Hull/mooring/riser coupled dynamic analysis and sensitivity study of a tanker-based FPSO

Hull/mooring/riser coupled dynamic analysis and sensitivity

study of a tanker-based FPSO

Arcandra Tahara, M.H. Kimb,*

aCSO Aker Engineering Inc., Houston, TX 77079, USAbDepartment of Civil Engineering, Ocean Engineering Program Texas A and M University,

Offshore Technology Research Center College Station, TX 77843, USA

Received 22 August 2002; revised 5 February 2003; accepted 7 February 2003

Available online 17 April 2004

Abstract

A computer program is developed for hull/mooring/riser coupled dynamic analysis of a tanker-based turret-moored FPSO (Floating

Production Storage and Offloading) in waves, winds, and currents. In this computer program, the floating body is modeled as a rigid body

with six degrees of freedom. The first- and second-order wave forces, added mass, and radiation damping at various yaw angles are calculated

from the second-order diffraction/radiation panel program WAMIT. The wind and current forces for various yaw angles of FPSO are

modeled following the empirical method suggested by OCIMF (Oil Company International Marine Forum).

The mooring/riser dynamics are modeled using a rod theory and finite element method (FEM), with the governing equations described in a

generalized coordinate system. The dynamics of hull, mooring lines, and risers are solved simultaneously at each time step in a combined

matrix for the specified connection condition. For illustration, semi-taut chain-steel wire-chain mooring lines and steel catenary risers are

employed and their effects on global FPSO hull motions are investigated. To better understand the physics related to the motion

characteristics of a turret-moored FPSO, the role of various hydrodynamic contributions is analyzed and assessed including the effects of hull

and mooring/riser viscous damping, second-order difference-frequency wave-force quadratic transfer functions, and yaw-angle dependent

wave forces and hydrodynamic coefficients. To see the effects of hull and mooring/riser coupling and mooring/riser damping more clearly,

the case with no drag forces on those slender members is also investigated. The numerical results are compared with MARIN’s wave basin

experiments.

q 2004 Elsevier Ltd. All rights reserved.

Keywords: FPSO; Hull-mooring-riser coupled dynamic analysis; Mooring and riser damping; Large yaw angles; Case studies; Quadratic transfer function;

Motion simulations

1. Introduction

FPSOs have been successfully installed and operated in

many places worldwide for oil and gas production. Besides

environmental risk related to oil spills, FPSOs have a

number of advantages compared to other platforms. The

biggest advantage is huge storage capacity (no pipeline

needed) and ample deck space giving better layout

flexibility. Recently, Minerals Management Service

(MMS) has approved the use of FPSOs with double hull

in the Gulf of Mexico.

Several researchers have done various studies about

the dynamic characteristics of FPSOs in winds, waves,

and currents. Wichers [22], for example, initiated a

comprehensive study for numerical simulations of a turret-

moored FPSO in irregular waves with winds and currents.

He derived the equation of motions of such model in the

time domain using an uncoupled method and solved rigid-

body and mooring-line dynamics separately. On the other

hand, several researchers [20,13] investigated the behavior

and stability of turret-moored FPSOs based on a set of

simplified ship-maneuvering equations.

Spars and TLPs are designed so that their natural

frequencies are away from the dominant wave frequencies

in all six degree-of-freedom modes. On the other hand, the

natural frequencies of FPSOs in heave and pitch are within

typical frequency band of sea energy spectra, and thus heave

and pitch wave-frequency motions tend to be large.

These wave frequency motions in turn can excite riser

0141-1187/$ - see front matter q 2004 Elsevier Ltd. All rights reserved.

doi:10.1016/j.apor.2003.02.001

Applied Ocean Research 25 (2003) 367–382

www.elsevier.com/locate/apor

* Corresponding author. Tel.: þ1-979-847-8710; fax: þ1-979-862-8162.

E-mail address: [email protected] (M.H. Kim).

http://www.elsevier.com/locate/apor

and mooring-line motions creating significant interaction

effects. Therefore, the hull/mooring/riser coupling effects of

FPSOs are expected to be more important compared with

spars and TLPs. The interactions between hull and slender

members of deepwater platforms are difficult to be

investigated by model testing because of the depth

limitation of wave basins. The mismatch of Reynolds

numbers between the slender members of model and

the prototype is another intrinsic problem in the physical

model testing.

On the other hand, the dynamic interactions between hull

and slender members can be evaluated numerically in

several ways. One simple approach is called uncoupled

analysis, which assumes that mooring lines and risers

respond statically (as a mass-less non-linear spring) to hull

motions [13]. With this assumption, the inertia effects and

hydrodynamic loading on mooring lines and risers are

neglected. After hull motions are calculated, the mooring

and riser dynamics can be evaluated independently by

inputting the fairlead responses. The reliability and

accuracy of this approach is expected to diminish as water

depth increases. Kim et al. and Ma and Lee [8,9,10,14]

showed that such uncoupled analysis of TLPs and spars may

be inaccurate when used in deepwater. Wichers et al. [25]

showed that the uncoupled analysis may give even larger

error in case of FPSO. Wichers et al. [25] concluded that

fully coupled dynamic models are necessary to estimate

realistic design values. Using hull/mooring/riser coupled

dynamic analysis tools [2,10], the effects of risers and

mooring lines on FPSO hull motions and vice versa can be

more accurately predicted, as illustrated in this paper.

Turret-moored FPSOs are free to rotate in the horizontal

plane, and thus may have large yaw motions in non-parallel

wind–wave–current environments or in multidirectional

waves. In such a case, the variation of wave forces and other

hydrodynamic coefficients against various yaw angles need

to be incorporated for more reliable motion analysis. To

include such effects, all the first- and second-order wave

forces and hydrodynamic coefficients have to be pre-

calculated and tabulated for various yaw angles, which

requires substantial computational effort. Instead, one can

use, as an approximation, the single set of hydrodynamic

quantities calculated at the mean yaw angle and use them

throughout entire simulation. So far, no researcher quanti-

tatively showed how important the effects of exact yaw-

angle dependent hydrodynamic forces are.

One of the most difficult factors in the numerical

simulation of a turret-moored FPSO is the reasonable

estimation of hull viscous damping. The low-frequency

viscous reaction forces on FPSOs were measured by

Wichers and Ji [24]. They also examined the coupling

terms due to the combined modes of motions in still water

and in current. A conclusion of this study is that the viscous

part in normal direction contributes significantly to the hull

dynamics, particularly in currents, and thus cannot be

neglected. In the present study, we simulated FPSO motions

with or without considering the empirical viscous forces on

FPSO hull, and quantitatively showed the difference

between them.

When simulating FPSO motions numerically, no

researcher so far used the exact second-order difference-

frequency wave force QTFs [11,7], which is computationally

very intensive. Instead, most researchers used the so-called

Newman’s approximation [15] which assumes that the

difference-frequency force QTFs can be approximated by

the mean-drift forces and moments (diagonal values) when

the system’s natural frequencies are small. In the present

study, both Newman’s approximation and the full-QTF

approach are used to illustrate the reliability of the

approximation method popular in the offshore industry.

Here, the wave force QTFs are calculated from the second-

order diffraction/radiation program WAMIT [12].

Recently, the extreme responses of a turret-moored

FPSO in the Gulf of Mexico were also studied experimen-

tally by Baar et al. [3]. They investigated the responses of a

FPSO in collinear and non-collinear winds, waves, and

currents of 100-year hurricane. A conclusion from their

study is that the responses of their turret-moored FPSO are

more sensitive to non-collinear environmental conditions. A

similar experimental study for a turret-moored FPSO

designed for 6000-ft water depth was also conducted by

Ward et al. [21] in the Offshore Technology Research

Center wave basin. They also showed that the responses of

the FPSO are more severe in non-collinear environmental

conditions. In this regard, a typical non-parallel 100-yr

storm in the Gulf of Mexico is used as an environmental

condition.

2. Description of FPSO, mooring system and Riser

The vessel used in this study is a 1,440,000 bbls storage

tanker and has an LBP of 310 m, a beam of 47.17 m, and a

depth of 28.04 m. At the full-load condition, the vessel has

an average draft of 18.9 m, with displacement of 240,869

MT. The internal turret mooring system is located 63.55 m

aft of the forward perpendicular of the vessel and has

15.85 m of diameter. The main particulars of the vessel are

summarized in Table 1. The body plan and its corresponding

isometric view are shown in Fig. 1. According to the

common rule of classification societies, transverse sections

are numbered from aft to forward perpendiculars. Thus,

station 0 is on the stern and station 20 is on the bow.

Hull viscous damping of the same vessel is obtained by

the model test conducted by Wichers and Ji [24] at MARIN.

He divided the hull into four parts and determined the

associated resistance coefficients. Those viscous damping

coefficients in sway (normal) direction are shown in Fig. 2.

The in-line hull viscous damping is expected to be small,

thus not considered in the present simulations.

The turret supports 12 chain-wire-chain mooring system

and 13 catenary risers. Tables 2 and 3 show the main

A. Tahar, M.H. Kim / Applied Ocean Research 25 (2003) 367–382368

particulars of the mooring system and the hydrodynamic

coefficients for them. The main particulars of the riser

system are shown in Table 4. The 12 anchor legs and

mooring lines are arranged in four groups, each group

having three anchor legs and 90-degree apart against each

other. The bottom-touching part of each leg consists of

studless Grade K4 chain terminating at the anchor pile. The

chain used for the top portion (Segment 3) is exactly the

same as that for Segment 1 except for the shortened length.

Schematic representation of the turret mooring system is

shown in Fig. 3. It follows from this figure that the x-axis

points to the East, y-axis points to the North, and z-axis

positive upwards. It is assumed that the initial ship

longitudinal axis coincides with the x-axis-bow to the East.

In the present study, 13 SCR are included. The prototype

riser particulars are shown in Table 4 while their

hydrodynamic coefficients are presented in Table 5. The

location of the risers is shown in Fig. 4 and angular

arrangement of each riser is listed in Table 6. It should be

noted that the risers are not symmetric with respect to the

x-axis. In the dynamic analysis for slender members,

tangential drag forces are expected to be negligible, thus

not included. The sea floor is modeled as elastic bed with

non-linear spring whose restoring force is proportional to

the displacement squared. The expected Coulomb friction

from the seabed and possible vortex induced vibration

(VIV) are not considered in the present simulations.

3. Environmental condition

A typical 100-year hurricane in the Gulf of Mexico with

significant wave height of 12.2 m and peak wave period of

14 s is selected as wave environment. As for wind, 1 h mean

wind speed (at 10 m height) of 41.1 m/s is used and the time

dependent wind velocity is generated from the correspon-

ding API wind spectrum. The wind direction is assumed to

be 30-deg left of waves. As for currents, a storm driven

Fig. 1. Body plan and isometric view of FPSO.

Table 1

Main particulars of turret-moored FPSO

Designation Symbol Unit Quantity

Vessel size KDWT 200

Length between perpendicular Lpp m 310

Breadth B m 47.17

Depth H m 28.04

Draft T m 18.90

Length beam ratio L=B 6.57

Beam draft ratio B=T 2.5

Displacement MT 240,869

Block coefficient Cb 0.85

Center of buoyancy forward of

section 10

FB M 6.6

Water plane area A m2 13,400

Water plane coefficient Cw 0.9164

Center of water plane area

forward of section 10

FA m 1.0

Center of gravity above base KG m 13.32

Metacentric height tranverse MGt m 5.78

Metacentric height longitudinal MGl m 403.83

Transverse radius of gyration

in air

Kxx m 14.77

Longitudinal radius of gyration

in air

Kyy m 77.47

Yaw radius of gyration in

air

KCC m 79.30

Wind area frontal Af m2 1,012

Wind area side Ab m2 3,772

Turret in center line behind

Fpp (20.5% LppÞ

m 63.55

Turret elevation below tanker base m 1.52

Turret diameter m 15.85

Fig. 2. Viscous damping coefficients over the length of FPSO hull.

A. Tahar, M.H. Kim / Applied Ocean Research 25 (2003) 367–382 369

shear current is assumed. The current is assumed to flow

from 30-deg right of wave direction. The non-collinear

environmental condition is summarized in Table 7.

The JONSWAP spectrum used here is the same as that

of Hasselman et al. (1973) with enhancement parameter

g ¼ 2:5 :

SðvÞ ¼ ag2v25 exp 21:25v

v0

� �24

" #g

exp 2ðv2v0Þ

2t2v20

� �ð1Þ

where g is the peakedness parameter, and t is the shape

parameter (0.07 for v # v0 and 0.09 for v $ v0Þ: The

value of a is related to a prevailing wind velocity of Uw and

a fetch of X; and can be written as

a ¼ 0:076ðXÞ20:22 ð2Þ

The shape of the JONSWAP spectrum used for the

present study is presented in Fig. 5.

The 1-hr wind speed used for the API wind spectrum is

based on the recurrence period of 100 years. The API wind

spectrum has the following expression.

SðvÞ ¼s2ðzÞ

2pfp 1 þ1:5v

2pfp

" #5=3ð3Þ

where fp is average factor derived from measured spectrum

and is given by

fp ¼0:025VwðzÞ

zð4Þ

The symbol sðzÞ is the standard deviation of wind speed

and related to turbulence intensity. The value of sðzÞ can be

expressed as

sðzÞ ¼ 0:15z

20

� �20:125

VwðzÞ ð5Þ

where VwðzÞ is the one hour mean wind speed (m/s) z meters

above water level. The corresponding wind velocity

spectrum used in the present study is plotted in Fig. 6.

The wind and current forces and moments on FPSO hull

are in general difficult to assess numerically, and thus have

to be estimated based-on empirical formulas. An

extensive experimental data set for wind and current forces

Table 4

Riser particulars

Designation Top tension

(kN)

OD

(mm)

AE

(kN)

Mass

(kg/m)

Dry/wet

(N/m)

Liquid prod. 1112.5 444.5 18.3 £ 106 196.4 1927/1037

Gas prod. 609.7 386.1 10.8 £ 106 174.1 1708/526

Water injection 2020.0 530.9 18.6 £ 106 285.7 2803/1898

Gas injection 1352.8 287.0 31.4 £ 106 184.5 1810/1168

Gas export 453.9 342.9 8.6 £ 106 138.4 1358/423

Cdn ¼ 1; except for water and gas injection ¼ 1.414.

Table 2

Main particulars of FPSO mooring system

Designation Unit Quantity

Pretension Kn 1201

Number of lines 4 £ 3

Degrees between the 3 lines deg. 5

Length of mooring lines m 2087.9

Radius of location of chain stoppers on turn table m 7.0

Segment 1 (ground section): chain

Length at anchor point m 914.4

Diameter mm 88.9

Dry weight kg/m 164.9

Wet weight kg/m 143.4

Stiffness AE kN 794841

Mean breaking load (MBL) kN 6515

Segment 2: wire

Length m 1127.8

Diameter mm 107.9

Dry weight kg/m 42.0

Wet weight kg/m 35.7

Stiffness AE kN 690168

Mean breaking load (MBL) kN 6421

Segment 3: chain

Length m 45.7

Table 3

Hydrodynamic coefficients for chains and wire

Coefficients to be used Symbol Chain Rope/wire

Drag normal Cdn 2.45 1.2

Added inertia coefficient normal Cin 2.0 1.15Fig. 3. General arrangement of mooring system.


on FPSO-type hulls are available, and the results are

documented in a book published by OCIMF [17].

4. Wind and current loads using OCIMF data

For FPSO dynamic analysis, wind induced surge–sway–

yaw responses can be large because wind loading contains

significant energy close to the natural frequencies of

horizontal plane motions. For static equilibrium, both

wind and current forces are important.

The wind and current loading coefficients for typical

VLCCs at various heading angles are presented in OCIMF

document (1994). The resultant wind force and moment

acting on a tanker are then calculated from the following

equations

FXw ¼1

2CXwrwV2

wAT ð6Þ

FYw ¼1

2CYwrwV2

wAL ð7Þ

MXYw ¼1

2CXYwrwV2

wALLpp ð8Þ

where AT and AL are transverse and longitudinal projected

areas of the tanker above MWL. FXw;FYw; and MXYw are the

surge, sway wind forces and yaw wind moments, respec-

tively. CXw;CYw; and CXYw are the longitudinal, lateral

force and yaw moment coefficients, respectively. These

coefficients can be read from the graphs given for the

following conditions

1. Wind angle of attack: 180 8 (bow) to 0 8 (stern).

2. Two drafts: representing fully loaded and ballasted

tanker condition.

3. Two bows configuration: cylindrical bow and conven-

tional bulbous bow.

Similarly, current loading coefficients CXw;CYw; and

CXYw are also given in the OCIMF data set as function of

current angle of attack, water depth to draft ratio ðd=TÞ; the

bow configuration, and loading condition. In the present

study, we assumed that the tanker is fully loaded and has

cylindrical bow.

It should be noted that all the force coefficients of the

OCIMF data are presented with respect to the body-fixed

coordinate whose origin is at the mid-ship, and the

corresponding sign convention is shown in Fig. 7. Since

the origin of our earth-fixed global coordinate system is on

the center of the turret, the wind and current forces

calculated from the OCIMF data need to be transferred to

those with respect to the global coordinate system as follows

~FX

~FY

( )¼

cos u 2sin u

sin u cos u

" #FX

FY

( )ð9Þ

~MXY ¼ FY Xtur þ MXY ð10Þ

Table 5

Hydrodynamic coefficients for risers

Designation Symbol Coefficient

Drag normal Cdn 1.0

Added inertia coefficient normal Cin 1.0

Connection level below tanker base 1.52 m

Total length 1829 m

Fig. 4. General arrangement of the risers.

Table 6

Earth bound azimuthal arrangement of risers

Designation

Riser location (8)

Liquid prod. (LP) 0 90 180 270Gas prod. (GP) 45 135 225 315Water injection (WI) 165 337.5Gas injection (GI) 30 210Gas export (GE) 300

Table 7

Design environmental condition

Designation Unit Value

Waves

Hs m 12.19

Tp sec. 14

Wave spectrum JONSWAP ðg ¼ 2:5Þ

Wave direction deg. 180 (to West)

Wind

Wind speed (1-hr) m/s 41.12@10 m

Wind spectrum API RP 2A-WSD

Wind direction deg. 210

Storm current profile (linear between points)

Depth: 0 m m/s 1.07

60.96 m m/s 1.07

91.44 m m/s 0.09

Seabed m/s 0.09

Current direction deg. 150


where ~FX ; ~FY ; and ~MXY are the surge, sway forces and yaw

moments with respect to the global coordinate system. The

symbol u is the instantaneous yaw angle of the vessel,

and Xtur is the distance between the center of turret and

the mid-ship.

5. Hydrodynamic computation using wamit

FPSO’s horizontal motions have smaller natural fre-

quencies than those of significant wave energy, and thus

more likely excited by second-order difference-frequency

wave forces. Therefore, the inclusion of the second-order

slowly varying wave forces in FPSO motion analysis is very

important. For the computation of hydrodynamic coeffi-

cients (added mass and radiation damping) and first- and

second-order wave forces, a second-order diffraction/radia-

tion program called WAMIT is used [12]. This program is

based on the three-dimensional panel method (boundary

integral equation method) and Green’s theorem with a free-

surface Green function.

The wetted surface of the vessel is discretized by

quadrilateral panels, as shown in Fig. 8. Smaller panels

were distributed near the waterline to enhance the accuracy

of waterline integrals used for second-order forces. For the

second-order computation, the free surface near the FPSO

also needs to be discretized, and it is shown in Fig. 9.

The FPSO hull, shown in Fig. 8, is symmetric

with respect to the x-axis. The symmetry is exploited in

the first- and second-order computation, and thus only half

domain is discretized, as shown in Fig. 9. For each half

domain, the body and free surface are discretized by 1843

and 480 panels, respectively. The truncation radius of the

free surface panels is 294 m from the turret. The

convergence of the difference-frequency QTFs is known

to be much faster than the sum-frequency QTFs. From

convergence test, the truncation radius and number of

panels are found to be sufficient for the desired accuracy of

the present study.

The wave force quadratic transfer functions are com-

puted for nine wave frequencies, ranging from 0.24 to

1.8 rad/sec and the intermediate values for other frequencies

are interpolated from them. In the present example and

frequency range of interest, the interpolated values were

found to be sufficiently accurate. For slowly varying surge

motions, only the QTF values near the diagonal are used.

The hydrodynamic coefficients and wave forces are

expected to vary appreciably with large yaw angles and the

effects should be taken into consideration for the reliable

prediction of FPSO global motions. Therefore, they are

calculated in advance for various yaw angles with 5-degree

interval and the data are tabulated as inputs.

The second-order diffraction/radiation computation for a

3D body is computationally very intensive especially when

it has to be run for various yaw angles. Therefore, many

researchers avoided such a complex procedure and have

instead used simpler approach called Newman’s approxi-

mation i.e. the off-diagonal components of the second-order

difference-frequency QTFs are approximated by their

Fig. 5. Wave spectrum.

Fig. 6. Wind velocity spectrum. Fig. 8. Grid modeling of body surface of FPSO.

Fig. 7. Sign convention and coordinate system for OCIMF data.


diagonal values (mean drift forces and moments). The

approximation can be justified only when the relevant

natural frequency is very small and the slope of QTFs near

the diagonal is not large. In this paper, the full QTFs are

calculated and the validity of Newman’s approximation is

tested against more accurate results with complete QTFs.

6. Hull/mooring/riser coupled dynamic analysis

The non-linear hull/mooring/riser coupling character-

istics of turret-moored FPSOs in irregular waves are

investigated in this section. When water depth is large,

hull/mooring/riser coupling effects are expected to be

appreciable, and their dynamics should be solved simul-

taneously as an integrated system.

For the static/dynamic analysis of mooring lines and

risers, an extension of the theory developed for slender rods

by Garrett [4,16] was used. Assuming that there is no torque

or twisting moment, one can derive a linear momentum

conservation equation with respect to a position vector ~rðs; tÞ

which is a function of arc length s and time t :

2ðB~r00Þ00 þ ðl~r0Þ0 þ ~q ¼ m€~r ð11Þ

l ¼ T 2 Bk2 ð12Þ

where primes and dots denote spatial s-derivative and time

derivative, respectively, B is the bending stiffness, T the

local effective tension, k the local curvature, m the mass per

unit length, and ~q the distributed force on the rod per unit

length. The scalar variable l can be regarded as a Lagrange

multiplier. The rod is assumed to be elastic and extensible,

thus the following condition is applied

1

2ð~r·~r 2 1Þ ¼

T

AtE<

l

AtEð13Þ

where E ¼Young’s modulus, At ¼ Ae 2 Aið¼outer 2 inner

cross sectional area). For these equations, geometric non-

linearity is fully considered and there is no special

assumption made concerning the shape or orientation of

lines. The benefit of this equation is that Eq. (11) is directly

defined in the global coordinate system and does not require

any transformations to the local coordinate system, which

saves overall computational time significantly.

The normal component of the distributed external force

on the rod per unit length, qn;, is given by a generalized

Morison equation:

qn ¼ CIrAe _vn þ CD

1

2rDlvnrlvnr þ CmrAe€rn ð14Þ

where CI;CD and Cm are inertia, drag, and added mass

coefficients, and _vn; vnr; and €rn are normal fluid acceleration,

normal relative velocity, and normal structure acceleration,

respectively. The symbols r and D are fluid density and

local diameter. In addition, the effective weight, or net

buoyancy, of the rod should be included in qn as a static

load.

A finite element method similar to Garrett [4] has

been developed to solve the above mooring dynamics

problem and the details of the methodology are given in

Refs. [18,19]. The FEM allows any combination of mooring

types and materials as long as their deformations are small

and within proportional limit. The upper ends of the

mooring lines and risers are connected to the hull fairlead

through generalized elastic springs and dampers. The

combination of linear and torsional springs can model

arbitrary connection conditions. The forces and moments

proportional to the relative displacements are transmitted to

the hull at the connection points. The transmitted forces

from mooring lines and risers to the platform are given by

~FP ¼ ~Kð ~T~uP 2 ~uIÞ þ ~Cð ~T_~uP 2 _~uIÞ; ð15Þ

where ~K; ~C are stiffness and damping matrices of connectors

at the connection point, and ~T represents a transformation

matrix between the platform origin and connection point.

The symbols ~uP; ~uI represent column matrices for the

displacements of the platform and connection point.

Then, the following hull response equation can be

combined into the riser/mooring-line equations in the time

domain:

ð ~M þ ~Mað1ÞÞ€~up þð1

0

~Rðt 2 tÞ_~up dtþ ~KH ~up

¼ ~FD þ ~Fð1Þ þ ~Fð2Þ þ ~Fp þ ~Fw þ ~Fc þ ~FWD ð16Þ

where ~M; ~Ma are mass and added mass matrix,~R ¼retardation function (inverse cosine Fourier transform

of radiation damping) matrix, ~KH ¼hydrostatic restoring

coefficient, ~FD ¼drag force matrix on the hull,~Fð1Þ; ~Fð2Þ ¼first- and second-order wave load matrix on the

hull, ~Fp ¼transmitted force matrix from the interface,~Fw ¼dynamic wind loading, ~Fc ¼current loading on hull,

Fig. 9. Grid modeling of body and free surface of FPSO.


and ~FWD ¼wave drift damping force matrix. To check the

relative importance, the diagonal components of the wave-

drift-damping matrix were calculated in the frequency

domain by using the mean-drift-gradient method suggested

by Aranha [1]. The time series of wave drift damping can

then be generated based on Newman’s (diagonal) approxi-

mation. In the present FPSO study, the wave drift damping

is found to be small compared to hull viscous damping and

mooring/riser damping.

The added mass at infinite frequency was obtained from

Kramers–Kronig relation. For the time series of ~Fð1Þ; ~Fð2Þ;

and ~FWD; a two-term Volterra series was used [6]. The hull

drag force in the normal direction was calculated with

respect to the instantaneous hull position based on the

Morrison drag formular with relative velocity squared. Care

should be taken not to double count the effects of steady

current, since it is already given by OCIMF data.

The static problem of the integrated system was solved

using Newton’s iterative method. The dynamic problem was

integrated using an efficient and reliable time marching

scheme similar to Adams–Moulton method [4]. In the

dynamic program, special consideration is required due to

the fact that the time derivatives of l do not appear in the

equations and the added mass matrix is a function of the

instantaneous position. In addition, the free-surface fluctu-

ation and possible contact of mooring lines and catenary

risers with the seafloor require special consideration.

7. Results and analysis

The time-domain simulations of hull/mooring/riser

coupled statics/dynamics are conducted by a computer

program WINPOST. All the response results presented in

this paper are with respect to the center of turret.

Each mooring line is equally divided into five elements

for bottom chain, eight elements for wire, and one element

for upper chain (line connected to the body). Since high-

order elements and interpolation functions are used in the

present FEM, convergence characteristics are found to be

excellent, which was checked by doubling the number of

elements. The foot-print of the mooring line at the sea floor

is carefully defined by performing static analysis of each leg

and matching the pretension at the fairlead with that

described in Table 2.

All risers are modeled as steel catenary risers (SCRs)

where a certain length of the risers touch the seafloor. The

risers are equally divided into 12 elements, and connected to

the hinged joint at turret. The total number of degrees of

freedom in the combined matrix for the coupled analysis is

2773. The inertia and drag coefficients used are listed in

Table 5.

For irregular wave simulations, 51 wave components

with random phases are used. The frequency interval is

randomly perturbed to avoid periodical repetition (Kim and

Yue, 1991). The simulation is started with the tanker system

in the equilibrium position. A ramping function is

introduced to gradually increase the wave loading from

zero to its actual value during the first 50 s of simulation.

8. Numerical static offset and free-decay tests

The results of numerical static offset tests are shown in

Fig. 10. The surge stiffness as well as line tension on each

mooring line match well with the target values provided by

Wichers and Devlin [23].

For free decay simulation of surge, initial displacement is

given by applying appropriate surge force and then the

system is released. Subsequently, the free-decay response is

recorded in the duration of 2000 s with the time interval of

0.01 s. From the time series and using the concept of

logarithmic decrement, damping coefficients can be calcu-

lated. Generally, the tanker hull will be exposed to four

kinds of damping forces; radiation, viscous, wave-drift, and

wind damping. Viscous and wind damping are difficult to be

analyzed by mathematical models, and thus have to be

determined empirically. On the other hand, the radiation and

wave drift damping may be obtained numerically.

In the present study, the radiation damping is computed

using the panel method and the results of surge damping

coefficients are shown in Fig. 11. The present results are

compared with the numerical and model test results of

Wichers [22,24]. In the same figure, wave drift damping in a

large wave ðH ¼12.2 m ¼ significant wave height of 100-yr

hurricane) are also presented for comparison. The wave drift

damping is proportional to the wave height squared, but it

turned out that its magnitude in surge is smaller than the

radiation damping even in such large waves. It is also seen

that the measured damping is much greater than the

calculated radiation and wave drift damping, which reveals

that the viscous damping from hull/mooring/riser is

dominant over the potential-flow-related contributions.

However, the radiation damping can be important to

wave-frequency motions.

A typical time history of the surge free-decay test is

presented in Fig. 12. All mooring lines and risers are

included in this example. Wichers and Ji [24] included the

current effects, and therefore, their damping is larger.

Fig. 10. Surge static offset curve of FPSO.


The surge natural periods (381 s) calculated from the free-

decay run are identical. It should be noted that all the free-

decay tests were carried out in calm water, therefore there is

no wave drift damping in it.

8.1. Simulation for 100-year Hurricane (non-parallel

wind and current)

In this section, the motion and tension characteristics of

the FPSO system are simulated in the time domain for a

non-parallel wind-wave-current environment (100-year

hurricane).

To better understand the role of each important

hydrodynamic contribution, several different cases inclu-

ding/excluding important hydrodynamic parameters are

selected as follows. As pointed out earlier, wind and current

force coefficients are stored for various yaw angles using

OCIMF data. Wave forces are calculated in advance and

tabulated with 5-degree yaw angle interval. The maximum

given in the tables is the actual maximum value obtained

from 3-h simulations. The time interval for the simulation

was 0.01 s. The contributions from wave drift damping and

wave-current interaction effects are shown to be small thus

not included in all cases considered here.

† Case 1: Newman’s approximation was used for second-

order difference frequency forces. Cross flow viscous

drag forces and moments on the hull were excluded.

Wave forces on hull are varied against yaw angles.

The hydrodynamic coefficients (added mass, radiation

damping, hydrostatic coefficients) are assumed to be

constant regardless of the change in yaw.

† Case 2: Same as Case 1, but cross-flow hull drag was

included. By comparing Case 1 and 2, the effects of hull

viscous drag can be directly observed.

† Case 3. Same as Case 2, but the full second-order

difference-force QTFs were used instead of Newman’s

approximation. The Newman’s approximation has been

widely used in offshore industry in FPSO motion

analyses, and its efficacy can be checked by comparing

against Case 2.

† Case 4. Same as Case 2 but the wind yaw moments were

increased by 19% to account for the effects of super

structures on deck. This case is similar to Wicher’s

[23,25] and direct comparison is possible.

† Case 5. Same as Case 4 but impose zero drag coefficients

on slender members. By comparing with Case 4, the

mooring/riser damping effects can be identified.

† Case 6. Same as Case 4 but the wave forces are

assumed to be constant (at the mean yaw angle) and

do not vary with yaw motions. The effects of

changing wave forces with yaw motions can be

directly observed.

In all cases, the vertical-plane motions (heave and pitch)

remain almost the same regardless of the variation of

methodology. Therefore, from now on, our discussions will

mainly be focused on horizontal-plane motions. In addition,

heave and pitch motions are dominated by wave-frequency

components and can be predicted pretty well even by linear

potential wave-body interaction theory.

8.2. Case 1 (Without hull viscous damping)

The hull-motion and riser/mooring tension results for

Case 1 are summarized in Table 8. Here, we can observe

that the slowly varying surge responses dominate wave-

frequency responses. It is also found that both wind forces

and wave forces are important to low-frequency responses.

Since hull viscous drag is not included, most of the

surge/sway damping is expected to come from risers and

mooring lines. However, their contributions to yaw mode

are small. The magnitudes of resonant slowly varying hull

responses are controlled by available damping. Neglecting

hull viscous damping, the low-frequency rms sway and yaw

motions of Case 1 are expected to be greatly overestimated.

Like surge and sway motions, slowly varying responses

dominate wave-frequency responses in yaw. The mean yaw

angle is primarily determined by the balance between

current and wind forces, and the contribution from the mean

wave drift yaw moment is relatively small. Therefore, the

mean yaw angle does not change much against different

methodologies.

Fig. 11. Surge radiation damping coefficients.

Fig. 12. The free decay test of FPSO.


8.3. Case 2 (With hull viscous damping)

In case 2, the cross-flow hull viscous drag is included.

Otherwise, the procedure would be the same as Case 1.

It should be noted that the hull drag in the normal

direction was calculated at the instantaneous yaw position

with Morison’s formula based on the relative velocity

(including wave kinematics) squared. The hull itself

was divided into several parts, and empirical damping

coefficients obtained from Wicher’s [23,25] experiment for

the same hull were assigned on each portion, which can be

seen in Fig. 2. The statistical results of Case 2 are

summarized in Table 8.

By including hull viscous drag, the rms values of sway

and yaw are greatly reduced, as expected. For sway, the

mean offset of Case 2 was increased by 65%, while the

standard deviation was decreased by factor of 3. For yaw,

the standard deviation was reduced by factor of 10.

Table 8

The results of Case 1,2 and 3

Vessel motions Case 1 Case 2 Case 3

Unit Mean Stdv Max Mean Stdv Max Mean Stdv Max

Surge total at turret m 244.0 17.0 283.3 238.2 10.6 261.3 240.2 10.9 264.2

Surge lf part m N/A 17.0 N/A N/A 10.6 N/A N/A 10.8 N/A

Surge wf part m N/A 0.6 N/A N/A 0.6 N/A N/A 0.6 N/A

Sway at turret m 8.4 11.6 35.1 13.9 3.3 23.8 17.0 6.0 36.0

Heave at turret m 0.0 2.3 7.8 0.0 2.3 7.8 0.0 2.4 7.4

Roll deg. 0.0 1.4 4.4 0.0 1.7 4.7 0.2 1.9 6.1

Pitch deg. 0.0 1.2 4.2 0.0 1.2 4.1 0.0 1.3 4.4

Yaw deg. 16.5 26.5 63.8 15.4 2.5 25.9 18.1 5.2 30.1

Mooring Tension (at fairlead)

Line#2 total kN 1779 290 2899 1653 169 2183 1683 176 2213

Line#2 lf part kN N/A 277 N/A N/A 151 N/A N/A 157 N/A

Line#2 wf part kN N/A 84 N/A N/A 77 N/A N/A 78 N/A

Line#8 total kN 847 184 2020 883 180 1921 871 183 1996

Riser Tension (at top)

Liquid prod. (line #13) kN 1315 184 2485 1298 181 2517 1301 184 2578

Water injection (line #22) kN 2275 278 4689 2254 276 4351 2264 279 4577

Gas export (line #25) KN 528 129 1494 528 128 1334 530 136 1850

Fig. 13. Surge and yaw difference frequency QTF of FPSO at 1708 of wave heading.


Hence, neglecting the hull viscous damping can result in

serious overestimation of the rms yaw motions. The rms

surge motions are also appreciably decreased. The decrease

is mainly due to the reduction of slowly varying compo-

nents. Therefore, to predict reasonable dynamic/maximum

sway and yaw motions of a FPSO, the use of proper em-

pirical hull viscous drag is very important. It is also interes-

ting to see that the mean surge offset is reduced, while the

mean sway offset is increased after including hull drag effects.

It is also observed in a separate analysis that the rms

values of surge, sway, and yaw are slightly decreased if

wave kinematics are additionally included in the hull

viscous drag formula.

For other vertical-plane motions, such as heave, roll, and

pitch, the effects of hull viscous damping are almost

negligible. As a result of the change in global hull

motions, the rms tension on weather-side mooring (#2) is

significantly reduced, while that of lee-side mooring (#8) is

only slightly reduced.

8.4. Case 3 (With full QTF)

As mentioned before, Case 3 is the same as Case 2, but

the full second-order wave force/moment QTFs were used

instead of Newman’s approximation. The QTFs were

computed from the WAMIT second-order diffraction/radia-

tion panel program for various combinations of component

waves, which is computationally very intensive. The results

of Case 3 are listed in Table 8.

Compared to Case 2, the increase of sway and yaw can be

noticed. The most significant influence of using full QTF is

in yaw motion. The yaw standard deviation of Case 3 is

about twice higher than that of Case 2. For other quantities,

Case 3 and Case 2 are similar.

Fig. 14. Time history of Case 4 Response.


To better understand the increase of rms yaw motions by

using full QTF instead of Newman’s approximation, the

difference-frequency surge-force and yaw-moment QTFs are

plotted in the bi-frequency domain in Fig. 13. It is seen that

some off-diagonal components of yaw are much larger than

the diagonal values, which may be the reason for

the appreciable increase in slowly varying yaw responses.

From this comparison, it can be concluded that care needs to

be taken using Newman’s approximation especially when the

input sea spectrum is not narrow-banded or double-peaked.

8.5. Case 4: (Comparisons with MARIN’s results)

Case 4 is the same as Case 2 except that the yaw wind-

drag coefficients from the OCIMF data are increased by

19%, as suggested by ABS rule (1996), to compensate

additional forces due to superstructure and equipment above

the deck, which are not included in the OCIMF data. For this

case, the time histories of the 6DOF hull motions and

mooring(line#2)/riser(line#22) tension are given in Fig. 14.

The corresponding spectra are also plotted in Fig. 15. From

those figures, we can directly see that slowly varying

components dominate wave-frequency components for

horizontal-plane motions (surge, sway, yaw), while

the opposite is true for vertical-plane responses (heave,

pitch, roll).

Wichers and Delvin [23] conducted a similar numerical

analysis for the same FPSO considered here and it was

based on the methodology similar to Case 4. However, their

drag coefficients Cd ¼ 1 for risers and mooring lines are

Fig. 14 (continued )


slightly different from the Cd value 1.2 used in this paper.

The statistical results of Case 4 and Wichers [23] are

presented in Table 9.

Wichers [23] also conducted a series of experiments for

the same environment and FPSO design in MARIN’s new

wave basin. However, the lab generated current and wind

field did not exactly match with the target values. The FPSO

model was built with extra bulwark at the bow to avoid

green water on the deck, so the resulting wind loading is

expected to be different from the OCIMF-data-based

prediction. In addition, VIV of slender members were

also reported and the resulting in-line drag coefficients

are expected to increase under such circumstances.

The turbulence or unsteadiness of the current generated in

the experiment may also influence the discrepancy. Even the

direction of the currents fluctuated case by case. It is also

questionable that the lab-generated wind force is close to

OCIMF-based numerical evaluation.

Despite these and other uncertainties, the comparisons

between the current tension/global-motion prediction and

the measured values are reasonable. In particular,

the agreement for heave and pitch motions is excellent.

Better agreement can be achieved by modeling the model

and adjusting empirical parameters, such as drag coeffi-

cients, through calibration. However, no effort is made in

this paper for that purpose. In the present study, the roll

Fig. 15. Response Spectrum of Case 4.


viscous damping is not added, and thus roll rms motions are

expected to be over-predicted. This can be easily fixed by

adding empirical roll viscous damping of FPSO hull.

One noticeable discrepancy between the present predic-

tion and MARIN’s experiment is the mean yaw angle,

which is determined mainly from the balance between wind

and current forces on the hull. Whereas, the yaw rms values

are reasonably predicted.

Similar comparisons can also be made for the statistics of

riser/line tension. For taut line #2, the rms values of Case 4

are lower than those of the experiment, but the mean values

are almost the same. There exists similar trend for the mean

and rms tension of risers. The discrepancy in rms values

may be associated with the observed VIV in the experiment.

8.6. Case 5 (No drag on mooring/riser)

In Case 5, the same simulation as Case 4 is repeated with

the zero drag coefficients on risers and mooring lines to see

their effects on hull motions. However, the inertia and added

mass effects of the slender members are taken into

consideration. Since those inertia effects are expected to

be less important than damping and drag forces, this case is

similar to the uncoupled analysis, where the slender

members are treated as massless non-linear springs. The

major difference of this case compared to Case 4 is the

significant increase in surge and sway rms values. It is

mainly due to the absence of riser/mooring viscous

damping. As a result, the corresponding riser/mooring

dynamic tension is greatly over-predicted. For instance,

the over-prediction of dynamic tension for #13 riser can be

as large as factor of 3. On the other hand, the mean surge

offset is appreciably decreased with Cd ¼ 0: This example

typically shows the importance of hull/mooring/riser

fully coupled analysis for the global motion analysis of

turret-moored FPSOs in deep water.

8.7. Case 6 (With yaw-independent wave forces)

In Case 6, the same methodology as Case 4 is repeated by

assuming that the wave forces do not change with yaw

angles, which is a typical assumption used for the motion

analysis of floating platforms other than turret-moored

FPSOs. Under that assumption, the first- and second-order

wave forces are calculated at the mean yaw angle, and the

same values are used for all the other yaw angles during the

entire simulation. Using this approximation, the rms surge

and sway are slightly increased, while rms yaw is decreased.

On the other hand, the mean sway and yaw are appreciably

over-predicted. The increase of mean yaw is particularly

noticeable. This phenomena can partly be explained by the

mean surge drift forces and mean yaw moments plotted for

several different wave headings in the neighborhood of the

mean yaw angle, as in Fig. 16. It can be seen that the mean

yaw moments are more sensitive to the change of yaw

angles than the mean surge drift forces near the

spectral peak. The yaw-dependent wave forces should be

taken into consideration when yaw motions are large,

particularly in multi-directional waves and crossed-wave

conditions (Table 10).

Up to this point (Case1–6), all the time-domain

simulations were carried out by assuming that the added

Table 9

The results of analysis of Case 4 and [23,25]

Vessel motions Unit Case 4 Stdv Max Refs. [23,25] Stdv Max

Mean Mean

Surge total at turret m 238.1a 10.6b 261.4a 242.1a 16.2c 296.4a

Surge lf part m N/A 10.6 N/A N/A N/A N/A

Surge wf part m N/A 0.6 N/A N/A N/A N/A

Sway at turret m 14.0a 3.2b 28.4a 16.6b 4.1b 30.5a

Heave at turret m 0.0a 2.3a 8.0a N/A N/A N/A

Roll deg 0.1 1.7 4.80 N/A N/A N/A

Pitch deg 0.0a 1.2a 4.2a N/A N/A N/A

Yaw deg 15.5c 2.4a 26.5b 12.0b 2.6a 21.2a

Mooring tension (at fairlead)

Line#2 total kN 1651a 170b 2149b 1743a 351b 3433a

Line#2 lf part kN N/A 151 N/A N/A N/A N/A

Line#2 wf part kN N/A 71 N/A N/A N/A N/A

Line#8 total kN 884 180 1774 876 201 1985


Liquid prod. (line #13) kN 1299a 181a 2653a 1182 þ 292 þ 3921d

Water injection (line #22) kN 2252a 276a 4179a 2149a 363a 5481b

Gas export (line #25) kN 528 þ 129a 1395a 482a 202b 2514d

a The difference between prediction and MARIN’s experiment can be described as within 25%.b The difference between prediction and MARIN’s experiment can be described as 25–50%.c The difference between prediction and MARIN’s experiment can be described as 100–200%.d The difference between prediction and MARIN’s experiment can be described as 50–100%.


mass and radiation damping of FPSO hull do not vary with

yaw angles. We also examined the validity of this

assumption by actually calculating and using the yaw-

angle dependent added mass and radiation damping. It is

seen that all the results are only slightly changed (rms

difference less than 5%), which implies that the variation of

added mass and radiation damping can be neglected unless

yaw angles are very large.

9. Summary and conculsions

Hull/mooring/riser coupled dynamics analysis of a

turret-moored FPSO in non-parallel winds, waves, and

currents was conducted for six different cases to better

understand the motion characteristics, coupling effects, and

the role of various hydrodynamic contributions. The first-

and second-order wave forces, added mass, and radiation

damping were computed by a second-order diffraction/ra-

diation panel program for various yaw angles. The two-term

Volterra series model was then adopted for the generation of

the time series of first- and second-order wave loads in

irregular waves. The current and wind loading for various

yaw angles of hull were obtained from OCIMF empirical

data. The mooring/riser dynamics were modeled using a

generalized-coordinate-based rod theory and finite element

method. Then, the dynamics of hull, mooring lines, and

risers were solved simultaneously in a combined matrix

(fully coupled analysis) at each time step.

From the six different cases of simulations, the following

conclusions can be drawn.

† The slowly varying responses dominate wave-frequency

responses in surge, sway, and yaw, while wave-frequency

responses are more important for heave, roll, and pitch.

† The mean yaw angle is mainly controlled by the balance

of current and wind forces on hull, and the contribution

of the wave drift mean yaw-moment is less important.

The reliable computation of yaw mean offset and rms

responses plays an important role in obtaining reasonable

surge and sway responses.

† The inclusion of proper empirical value for hull viscous

damping is very crucial to reasonable estimate of

Table 10

The results of analysis of Case 5 and Case 6

Vessel motions Unit Case 5 Case 6

Mean Stdv Max Mean Stdv Max

Surge total at turret m 233.4 27.1 2102.7 241.4 10.9 266.22

Surge lf part m N/A 27.1 N/A N/A 10.9 N/A

Surge wf part m N/A 0.9 N/A N/A 0.6 N/A

Sway at turret m 12.6 9.3 35.3 16.8 3.8 25.7

Heave at turret m 0.0 2.4 8.7 0.0 2.3 7.9

Roll deg. 0.1 1.8 6.1 0.1 1.8 4.9

Pitch deg. 0.0 1.3 4.8 0.0 1.2 4.3

Yaw deg. 16.8 4.5 28.8 22.9 2.1 28.3

Mooring Tension (at fairlead)

Line#2 total kN 1644 547 4235 1701 179 2212

Line#2 lf part kN N/A 383 N/A N/A 161 N/A

Line#2 wf part kN N/A 361 N/A N/A 78 N/A

Line#8 total kN 987 324 3183 864 179 1972


Liquid prod. (line #13) kN 1328 540 5958 1304 187 2335

Water injection (line #22) kN 2272 560 6244 2266 277 4097

Gas export (line #25) kN 554 294 3386 533 135 1491

Fig. 16. Surge and yaw mean drift forces LTF of FPSO.


horizontal plane motions. For slowly varying horizontal-

plane responses, viscous damping from hull, mooring,

and risers is much greater than radiation and wave drift

damping.

† The use of full second-order force QTFs increases sway

and yaw rms (slowly varying) responses. The influence

will be more important for broader or double-peaked

sea energy spectra. Otherwise, the motion prediction

based on the Newman’s approximation is pretty

reasonable.

† When uncoupled analyses are used and drag forces on

mooring/riser are neglected, surge and sway rms

responses can be significantly increased and riser/moor-

ing dynamic tension can be greatly over-predicted.

The use of yaw-independent wave forces (at mean yaw)

may be reasonable when yaw motions are not large.

Otherwise, or in multi-directional waves and crossed

waves, yaw dependent wave forces have to be used.

The present numerical results are compared with

MARIN’s experiments. The overall comparisons for

6DOF motions and mooring/riser tensions are reasonable

considering the various uncertainties involved with physical

model testing. In particular, the wave-frequency responses

match very well against measured values. The comparison

of slowly varying responses could be further improved by

using the actual lab-generated environment and calibrating

the empirical coefficients.

The present study has clearly demonstrated the role of

various hydrodynamic contributions in the prediction of the

motions of a turret-moored FPSO in non-parallel winds,

waves, and currents, and thus will help researchers and

engineers to develop their own methodology and numerical

tools in the future.

Acknowledgements

The present FPSO research was financially supported by

Minerals Management Service (MMS), Offshore Technol-

ogy Research Center (OTRC), and also partly by a Joint

Industry Project (BP-Amoco, CSO-Aker, Conoco, Brown

and Root, Sea Engineering).

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Documents

Hull/mooring/riser coupled dynamic analysis and sensitivity study of a tanker-based FPSO