INTRODUCTION

The mooring system is one of the main limitations for theoffshore oil exploration in ultra deep waters, being one ofthe most relevant themes on the studies conducted for thedevelopment of any floating unit intended to operate onsuch water depths.

From the high level of importance of this subject, thepossible calculation methods, the types of analysis, thedesign of mooring systems and the tools available on themarket for their calculations became a joint of factorsthat motivated this paper, what is in fact a synthesis ofthe lessons learned on the last 10 (ten) years, time thatPROJEMAR is involved on the development of deepwater mooring systems.

THE NATURE OF THE PROBLEM

As the floating offshore systems extend to even deeperwaters, the various effects involved on the dynamics ofthe system, especially the effects of mooring and risers,became more significantly when predicting the responseof the floater.

A moored structure exposed to the various environmentalagents can be characterized as a dynamical system withlow and high frequency excitation. The excitation forcesare a result of the wind, waves and current acting on the

A Practical Overview of Various Calculation Methods

Ricardo Barreto PorteIla, PROJEMAR, Brazil (ricardo.portella@projemar.com)Marcio de Abreu Grove, PROJEMAR, Brazil (marcio.grove@projemar.com)

Inga Lehmann, Delft University, Netherlands (I.Lehmann@student.TUDelft.n1)

SUMMARY

This paper presents an overview of different calculation methods available, from simple static and frequency domainanalysis to complex coupled full time domain simulations. This overview is based on the practical PROJEIVIARexperience developed during the past 10 years. During this time, PROJEMAR was involved in different mooringsystems designs and analysis for different kinds of floating structures, from semi-submersible drilling units up to FPSOswith 2,000,000 barrels storage capacity.The background of this paper is based on the fact that results derived by using new tools with more advanced calculationmethods are compared with the methodology previously used as well as model tests data. This comparison makes itpossible to observe the differences, advantages and disadvantages between the methods. Using this information, for eachcase the best methodology can be found.Finally, the paper gives an overall picture about the calculation methods and their optimal utilization depending on thecomplexity of the system intended to be analyzed.

Deepwater Mooring Systems Design and Analysis

system. The reaction forces are a combination of addedmass forces, wave drift damping forces, viscous forcesand combined restoring and damping forces to theinteraction with the mooring lines and risers. Figure 1bellow exemplify the complexity of a deep watermooring system.

Figure 1 Deep Water Mooring System

Delft University of TechnologyShip Hydromechanics Laboratory

Library1

Mekelweg 2, 2628 CD DelftThe Netherlands

Phone: +31 15 2786873 - Fax: +31 15 2781836-

t

In accordance with the API RP 2SK [1]: -Permanentmooring systems should be designed for two primaryconditions: system overloading and fatigue. Therefore,analysis for extreme response and fatigue damage shouldbe performed. For mobile moorings, only analysis forextreme response is required."

On deepwater mooring systems, the station keepinganalysis must consider the mooring, risers and vesselresponse and the significance of coupled effects thatincrease with the water depth, as shown on table I bellowobtained on reference [2].

* Limited information available

Table I: Significance of Low Frequency (LF) couplingeffects for moored floaters

The key issue is to obtain a realistic prediction of thefloater response under the simultaneous action of thevarious environmental excitations.

One of the most useful tools to obtain the expectedmotion response of complex floating systems are themodel tests, but due to the limitations of model basins forultra deep water floaters and due to the necessity ofaccurate estimations during the initial phase of thedesign, mathematical modeling and numericalsimulations shall be extensively used to predict theexpected global motion behavior of the floater.

At this point is of fundamental importance to understandeach kind of analysis possible to be performed, itsadvantages and limitations in terms of quality of theresult and time required. And, for the designer point ofview, what are the tools available on the market able tosolver his specific problem. The essential requirementsfor a mooring analysis are

To determine the equilibrium position of thebodies of the system, under the influence ofexternal forces:

Calculate for all bodies positions, the resultantmooring line tensions and thus all the parameterswhich characterize the behavior of the individuallines.

STATUTORY REQUIREMENTS

Classification Societies and other regulatory agenciesresponsible for the standardization of the design andconstruction of offshore floating units usually dedicate apart of these rules for the design and analysis of thestationkeeping systems.

On the guidance codes is possible to find that requireddesign safety factor depend on the design conditions ofthe system, as well as the level of analysis performed oreven on the type of software used for the calculations.

References [1], [3] and [4] propose minimum safetyfactors with respect to the mooring lines tensions, thatare dependent of the analysis method used. References[I] and [3] depends on the calculation method only.Reference [4] is related to a software, based on acalculation methodology wich can be considered anintermediate stage between a quasi-static analysis and afull dynamic analysis, denominated quasi-dynamicanalysis.

Table 2 present the required mooring lines tensions limitand equivalent safety factors obtained on references [1],[3] and [4] for various conditions and analysis methods.

Table 2: Mooring lines tension limits and equivalentsafety factors required for various conditions andanalysis methods

TYPES OF MOORING ANALYSIS

The first kind of mooring evaluation and the simplest isbased on the equilibrium between the meanenvironmental loads of wind, waves and current on thevessel and the static horizontal restoring characteristicsof mooring system.

When the low frequency variations are taken intoaccount due to the low frequency drift forces in wavegroups, then time domain simulations are required.Normally this type of analysis is based on static restoring

FloatingSystem

Water DepthShallow Moderate Deep Ultra Deep

FPSO Small Moderate High HighTLP ---- Small Moderate Moderate*Spar ---- ---- Moderate Moderate-

High*Semi ----

Cond. Anal.Method

API ABS BVTens.Limit

S.F. Tens.Limit

S.F. Tens.Limit

S.F.

Intact Quasi-static 50% 2.00 50% 2.00 -

Intact Dynamic 60% 1.67 60% 1.67 57% 1_75

Dam. Quasi-static

70% 1.43 70% 1.43 -

Dam.Dynamic 80% 1.25 80% 1.25 80% 1.25

Trans. Quasi-static

85% 1.18 85% 1.18 - -

Trans. Dynamic 95% 1.05 95% 1.05 n.a. n.a.

:

2

-------- ----

'

- -

- -

-

characteristics of the mooring system, considered as aspring.

When the dynamics of the mooring system have a largeeffect on the low frequency damping of the system andthe wave frequency mooring line dynamics increase themooring line loads significantly, calculations needs to becarried out to determine this effects.

The mooring lines dynamic can be estimated using themotions from the quasi-static analysis as input on risersanalysis codes, however, this is an uncoupled approachdue to the fact that the dynamics of the mooring systemare calculated separately from the dynamics of thefloater.

For more sophisticated coupled mooring analysis, thecombined wave and low frequency motions of the floaterneed to be coupled in time domain to the dynamics of themooring lines.

As seen, statutory codes normally define 3 (three) typesof mooring analysis depending on the calculationmethodology, however sub-variations are possible inaccordance to the complexity of the calculations and thenumber of physical effects included.

4.1 STATIC ANALYSIS

The simplest mooring analysis can be done using theequilibrium between the mean environmental loads andthe restoring force of the lines, where the static loadrestoring curves are calculated by means of a catenaryformulation.

The results are relationships between the excursion fromthe equilibrium position of the floater and the restoringforce of the mooring system in a simple equilibrium offorces. The heel and trim angles changes due to themooring system are also determined as part of a detailedstatic analysis.

Parameters which can be varied in the static loadcalculation are

the number of mooring lines;

the layout of the individual mooring lines (chain/wire / synthetic line );

the mooring lines pretension;

the length and size of the mooring lines;

Normally static analysis is used to calculate the tensionin mooring lines of ships in harbors, when the action ofthe waves are not critical.

In some cases it is admitted to use static analysis duringthe initial phases of the design of offshore mooringsystems, but due to the easy obtaining and use of moresophisticated calculation software and due to the poorresults obtained by static analysis this option is becameto be rare on the offshore industry. When static analysisis used, a great safety factor is required to accommodatethe effects of the system dynamics, but this is not beingconsidered a good engineering procedure anymore.

4.2 QUASI-STATIC ANALYSIS

On the quasi-static approach the dynamic wave loads areincluded on the calculations by offseting the floater by aquantity derived from statistical analysis of the waveinduced motions.

The principle of the quasi-static analysis is that a meanposition of the floater is calculated based on a staticequilibrium position under the action of the meanexternal loads. About this mean position, the wavefrequency motion is determined based on a spectralanalysis performed with the transfer functions of thehorizontal movement of the floater and the sea statespectrum. The responses of the horizontal movementsand the corresponding tensions on the mooring lines arethen determined based on the final position of the floater,what takes into account the mean position and the wavefrequency motion.

On this type of analysis, the vertical fairlead motions andthe dynamic effects associated with mass, damping andfluid acceleration are completely neglected.

Once more, due to the easy obtaining and use of moresophisticated calculation software able to compute bothlow frequency and high frequency wave loads on timedomain simulation, this type of calculation is now rare onthe offshore industry, but it is still a valid tool if it isdesirable to obtain only the effect of the high frequencywave loads.

For a mooring system where the horizontal movementnatural frequency is considerable distant of the wavefrequency, the high and low frequency phenomena canbe considered independent and the high frequencymovements of the floater can be estimated based on aquasi-static approach.

4.3 DYNAMIC ANALYSIS

Dynamic Analysis accounts for the time varying effectsdue to mass, damping and fluid acceleration and thefloater movements are calculated for the 6 (six) degreesof freedom.

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Different dynamic analysis techniques are available onthe literature and on the analysis software market.Basically the distinguish feature among the variousdynamic analysis techniques is how the non-linearitiesand the interactions between the bodies on the system aretreated on the calculation.

Depending on the calculation methodology used to solvethe movement equation and the level of approximationsused, different nomenclatures are used to nominatedifferent dynamic analysis techniques. In order to have aparallel with the softwares we are used to deal, we willassume the same nomenclature adopted for the softwaremakers to analyze 3 (three) different techniques toperform dynamic mooring analysis.

4.3.1 Quasi-Dynamic Analysis

The basic principle of the quasi-dynamic analysis [6] ofmooring systems is time domain simulation of themooring system response in specified environmentalconditions. The mooring lines are considered as non-linear massless springs, which can freely follow theirfairleads without inducing to the vessel any other loadthan their static spring reactions in the instantaneousvertical plane of the mooring line.

The calculation procedure consists of the determinationof the low frequency response of the moored vesselunder the effect of waves, wind and current, followed bysuperimposition of the wave frequency motions. It isassumed that low and wave frequency components donot significantly interfere with each other because of thevery different time scales. As a consequence, they areassessed separately in the framework of thisapproximation and added together at the end of each timestep of the simulation.

The mooring line tension is derived at the end of eachtime step from the static catenary response obtained forthe instantaneous position of the fairlead.

The low frequency component is obtained by solving ateach time step a vectorial differential equation of thefollowing form:

}= E{F(t)}

where:

{X} is the three-component vector characterising thehorizontal postion of the vessels centre ofgravity

[m] is the horizontal mass matrix of the vesselcalculated at its centre of gravity

{F(0} is the three-component vector of thosehorizontal loads, applied to the center of gravityat instant t

For the wave frequency response is assumed that thewave frequency motions of the vessel are notsignificantly disturbed by the variation of the mooringstiffness with the low frequency offset. An averagemooring stiffness can therefore be used for pre-determining the Response Amplitude Operators (RAO's)of the vessel.

At each time step, the six wave frequency motions of thevessels center of gravity are added to its low frequencyposition. To do so, the amplitude of each component ofthe wave signal is multiplied by the RAO's of the centerof gravity of the vessel and the summation is carried outwith due account for time and space phases.

Equations of rigid body motions lead to the instantaneousposition of each fairlead. The tension in the mooring lineis then computed by interpolation of the static catenaryresponses precalculated for different vertical positions ofthe fairlead.

The previous assumptions make that the load induced byany mooring line to the moored vessel in its lowfrequency motion depends only on the anchor-to-fairleaddistance, the so-called static line response. The mooringline response can therefore be pre-computed to build upthe mooring line characteristics. The method also allowsto include material elastic behavior, suspended, laiddown and buoyant segments, and buoys and sinkers intothe calculations.

For external loads contributing to the low frequencyresponse on the system holds that their are computedreferring to an axis system in the center of gravity of thevessel. These loads include hydrodynamic, wave drift,current, wind, mooring and damping loads. All equationsand moments are projected to this point of the systembecause its position within the vessel does not depend onthe loading condition. The same axis system can thus beused to record permanent data (fairlead coordinates,lightship center of gravity, etc.) as well as other datawhich vary with the vessel or site conditions (RAO, loadtransfer functions, etc.).

This methodology has certain limits, too. The mooringsystem is assumed not to be subjected to resonance at thewave frequency. In addition, out-of horizontal-plane lowfrequency motions are supposed to be negligible. Hence,this methodology may not be appropriate to Tension LegPlatforms or certain types of semi-submersiblesoperating at very large draughts.

It is supposed that horizontal and wave frequencyphenomena do not interfere. Compliance with this

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assumption is reasonably satisfied if the natural period ofthe mooring system in surge, sway and yaw is greaterthan five times the zero-up crossing period of the wave.

4.3.2 Time Domain Analysis

An analysis called as a time domain simulation [7]allows non-linearities for all terms in the equation ofmotion. Both high frequency and low frequency waveexcitations are applied simultaneously in the timedomain, what allows non-linear coupling effects in theloading and in the equation of motions.

Time series of wave excitation forces are calculatedassuming that the floater has the same position during thesimulation, but heading changes are accounted for bygeneration of time series heading and interpolations onthe time series in the time domain.

In a simplified form, the equation of motions for amooring system can be written as:

Mi + Ci + D1X + D, f() + Kx q(t, x, 1)

where:

M = m + A(co)

A(co) = A a(co)

= A(w = oo)

C (co) = C +c(w)

C,, = C( = _= 0

M - frequncy-dependent mass matrix

m - body mass matrix

A frequency-dependent added mass

C=c frequncy-dependent potential damping

matrix

DI linear damping matrix

quadratic damping matrix

f - vector function

K position-dependent hydrostatic stiffness

matrix

q exciting force vector

The exciting forces on the right-hand side of the equationof motions are given by:

q(t , x , = qw, + q + + q1 + q

where:

qwi wind drag force

qwA - 1s order wave excitation force

qwA - 2nd order wave exitation force

qcu - current drag force

(lex, - external specified forces

The equation of motions can be written as:

(m+ A001+ D1 +D2f(Z)+Kx+ f h(t r)±(r)dr = g(t,x,i)

On this form, the convolution integral are forces due tothe frequency-dependent added-mass and damping and his called the retardation function given by:

h(r) = 2 'sc coswr dr = - -2 °J.co asin co r dz- for r >0

This methodologymethodology is attractive when the couplingbetween the bodies is small. The motions of each bodyare calculated separately and the coupling betweenbodies treated as excitation forces. The advantage of thismethod is that the computation time increases onlylinearly with the number of bodies in the simulation.

4.3.3 Coupled Analysis

In coupled numerical mooring analysis [8] there is adirect coupling between all dynamic loads inmooring/riser systems and the wave frequency and lowfrequency vessel motions. In this way the mooringsystem is not only applying a restoring force, but it canalso apply damping and inertia type of loads on vessel.

In this calculation method is possible to take into accountaspects that can be significant for deep water, as a resultof the long mooring lines lengths and relatively largeweights. These are:

Mooring line/riser dynamics due to vessel velocitiesand accelerations, on top of the quasi-static loading asa result of the displacements. These are one of themost important aspects to include in a coupledmooring analysis, since the low frequency mooringline loads are caused by the restoring force of theeach mooring line.

- The analysis using a coupled model offers thepossibility of to investigate the low frequencycontribution, besides the wave frequency variations,

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=

-

-

-

co)

that came from the dynamic mooring line behavior incombination with vessel motions.

Good correspondence between the calculated andmeasured line loads in a model test.

Direct current (drag) and loads on risers/mooringlines, what can have a considerable effect, especiallyon the mean vessel displacements, considering themooring lines as members of the system with lift anddrag properties, rather than as a catenary system only.

Mooring/riser system damping on the low frequencymotions.

A correct damping level is a critical issue for obtaining arealistic prediction of the vessel and mooring responseand all the damping terms should be included on thecalculation in order to obtain realistic results. Mooringand risers damping derivation in an interactive processwhere the damping depends on the amplitude, whichitself depends on the damping.

As shown on table 1, fully coupled analysis is essentiallya requirement for deepwater systems. Table 3 bellow,obtained on reference [2] presents the various dampingterms on a mooring system and their physical source,showing the calculation methodology where it can beincluded.

Table 3: The various damping terms and their sourcespresent in a station keeping analysis

5. RESULTS OBTAINED FOR DIFFERENTCALCULATION METHODS

On reference [9] it is presented some comparisonsbetween mooring analysis calculation and model tests.Both the analysis and the model tests were conducted byPROJEMAR for a complex semi-submersible floatingproduction unit moored in 1080 m of water depth with16 mooring lines and 103 flexible risers [10].

Comparisons were done first between the model testsresults and the initial mooring calculations performedwithout calibration on the mooring analysis programs.After that the programs were calibrated using lineardamping and slight modifications on the added massmatrix in order to obtain closer results in terms of themost tensioned mooring lines and the platform offsets.

Figures 2 and 3 show the results obtained for thecomparisons made with the tests results and the programswithout calibration, where it is easy to see that there aresignificant differences between the analytical results andthe tests.

3500ze_ 3000

g 2500g 2000

1500/000

e. 5000u. 0

Ll 12 L3 L4 L5 L9 L13 L14 L15 LIB

linha no.

testa

ARANE

SIIVO

Figure 2 Mean tension on the top of the line

0

5000

4000

3000

2000

a 1000

0

L1 12 L3 L4 L5 L9 L13 L14 L15 L16

linha no.

Gates%

AR1ANE

OSIMD

Figure 3 Max tension on the top of the line

The results after the calibration show a very goodagreement for the tension of the most tensioned lines andfor the platform offsets, but we still found considerabledifferences for the slacked lines, indicating the -adjust"made was good to check the dimensioning of themooring system. but was not well done in terms ofreproduce the correct dynamics of the system.

6. MODERN DEVELOPMENT OF THENUMERICAL TANKS

As physical wave basins cannot be of infinite extent, as isthe ocean they are intended to model, all around theworld, several hydrodynamics teams are activelyinvolved in research to develop Numerical Wave Tanks.The main feature of these tanks is to calculate fullNavier-Stokes equations taking account the viscosity andfree surface conditions.

With the idea to extend the limits of the ocean basinsanalysis a numerical simulator laboratory calledNumerical Offshore Tank is being developed on theUniversity of Sao Paulo in Brazil [11].

The intent of this development is to take into accountalmost all physical phenomena acting on the floatingbodies and mooring and risers lines. Since full non-linear solution is not available, several numerical,

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Source,

PhysicsQuasi-static

derivation

Fullycoupled

derivationMooring line/riser Drag No YesCurrent Drag Yes YesWind Drag Yes YesWave drift Potential Yes YesRadiation Potential Yes Yes'

empirical and analytical models are being considered andintegrated to the numerical simulator.

On the Numerical Offshore Tank the time domainpotential problem is solved to wave forces acting on thebodies and empirical models are used to simulate currentand wind forces. Mooring and riser lines are representedby finite element models where realistic hydrodynamicforce models are used.

To deal with a very high processing time for thesimulations due to floating bodies with several mooringlines and risers the numerical tank uses a special clusterwith 60 PC based computer running the code in parallel.

In order to work as a "ocean basin- a 3D visualisation ofthe simulation tests is presented in a virtual reality roomwith stereoscopic projection, given to designer theimpression of a physical model test.

MAIN CONCLUSIONS

There are several computer programs in the market ableto perform complex calculations to deal with mooringanalysis of complex offshore units. But, so as to developthe mooring design is fundamental for the designer tounderstand the physical of the problem he wants to solveand know the capabilities and limitations of the softwarehe intends to use.

Due to the actual easy obtaining and use of adequatesoftware able to perform dynamic analysis , simplifiedmooring analysis like static or quasi static calculationsshall not be used anymore for offshore applications.Time domain simulation are now almost a requirement.

Fully coupled analysis is essentially a requirement fordeep water systems. Dealing with deepwater mooringanalysis it is of fundamental importance to dealadequately with coupling effects and its non-linearities.That can be done by the use of model tests results,complex mooring analysis softwares, numerical wavetanks or, preferable, a suitable combinations of thesetools in order to obtain the most effective result.

REFERENCES

American Petroleum Institute (API):"Recommended Practice for Design and Analysis ofStationkeeping Systems for Floating StructuresAPI RP 2SK second edition, 1997Det Norske Veritas: -DNV Software News",January, 2003 pp 14-15American Bureau of Shipping: "Guide for Buildingand Classing Floating Production Installations", June2000

Bureau Veritas: -Guidance Note for the Quasi-Dynamic Analysis of Mooring Systems UsingARIANE Software", tentative issue, May 1998Buchner B., Cozijn J.L., and Boer G. de, "CoupledMooring Analysis", Maritime Research InstituteNetherlands MARIN Training Course, March2001Bureau Veritas: -ARIANE-3Dynamic TheoryManual General and Time Domain Simulation"Norwegian Marine Technology Research InstituteMARINTEK: "Simulation of Complex MarineOperations SIMO Theory Manual" , January2001Det Norske Veritas and MARINTEK: -Deep WaterCoupled Floater Motion Analysis DEEP C UserManual", March 2002Portella R. B: "Numerical Simulations and ModelTests for the Mooring Calculation of a FloatingSystem", Master of Science Degree Thesis, COPPE,Federal University of Rio de Janeiro, Brazil, March2001 (in portuguese).Portella R. B.: "Mooring System: From InitialDesign to Offshore Installation", OffshoreTechnology Conference paper no. OTC 12174, May2000Nishimoto K., Donato M., Masetti I., Jacob B. P.,Martins M., Menezes I., Hirata K.: -Development ofNumerical Offshore Tank for Ultra Deep Water OilProduction Systems", International Conference onOffshore Mechanics and Arctic Engineering paperno. OMAE2003-37381, June 2003

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8.