000

Time (s)

4700

4900

5100

5300

5500

5700

5900

4500

Time (s)

1700

1900

2100

2300

2500

2700

2900

1500

Time (s)40

0

600

200

800

1200

1400

1000

Time (s)

4700

4900

5100

5300

5500

5700

5900

4500

Time (s)

3200

3400

3600

3800

4000

4200

4400

3000

Time (s)

400420440460480500520540560580600

1700

1900

1500

2300

2500

2700

2900

2100

Time (s)


Let

119860 (119894+1) = [119860 119894+1 119861119894+1 119862119894+1 119863119894+1]119879 119860 (119894) = [119860 119894 119861119894 119862119894 119863119894]119879

119885(119894) =[[[[[[[[[[

11989941198981 11989931198981 minus11989921198982 minus11989911198982120573119894120573119894+1 1198993 (1198981 + V1)120573119894120573119894+1 1198994 (1198981 + V1)


minus120573119894120574119894+1 1198994 (1198983 + V1)minus120574119894120574119894+1 1198991 (1198984 + V2)

120574119894120574119894+1 1198992 (1198984 + V2)

]]]]]]]]]]

(20)

Then (18) becomes

119860 (119894+1) = 119885(119894)119860 (119894) (21)

From (21)

119860 (119873) = 119885119860 (1) (22)

where




0

15

30

45

60

75

90

105

120


5

55

6

65

7

75


780

800

820

840

860

880

900

920

940

960


WellheadMudline

0

100

200

300

400

500

600




Bend

ing

mom

ent (

kNmiddotm

)Be

ndin

g m

omen

t (kN

middotm)

Bend

ing

mom

ent (

kNmiddotm

)Be

ndin

g m

omen

t (kN

middotm)





1198601 = 1198621 = 0


[[[[[[

119860119873119861119873119862119873119863119873

]]]]]]= 0

(24)



























minus200 0 200 400 600 800 10000

10

20

30

40

50

60


005

115

225

335

445

5

0005 0015 0025 0035

0123456789

10

400 600 800 1000 3

4

5

6

7

8

9

12 16 20 24 28

60

50

40

30

20

10

0

Dep

th (m

)

60

50

40

30

20

10

0

Dep

th (m

)

60

50

40

30

20

10

0D

epth

(m)

Dep

th (m

)

t = 0 st = 2325 st = 2605 st = 2974 s

t = 3300 st = 3624 st = 3949 st = 4273 s

t = 200 st = 2325 st = 2605 st = 2974 s

t = 3300 st = 3624 st = 3949 st = 4273 s

t = 200 st = 2325 st = 2605 st = 2974 s

t = 3300 st = 3624 st = 3949 st = 4273 s










0005

01015

02025

03035

04045

05

0 1 2 3 4 5 6 7 8Offset ( WD)

Late

ral d

ispla

cem

ent a

t WH

(m)

400500600700800900

10001100120013001400

0 1 2 3 4 5 6 7 8Offset ( WD)

0

002

004

006

008

01

012

0 1 2 3 4 5 6 7 8Offset ( WD)

400500600700800900

1000110012001300

0 1 2 3 4 5 6 7 8Offset ( WD)




Bend

ing

mom

ent a

t WH

(kNmiddotm

)

Late

ral d

ispla

cem

ent a

tminus5

m (m

)

Bend

ing

mom

ent a

tminus5

m (k

Nmiddotm

)

WH)



0 5 10 15 20 25Offset (m)

400

600

800

1000

1200

1400

1600

0 5 10 15 20 25Offset (m)

Bend

ing

mom

ent (

kNmiddotm

)

0001002003004005006007

Late

ral d

ispla

cem

ent (

m)









00005

0010015

0020025

0030035

004


Late

ral d

ispla

cem

ent (

m)

WHML(minus5m)

WHML(minus5m)

0100200300400500600700800900

1000


Bend

ing

mom

ent (

kNmiddotm

)




5 Conclusion





Acknowledgments


References



























RoboticsJournal of




VLSI Design



Shock and Vibration







Journal of



Volume 2018



Volume 2018


Journal of




SensorsJournal of



RotatingMachinery





Propagation






Hindawi


Advances in

Multimedia






(7)



1198951 + 4(2 + 119870rΔℎ11986401198680)119910

1198950

(8)




1199100119894 = 119910static (119894) 119894 = minus1 0 1 119873 + 2 (10)




12059721205971199092 [119864 (119909) 119868 (119909) 1205972119910 (119909 119905)1205971199092 ] + 120597120597119909 [119875 (119909) 120597119910 (119909 119905)120597119909 ]

+ 119898 (119909) 1205972119910 (119909 119905)1205971199052 + 119896 (119909) 119910 (119909 119905) = 0(11)


119864 (119909) 119868 (119909) 1205974119910 (119909 119905)1205971199094 + 119875 (119909) 1205972119910 (119909 119905)1205971199092+ 119898 (119909) 1205972119910 (119909 119905)1205971199052 + 119896 (119909) 119910 (119909 119905) = 0

(12)


119884 (119909) = 119860 cosh (120573119909) + 119861 sinh (120573119909) + 119862 cos (120574119909)+ 119863 sin (120574119909)

120573 = ( 1198752119864119868 + ( 1198752

411986421198682 + 1198981205962

119864119868 minus119896119891119864119868)12)12

120574 = ( minus1198752119864119868 + ( 1198752

411986421198682 + 1198981205962

119864119868 minus119896119891119864119868)12)12

(13)




Node 0

Node 1

Node 2

Segment 1

Segment 2

Segment

Segment i + 1

i minus 1

Segment i Li

Segment Nminus 1

Segment N

Node N

Node Nminus 1

Node i + 1

Node i minus 1

Node i

x

y




(14)


119884119894 (119909) = 119860 119894 cosh (119883119894) + 119861119894 sinh (119883119894) + 119862119894 cos (119883119883119894)+ 119863119894 sin (119883119883119894) (15)


119884119894+1 (119909) = 119860 119894+1 cosh (119883119894+1) + 119861119894+1 sinh (119883119894+1)+ 119862119894+1 cos (119883119883119894+1) + 119863119894+1 sin (119883119883119894+1) (16)


119884119894+1 (119909119894) = 119884119894 (119909119894)1198841015840119894+1 (119909119894) = 1198841015840119894 (119909119894)(119864119868)119894+1 11988410158401015840119894+1 (119909119894) = (119864119868)119894 11988410158401015840119894 (119909119894)(119864119868)119894+1 119884101584010158401015840119894+1 (119909119894) minus 119875119894+11198841015840119894+1 (119909119894)= (119864119868)119894 119884101584010158401015840119894 (119909119894) minus 1198751198941198841015840119894 (119909119894)

(17)


[[[[[[

119860 119894+1119861119894+1119862119894+1119863119894+1

]]]]]]=[[[[[[[[[[[

11989941198981 11989931198981 minus11989921198982 minus11989911198982120573119894120573119894+1 1198993 (1198981 + V1)

120573119894120573119894+1 1198994 (1198981 + V1)120574119894120573119894+1 1198991 (1198982 + V2)


minus120573119894120574119894+1 1198994 (1198983 + V1)minus120574119894120574119894+1 1198991 (1198984 + V2)

120574119894120574119894+1 1198992 (1198984 + V2)

]]]]]]]]]]]

[[[[[[[

119860 119894119861119894119862119894119863119894

]]]]]]] (18)

where

1198991 = sin (120574119894119897119894) 1198992 = cos (120574119894119897119894) 1198993 = sinh (120573119894119897119894)

1198994 = cosh (120573119894119897119894)1198981 = (119864119868)119894 1205732119894 + (119864119868)119894+1 1205742119894+1(119864119868)119894+1 (1205732119894+1 + 1205742119894+1) 1198982 = (119864119868)119894 1205742119894 minus (119864119868)119894+1 1205742119894+1(119864119868)119894+1 (1205732119894+1 + 1205742119894+1)


200 400 600 800 1000 1200 1400

Mode 1Mode 2Mode 3

Mode 4Mode 5Mode 6

Mode 1Mode 2Mode 3

Mode 4Mode 5Mode 6

x (m)x (m)


010305070911

Nor

mal

ized

ampl

itude


51

0001002003

Nor

mal

ized

ampl

itude

1381

1379

1377

1375

1373

1371

1369

1367

1365

1363

1361

1359

1357

1355

1353


200 400 600 800 1000 1200 1400x (m)

x (m)

x (m)

x (m)

Nor

mal

ized

ampl

itude


010305070911

0 200 400 600 800 1000 1200 1400N

orm

aliz

ed a

mpl

itude


010305070911

0 200 400 600 800 1000 1200 1400

Nor

mal

ized

ampl

itude


0 200 400 600 800 1000 1200 1400

Nor

mal

ized

ampl

itude


010305070911









000501

01502

02503

035

1 2 3 4 5 6Mode

Nat

ural

Fre

quen

cy (H

z)

0

00501

015

02025

03

1 2 3 4 5 6Mode

Nat

ural

Fre

quen

cy (H

z)

Clay 1Clay 2Clay 3





200 300 400 500 600 700 800 900 1000


0102030405060

Plat

form

mot

ion

ampl

itude

(m)


DriftWave


Bottom of USJ

Top of LSJ

00280029

003003100320033003400350036

Wellhead

0010011001200130014001500160017


202530354045505560

Late

ral d

ispla

cem

ent

(m)

45505560657075

Late

ral d

ispla

cem

ent

(m)

1700

1900

1500

2300

2500

2100

2900

270040

0

600

800

200

1200

1400

1000

Time (s)

6062646668707274767880

Late

ral d

ispla

cem

ent

(m)

3400

3200

3000

3800

4000

4200

4400

3600

Riser (minus300 m)

2527293133353739

Late

ral d

ispla

cem

ent

(m)

Late

ral d

ispla

cem

ent

(m)

Late

ral d

ispla

cem

ent

(m)

5900

5700

5500

5300

5100

4900

4700

4500 40

0

600

800

1000

1200

140020

0

1700

1900

2100

2300

2500

2700

2900

1500

000001000020000300004000050000600007

Late

ral d

ispla

cem

ent

(m)

Late

ral d

ispla

cem

ent

(m)

3000

3400

3600

3800

4000

4200

4400

3200

5900

5700

5500

5300

5100

4900

4700

4500


Mudline

Riser (minus600 m)

Time (s) Time (s)


Time (s)

Time (s)


1198983 = (119864119868)119894 1205732119894 minus (119864119868)119894+1 1205732119894+1(119864119868)119894+1 (1205732119894+1 + 1205742119894+1)

1198984 = (119864119868)119894 1205742119894 + (119864119868)119894+1 1205732119894+1(119864119868)119894+1 (1205732119894+1 + 1205742119894+1)

V1 = 119875119894+1 minus 119875119894(119864119868)119894+1 (1205732119894+1 + 1205742119894+1) V2 = 119875119894 minus 119875119894+1(119864119868)119894+1 (1205732119894+1 + 1205742119894+1)

(19)


Wellhead Mudline

23235

24245

25255

26

662646668

77274

889092949698

100102104106

Top of LSJ

780800820840860880900920940960

800820840860880900920940

0102030405060

Bottom of USJ

400

600

800

200

1200

1400

1000

Time (s)



Bend

ing

mom

ent

(KNmiddotm

)

4547495153555759

Bend

ing

mom

ent

(KNmiddotm

)

Bend

ing

mom

ent

(KNmiddotm

)

Bend

ing

mom

ent

(KNmiddotm

)

Bend

ing

mom

ent

(KNmiddotm

)

Bend

ing

mom

ent

(KNmiddotm

)

Bend

ing

mom

ent

(KNmiddotm

)

Bend

ing

mom

ent

(KNmiddotm

)

3200

3400

3600

3800

4000

4200

4400

3000

Time (s)

4700

4900

5100

5300

5500

5700

5900

4500

Time (s)

1700

1900

2100

2300

2500

2700

2900

1500

Time (s)40

0

600

200

800

1200

1400

1000

Time (s)

4700

4900

5100

5300

5500

5700

5900

4500

Time (s)

3200

3400

3600

3800

4000

4200

4400

3000

Time (s)

400420440460480500520540560580600

1700

1900

1500

2300

2500

2700

2900

2100

Time (s)


Let

119860 (119894+1) = [119860 119894+1 119861119894+1 119862119894+1 119863119894+1]119879 119860 (119894) = [119860 119894 119861119894 119862119894 119863119894]119879

119885(119894) =[[[[[[[[[[

11989941198981 11989931198981 minus11989921198982 minus11989911198982120573119894120573119894+1 1198993 (1198981 + V1)120573119894120573119894+1 1198994 (1198981 + V1)


minus120573119894120574119894+1 1198994 (1198983 + V1)minus120574119894120574119894+1 1198991 (1198984 + V2)

120574119894120574119894+1 1198992 (1198984 + V2)

]]]]]]]]]]

(20)

Then (18) becomes

119860 (119894+1) = 119885(119894)119860 (119894) (21)

From (21)

119860 (119873) = 119885119860 (1) (22)

where




0

15

30

45

60

75

90

105

120


5

55

6

65

7

75


780

800

820

840

860

880

900

920

940

960


WellheadMudline

0

100

200

300

400

500

600




Bend

ing

mom

ent (

kNmiddotm

)Be

ndin

g m

omen

t (kN

middotm)

Bend

ing

mom

ent (

kNmiddotm

)Be

ndin

g m

omen

t (kN

middotm)





1198601 = 1198621 = 0


[[[[[[

119860119873119861119873119862119873119863119873

]]]]]]= 0

(24)



























minus200 0 200 400 600 800 10000

10

20

30

40

50

60


005

115

225

335

445

5

0005 0015 0025 0035

0123456789

10

400 600 800 1000 3

4

5

6

7

8

9

12 16 20 24 28

60

50

40

30

20

10

0

Dep

th (m

)

60

50

40

30

20

10

0

Dep

th (m

)

60

50

40

30

20

10

0D

epth

(m)

Dep

th (m

)

t = 0 st = 2325 st = 2605 st = 2974 s

t = 3300 st = 3624 st = 3949 st = 4273 s

t = 200 st = 2325 st = 2605 st = 2974 s

t = 3300 st = 3624 st = 3949 st = 4273 s

t = 200 st = 2325 st = 2605 st = 2974 s

t = 3300 st = 3624 st = 3949 st = 4273 s










0005

01015

02025

03035

04045

05

0 1 2 3 4 5 6 7 8Offset ( WD)

Late

ral d

ispla

cem

ent a

t WH

(m)

400500600700800900

10001100120013001400

0 1 2 3 4 5 6 7 8Offset ( WD)

0

002

004

006

008

01

012

0 1 2 3 4 5 6 7 8Offset ( WD)

400500600700800900

1000110012001300

0 1 2 3 4 5 6 7 8Offset ( WD)




Bend

ing

mom

ent a

t WH

(kNmiddotm

)

Late

ral d

ispla

cem

ent a

tminus5

m (m

)

Bend

ing

mom

ent a

tminus5

m (k

Nmiddotm

)

WH)



0 5 10 15 20 25Offset (m)

400

600

800

1000

1200

1400

1600

0 5 10 15 20 25Offset (m)

Bend

ing

mom

ent (

kNmiddotm

)

0001002003004005006007

Late

ral d

ispla

cem

ent (

m)









00005

0010015

0020025

0030035

004


Late

ral d

ispla

cem

ent (

m)

WHML(minus5m)

WHML(minus5m)

0100200300400500600700800900

1000


Bend

ing

mom

ent (

kNmiddotm

)




5 Conclusion





Acknowledgments


References



























RoboticsJournal of




VLSI Design



Shock and Vibration







Journal of



Volume 2018



Volume 2018


Journal of




SensorsJournal of



RotatingMachinery





Propagation






Hindawi


Advances in

Multimedia



Node 0

Node 1

Node 2

Segment 1

Segment 2

Segment

Segment i + 1

i minus 1

Segment i Li

Segment Nminus 1

Segment N

Node N

Node Nminus 1

Node i + 1

Node i minus 1

Node i

x

y




(14)


119884119894 (119909) = 119860 119894 cosh (119883119894) + 119861119894 sinh (119883119894) + 119862119894 cos (119883119883119894)+ 119863119894 sin (119883119883119894) (15)


119884119894+1 (119909) = 119860 119894+1 cosh (119883119894+1) + 119861119894+1 sinh (119883119894+1)+ 119862119894+1 cos (119883119883119894+1) + 119863119894+1 sin (119883119883119894+1) (16)


119884119894+1 (119909119894) = 119884119894 (119909119894)1198841015840119894+1 (119909119894) = 1198841015840119894 (119909119894)(119864119868)119894+1 11988410158401015840119894+1 (119909119894) = (119864119868)119894 11988410158401015840119894 (119909119894)(119864119868)119894+1 119884101584010158401015840119894+1 (119909119894) minus 119875119894+11198841015840119894+1 (119909119894)= (119864119868)119894 119884101584010158401015840119894 (119909119894) minus 1198751198941198841015840119894 (119909119894)

(17)


[[[[[[

119860 119894+1119861119894+1119862119894+1119863119894+1

]]]]]]=[[[[[[[[[[[

11989941198981 11989931198981 minus11989921198982 minus11989911198982120573119894120573119894+1 1198993 (1198981 + V1)

120573119894120573119894+1 1198994 (1198981 + V1)120574119894120573119894+1 1198991 (1198982 + V2)


minus120573119894120574119894+1 1198994 (1198983 + V1)minus120574119894120574119894+1 1198991 (1198984 + V2)

120574119894120574119894+1 1198992 (1198984 + V2)

]]]]]]]]]]]

[[[[[[[

119860 119894119861119894119862119894119863119894

]]]]]]] (18)

where

1198991 = sin (120574119894119897119894) 1198992 = cos (120574119894119897119894) 1198993 = sinh (120573119894119897119894)

1198994 = cosh (120573119894119897119894)1198981 = (119864119868)119894 1205732119894 + (119864119868)119894+1 1205742119894+1(119864119868)119894+1 (1205732119894+1 + 1205742119894+1) 1198982 = (119864119868)119894 1205742119894 minus (119864119868)119894+1 1205742119894+1(119864119868)119894+1 (1205732119894+1 + 1205742119894+1)


200 400 600 800 1000 1200 1400

Mode 1Mode 2Mode 3

Mode 4Mode 5Mode 6

Mode 1Mode 2Mode 3

Mode 4Mode 5Mode 6

x (m)x (m)


010305070911

Nor

mal

ized

ampl

itude


51

0001002003

Nor

mal

ized

ampl

itude

1381

1379

1377

1375

1373

1371

1369

1367

1365

1363

1361

1359

1357

1355

1353


200 400 600 800 1000 1200 1400x (m)

x (m)

x (m)

x (m)

Nor

mal

ized

ampl

itude


010305070911

0 200 400 600 800 1000 1200 1400N

orm

aliz

ed a

mpl

itude


010305070911

0 200 400 600 800 1000 1200 1400

Nor

mal

ized

ampl

itude


0 200 400 600 800 1000 1200 1400

Nor

mal

ized

ampl

itude


010305070911









000501

01502

02503

035

1 2 3 4 5 6Mode

Nat

ural

Fre

quen

cy (H

z)

0

00501

015

02025

03

1 2 3 4 5 6Mode

Nat

ural

Fre

quen

cy (H

z)

Clay 1Clay 2Clay 3





200 300 400 500 600 700 800 900 1000


0102030405060

Plat

form

mot

ion

ampl

itude

(m)


DriftWave


Bottom of USJ

Top of LSJ

00280029

003003100320033003400350036

Wellhead

0010011001200130014001500160017


202530354045505560

Late

ral d

ispla

cem

ent

(m)

45505560657075

Late

ral d

ispla

cem

ent

(m)

1700

1900

1500

2300

2500

2100

2900

270040

0

600

800

200

1200

1400

1000

Time (s)

6062646668707274767880

Late

ral d

ispla

cem

ent

(m)

3400

3200

3000

3800

4000

4200

4400

3600

Riser (minus300 m)

2527293133353739

Late

ral d

ispla

cem

ent

(m)

Late

ral d

ispla

cem

ent

(m)

Late

ral d

ispla

cem

ent

(m)

5900

5700

5500

5300

5100

4900

4700

4500 40

0

600

800

1000

1200

140020

0

1700

1900

2100

2300

2500

2700

2900

1500

000001000020000300004000050000600007

Late

ral d

ispla

cem

ent

(m)

Late

ral d

ispla

cem

ent

(m)

3000

3400

3600

3800

4000

4200

4400

3200

5900

5700

5500

5300

5100

4900

4700

4500


Mudline

Riser (minus600 m)

Time (s) Time (s)


Time (s)

Time (s)


1198983 = (119864119868)119894 1205732119894 minus (119864119868)119894+1 1205732119894+1(119864119868)119894+1 (1205732119894+1 + 1205742119894+1)

1198984 = (119864119868)119894 1205742119894 + (119864119868)119894+1 1205732119894+1(119864119868)119894+1 (1205732119894+1 + 1205742119894+1)

V1 = 119875119894+1 minus 119875119894(119864119868)119894+1 (1205732119894+1 + 1205742119894+1) V2 = 119875119894 minus 119875119894+1(119864119868)119894+1 (1205732119894+1 + 1205742119894+1)

(19)


Wellhead Mudline

23235

24245

25255

26

662646668

77274

889092949698

100102104106

Top of LSJ

780800820840860880900920940960

800820840860880900920940

0102030405060

Bottom of USJ

400

600

800

200

1200

1400

1000

Time (s)



Bend

ing

mom

ent

(KNmiddotm

)

4547495153555759

Bend

ing

mom

ent

(KNmiddotm

)

Bend

ing

mom

ent

(KNmiddotm

)

Bend

ing

mom

ent

(KNmiddotm

)

Bend

ing

mom

ent

(KNmiddotm

)

Bend

ing

mom

ent

(KNmiddotm

)

Bend

ing

mom

ent

(KNmiddotm

)

Bend

ing

mom

ent

(KNmiddotm

)

3200

3400

3600

3800

4000

4200

4400

3000

Time (s)

4700

4900

5100

5300

5500

5700

5900

4500

Time (s)

1700

1900

2100

2300

2500

2700

2900

1500

Time (s)40

0

600

200

800

1200

1400

1000

Time (s)

4700

4900

5100

5300

5500

5700

5900

4500

Time (s)

3200

3400

3600

3800

4000

4200

4400

3000

Time (s)

400420440460480500520540560580600

1700

1900

1500

2300

2500

2700

2900

2100

Time (s)


Let

119860 (119894+1) = [119860 119894+1 119861119894+1 119862119894+1 119863119894+1]119879 119860 (119894) = [119860 119894 119861119894 119862119894 119863119894]119879

119885(119894) =[[[[[[[[[[

11989941198981 11989931198981 minus11989921198982 minus11989911198982120573119894120573119894+1 1198993 (1198981 + V1)120573119894120573119894+1 1198994 (1198981 + V1)


minus120573119894120574119894+1 1198994 (1198983 + V1)minus120574119894120574119894+1 1198991 (1198984 + V2)

120574119894120574119894+1 1198992 (1198984 + V2)

]]]]]]]]]]

(20)

Then (18) becomes

119860 (119894+1) = 119885(119894)119860 (119894) (21)

From (21)

119860 (119873) = 119885119860 (1) (22)

where




0

15

30

45

60

75

90

105

120


5

55

6

65

7

75


780

800

820

840

860

880

900

920

940

960


WellheadMudline

0

100

200

300

400

500

600




Bend

ing

mom

ent (

kNmiddotm

)Be

ndin

g m

omen

t (kN

middotm)

Bend

ing

mom

ent (

kNmiddotm

)Be

ndin

g m

omen

t (kN

middotm)





1198601 = 1198621 = 0


[[[[[[

119860119873119861119873119862119873119863119873

]]]]]]= 0

(24)



























minus200 0 200 400 600 800 10000

10

20

30

40

50

60


005

115

225

335

445

5

0005 0015 0025 0035

0123456789

10

400 600 800 1000 3

4

5

6

7

8

9

12 16 20 24 28

60

50

40

30

20

10

0

Dep

th (m

)

60

50

40

30

20

10

0

Dep

th (m

)

60

50

40

30

20

10

0D

epth

(m)

Dep

th (m

)

t = 0 st = 2325 st = 2605 st = 2974 s

t = 3300 st = 3624 st = 3949 st = 4273 s

t = 200 st = 2325 st = 2605 st = 2974 s

t = 3300 st = 3624 st = 3949 st = 4273 s

t = 200 st = 2325 st = 2605 st = 2974 s

t = 3300 st = 3624 st = 3949 st = 4273 s










0005

01015

02025

03035

04045

05

0 1 2 3 4 5 6 7 8Offset ( WD)

Late

ral d

ispla

cem

ent a

t WH

(m)

400500600700800900

10001100120013001400

0 1 2 3 4 5 6 7 8Offset ( WD)

0

002

004

006

008

01

012

0 1 2 3 4 5 6 7 8Offset ( WD)

400500600700800900

1000110012001300

0 1 2 3 4 5 6 7 8Offset ( WD)




Bend

ing

mom

ent a

t WH

(kNmiddotm

)

Late

ral d

ispla

cem

ent a

tminus5

m (m

)

Bend

ing

mom

ent a

tminus5

m (k

Nmiddotm

)

WH)



0 5 10 15 20 25Offset (m)

400

600

800

1000

1200

1400

1600

0 5 10 15 20 25Offset (m)

Bend

ing

mom

ent (

kNmiddotm

)

0001002003004005006007

Late

ral d

ispla

cem

ent (

m)









00005

0010015

0020025

0030035

004


Late

ral d

ispla

cem

ent (

m)

WHML(minus5m)

WHML(minus5m)

0100200300400500600700800900

1000


Bend

ing

mom

ent (

kNmiddotm

)




5 Conclusion





Acknowledgments


References



























RoboticsJournal of




VLSI Design



Shock and Vibration







Journal of



Volume 2018



Volume 2018


Journal of




SensorsJournal of



RotatingMachinery





Propagation






Hindawi


Advances in

Multimedia



200 400 600 800 1000 1200 1400

Mode 1Mode 2Mode 3

Mode 4Mode 5Mode 6

Mode 1Mode 2Mode 3

Mode 4Mode 5Mode 6

x (m)x (m)


010305070911

Nor

mal

ized

ampl

itude


51

0001002003

Nor

mal

ized

ampl

itude

1381

1379

1377

1375

1373

1371

1369

1367

1365

1363

1361

1359

1357

1355

1353


200 400 600 800 1000 1200 1400x (m)

x (m)

x (m)

x (m)

Nor

mal

ized

ampl

itude


010305070911

0 200 400 600 800 1000 1200 1400N

orm

aliz

ed a

mpl

itude


010305070911

0 200 400 600 800 1000 1200 1400

Nor

mal

ized

ampl

itude


0 200 400 600 800 1000 1200 1400

Nor

mal

ized

ampl

itude


010305070911









000501

01502

02503

035

1 2 3 4 5 6Mode

Nat

ural

Fre

quen

cy (H

z)

0

00501

015

02025

03

1 2 3 4 5 6Mode

Nat

ural

Fre

quen

cy (H

z)

Clay 1Clay 2Clay 3





200 300 400 500 600 700 800 900 1000


0102030405060

Plat

form

mot

ion

ampl

itude

(m)


DriftWave


Bottom of USJ

Top of LSJ

00280029

003003100320033003400350036

Wellhead

0010011001200130014001500160017


202530354045505560

Late

ral d

ispla

cem

ent

(m)

45505560657075

Late

ral d

ispla

cem

ent

(m)

1700

1900

1500

2300

2500

2100

2900

270040

0

600

800

200

1200

1400

1000

Time (s)

6062646668707274767880

Late

ral d

ispla

cem

ent

(m)

3400

3200

3000

3800

4000

4200

4400

3600

Riser (minus300 m)

2527293133353739

Late

ral d

ispla

cem

ent

(m)

Late

ral d

ispla

cem

ent

(m)

Late

ral d

ispla

cem

ent

(m)

5900

5700

5500

5300

5100

4900

4700

4500 40

0

600

800

1000

1200

140020

0

1700

1900

2100

2300

2500

2700

2900

1500

000001000020000300004000050000600007

Late

ral d

ispla

cem

ent

(m)

Late

ral d

ispla

cem

ent

(m)

3000

3400

3600

3800

4000

4200

4400

3200

5900

5700

5500

5300

5100

4900

4700

4500


Mudline

Riser (minus600 m)

Time (s) Time (s)


Time (s)

Time (s)


1198983 = (119864119868)119894 1205732119894 minus (119864119868)119894+1 1205732119894+1(119864119868)119894+1 (1205732119894+1 + 1205742119894+1)

1198984 = (119864119868)119894 1205742119894 + (119864119868)119894+1 1205732119894+1(119864119868)119894+1 (1205732119894+1 + 1205742119894+1)

V1 = 119875119894+1 minus 119875119894(119864119868)119894+1 (1205732119894+1 + 1205742119894+1) V2 = 119875119894 minus 119875119894+1(119864119868)119894+1 (1205732119894+1 + 1205742119894+1)

(19)


Wellhead Mudline

23235

24245

25255

26

662646668

77274

889092949698

100102104106

Top of LSJ

780800820840860880900920940960

800820840860880900920940

0102030405060

Bottom of USJ

400

600

800

200

1200

1400

1000

Time (s)



Bend

ing

mom

ent

(KNmiddotm

)

4547495153555759

Bend

ing

mom

ent

(KNmiddotm

)

Bend

ing

mom

ent

(KNmiddotm

)

Bend

ing

mom

ent

(KNmiddotm

)

Bend

ing

mom

ent

(KNmiddotm

)

Bend

ing

mom

ent

(KNmiddotm

)

Bend

ing

mom

ent

(KNmiddotm

)

Bend

ing

mom

ent

(KNmiddotm

)

3200

3400

3600

3800

4000

4200

4400

3000

Time (s)

4700

4900

5100

5300

5500

5700

5900

4500

Time (s)

1700

1900

2100

2300

2500

2700

2900

1500

Time (s)40

0

600

200

800

1200

1400

1000

Time (s)

4700

4900

5100

5300

5500

5700

5900

4500

Time (s)

3200

3400

3600

3800

4000

4200

4400

3000

Time (s)

400420440460480500520540560580600

1700

1900

1500

2300

2500

2700

2900

2100

Time (s)


Let

119860 (119894+1) = [119860 119894+1 119861119894+1 119862119894+1 119863119894+1]119879 119860 (119894) = [119860 119894 119861119894 119862119894 119863119894]119879

119885(119894) =[[[[[[[[[[

11989941198981 11989931198981 minus11989921198982 minus11989911198982120573119894120573119894+1 1198993 (1198981 + V1)120573119894120573119894+1 1198994 (1198981 + V1)


minus120573119894120574119894+1 1198994 (1198983 + V1)minus120574119894120574119894+1 1198991 (1198984 + V2)

120574119894120574119894+1 1198992 (1198984 + V2)

]]]]]]]]]]

(20)

Then (18) becomes

119860 (119894+1) = 119885(119894)119860 (119894) (21)

From (21)

119860 (119873) = 119885119860 (1) (22)

where




0

15

30

45

60

75

90

105

120


5

55

6

65

7

75


780

800

820

840

860

880

900

920

940

960


WellheadMudline

0

100

200

300

400

500

600




Bend

ing

mom

ent (

kNmiddotm

)Be

ndin

g m

omen

t (kN

middotm)

Bend

ing

mom

ent (

kNmiddotm

)Be

ndin

g m

omen

t (kN

middotm)





1198601 = 1198621 = 0


[[[[[[

119860119873119861119873119862119873119863119873

]]]]]]= 0

(24)



























minus200 0 200 400 600 800 10000

10

20

30

40

50

60


005

115

225

335

445

5

0005 0015 0025 0035

0123456789

10

400 600 800 1000 3

4

5

6

7

8

9

12 16 20 24 28

60

50

40

30

20

10

0

Dep

th (m

)

60

50

40

30

20

10

0

Dep

th (m

)

60

50

40

30

20

10

0D

epth

(m)

Dep

th (m

)

t = 0 st = 2325 st = 2605 st = 2974 s

t = 3300 st = 3624 st = 3949 st = 4273 s

t = 200 st = 2325 st = 2605 st = 2974 s

t = 3300 st = 3624 st = 3949 st = 4273 s

t = 200 st = 2325 st = 2605 st = 2974 s

t = 3300 st = 3624 st = 3949 st = 4273 s










0005

01015

02025

03035

04045

05

0 1 2 3 4 5 6 7 8Offset ( WD)

Late

ral d

ispla

cem

ent a

t WH

(m)

400500600700800900

10001100120013001400

0 1 2 3 4 5 6 7 8Offset ( WD)

0

002

004

006

008

01

012

0 1 2 3 4 5 6 7 8Offset ( WD)

400500600700800900

1000110012001300

0 1 2 3 4 5 6 7 8Offset ( WD)




Bend

ing

mom

ent a

t WH

(kNmiddotm

)

Late

ral d

ispla

cem

ent a

tminus5

m (m

)

Bend

ing

mom

ent a

tminus5

m (k

Nmiddotm

)

WH)



0 5 10 15 20 25Offset (m)

400

600

800

1000

1200

1400

1600

0 5 10 15 20 25Offset (m)

Bend

ing

mom

ent (

kNmiddotm

)

0001002003004005006007

Late

ral d

ispla

cem

ent (

m)









00005

0010015

0020025

0030035

004


Late

ral d

ispla

cem

ent (

m)

WHML(minus5m)

WHML(minus5m)

0100200300400500600700800900

1000


Bend

ing

mom

ent (

kNmiddotm

)




5 Conclusion





Acknowledgments


References



























RoboticsJournal of




VLSI Design



Shock and Vibration







Journal of



Volume 2018



Volume 2018


Journal of




SensorsJournal of



RotatingMachinery





Propagation






Hindawi


Advances in

Multimedia



200 300 400 500 600 700 800 900 1000


0102030405060

Plat

form

mot

ion

ampl

itude

(m)


DriftWave


Bottom of USJ

Top of LSJ

00280029

003003100320033003400350036

Wellhead

0010011001200130014001500160017


202530354045505560

Late

ral d

ispla

cem

ent

(m)

45505560657075

Late

ral d

ispla

cem

ent

(m)

1700

1900

1500

2300

2500

2100

2900

270040

0

600

800

200

1200

1400

1000

Time (s)

6062646668707274767880

Late

ral d

ispla

cem

ent

(m)

3400

3200

3000

3800

4000

4200

4400

3600

Riser (minus300 m)

2527293133353739

Late

ral d

ispla

cem

ent

(m)

Late

ral d

ispla

cem

ent

(m)

Late

ral d

ispla

cem

ent

(m)

5900

5700

5500

5300

5100

4900

4700

4500 40

0

600

800

1000

1200

140020

0

1700

1900

2100

2300

2500

2700

2900

1500

000001000020000300004000050000600007

Late

ral d

ispla

cem

ent

(m)

Late

ral d

ispla

cem

ent

(m)

3000

3400

3600

3800

4000

4200

4400

3200

5900

5700

5500

5300

5100

4900

4700

4500


Mudline

Riser (minus600 m)

Time (s) Time (s)


Time (s)

Time (s)


1198983 = (119864119868)119894 1205732119894 minus (119864119868)119894+1 1205732119894+1(119864119868)119894+1 (1205732119894+1 + 1205742119894+1)

1198984 = (119864119868)119894 1205742119894 + (119864119868)119894+1 1205732119894+1(119864119868)119894+1 (1205732119894+1 + 1205742119894+1)

V1 = 119875119894+1 minus 119875119894(119864119868)119894+1 (1205732119894+1 + 1205742119894+1) V2 = 119875119894 minus 119875119894+1(119864119868)119894+1 (1205732119894+1 + 1205742119894+1)

(19)


Wellhead Mudline

23235

24245

25255

26

662646668

77274

889092949698

100102104106

Top of LSJ

780800820840860880900920940960

800820840860880900920940

0102030405060

Bottom of USJ

400

600

800

200

1200

1400

1000

Time (s)



Bend

ing

mom

ent

(KNmiddotm

)

4547495153555759

Bend

ing

mom

ent

(KNmiddotm

)

Bend

ing

mom

ent

(KNmiddotm

)

Bend

ing

mom

ent

(KNmiddotm

)

Bend

ing

mom

ent

(KNmiddotm

)

Bend

ing

mom

ent

(KNmiddotm

)

Bend

ing

mom

ent

(KNmiddotm

)

Bend

ing

mom

ent

(KNmiddotm

)

3200

3400

3600

3800

4000

4200

4400

3000

Time (s)

4700

4900

5100

5300

5500

5700

5900

4500

Time (s)

1700

1900

2100

2300

2500

2700

2900

1500

Time (s)40

0

600

200

800

1200

1400

1000

Time (s)

4700

4900

5100

5300

5500

5700

5900

4500

Time (s)

3200

3400

3600

3800

4000

4200

4400

3000

Time (s)

400420440460480500520540560580600

1700

1900

1500

2300

2500

2700

2900

2100

Time (s)


Let

119860 (119894+1) = [119860 119894+1 119861119894+1 119862119894+1 119863119894+1]119879 119860 (119894) = [119860 119894 119861119894 119862119894 119863119894]119879

119885(119894) =[[[[[[[[[[

11989941198981 11989931198981 minus11989921198982 minus11989911198982120573119894120573119894+1 1198993 (1198981 + V1)120573119894120573119894+1 1198994 (1198981 + V1)


minus120573119894120574119894+1 1198994 (1198983 + V1)minus120574119894120574119894+1 1198991 (1198984 + V2)

120574119894120574119894+1 1198992 (1198984 + V2)

]]]]]]]]]]

(20)

Then (18) becomes

119860 (119894+1) = 119885(119894)119860 (119894) (21)

From (21)

119860 (119873) = 119885119860 (1) (22)

where




0

15

30

45

60

75

90

105

120


5

55

6

65

7

75


780

800

820

840

860

880

900

920

940

960


WellheadMudline

0

100

200

300

400

500

600




Bend

ing

mom

ent (

kNmiddotm

)Be

ndin

g m

omen

t (kN

middotm)

Bend

ing

mom

ent (

kNmiddotm

)Be

ndin

g m

omen

t (kN

middotm)





1198601 = 1198621 = 0


[[[[[[

119860119873119861119873119862119873119863119873

]]]]]]= 0

(24)



























minus200 0 200 400 600 800 10000

10

20

30

40

50

60


005

115

225

335

445

5

0005 0015 0025 0035

0123456789

10

400 600 800 1000 3

4

5

6

7

8

9

12 16 20 24 28

60

50

40

30

20

10

0

Dep

th (m

)

60

50

40

30

20

10

0

Dep

th (m

)

60

50

40

30

20

10

0D

epth

(m)

Dep

th (m

)

t = 0 st = 2325 st = 2605 st = 2974 s

t = 3300 st = 3624 st = 3949 st = 4273 s

t = 200 st = 2325 st = 2605 st = 2974 s

t = 3300 st = 3624 st = 3949 st = 4273 s

t = 200 st = 2325 st = 2605 st = 2974 s

t = 3300 st = 3624 st = 3949 st = 4273 s










0005

01015

02025

03035

04045

05

0 1 2 3 4 5 6 7 8Offset ( WD)

Late

ral d

ispla

cem

ent a

t WH

(m)

400500600700800900

10001100120013001400

0 1 2 3 4 5 6 7 8Offset ( WD)

0

002

004

006

008

01

012

0 1 2 3 4 5 6 7 8Offset ( WD)

400500600700800900

1000110012001300

0 1 2 3 4 5 6 7 8Offset ( WD)




Bend

ing

mom

ent a

t WH

(kNmiddotm

)

Late

ral d

ispla

cem

ent a

tminus5

m (m

)

Bend

ing

mom

ent a

tminus5

m (k

Nmiddotm

)

WH)



0 5 10 15 20 25Offset (m)

400

600

800

1000

1200

1400

1600

0 5 10 15 20 25Offset (m)

Bend

ing

mom

ent (

kNmiddotm

)

0001002003004005006007

Late

ral d

ispla

cem

ent (

m)









00005

0010015

0020025

0030035

004


Late

ral d

ispla

cem

ent (

m)

WHML(minus5m)

WHML(minus5m)

0100200300400500600700800900

1000


Bend

ing

mom

ent (

kNmiddotm

)




5 Conclusion





Acknowledgments


References



























RoboticsJournal of




VLSI Design



Shock and Vibration







Journal of



Volume 2018



Volume 2018


Journal of




SensorsJournal of



RotatingMachinery





Propagation






Hindawi


Advances in

Multimedia



Wellhead Mudline

23235

24245

25255

26

662646668

77274

889092949698

100102104106

Top of LSJ

780800820840860880900920940960

800820840860880900920940

0102030405060

Bottom of USJ

400

600

800

200

1200

1400

1000

Time (s)



Bend

ing

mom

ent

(KNmiddotm

)

4547495153555759

Bend

ing

mom

ent

(KNmiddotm

)

Bend

ing

mom

ent

(KNmiddotm

)

Bend

ing

mom

ent

(KNmiddotm

)

Bend

ing

mom

ent

(KNmiddotm

)

Bend

ing

mom

ent

(KNmiddotm

)

Bend

ing

mom

ent

(KNmiddotm

)

Bend

ing

mom

ent

(KNmiddotm

)

3200

3400

3600

3800

4000

4200

4400

3000

Time (s)

4700

4900

5100

5300

5500

5700

5900

4500

Time (s)

1700

1900

2100

2300

2500

2700

2900

1500

Time (s)40

0

600

200

800

1200

1400

1000

Time (s)

4700

4900

5100

5300

5500

5700

5900

4500

Time (s)

3200

3400

3600

3800

4000

4200

4400

3000

Time (s)

400420440460480500520540560580600

1700

1900

1500

2300

2500

2700

2900

2100

Time (s)


Let

119860 (119894+1) = [119860 119894+1 119861119894+1 119862119894+1 119863119894+1]119879 119860 (119894) = [119860 119894 119861119894 119862119894 119863119894]119879

119885(119894) =[[[[[[[[[[

11989941198981 11989931198981 minus11989921198982 minus11989911198982120573119894120573119894+1 1198993 (1198981 + V1)120573119894120573119894+1 1198994 (1198981 + V1)


minus120573119894120574119894+1 1198994 (1198983 + V1)minus120574119894120574119894+1 1198991 (1198984 + V2)

120574119894120574119894+1 1198992 (1198984 + V2)

]]]]]]]]]]

(20)

Then (18) becomes

119860 (119894+1) = 119885(119894)119860 (119894) (21)

From (21)

119860 (119873) = 119885119860 (1) (22)

where




0

15

30

45

60

75

90

105

120


5

55

6

65

7

75


780

800

820

840

860

880

900

920

940

960


WellheadMudline

0

100

200

300

400

500

600




Bend

ing

mom

ent (

kNmiddotm

)Be

ndin

g m

omen

t (kN

middotm)

Bend

ing

mom

ent (

kNmiddotm

)Be

ndin

g m

omen

t (kN

middotm)





1198601 = 1198621 = 0


[[[[[[

119860119873119861119873119862119873119863119873

]]]]]]= 0

(24)



























minus200 0 200 400 600 800 10000

10

20

30

40

50

60


005

115

225

335

445

5

0005 0015 0025 0035

0123456789

10

400 600 800 1000 3

4

5

6

7

8

9

12 16 20 24 28

60

50

40

30

20

10

0

Dep

th (m

)

60

50

40

30

20

10

0

Dep

th (m

)

60

50

40

30

20

10

0D

epth

(m)

Dep

th (m

)

t = 0 st = 2325 st = 2605 st = 2974 s

t = 3300 st = 3624 st = 3949 st = 4273 s

t = 200 st = 2325 st = 2605 st = 2974 s

t = 3300 st = 3624 st = 3949 st = 4273 s

t = 200 st = 2325 st = 2605 st = 2974 s

t = 3300 st = 3624 st = 3949 st = 4273 s










0005

01015

02025

03035

04045

05

0 1 2 3 4 5 6 7 8Offset ( WD)

Late

ral d

ispla

cem

ent a

t WH

(m)

400500600700800900

10001100120013001400

0 1 2 3 4 5 6 7 8Offset ( WD)

0

002

004

006

008

01

012

0 1 2 3 4 5 6 7 8Offset ( WD)

400500600700800900

1000110012001300

0 1 2 3 4 5 6 7 8Offset ( WD)




Bend

ing

mom

ent a

t WH

(kNmiddotm

)

Late

ral d

ispla

cem

ent a

tminus5

m (m

)

Bend

ing

mom

ent a

tminus5

m (k

Nmiddotm

)

WH)



0 5 10 15 20 25Offset (m)

400

600

800

1000

1200

1400

1600

0 5 10 15 20 25Offset (m)

Bend

ing

mom

ent (

kNmiddotm

)

0001002003004005006007

Late

ral d

ispla

cem

ent (

m)









00005

0010015

0020025

0030035

004


Late

ral d

ispla

cem

ent (

m)

WHML(minus5m)

WHML(minus5m)

0100200300400500600700800900

1000


Bend

ing

mom

ent (

kNmiddotm

)




5 Conclusion





Acknowledgments


References



























RoboticsJournal of




VLSI Design



Shock and Vibration







Journal of



Volume 2018



Volume 2018


Journal of




SensorsJournal of



RotatingMachinery





Propagation






Hindawi


Advances in

Multimedia



0

15

30

45

60

75

90

105

120


5

55

6

65

7

75


780

800

820

840

860

880

900

920

940

960


WellheadMudline

0

100

200

300

400

500

600




Bend

ing

mom

ent (

kNmiddotm

)Be

ndin

g m

omen

t (kN

middotm)

Bend

ing

mom

ent (

kNmiddotm

)Be

ndin

g m

omen

t (kN

middotm)





1198601 = 1198621 = 0


[[[[[[

119860119873119861119873119862119873119863119873

]]]]]]= 0

(24)



























minus200 0 200 400 600 800 10000

10

20

30

40

50

60


005

115

225

335

445

5

0005 0015 0025 0035

0123456789

10

400 600 800 1000 3

4

5

6

7

8

9

12 16 20 24 28

60

50

40

30

20

10

0

Dep

th (m

)

60

50

40

30

20

10

0

Dep

th (m

)

60

50

40

30

20

10

0D

epth

(m)

Dep

th (m

)

t = 0 st = 2325 st = 2605 st = 2974 s

t = 3300 st = 3624 st = 3949 st = 4273 s

t = 200 st = 2325 st = 2605 st = 2974 s

t = 3300 st = 3624 st = 3949 st = 4273 s

t = 200 st = 2325 st = 2605 st = 2974 s

t = 3300 st = 3624 st = 3949 st = 4273 s










0005

01015

02025

03035

04045

05

0 1 2 3 4 5 6 7 8Offset ( WD)

Late

ral d

ispla

cem

ent a

t WH

(m)

400500600700800900

10001100120013001400

0 1 2 3 4 5 6 7 8Offset ( WD)

0

002

004

006

008

01

012

0 1 2 3 4 5 6 7 8Offset ( WD)

400500600700800900

1000110012001300

0 1 2 3 4 5 6 7 8Offset ( WD)




Bend

ing

mom

ent a

t WH

(kNmiddotm

)

Late

ral d

ispla

cem

ent a

tminus5

m (m

)

Bend

ing

mom

ent a

tminus5

m (k

Nmiddotm

)

WH)



0 5 10 15 20 25Offset (m)

400

600

800

1000

1200

1400

1600

0 5 10 15 20 25Offset (m)

Bend

ing

mom

ent (

kNmiddotm

)

0001002003004005006007

Late

ral d

ispla

cem

ent (

m)









00005

0010015

0020025

0030035

004


Late

ral d

ispla

cem

ent (

m)

WHML(minus5m)

WHML(minus5m)

0100200300400500600700800900

1000


Bend

ing

mom

ent (

kNmiddotm

)




5 Conclusion





Acknowledgments


References



























RoboticsJournal of




VLSI Design



Shock and Vibration







Journal of



Volume 2018



Volume 2018


Journal of




SensorsJournal of



RotatingMachinery





Propagation






Hindawi


Advances in

Multimedia






















minus200 0 200 400 600 800 10000

10

20

30

40

50

60


005

115

225

335

445

5

0005 0015 0025 0035

0123456789

10

400 600 800 1000 3

4

5

6

7

8

9

12 16 20 24 28

60

50

40

30

20

10

0

Dep

th (m

)

60

50

40

30

20

10

0

Dep

th (m

)

60

50

40

30

20

10

0D

epth

(m)

Dep

th (m

)

t = 0 st = 2325 st = 2605 st = 2974 s

t = 3300 st = 3624 st = 3949 st = 4273 s

t = 200 st = 2325 st = 2605 st = 2974 s

t = 3300 st = 3624 st = 3949 st = 4273 s

t = 200 st = 2325 st = 2605 st = 2974 s

t = 3300 st = 3624 st = 3949 st = 4273 s










0005

01015

02025

03035

04045

05

0 1 2 3 4 5 6 7 8Offset ( WD)

Late

ral d

ispla

cem

ent a

t WH

(m)

400500600700800900

10001100120013001400

0 1 2 3 4 5 6 7 8Offset ( WD)

0

002

004

006

008

01

012

0 1 2 3 4 5 6 7 8Offset ( WD)

400500600700800900

1000110012001300

0 1 2 3 4 5 6 7 8Offset ( WD)




Bend

ing

mom

ent a

t WH

(kNmiddotm

)

Late

ral d

ispla

cem

ent a

tminus5

m (m

)

Bend

ing

mom

ent a

tminus5

m (k

Nmiddotm

)

WH)



0 5 10 15 20 25Offset (m)

400

600

800

1000

1200

1400

1600

0 5 10 15 20 25Offset (m)

Bend

ing

mom

ent (

kNmiddotm

)

0001002003004005006007

Late

ral d

ispla

cem

ent (

m)









00005

0010015

0020025

0030035

004


Late

ral d

ispla

cem

ent (

m)

WHML(minus5m)

WHML(minus5m)

0100200300400500600700800900

1000


Bend

ing

mom

ent (

kNmiddotm

)




5 Conclusion





Acknowledgments


References



























RoboticsJournal of




VLSI Design



Shock and Vibration







Journal of



Volume 2018



Volume 2018


Journal of




SensorsJournal of



RotatingMachinery





Propagation






Hindawi


Advances in

Multimedia








minus200 0 200 400 600 800 10000

10

20

30

40

50

60


005

115

225

335

445

5

0005 0015 0025 0035

0123456789

10

400 600 800 1000 3

4

5

6

7

8

9

12 16 20 24 28

60

50

40

30

20

10

0

Dep

th (m

)

60

50

40

30

20

10

0

Dep

th (m

)

60

50

40

30

20

10

0D

epth

(m)

Dep

th (m

)

t = 0 st = 2325 st = 2605 st = 2974 s

t = 3300 st = 3624 st = 3949 st = 4273 s

t = 200 st = 2325 st = 2605 st = 2974 s

t = 3300 st = 3624 st = 3949 st = 4273 s

t = 200 st = 2325 st = 2605 st = 2974 s

t = 3300 st = 3624 st = 3949 st = 4273 s










0005

01015

02025

03035

04045

05

0 1 2 3 4 5 6 7 8Offset ( WD)

Late

ral d

ispla

cem

ent a

t WH

(m)

400500600700800900

10001100120013001400

0 1 2 3 4 5 6 7 8Offset ( WD)

0

002

004

006

008

01

012

0 1 2 3 4 5 6 7 8Offset ( WD)

400500600700800900

1000110012001300

0 1 2 3 4 5 6 7 8Offset ( WD)




Bend

ing

mom

ent a

t WH

(kNmiddotm

)

Late

ral d

ispla

cem

ent a

tminus5

m (m

)

Bend

ing

mom

ent a

tminus5

m (k

Nmiddotm

)

WH)



0 5 10 15 20 25Offset (m)

400

600

800

1000

1200

1400

1600

0 5 10 15 20 25Offset (m)

Bend

ing

mom

ent (

kNmiddotm

)

0001002003004005006007

Late

ral d

ispla

cem

ent (

m)









00005

0010015

0020025

0030035

004


Late

ral d

ispla

cem

ent (

m)

WHML(minus5m)

WHML(minus5m)

0100200300400500600700800900

1000


Bend

ing

mom

ent (

kNmiddotm

)




5 Conclusion





Acknowledgments


References



























RoboticsJournal of




VLSI Design



Shock and Vibration







Journal of



Volume 2018



Volume 2018


Journal of




SensorsJournal of



RotatingMachinery





Propagation






Hindawi


Advances in

Multimedia



0005

01015

02025

03035

04045

05

0 1 2 3 4 5 6 7 8Offset ( WD)

Late

ral d

ispla

cem

ent a

t WH

(m)

400500600700800900

10001100120013001400

0 1 2 3 4 5 6 7 8Offset ( WD)

0

002

004

006

008

01

012

0 1 2 3 4 5 6 7 8Offset ( WD)

400500600700800900

1000110012001300

0 1 2 3 4 5 6 7 8Offset ( WD)




Bend

ing

mom

ent a

t WH

(kNmiddotm

)

Late

ral d

ispla

cem

ent a

tminus5

m (m

)

Bend

ing

mom

ent a

tminus5

m (k

Nmiddotm

)

WH)



0 5 10 15 20 25Offset (m)

400

600

800

1000

1200

1400

1600

0 5 10 15 20 25Offset (m)

Bend

ing

mom

ent (

kNmiddotm

)

0001002003004005006007

Late

ral d

ispla

cem

ent (

m)









00005

0010015

0020025

0030035

004


Late

ral d

ispla

cem

ent (

m)

WHML(minus5m)

WHML(minus5m)

0100200300400500600700800900

1000


Bend

ing

mom

ent (

kNmiddotm

)




5 Conclusion





Acknowledgments


References



























RoboticsJournal of




VLSI Design



Shock and Vibration







Journal of



Volume 2018



Volume 2018


Journal of




SensorsJournal of



RotatingMachinery





Propagation






Hindawi


Advances in

Multimedia



00005

0010015

0020025

0030035

004


Late

ral d

ispla

cem

ent (

m)

WHML(minus5m)

WHML(minus5m)

0100200300400500600700800900

1000


Bend

ing

mom

ent (

kNmiddotm

)




5 Conclusion





Acknowledgments


References



























RoboticsJournal of




VLSI Design



Shock and Vibration







Journal of



Volume 2018



Volume 2018


Journal of




SensorsJournal of



RotatingMachinery





Propagation






Hindawi


Advances in

Multimedia





















RoboticsJournal of




VLSI Design



Shock and Vibration







Journal of



Volume 2018



Volume 2018


Journal of




SensorsJournal of



RotatingMachinery





Propagation






Hindawi


Advances in

Multimedia




RoboticsJournal of




VLSI Design



Shock and Vibration







Journal of



Volume 2018



Volume 2018


Journal of




SensorsJournal of



RotatingMachinery





Propagation






Hindawi


Advances in

Multimedia


Documents

Coupled Dynamic Analysis for the Riser-Conductor of Deepwater Surface BOP Drilling Systemdownloads.hindawi.com/journals/sv/2018/6568537.pdf · 2019-07-30 · method for the riser-conductor