Aspects of mooring in deep water

kSPECTS

OF

MOORING IN DEEP WATER

By P.

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A Thesis submitted to the Faculty and the Board o f Trustees of the Colorado School of Mines in p a rtia l fu lfillm e n t of the requirements fo r the degree of Master of Science in Petroleum Engineering. Golden, Colorado Date: 7 ,1 9 7 6 Signed:

k

... ) - f Q-tJ

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ABSTRACT

The purpose o f th is thesis is to cover various aspects of deep water mooring with sp ecific application fo r o il and gas d r illin g and production from ships and semisubmersibles. The th eo retical aspect is presented and the work is i l l u s ­ tra te d by example calcu latio ns.

Throughout the work, emphasis has been placed on mooring in deep water - i . e . , to 1500 f t of water.

TABLE OF CONTENTS

Page

ABSTRACT... . i i i

LIST OF TABLES AND FIGURES... . . . vi

ACKNOWLEDGMENTS... . ... vi i i INTRODUCTION... . ... 1 i ENVIRONMENTAL FORCES. . . 3 Wind F o rc e ... . 4 Wave D r if t F o rc e ... . 6 Current Force... 9 MOORING PATTERN ... 12 MOORING LINES ... 14 ANCHORS... : ... 17 MOORING... 19 MATHEMATICAL DEVELOPMENT... 21 EXAMPLE CALCULATION . . . . ... 27 Pretensioning... 27

High Line Tension... 29

Total Cable Length and Maximum Reel-Up ... 30

Restoring Force... 31

Example Summary... 32

DISCUSSION OF DATA... 35

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m in e s

,i .

Page

APPENDICES... 36

A - Procedure fo r Pretensioning C alculation... 36

B - Calculation Procedure fo r Combination Wirerope-Chain Mooring Line . . ... 39

C - Data fo r Example C a lc u la tio n ... 42

D - Nomenclature... 43

SELECTED LITERATURE ... 46

LIST OF TABLES AND FIGURES

Table Page

1. Height C oefficien t (C^) fo r Wind Force C alculation. . . 48

2. Shape C o e ffic ien t (C$ ) fo r Wind Force Calculation . . . 48

3. Comparative Data of Wirerope Configuration. . . . 49

4. High Line Tension, Anchor P u ll, and Bottom Line Amount... 50

5. High Line Tension, Anchor P u ll, and Bottom Line Amount... 51

6. Restoring Force - Beam Attack . . ... 52

7. Restoring Force - Beam A t t a c k ... 53

Figure 1. Types of Vessel Movements Defined ... 54

2. Wind, Wave D r if t and Current Load on Vessel fo r Beam Attack and Specific D ra ft... 55

3. Mooring Line Configurations fo r Deep Water Spread Mooring Systems . . . ... 56

4. Anchor Types... 57

5. Anchor B e h a v io r... 57

6. Cable Deployment Geometry ... 58

7. Cable Segment with S ta tic Forces... 59

8. Catenary Geometry ... 59

9. High Line Tension and Anchor P u l l ... 60

TO. High Line Tension and Anchor P u l l ... 61

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Page

11. Mooring Pattern and O ffset Configuration... . . 62

12. Restoring Force - Beam A t t a c k . 63

13. Restoring Force - Beam Attack . . . ^ ...64

14. Pretensioning Geometry. ... 65

15. Combination Wirerope - Chain Geometry . . . 66

ACKNOWLEDGMENTS

The author expresses his g ratitu d e to:

Dr. B.J. M itchell fo r serving as thesis advisor and fo r suggesting th is subject;

Dr. D.M. Bass fo r serving on the thesis committee; Dr. D .I. Dickinson fo r serving on the thesis committee.

The author is indebted to Greenland .Petroleum Consortium K/S fo r providing fin a n c ia l support.

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INTRODUCTION

Offshore d r illin g a c tiv ity fo r o il and gas has been on a steady increase since the early 5 0 's and has only temporarily

leveled out in recent years. For offshore d r illin g operations

various d r illin g vessel configurations have been employed, e .g .,

ja c k -u p 's , d r i l l ships and barges, and semisubmersibles. Jack-up's

are in contact with the seafloor while d r illin g and th erefo re, station-keeping is not a problem; however, they have a lim ited water depth in which they can operate - namely less than 300 f t .

For many years offshore d r illin g has been conducted in water depth o f more than 300 f t using d r i l l ships and semisubmersibles. Station-keeping fo r these flo a tin g vessels is handled with mooring

systems and more rece n tly , dynamic positioning. The combination

o f vessel-type and station-keeping systems w ill give four combinations, each with it s application as determined from operational and economical considerations.

This thesis deals with mooring of ships and semisubmersibles and tre a ts the special problems encountered in deep water (1500 f t ) mooring systems design.

D r illin g operations conducted from flo a tin g vessels are

dependent upon the mooring system to keep the vessel w ithin certain horizontal lim its o f the entry in the seafloor, e n title d excursion,

as measured in fe e t or in percent of to ta l water depth. They are fu rth e r dependent upon the system to maintain the position in such a fashion th a t d r illin g operations can be successfully carried out.

The distance of excursion is usually kept to w ithin 6 to 10% 1-3

o f to ta l water depth even under the most severe environmental

conditions. For normal operations the excursions should not amount

1-3

to more than 2 to 3% provided that the mooring system is designed

to give maximum restoring fo rc e, which is the sum o f the horizontal forces from the mooring system on the vessel as defined la te r in

th is thesis. Only in rare occasions should the excursion be more

than 5 to 6%. Depending on the marine r is e r design, d r illin g w ill

normally be suspended fo r excursions of a magnitude of 5 to 6%. Even though the marine r is e r normally can sustain these imposed stresses, the d r ill- p ip e w ill cause excessive wear on the ris e r a t these excursions.

1-3 Survival conditions e x is t fo r excursions beyond 8 to 10% again depending on the marine r is e r design, and in th is mode the r is e r is normally disconnected from the blowout preventer a t the

seafloor. During survival conditions, the excursion is only

T-1850 3

ENVIRONMENTAL FORCES

The f i r s t parameters to be considered in a mooring systems design are the environmental forces acting on the vessel and the vessel's response to these forces, in order to establish the

vessel's environmental resistance ch a ra c te ris tic s . The vessel's

response w ill d if f e r depending on the angle of attack of the environmental forces, and each of the attack angles causing large vessel movements must be studied.

The environmental forces to be studied are wind fo rc e, wave

d r i f t fo rce, and current force. In obtaining data on these forces,

the d ire c tio n and occurrence frequency should be determined in addition to the amplitude.

The environmental forces w ill induce s ta tic forces to the vessel, however, due to the o s c illa to ry changes in the wave d r i f t fo rc e, and the gusty nature of the wind fo rce, a certain amount of dynamic force are transferred to the mooring system.

The above mentioned forces w ill cause horizontal motion o f the

vessel. In addition hereto the waves w ill cause a heave motion

which is the v e rtic a l up and down motion of the vessel and a t the same time r o llin g motions around the three a x is , respectively yaw,

p itc h , and r o ll as shown in Figure 1. These motions of the vessel

system. Each o f the forces w ill be dealt with separately and both

the s ta tic and the dynamic forces w ill be treated . Equations are given

fo r an approximate calcu latio n of the load on the vessel from the s ta tic forces.

In most cases, the forces and the vessel response is obtained in connection with model tank tests in order to give a higher degree o f accuracy, since one the greatest uncertainties in analyzing a

mooring system is in determining the s ta tic and dynamic forces. This

is esp ecially true where more complicated vessel shapes are involved, as in the case of most semisubmersibles.

For mooring in deep water the e ffe c t of the dynamic forces is

much less important than the s ta tic forces^, since the vessel's

heave and r o llin g motions fo r a given type of sea as a percent of to ta l water depth becomes r e la t iv e ly smaller with increased water

depth. For th is reason emphasis in th is thesis is placed on the

s ta tic forces.

Mind Force

The wind force is a re s u lt o f a change in directio n of an a ir

mass and thereby its momentum as i t contacts a surface. A part of

the to ta l force is due to the drag of the a ir as i t slips past a surface.

A prop eller-typ e instrument is used to measure the wind v e lo c ity

and is simultaneously measuring the wind d ire c tio n . The wind v e lo c ity

is reported as an e ffe c tiv e v e lo c ity measured in knots and equal to 1.096 times a 5 min. mean reading^.

T-1850 5

The force induced onto a surface is a function o f the wind

v e lo c ity , and fo r simple shapes the force can be calculated. Below

are lis te d equations fo r calcu lating the s ta tic wind force as o

published by The American Bureau of Shipping :

F = 0.00338 • A • Vk • Ch • Cs (1)

v/here F = wind fo rce, Ib f

A = the projected area of a ll exposed surfaces in e ith e r the upright or the t i l t e d position, sq f t

= wind v e lo c ity , knots

Ch = height c o e ffic ie n t (see Table 1) C$ = shape c o e ffic ie n t (see Table 2)

Consideration should be given to the mean wind v e lo c ity as well 3

as the maximum gust v e lo c ity . Harris suggest calculation o f the

wind v e lo c ity from equation (2 ):

Vk = 0,6 ’ Va + 0,4 ' Vg ^

where = average wind v e lo c ity , knots

a

V = gust wind v e lo c ity , knots y

In most cases, fo r mooring system design, only the s ta tic wind forces are considered, based upon the wind force calculated using equation (1 ).

An example relatio n ship between the wind v e lo c ity and force on the vessel is given in Figure 2.

the s ig n ific a n t wave height is defined as the average of the highest one-third o f the waves (Hg), and the wave period is defined as the tim e, in seconds, fo r a wave crest to traverse a distance equal to one wave length where the wave length again is the horizontal distance between adjacent crests.

Wave action w ill cause horizontal motion of the vessel involving both a d ire c t wave induced short period motion and a gradual long

period o s c i l l a t i o n ^ . This motion is caused by the wave d r i f t

force resu ltin g from the change in wave momentum as part of the

wave is refle cte d from the vessel. In regular seas the re fle cte d

waves causes a steady d r i f t force resu ltin g in a s ta tic force

o

applied to the vessel. American Bureau of Shipping has published

equations fo r calculation o f the s ta tic forces. For ship-shaped

vessels the equations are:

Bow forces:

Fbow = ( 0 ' 273 * H2 • B2 • D / T 4

where T > 0.332 • vT

Fbow = ( ° - 273 • H2 • B2 • L ) / ( 0.664 • - T )4

T-1850 7 Beam forces: (5) where T > 0.642(B + 2D)1/2 Fbeam = (2.10 • H2 • B2 • L )/(1 .2 8 (B + 2D)1/2 - T )4 where T < 0.642(B + 2D)1/2 ( 6 )

where F = wave force, lb f

T = wave period, sec

H = s ig n ific a n t wave height, f t B = beam of vessel, f t

D = d ra ft of vessel, f t L = length of vessel, f t

Due to the complex shape of semisubmersible vessels, the wave d r i f t force are normaliv determined by model tank te s ts .

Irre g u la r waves cause a varying sequence o f d r i f t forces

because o f the changes in wave height and period. In v e s tig a tio n s ^

have shown th a t the long period o s c illa tio n s of the vessel can be of dominating influence in determining mooring lin e tension.

In determining the peak mooring forces from slow o s c illa to ry motions, Hsu et aj_.^ describe the basic method o f calcu latin g wave

q

d r i f t forces based on the concept of "Radiation Stress" as explained

with the following quotation: " I t is well known that surface waves

possess momentum which is directed p a ra lle l to the d irectio n of propagation and is proportional to the squares of wave amplitude. Now i f a wave tra in is re fle cte d from an obstacle it s momentum must

be reserved. Conservation of momentum requires th at there be a force exerted on the obstacle equal to the rate o f change o f a wave

momentum." They found th at the resulting average d r i f t force per

u n it length as a function of incident wave height and the vessel's heave and sway motion fo r a ship-shaped vessel based on beam atta c k , is given by:

Q = 2 • cosh(k(h-D))/cosh(k • h)

Ah»As = amplitude of vessel heave and sway, respectively

9h,8 s = P^ase an9^e vessel heave and sway motion with

respect to incident wave tra in surface elevation at the average position of the center lin e of the vessel, i . e . , x = 0 h = water depth 2tt k = wave number = — X = wave length p = water density g = acceleration of g ravity D = vessel d ra ft

In setting up a mathematical model to describe the vessels motion in

c

irre g u la r seas Hsu et a l . combine the above equation (7) with a

time-(7)

T-1850 tAKES LIBRARY 9

COLORADO SCHOOL of MINFS GOLDEN. COLORADO 80401*

dependent wave d r i f t force on a mass and non-linear spring system to illu s t r a t e the moored vessel, fu rth e r assuming that "each o f the waves o f an irre g u la r sea w ill impart to the moored vessel the same wave d r i f t force which i t would were i t merely one of a sequence of regular waves," as follows:

m • x" + c • x' + f ( x ) = g ( t ) (8)

where m = v irtu a l mass

c = damping

f ( x ) = non-linear spring function

g ( t ) = the wave d r i f t force function th at gives the average wave force over each wave cycle

The application of th is model, however, must be used in connection w ith comprehensive testing in a model tank to obtain amplitudes of vessel heave and sway, and phase angles fo r d iffe re n t types of sea or knowledge of the vessel's response must be known.

The above model was developed fo r ship-shaped vessels. The

slow d r i f t o s c illa tio n force can be expected to have much less

influence fo r open constructions, as is the case with semisubmersibles^. S ta tic induced wave d r i f t force is shown in Figure 2 fo r an

example vessel as a function of s ig n ific a n t wave height.

Current Force

The current force is a re s u lt o f the change in the v e lo c ity of a

dracj force of water is a much greater part of the to ta l water force than is a ir drag with respect to to ta l wind force.

Measurement o f the current v e lo c ity is made in a fashion s im ila r

to the wind measurement. The current v e lo c ity is expressed in knots.

Ocean currents are normally f a i r l y stable w ithin a given water depth, however, they w ill change d irectio n and v e lo c ity with

increased water d e p th ^ . Current ve lo c ity p ro file s are often

constant in a region, but can d if f e r widely from region to re g io n ^ . The current force constitutes a combination o f the current as i t a ffe c ts the vessel, the marine r is e r , and the mooring system, mathematically represented by equation (9 ):

Ft = Fves + Fr is + Fmor {9)

The following equations can be used to calcu late current forces

9

fo r ship-shaped vessels :

Fbeam = 0.30 • A • V2 (10)

Fbow - ° - 016 • A • Vc (11)

For semisubmersibles current forces on the bow or beam can be 12

calculated from :

F = (2 .4 • Ac + 5.7 • Af ) • V2 (12)

where F = current fo rc e, lb f

A = the wetted area, sq f t

A = to ta l projected area o f a ll c y lin d ric a l members below

V

T-1850 11

Af = to ta l projected area of a ll other members below the surface, sq f t

V = current v e lo c ity , knots

V

The current force on the marine ris e r can be calculated from the following hydrodynamic equation^’

Fr is = ° - 0457 ' C ' Ac ' p - Vc ( 13>

where C = drag c o e ffic ie n t

p = flu id density, lbm /ft^

For a lin e a r current d is trib u tio n of 2 knots at the surface and 1 knot a t the seabottom, and a drag c o e ffic ie n t o f 1 ^ * ^ , the ho ri­ zontal reaction force at the vessel is no more than 12,800 Ib f fo r a

24" diameter ris e r . This force is small compared to the other

environmental forces and therefore normally disregarded.

Current forces acting on the mooring system are normally very small and hence they are disregarded in most mooring systems design.

An example relation ship between the load on a vessel and the magnitude of the current v e lo c ity is shov/n in Figure 2.

MOORING PATTERN

Mooring patterns refers to the layout of the mooring lin e s , th e ir number, and th e ir position in terms o f the vessel and in

terms o f each other (see Figure 11). The most common type of

mooring system is the spread system with from one to twelve lin e s . These lin es can be la id out in many d iffe re n t fashions and a great

number o f mooring patterns do e x is t. The reason fo r the d iv e rs ity

i

o f patterns stems from an e ffo r t to gain the maximum restoring force from the mooring system in combination with the vessel's environmental resistance ch a ra c te ris tic s .

Some patterns are u n i-d ire c tio n a l where the mooring system is

designed to take the biggest load in one given d ire c tio n . This

pattern is preferred in areas with a prevailing environmental

force d ire c tio n . Other patterns are om ni-directional, constituting

an often symmetrical pattern able to withstand loads from any d ire c tio n . Normally the vessel is moored with the bow toward the d irection

o f the predominant environmental force. The mooring pattern

selection is influenced by the magnitude, direction and occurrence frequency of the design forces and by the design of the vessel deck machinery.

Design of the optimum pattern is a very important facto r in the to ta l mooring system design, p a rtly in reducing mooring lin e

T-1850 13

tension, and p a rtly in order to minimize horizontal displacement and

thus to stay w ithin given excursion tolerances. The s t i f f e r the

system the more optimal i t is in absorbing dynamic forces and thus reducing the vessel's potential fo r gaining momentum which can re s u lt in peak mooring forces, as the momentum must be absorbed in

the mooring system . I f the system, however, becomes too s t i f f , i t

w ill cause excessive mooring lin e tension, even fo r small o ffs e ts , and the l i f e of the mooring lin e w ill be severely impaired because o f fatigue

1 15

c h aracteristics * .

The optimum pattern is found by evaluating the vessel o ffs e t and the maximum tension in the mooring lines fo r d iffe re n t patterns fo r

the environmental forces acting. Eventually, however, i t w ill be a

matter of economic "tra d e -o ff" where additional expenditure are not warrented by increase in mooring system e ffic ie n c y .

MOORING LINES

The mooring lin e is the t i e between the vessel and the anchor. There are several mooring lin e types and configurations, a ll employing chains, cables or combinations th ereof, as lis te d below (see Figure 3):

1. A ll wirerope or a ll chain,

2. Clumb weight system; where a secondary anchor is

placed a t the end o f the chain and the primary anchor attached to the secondary anchor by wirerope (not "piggy-back"),

3. Combination a ll chain; where one type of chain is

used fo r the main portion of lin e from the vessel and a heavier chain used along the sea bottom and attached to the anchor,

4. Combination wirerope-chain; where wirerope is used

from the vessel to the seafloor and chain is used to t i e into the anchor.

Presently the two most common configurations are (a) all-w ire ro p e

or (b) a ll-c h a in . These systems are suitable in water depth to

1500 f t . The actual choice between wirerope and chain depends on

expected mooring lin e load, water depth, environmental forces, handling equipment, storage f a c i l i t i e s on board the vessel, and economics.

T-1850 15

, For low water depth applications wirerope moorinq lines re s u lt in too s t i f f o f a mooring system^ and the a ll-c h a in mooring lin e

system is generally preferred. However, with increased water depth

the a ll-c h a in system is in the most cases too loose and requires very high pretensioning.

Beyond 1500 f t water depth - in what is normally called u ltra deep water - problems arises with the all-w irero p e system as well as

the a ll-c h a in system. This is mainly due to problems in the manu­

facturin g of these long lines and in handling. Of the four mooring

lin e systems lis te d above, the fourth - combination wirerope-chain system - seems to be the most suited as reported by Childers^. Drawbacks with th is system is due to the-necessity of a new design of the deck handling equipment to give reasonable deployment and

re trie v in g o f the mooring lin e . In addition to the handling problems

Atwood, e t a l . , reported t)tat. the varying vib ra tio n ch aracteristics

o f the wirerope and the chain can re s u lt in severe connection problems between the two.

Working load of wirerope and chain are normally one-third of

1-2 . 2 14

th e ir rated breaking strengths . Fatigue ch aracteristics 9 is

a major problem o f the wirerope as well as the chain mooring lin e s , necessitating th a t the working load be kept low, as i t w ill reduce

the strength and the l i f e o f the lin e s . The higher the peak load and

the longer the spread between high and low tension the shorter the

l i f e . Fatigue o f the lin es is one.of the biggest reasons fo r lin e

2 13

breakage 9 . For reasons of fa tig u e most mooring systems are designed

The correct pretensioning of the mooring lines is very important

in the to ta l mooring lin e design. Optimum pretensioning load has

1 - 2

been reported to be such th a t the maximum working load (one-third

o f the breaking strength) occurs at the maximum allowable excursion

fo r continued d r illin g , normally 5 to 6% of to ta l water depth. Hsu

e t a l .^ shows in connection with his development o f the peak forces from o s c illa to ry d r i f t how the optimum pretensioning can be obtained;

however, th is procedure requires model tank te s ts . I f the pretensioning

is too high, i t w ill re s u lt in excessive tension in the mooring lines even fo r small vessel offsets and causes the wirerope or chain l i f e

to be severely impaired due to th e ir fatigue c h a ra c te ris tic s . I f the

pretensioning is too small, the mooring system w ill be too loose and th is results in large excursions before s u ffic ie n t restoring force is

obtained. This could cause undue d r illin g stoppage and i t could re s u lt

in an increase in peak mooring lin e tension since the vessel would be allowed to obtain momentum, which eventually must be absorbed in the mooring lin e s .

T- ,8S 0

' W m S S i

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GOLDEN. COLORADO 80401

ANCHORS

The anchor w ill supply the main holding power fo r the mooring

system. The most common anchors used are one of the following lig h t

1 -3

weight type anchors : (see Figure 4)

1. The U.S. Navy Light Weight Type Anchor (LWT)

2. The Danforth Anchor

3. The Stockless Anchor

Beck^ reports on testin g with an even la t e r design e n title d "Boss" (see Figure 4) which generally outperformes the others by digging in fa s te r in sand as well as mud bottoms, by giving increased p u llin g power and by avoiding ball-up (see Figure 5 ).

A ll o f the anchors are b f the dynamic type which w ill increase

th e ir holding power with increased p u ll. I t should be pointed out

th at th is is true only i f there is no u p liftin g force on the anchor. Even a t a pull of 6° between the force and the sea bottom the holding

1 o

power w ill decrease rapid ly . For th is reason, i t is important

th a t s u ffic ie n t mooring lin e is deployed at a ll times to insure th at the end of the mooring lin e is p a ra lle l with the ocean flo o r a t or before the anchor.

The anchor sizes normally used are in the range of 20,000 - 30,000 1-3

Ib f dry weight . The maximum holding power w ill among other thinqs

2

holding power is in the range of 12 to 17 times th e ir dry weight.

For larg e r anchors to 30,000 Ib f the holding power w ill decline and

2

be around 8 to 11 times th e ir dry weight. For even larg e r anchors

2

the holding power w ill generally be below 10 times th e ir dry weight.

The holding power w ill vary with the type of sea bottom depending on

whether i t is sand or mud. However, the above indicated holding

powers seem to be good average values.

The type of sea bottom w ill a ffe c t the anchors tendency to dig

in . Here i t is important to adjust the fluke angle - which is the

angle between the anchor shaft and blade as seen from Figure 5 - to 1-3

correct value . Even though the correct flu ke angle w ill vary fo r

d iffe re n t types of anchors, 50° seems to be a good value in mud and 3

30° to 35° best in sand . For unknown or mixed sea bottom, a fluke

angle of 35° to 40° should be used^.

As seen from above, an increase in anchor dry weight does not \

s ig n ific a n tly increase holding pov/er because the power fa c to r w ill

decrease. For very large semisubmersibles and a combination of

severe environmental conditions and deep water operation, single

anchors w ill not give s u ffic ie n t holding power. Very often therefore

"piggy-backing" is necessary, which is two anchors used in series^. In addition to the common type anchors other anchor types are

used. Of in te re s t is anchor piles sp ecially used in areas where

3

hard rock bottom precludes the use of conventional anchors . A more

recent innovation is explosive-set anchors. This anchor type is embedded

into the seafloor by the aid of an explosive charge. The advantage of

these anchors is a very large holding power; however, the anchor is not recoverable.

T-1850 19

MOORING

Mooring is here meant in a p rac tica l sense to constitute a ll the operations necessary from the time the vessel is on location to the time when the spudding fo r the d r illin g operation can take place.

In practice the mooring o f a large d r illin g vessel is aided by

work boats or anchor handling boats. I t is necessary th at the

proper size boat be used; one th a t can handle the weight of anchor(s) and mooring lin e during deployment of the anchor and at the same time output s u ffic ie n t amount of bollard pull to overcome the drag from

the mooring lin e through the water and along the sea bottom. These

drag forces are especially severe i f chain mooring lines are being

deployed. In Figure 6 is shown the normal deployment geometry (ABCD).

S u ffic ie n t length of mooring lin e must be deployed. As previously

!

stated, enough lin e must be deployed to insure th at the mooring lin e is tangent to the seafloor at or before the anchor(s), even a t

maximum o ffs e t.

The cable length required fo r mooring is determined by the calculated mooring lin e length, plus a length fo r settin g of the

anchor and a length fo r vessel positioning. The amount of drag

while setting the anchor can be as much as 300 f t in so ft mud bottom^3.

The solution of the mooring lin e configuration equations

requires knowledge of the amount o f mooring lin e remaining deployed a f te r r e e l-in during anchor setting operation, the pretensioning of the lin e and the fin a l positioning o f the vessel; thus i t is necessary

to keep track of the amount of lin e . I f the balance o f lin e remaining

deployed is not s u ffic ie n t to f u l f i l l the requirement o f some lin e a t the seafloor even fo r maximum o ffs e t, new deployment of the anchor must be attempted.

I f the anchor w ill not dig in , i t could be an indication of

fa u lty flu ke angle s e ttin g . However, additional information could

be necessary in terms o f the sp ecific anchor used and in terms of the nature o f the sea bottom in order to define the correct flu ke angle s e ttin g .

T-1850 21

MATHEMATICAL DEVELOPMENT

A segment o f a cable system is shown in Figure 7 with the s ta tic

forces to be considered: namely, lin e tensions, g ra v ity fo rc e, and

hydrodynamic forces induced by a constant or slowly varying current.

Under the assumed condition o f co -p la n a rity between the cable and

the c u rre n t, the problem becomes two dimensional, as shown in Figure 7.

Equilibrium conditions applied to the system leads to :

dT = [w • sin <f> - F (R ,$ )(1 + ^ ) 1 dS

d<j> = (1 /T ) f D(R,<f>) (1 + + w • cos <j> 1 dS

dX = cos ^ ^

/ T

dY = + sin ^ • dS

These equations are non-linear and a general e x p lic it^ s o lu tio n can not be derived; thus, in order to solve the equations various

s im p lific a tio n s are necessary. I f the mooring cable is considered heavy

and s t i f f the stretch can be neglected and the equations reduce to:

dT = (w • sin <f> -F(R,<f>)) dS (18) d<t> = 0/T)(D(R,<t>) + w • cos <fr) dS (19) (14) (15) (16) (17)

and

dX = cos <j> • dS (20)

dY = sin <f> • dS (21)

Solution fo r th is equation set by e x p lic it solution would y ie ld :

dS ” w • sin f - F ( M ) ^ ds - , T • d<t> D(RV<J>) + w * cos <|>' r ^ <J> ! T ( * ) = TQ • exp d * J (23) 0 where P(cj>) = w • sin <f> - F(R,$) Q(<t>) = D(R,<f)) + w • cos <j> Further dS = TM 'y d^ 4 T r /-e s = / _a o i t r • e x p [ / A _$ W « ] de (2 4 ) vo o and fo r X and Y n •<p t r /-y * = / Q l e l • exp [ I ® d* ] cos 0 ‘ de (2 5 ) *0 0 y m J * < m

•exp [ / e-

ft}d* ] s1n

’ de

(26)

Special form of F(R,<j>) and D(R, <f>) can be evaluated, however, the current forces are r e la t iv e ly very small in the case considered and thus can be neglected giving:

cos <j>( c o F T cos 0,

T - To • » r r (27)

T-1850 23 X = T 0 w 1 n (—X O S (j> j — + s i n $ \CO S <f> ' 1n( cos<}> cos <P o o ) cos 4>n , , Y = j ---° '--- — ) + y o w vcos (p cos 0 ' o = • cos^ <f> + (S • w + T • sin * ) ' I o o o o 1 / 2

For the p a rtic u la r case of* XQ = YQ = = 0 the equations fo r a

catenary w ill be obtained, since:

exp f X • w l

I

.

giving the equations as follows:

C _ TD c4nh X • w 5 = — • sinh —t ---w T, D Y = . (cosh — - 1) w I D T = [ T2 + (S • w)2 ] V 2

In the development of the caternary curve and it s solutions the follow ing w ill be defined (see Figure 8 ):

(29) (30) (31) (32) (33) (34)

Ya = Y + C where C = — (35) Ya = - j j • cosh (36) w Td and fu rth er Ya = C - J — _{cos <p} (37)

J

In Appendix A the catenary equations are used in the solution of the catenary-shaped mooring lin e to find lin e tension, catenary length, anchor p u ll, and cable angle with horizontal at the vessel; subsequently the pretensioning and reel-up are determined.

In Appendix B the basic mathematical development is used to w rite the equation set fo r the solution of combination wirerope- chain mooring lin e configuration.

In the following mathematical development the stretch o f the

cable is considered. The current forces are neglected as in the

above development since they are r e la tiv e ly small sp ecially in the

case of wirerope. Based on these assumptions, equations 14 to 17

are reduced to:

T-1850 25

d<f> = 1 • w • cos <f> • dS (42)

dX = (1 + • cos <t> • dS (43)

dY * (1 + ^7^*) * sin <f> • dS (44)

The above equation set must be solved by numerical methods

and computer in teg ration . The equations would be:

These equations are lis te d fo r completeness. Cable configurations

are calculated fo r a wirerope and in Table 3 comparisons are made between the cable configurations based on the catenary development

and the above development which includes stretch. As w ill be seen

from the data the e ffe c t of the stretch is s ig n ific a n t fo r the larg e r tensions (la rg e r o ffs e ts ).

(45)

V i = Tj + w • s1n V i • dS (46)

(47)

(48)

and the boundry conditions ( j = 0):

For subsequent calcu latio ns, used in a ll cases, as the w ritin g computer program fo r the solution the scope o f th is thesis.

however, the catenary solution is and implementing of a comprehensive including cable stretch is outside

T-1850 27

EXAMPLE CALCULATION

In th is paragraph is presented an example to show the method

o f c a lcu latio n . Of special in te re s t is :

Line pretensioning, High lin e tension,

! Total cable length and maximum re e l-u p ,

Restoring force.

The calculations w ill be made fo r two water depths - 1000 f t and

1500 f t . For each water depth the calculations w ill be made fo r two

mooring lin e configurations - namely, all-w ire ro p e and a ll-c h a in .

This number of calculations are performed to illu s t r a t e the differences

and ch aracteristics of the two systems. Only the calculation fo r the

a ll-w ire ro p e configuration a t 1500 f t of water depth is carried through in the following example; however, a ll calculated values are shown in graphs.

The data used fo r the example calculation are given in Appendix C.

Pretensioning

The equations used fo r calcu latin g the pretensioning force at zero o ffs e t are those developed in the paragraph e n title d "Mathematical

Development". The solution procedure fo r the data set given is a t r i a l

and erro r procedure due to the complexity of the equations. The calcu­

The maximum working load fo r the mooring lin e cable or chain 1-3

should be one-third its break strength given in the data, Appendix C.

For the case of the wirerope th is would be 350,000 Ib f.

From the te x t the maximum working load in the mooring lin e should occur a t a maximum o ffs e t fo r continued d r illin g , which is given in

the data as 6% o ffs e t, and the data set a t th is o ffs e t then:

Line tension 350,000 lb f

Cable buoyant weight 17.0 l b f / f t

! Water depth 1,500 f t

Cable length a t the sea bottom 3,450 f t

The length of cable a t the sea bottom must i n i t i a l l y be estimated, and la te r calculations w ill show whether or not the amount o f cable is s u ffic ie n t to f u l f i l l the requirement o f some amount of cable at

the sea bottom even a t maximum o ffs e t. For each o f the cases calculated,

the to ta l lin e length is selected so that between 50 and 100 f t remains on the sea bottom a t maximum o ffs e t.

Solving the equation set fo r the catenary with the data given - as described in Appendix A - y ie ld in g :

Srem = 11*165 f t

and the distance from the entry in the seafloor to the anchor (zero o ffs e t):

L = 10,879 f t

At zero o ffs e t the solution to the catenary can be calculated by t r i a l and e rro r, yie ldin g:

T-1850 29

C = 8,727 f t

and other data are a t zero o ffs e t:

Ya = 10,227 f t X = 5,046 f t Scat = 5,332 f t Sbot = 5>833 f t Td « 148,359 lb f = 31.4 deg

F in a lly the pretensioning a t zero o ffs e t is

T a « 173,859 lb f pre

High Line Tension

The high lin e is defined as the lin e in the mooring system with

the most tension fo r any given amount of o ffs e t. Any one lin e w ill

have the highest tension i f the displacement o f the vessel is in the plane of the mooring lin e away from the anchor.

The calculation procedure used to obtain the high lin e tension

fo r various displacements is outlined in Appendix A. For th is example

the known data fo r each vessel o ffs e t would be; w l b f / f t , Y f t , Srem f t and L + d f t where d is the additional length due to o ffs e t. The solution is found by t r i a l and erro r and explained in Appendix A.

For th is example, values o f lin e tension, anchor p u ll, and

data are lis te d in Table 4 and graphs are shown in Figure 9. In Table 5 the data fo r the other three cases are lis te d , and the data fo r these cases are shown in Figure 10.

Total Cable Length and Maximum Reel-Up

The to ta l amount of cable to be deployed is calculated from the deployment geometry (Figure 6) and the catenary data fo r zero vessel o ffs e t, as follows:

S = Y + X + + Anchor setting + Vessel positioning

= 1,500 f t + 5 ,0 4 6 f t + 5,833 f t + 300 f t + 200 f t

S = 12,880 f t

The maximum cable reel-up is calculated from equation (50):

Cable reel-up = Y + X - S + Anchor setting +

Vessel positioning

= 1,500 f t + 5,046 f t - 5,332 f t + 300 f t + 200 f t

Cable reel-up = 1,714 f t

As stated previously i t is necessary to observe and keep track

of the amount of cable reel-u p. Should the reel-up fo r any lin e

exceed the above amount a new deployment o f th at lin e should be attempted.

T-1850 31

Restoring Force

The restoring force is defined as the resulting force acting on the vessel from a ll mooring lines fo r a given o ffs e t in a given d ire c tio n .

Vessel o ffs e t in one d irection w ill cause stretching and increased tension in the lines away from the directio n o f the o ffs e t and slacking and decreased tension in the lin es in the

d irec tio n of the o ffs e t; th is is shown in Figure 11. The method of

calcu lation fo r each lin e is the same as used fo r the high lin e tension calcu latio n above, except in th is case L + a is calculated from the mooring pattern geometry, d irectio n of movement and amount

o f o ffs e t: (see Figure 11)

1 / 2

9 9

(L • cos a + Offs • D) + (L • sin a ) '

where L = the distance between the entry in the seafloor

and the anchor, f t

a = the angle between the o ffs e t plane and the mooring lin e , deg

Offs = vessel o ffs e t D = water depth, f t

a = additional length due to o ffs e t, f t

For movements away from the cable, a^w ill be p o s itiv e , and fo r movements in the d irection of the cable, a_ is negative.

From the above length and the data at zero o ffs e t the lin e tension and anchor pull fo r each lin e can be calculated and the

restoring force then is the summation of the lin e tension forces as projected on the horizontal plane (same as anchor p u ll) and subsequently projected in the plane of the movement (o ffs e t):

n

Restoring force = Y. • cos a .

j= l 3 3

The leeward lin es are working against the restoring force. Under

very severe weather conditions i t can be necessary therefore to slack o f f the leeward lines to increase the restoring force and decrease the o ffs e t.

For the example ca lc u la tio n , the restoring force is calculated

fo r vessel offsets in the beam d irectio n (see Figure 11). Also

calculations are made o f the restoring force fo r the two leeward

lin es completely slacked. The calculations were performed on

computer and the results are lis te d in Table 6 and graphs shown in

Figure 12. In Table 7, the data fo r the other three cases are lis te d

and the data fo r these cases are graphed in Figure 13.

Example Summary

The example mooring system has been designed and the high lin e tension and restoring force determined fo r vessel offsets in the beam

d ire c tio n . The data are lis te d respectively in Table 4 and Table 6

and graphed in respectively Figure 9 and Figure 12.

The maximum tension in the mooring lin e should be no more than

one-half the rated break strength. Based on th is lim ita tio n the

T-1850 33

depth as determined from Figure 9. This is provided th at the o ffs e t

takes place in the plane of one of the mooring lin es away from the anchor,

Somewhat greater amount of o ffs e t can be tolerated in the beam or bow d irec tio n fo r the mooring pattern in the example because a

mooring lin e is not in the plane o f these movements. For o ffs e t

determination, however, the maximum allowed o ffs e t fo r the high lin e should be used fo r offsets in a ll directions.

For the example environmental forces given in the data in Appendix C, the load on the vessel can be determined in the beam

direc tio n from Figure 2. Assuming th a t a ll the forces are acting in

the same d ire c tio n , the load is :

Wind force 186,000 lb f

Wave d r i f t force 58,000 lb f

Current force 199,000 lb f

Total force 443,000 lb f

From Figure 12 the o ffs e t due to th is load (which is a severe environmental load) can be determined as 90 f t or 6.0% o f water

depth. This o ffs e t is well below the maximum allowed, and the

mooring system under th is s ta tic load would be able to absorb s ig n ific a n t o s c illa tio n s due to dynamic forces around the s ta tic o ffs e t and would fu rth e r be able to sustain additional environmental conditions.

I f any condition should cause nearly maximum vessel o ffs e t as lis te d above, the slacking of two leeward lines w ill add s ig n ific a n t

restoring force fo r two leeward lines completely slacked at 8 1/2%

o ffs e t is 855,000 lb f (from Figure 12). Thus the slacking of leeward

lines allows fo r additional safety margin.

Other angles o f attack of the environmental forces and other mooring system designs should be studied in order to determine

the optimum system to match the vessel's environmental c h a ra c te ris tic s ; however, the procedure is sim ila r to the one outlined in th is example.

T-1850 35

DISCUSSION OF DATA

Calculation of high lin e tension, high lin e anchor pull and restoring force have been performed fo r all-w ire ro p e and a ll-c h a in mooring lin e configuration fo r water depth of 1000 f t and 1500 f t fo r comparative purposes and the calculated values are lis te d in Table 5 and Table 7 and they are graphed in Figure 11 and Figure 13.

As is seen from these data, the two d iffe re n t mooring lin e con­

fig u ra tio n s re s u lt in s ig n ific a n tly d iffe re n t mooring systems. The

a ll-c h a in system requires large pretensioning in order to meet the design c r it e r ia of a lin e tension of one-third the break strength a t maximum o ffs e t fo r continued d r illin g and yet th is system is rath er loose as seen from the slope in Figure 10 and Figure 13.

The all-w ire ro p e system requires low pretensioning but w ill give rapid increase in restoring fo rce, as seen from the steeper

slope of these graphs. This fa c t makes the wirerope system a much

more desirable system fo r deep water applicatio n; however, fo r low water application th is system is much too s t i f f as previously stated.

For ultra-deep water a combination wirerope-chain system should be selected.

APPENDIX A

Procedure fo r Pretensioning Calculation

The mooring lin e geometry fo r pretensioning is shown in Figure 14 where the lin e pretensioning would take place in position ABC.

The lin e tension, however, would be known in position A'B'C indicating the o ffs e t at which the maximum working load in the mooring lin e

should take place.

From the mathematical development the following equations are obtained: Ya = C • cosh £- (A—1) (A-2) T = w • C • cosh jr = Ya • w (A—3) Y = Ya - C (A-4) (A-5) (A—6)

and from the geometry A'B'C in Figure 14:

bot (A-7)

T-1850 37

For the problem at hand the following data a t A'B'C would be known:

T = tension which is the maximum working cable load, lb f

w = cable weight (buoyant), I b f / f t

Y = water depth, f t

Sfoot = l ine laying on seafloor (s e le c te d ), f t

From equation (A-3)

Ya = T/w Now: C = Y a - Y = — - Y w

x

_I

and from equation (A-7) and (A-8)

L - X + Sbot - d

S = S * + S . .

rem cat bot

The pretensioning of the mooring lin e cables would take place

in the geometry ABC, in dicating zero o ffs e t. For th is geometry the

following data would be known:

Y = water depth, f t

^rem = am0Lmt ° f cable out a fte r pretensioning, f t

L = distance between entry in seafloor and anchor, f t

The solution to th is problem is found by a t r i a l and error c a l­ culation procedure using equations (A -l) thru (A -8 ), except th at equation (A-7) would be:

L = X + Sbot (A-10)

Guessing a value fo r C the following can be calculated:

Ya = Y + C

X = C • cosh'1

Scat = C ’ s1nh ₎ §

and from equation (A-10)

and

Sbot - L ' X

S"

= s . + s. .

This value of S" must equal the S _ given in the data. New value

rem rem

of C must be guessed u n til the eq u ality S” m _{3 rem rem} holds.

When the correct value o f C is found, the catenary at geometry ABC is known and a ll necessary data can be calculated.

In specific the pretensioning o f the mooring lin e at zero o ffs e t is :

T_„a = Ya • w pre

T-1850

ARTHUR EAKES LIBRARY

COLORADO SCHOOL of MINES GOLDEN. COLORADO 80401

39

APPENDIX B

Calculation Procedure fo r Combination Wirerope-Chain Mooring Line

The equation set fo r mooring lin e calculatio n o f a combination

lin e of wirerope-chain is lis te d below. The mooring lin e geometry

is shown in Figure 15.

The equations are those developed in the paragraph e n title d "Mathematical Development", and fo r the wirerope portion o f the mooring lin e the following equations can be w ritte n :

2 1 1 / 2 ( L ‘ v* + T • sin <f> ) wr wr o o ( B - l) cos <t> — (tan <{> - tan <J)Q) S = T • — wr o w._wr (B-2) + X o (B-3) (B-4)

and fo r the chain part:

Sch=

'sinh t

{B

'6)

and from the geometry in Figure 15 the following equations are derived: X = X 1 + X2 ( B - 1 0 ) Y = y1 + y2 ( B - 1 1 ) Srem = Sbot + Sch + Swr L = Sbot + X1 + X2 (B' 13) Schtot “ Sch + Sbot «B- 14)

For problem solving - finding pretensioning and lin e configuration s u ffic ie n t data would not be known fo r e ith e r the wirerope portion or

the chain portion to solve any one separately, since X-j, X^, , and

remain unknown fo r any one problem.

The solution is done grap hically as a t r i a l and error procedure on the e n tire system of equations, s tartin g with the chain portion

(catenary) fo r estimated values in order to fin d X^', X£, Y p and Y£. The sought solution would be one where:

T-1850 41 Y" = YIj + Y £ = Y L" - Xf + + SbQt = L (or L + d) and S n = S rem rem

APPENDIX C

Data fo r Example Calculation

Environmental data - maximum expected forces:

Wind v e lo c ity (V^) 59 knots

Wave height (s ig n ific a n t) 35 f t

Current v e lo c ity 2.0 knots

Mooring pattern: 8 lines

30° - 60° configuration as measured from the bow (see Figure 11)

Water depth: 1500 f t and 1000 f t Mooring lin e : Size Dry weight, I b f / f t Buoyant weight, I b f / f t Break strength, lb f WIREROPE 3 1/4" 19.5 17.0 1,050,000 CHAIN 3" 89.3 78.0 ,045,000 O ffset: Maximum, d r illin g Maximum 6% of water depth 10% o f water depth

T-T850 43

APPENDIX D Nomenclature

ENGLISH SYMBOLS UNIT

in 2 f t 2 f t 2 f t f t f t f t f t f t A - Area Ac Projected area Af - Projected area

Ah - Vessel heave amplitude

As - Vessel surge amplitude.

a - Wave amplitude

a - Length due to vessel o ffs e t

B - Beam o f vessel C - C oefficien t C - Calculation length ch - Height c ie ffic ie n t Cs - Shape c o e ffic ie n t c - Damping D - D raft of vessel

d - Length due to vessel o ffs e t

E - Modulus o f e la s t ic it y

F - Force

g - Acceleration o f g rav ity

H - Wave height s ig n ific a n t

f t f t l b / i n 2 lb f f t /s e c 2 f t

ISH SYMBOLS UNIT

h Water depth f t

k Wave number = —

L Length f t

L Length of vessel f t

m V irtu a l mass Ibm

Offs - O ffset (fra c tio n )

R Radius f t

S Length (cable) f t

T Tension force lb f

I Wave period sec

t d - Anchor pull lb f

V Velocity ft/s e c

Va ’ V e lo c ity , average knots

Vc - V e lo c ity , current knots

V g * Velocity knots

vk - Velocity knots

W Weight per foot I b f / f t

X Length in x -d irectio n f t

X Coordinant

Y Water depth f t

Y Length in y -d ire c tio n f t

T - 1850 45

GREEK SYMBOLS UNIT

a Angle deg

P Density g/cm^

0 Angle deg

9s - Phase angle deg

eh

-Phase angle deg

Angle deg

SUBSCRIPT

beam •- Vessel beam

bot - Sea bottom

bow - Vessel bow

cat - Catenary ch - Chain chtot - Chain to ta l mor - Mooring 0 - I n i t i a l condition pre - Pretensioning rem - Remaining ris - Ri ser T - Total ves - Vessel wr - Wirerope