Computation of relative motion effects in offshore supply operations

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OFFSHORE TECHNOLOGY CONFERENCE 6200 North Central Expressway

Dallas, Texas 75206

Computation of Relative

_{Motion Effects in}

Offshore Supply Operations

c Copyright 1976

Offshore Technology Conference on behalf of the American Institute of Mining. _{Metallurgical. and Petroleum} Engineers, Inc. (Society of Mining Engineers, _{The Metallurgical Society and Society of Petroleum Engineers!.} American Association of Petroleum Geologists,

American Institute of Chemical Engineers, American Society

of Civil Engineers, American Society of Mechanical Engineers, _{Institute of Electrical and Electronics} En-gineers, Marine Technology Society,_{Society of Exploration Geophysicists,}

and Society of Naval Architects

and Marine Engineers.

This paper was prepared for presentation at the Eighth Annual Offshore

Technology Conference, Houston, Tex., May 3-6, 1976. Permission

to copy is restricted to an abstract of not more than 300 words Illustrations may not be copied. Such use of an abstract should contain conspicuous_{acknowledgment of where and by} whom the paper is presented.

ABSTRACT

One of the more important

_effects

in limiting the

_{replenishment of a}

floating drilling platform is the

wave induced relative motion

_betwoen

the platform and the supply vessel.

Using currently available

_computer

programs for computing the

wave

induced motion of ships and stable

platforms it is now possible

_to

make good predictions

_{of vessel}

absolute and relative

_{motion in}

both regular and random

_waves.

_It

remains only to combine

_these

com-putations for typical

_{supply boats}

and semi-submersibles

_{to determine}

the relative vertical

_motion

between, say the platform

_crane

boom and the after deck

_{of the}

Supply

_boat.

_{The computation,}

then provides a good

_{basis for}

answering several design and opera-

_{tional questions such}

as:

Cc)

_{Comparisons of conventional}

Tteferences and illustrations at

end of oaoer.

By

J. _{R. Pau)ling and P. D.}

Wood, U. of California

THIS PAPER IS SUBJECT TO_CORRECTION

vs. semi-submersible supply

_vessels.

Some results are given

_of

para-metric studies of suoply

_vessel

size and proportions

_{ir', various sea}

otates.

INTRODUCTION

PAPER

NUMBER

OTO 2634

Continuous operation of an offshore

drilling rig in an

_{area of severe}

sea conditions may frequently

_be

hampered or even disrupted

_by

prob-lems of resupply caused by the

inability to transfer

_{supplies to}

the rig from a supply

_boat

along-side.

_{The relative motions}

_{of rig}

arid supply boat as well

_{as the}

absolute motions of the

_working

decks play an important

_{role in}

limiting the transfer

_operations.

In comparing the merits of various

designs of supply vessels and

methods of material

_{transfer, it}

is useful to have

_{estimates of the}

Both the relative

_{motion amplitude}

and relative velocity

_{between the}

two vessels affect the

_material

RECOMMENDED MINIMUM FOR UTILITY VESSEL

TYPICAL FOR EXISTING UP VESSELS

(a) Optimum

vessel.

size of supply

relative motions of the two (rig

and supply boat) when carrying out

(b) Limiting sea

_states.

_{the resupply operation.}

10

COMPUTATION OF RELATIVE MOTION EFFECTS IN OFFSHORE SUPPLY OPERATIONS OTC 2634

transfer operations, especially the

response rate capability of the

hoisting equipment.

In addition,

i15so1ute accelerations on the

working decks of both of the

ves-sels strongly influence the general

working environment of these areas

especially with respect to the

ability to transfer material

later-ally.

Knowledge of the relative

motion between supply vessel deck

and water surface may be of

impor-tance since this provides a measure

of the frequency of occurence of

water on deck, which, in turn, may

limit operations in certain cases.

A number of linearized analytical

procedures for predicting Ship or

platform motion have been developed

and shown, by coi1parison with model

or full scale test results, to give

acceptably reliable results.

Such

procedures have been described in

considerable detail elsewhere, and

the interested reader is referred

to papers by Salvesen, et.

al

(1970)

and Paulling (1970) for details of

procedures typical of the current

state of the art.

These procedures are of a linear

nature in several respects:

They

use linearized water wave theory in

predicting the fluid exciting forces

and they express the response of

the vessel in terms of linearized

dynamic equations of motion in

either the frequency or time

do-main.

The principal reason for

this linearization, aside from ease

of solution of the resulting

equa-tions, is the ability to make use

of the principle of superposition,

i.e.; to combine elementary

solu-tiens by simple addition in order

to arrive at more complex solutions

representing the response of the

vessel to a realistic random

sea-way.

In principle, the results of such

linear analyses should be

applic-able only in predicting vessel

performance in low sea states.

ExperimeOtal comparisons have shown

quite good agreement, however, in

what would be termed "moderately

severe sea States in many instanced.

Moreover, some essentiall'

Iron-linear phenomena such as the

fre-quency of bow deck immersion and

forefoot emergence are predicted

with reasonably good accoracy.

It

r L

is only when dealing with extreme

motion phenomena such as heavy

roll-ing and capsizroll-ing in "survival" sea

states that the linear predictions

are clearly deficient.

In the case of drilling rig resupply,

operations are curtailed in sea

states well below the level of

severity at which the linear

pro-cedures begin to become deficient.

It may be expected, therefore, that

vessel performance predictions of

both relative and absolute motions

may be used with confidence in this

case.

Such predictions, when

aug-mented with properly interpreted

experience in vessel operation,

should prove to be a useful tool

to the person engaged in the design

or selection of new vessels and

material handling equipment for

resupply operations.

ANALYSIS OF SHIP/PLATFORM RELATIVE

MOTION

Description of vessel motion in

regular waves.

Associated with each

vessel is e body coordinate system

oxyz which remains fixed in the

vessel.

For convenience, the orgin

is located at the vessel's C.G. and

the coordinate axes coincide with

the principal axes of inertia.

The

Cotions of each vessel are described

as small oscillations about a fixed

mean position anda second

.coord-mate system, oxyz, may be thought

of as occupying the mean position

of oxyz in each case.

Let oxyz

be associated with vessel 1, the

platform, and o x y z

₂₂₂₂

be associated

with the supply boat.

These are

shown in Figure 1.

i1

will be assumed that the mean

position of the supply boat relativa

to the rig remains unchanged with

time.

This implies that some kind

of mooring system is used which

maintains the supply boat in a

--,

constant average position re1ative.

to the rig.

We shall also as5umne1

however, that neither this moorings

system nor the rig's position keep'.1'.»

Ing systm appreciably inflUafle.-;

the wave-induced motjoa of -eihe

vessel.

_-i'-i

_-':',

A frequency-domain o1uUon fr

vessel motions will- be enpl4

OTC 2634

_{J. R. Paulling & P. D. Mood}

which we obtain the motions in

ran-dom seas by superimposing regular

wave component solutions.

Using

this concept, we consider first the

response of each vessel separately

to the same incident regular wave

train.

The relative motion response

functions may then be obtained for

such regular waves, and may be

superimposed in the same manner as

individual vessel motions to find

the random sea response.

This is

permissible because the relative

motion may be expressed as a linear

combination

of the individual

ves-sel motions and, therefore, it is

also a linear function of the

inci-dent wave excitation.

The incident regular wave train is

expressed as the real part of

-i(w t-r

n_w

ae

_n n n

where a

_n

= amplitude of wave train

= frequency

r

= phase angle

The equations of motion for each

vessel may be written in matrix

form as

rnx+bx+kx=Fen

(2)

X

X (t) is the complex response

-n

-n

vector, in which x

X , x

n1'

n n

are the displacements of the vessel

C.G. from its mean position, and

X , x , x

are the rotational

n n5 fl6

displacements about the coordinate

axes.

e

is a 6 x 6 matrix of mass and

added mass properties

b

is the hydrodynamic damping

-n matrix

k

_{is the hydrostatic restoring}

'

force matrix

-is the Vector of wave-induced

exciting forces

(1)

Details of the computation of the r,

b

_-n

,

_-n

k

and F

_-n

matrices may be found

in Salvesen (1970) and Paulling

(1970) for ships and semisubmersible

platforms respectively and ne-od not

be repeated here.

Only the real part of the solution

is used and the subscript "n"

is retained here to emphasize that

all terms in (2) pertain to regular

wave component n.

This solution

will be of the form

-iw t

x(t)=xe

n

-n

-n

For unit incident wave amplitude,

the complex response amplitude will

be X(ui), and this may be thought of

as a complex frequency-dependent

transfer function for the vessel

rigid body motions.

The transfer

functions for the motion components

at specific locations within the

vessel may be found if x(u) is

known.

Thus, recalling the order

of the terms of x

,

the absolute

-n

vertical motion of a location within

-one vessel whose coordinates are

)X, 'ia'

Za)

is given by

Y (uf) =x (5)-4-xx (w)-Z x (a)

_{A n}

2 n

a6

n

a6

n

The absolute vertical velocity and

acceleration are given respectively

by Y

and -w2 Y

_.

The relative

nA

n A

vertical motion between point a on

the rig representing, e.g.

,

the end

of the crane boom, and point b

rep-resenting a location on the deck of

the supply boat is given by

Y(w)=Y (s)-Y (w)

R n Aa n Ab n

= X

4-X x

-Z X

2

a ua a 4a

2b

Xbxb4 Zbxb

rrUiJ%riON OF _{RELATIVE MOTION EFFECTS IN}

OFFSHORE SUPPLY OPERATIONS _{OTC 2634}

Since YR(w) is a linear _combination

of motion transfer

functions for the two vessels, it _{may also be} treated as a linear

transfer func-tion relating the _{relative motion} response to the excitation, _{in this} example, the incident _waves.

The relative velocity and _acceleration

re given by

r'R(mn) and -w2Y (w

nR n

uìe individual vessel _responses

and ç(w) must be

_computed

taking into consideration

the loca-tions of the two _{vessels relative} to each other and _{to the origin of} the wave coordinate

system. In most vessel motion _{computational} procedures, the origin of _coordinates

is located at _{the vessel center of} ;ravity and the wave coordinates are usually referred to the mean )osition of the 0.0. _{If the wave}

oordinate origin is to be located slsewhere, the result will _{be only} a change in the phase,

Cn of the dave at the position of _{the Ship}

oordinate origin. _{This in turn} results in a chango in _{phase of the} -ransfer function; i.e., a change

n the relative magnitude _{of the} -eal and imaginary _{parts of the}

:(w )

vector for one or both

ves-- n

als. Once the response _transfer unctions have been computed for a iven initial position of the wave md ship coordinate _{systems, a} hange in the relative

position of he two vessels _{does not require}

a ecomputatìon of the _{response but} orely a revision of the real and

1aginary parts of the transfer

.anct ions.

asponse to random waves. Having e system transfer _function, uation (6) , we may determine the :ergy spectrum of the _{response to}

unidirectional random seaway whose

'ectrum is S(w) _by

Syy(s) R(w) .yR(a). Srir (w)

re i(w).i8 the Complex _conjugate

thé

relative motion transfer ifljon TR(e) _{If the input wave}

egtrum

j8

_{a ç,aussian random}

pro-s, the output

_{will be}

_Gaussian

also, and the

mean square value of the output,or

variance,of the pro-cess will be given _{by the integral} of S (w).

If, further, the process yy

is narrow-banded,

the peak values of the process, _{e.g. the average of}

the highest 1/n of the _{excursions, will} be given by 1/n (8) where a2 f S _{(w) du.} o and C is a function of n. Equations (6), _{(7) and (8) provide} a means of obtaining _{the relative} motion response in random seas bet-ween two vertically

coliner points on the two vessels.

The procedure has the drawback _{of requiring}

the reevaluation of the _transfer func-tion, equation _{(6), for each position} of the point _{at which relative} motion is desired.

A somewhat more _efficient pro-cedure may be developed if we are interested in only the _statistical description of the _{relative motion} and not its tise _history.

We note that the time history _{of the relative} motion is given by

Y (t)=xR a(t)-Z x

aa

(t)+x

aa

X (t)

-x

(t)+zbxb(t)_xbxb(t)

2b

If we now square _{this expression} and take the time _{average over a} long time interval, _{we obtain}

y' = X2 + Z2 x2 _{+X2 x2} + R a a a a a 2b.. (9) + Z2 x

+Xx2

-k: X b .b b b _a

_2a2a

+ 2X X X _{+ other croes} a

aa

product terms (10) OTC 2634

3. R. Paulling & p. 0. Wood For an ergodic

Gaussian random process, the time averages repre-sented by the first _{six terms are} equal to the spectral _{areas of} the six individual

motion com-ponents, and the _{remaining terms are} equal to the _{cross spectral areas} corresponding to pairs _{of the} ves-sel motions. _{The result may be} expressed most conveniently

in

matrix form, Price _{and Bishop (1974),} as follows. _{Let us write the}

relevant individual motion transfer functions for the two vessels in the form of a vector:

H(w) , where

U(a) = (x(w)

,ab

x(w)

(w) , X (w) , x (w)) (11)

zb 4b

Also, define the vector of coor-dinates of the points _whose rela-tive motion is desired:

Z = {l, -Z , X , _{- 1, z}

_{, x}

- _{a a}

b b

(12)

The cross spectral

density response matrix corresponding to the six motion components _{appearing in il(a)} for the input

wave spectrum S(u)

is then given by

= iÏ(w).FiT(w).S

(a) (13)

rin

where Ïl(w) is the complex _conjugate of I-I(a) and HT is

its transpose. Furthermore, the _{spectral density} of the relative

motion response may be written in the form

S (w) = zT.5 _(w)Z

yy - -xx

-= zT_T()

ZS(w)

(14) Now, _{note that the}

innermost matrix product - -'P

H(s) H (w) = A(a) depends only on the

characteristics of the two vessels _{and their} posi-tions relative to each other. The

points in the vessels

whose rela-tive motion is required is contained Only in the Z vector. _{Thus, the} relative mctron may be _{found for} several pairs of points by merely reevaluating an expression of the form zT.A.z for each pair of locations.

Equation (14) gives the spectrum of the relativo motion _responae as the sum of spectra _{and cross} spectra of the two vessels' motion components weighted by the coor-dinates of the locations at which is desired. _{The mean square value} of the response,

Equation (10), is thus given by the sum of corres-ponding spectral and _{cross-spectral}

areas _{similarly weighted}

by the coordinates of the points of rela-tive motion.

RESULTS FOR DRILLING PLATFORM/SUPPLY BOAT

Computations have been made using this procedure for _several configu-rations of drilling _{platform and} supply boats. _{The platform was} given a simple

configuration consIs-ting of two main _{lower hulls and} four surface piercing _{legs each} side and had

characteristics goner-oiiy similar to a modern large North Sea-semisubsersible. _{Three size} variations on a basic offshore supply boat and a semisubmersible platform simulating a semisubmer-sible supply vessel were

_used.

_The principal characteristics _{of the} vessels are given in _{Table 1.}

Results were computed _{using the} Pierson-Moskowitz wave spectral

_?mu1cEn.

ave directionality was included by means of a

cosine-ga.iatd spreading

_{function. 'Tot}

otT1ia spectral _{family is shown in}

Figure 2. _{The supply vessel was} located in approximately

the posi-L tion shown in Figure 1, i.e.,

on ,Jjthe platforms

centerline with the iletern clear by about fifty feet.

j Predominant wave

directions of zer degrees (following) _{and forty-five} degrees (quartering) _were consid-ered.

Relative motions were compu-ted for a position

at the stern of the supply boat _{and fifty feet} inboard corresponding te a crane outreach of one _{hundred feet.}

In general, the superior _relative motion characteristics of the semi-submersible are shown quite clearly. Her motioD characteristics _are about the sanie at the _{stern and at} the inboard location. The'conven-tional supply boats _{on the other} hand display considerably _more relative motion at the stern than

inboard.

CONCLUS IONS

While theresults _{presented here}

are insufficient to draw _final con-clusions, some indication is given of the ?Otefltial _{advantages in} material handling and _{transfer that} may be gained through the use of a aemisubmersible supply vessel. In

comparing conventional _{supply boats} of different size, the relative motions are shown to _{vary but little} for the range of size _{investigated.}

It would be of considerable _value tc correlate the motion characteris-tics computed here with _{real sea} sate conditions in which resupply operations are found to be _limited.

ACKNOWLEDGEMENT

This work was supported by Offshore DeveloDment Engineering, Inc.

REFERENCES

Fis, i -

PLATFORM AND SUPPLY BOAT _{COORDINATE SYSTEM,}

1. _{N. Salvesen, E. O.}

Tuck, O. Faltinsen; _{'Ship Motions}

._

and Sea Loads', Trans _SHAME y 78, 1970.

2. _{J. R. Paulling;}

'Nave Induced

TABLE i - VESSEL C6000CTEAISTICS

-Forces and Motions of Tubular

Length _Bean

D ra f t (li spI ece,,r

Structures", Proc. _Eighth Symposium on Naval _{Hydrodynamics,} Pasadena. _{ONN ACR 179, 1970.}

Feet Feet _{FB t}

I.. Ttns 3. _{J. L. Arps:}

"The Role of the Semisubmersible Work Vessel in

Offshore Production _Operations",

Drilling Sig ₃₂₀

2 16 BO ₁₀₉₀₄

Proc. 1973 Offshore _Technology Conference, Paper No. OTC 1867

Semjsubmersible 190 ₁₁₅

30 ₇₁₀₄

4. _{Donald M. Taylor;}

Aker H-3 - Supply Vessel

The $750 Million Rig', _Ocean

Supply Boat A 150 31.4 8.68 712 Industry, April 1974. 5. _{Allan McClure:}

"Semisubmer-cible Supply Vessel Design for Supply Boat S ₂₀₀

41.9 _11.0

1087

North Sea Operations", _Ocean

Supply Boat C ₂₅₀

52.3 14.5 ₇₁₉₄

Industry, Feb. 1975.

6. _{0. G. Keefer:}

Development of Motion-Compensating Cranes for Rough Sea Operations", Ocean-ology International 75,

i3riqhtnn, Enclansd, 1Ier. 1975.

7. 11. _{C. Price & R. E. D.}

Bishop: "Probabilistic Theory of Ship Dynamics", Chapman and Hall, London, 1974.

COMPUTATION OF RELATIVE MOTION

EFFECTS IN OFFSHORE SUPPLY OPERATIONS OTC 2634

PIRSO - MOSKOWLIL sP[CrA 40 Feet _{Sigeificani Height}

35 Ft 30 Ft 25 Ft 20 Ft 15 Ft 10 Ft .6_fREQUENCY.8 1.0 .2

Ii

1.6 1.8

2.0

2.2 C _{RAOIANS/StC060} I

Fis, 2 - PIERSOF4 MOSKOWITZ

UNIDIRECTIONAL SPECTRA,

{AVE _{[SPUNSE AMPLITUDE OPERATORS}

SUPPLY VESSELS

REAVE RESPONSE AMPLITUDE OPERATORS

Fie. 5

PITEN RESPONSE AMPLITUDE OPERATORS

8 .0 .2 1.1 .6 .8 S

FREQUCNCY I RAIINS1SfCQ0 I

.6_{F Pf O UE#C}.8 1.0 .2

I. i

.6 2.0

Supply Boats A, B, C

RELAtIVE V[RTICAL VELOCITY SO FEET INDOAPO

Semisubmersibie

Following _SeaR

Quartering

IO 15 20 25 30

SIGNIFICANT uHU HEISST I rIET J

Supply Boats A, B, C

RELATIVE VERTICAL MOTION 50 FEET INSOARO

'Foliowing

ßeaS

Qua r te r ing

S.erniauSusersi bie

RELATIVI VERTICAL MOTION

AT TIlE STERN

o

RELATIVE VERTICAL VELOCITY AT THE STERN

01 ₃₀

20 IICRIFICANT WANE HEISST 1 FEET J

30 75

IO IS ₂₀

SIGNIFICANT uANE REICHT I FEET

FIG, 6