Slow motion dynamics of dicas mooring systems under steady current, wind, and steady drift excitation

SLOW MOTION DYNAMICS OF DICAS MOORING

SYSTEMS UNDER STEADY CURRENT, WIND, AND

STEADY DRIFT EXCITATION

Luis 0. Garza-Rios, Research Fellow, and Michael M. Bernitsas, Professor Department of Naval Architecture and Marine Engineering

University of Michigan, Ann Aibor, Michigan 48109-2145, U.S.A.

Kazuo Nishimoto, Professor

Department of Naval Architecture and Ocean Engineering University of São Paulo, São Paulo, Brazil

Abstract - The slow motion dynamical behavior of a Differentiated Compliance

Anchoring System (DICAS), subject to a range of directions of eAternal excitation of constant magnitude, located in the Marlin Field, Campos Basin, Brazil, is assessed based on nonlinear stability analysis and bifurcation theoi7. Excitation consists of steady current, wind, and second order mean wave drift forces. Catastrophe sets are constructed in a two-dimensional parametric design space, separating regions of qualitatively differentdynamics. Stability analysis deflnes the morphogeneses occurring across bifurcation boundaries to find stable and limit cycle response near the principal equilibrium position. The resulting design graphs allow the designer to select an appropriate orientation and other design variables for DICAS without resorting to trial and error, or extensive nonlinear time simulations. The position of the vessel at equilibrium

defines the mean horizontal displacement of the system with respect. to a prescribed initial orientation. This position depends on the magnitude and direction of the external excitation. Limited simulations or further nonlinear analysis enable the designer to investigate whether or nOt DICAS slow motions comply with the allowable limits of motions for safe operations. The effect of water depth variation on the stability of DICAS is considered, and it is shown that the dynamics of the system change considerably with relatively small variation in water depth. The DICAS mathematical model consists of the nonlinear, horizontal plane fifth-order, large drift, low-speed maneuvering equations. Mooring lines are modeled by catenaries with touchdown and

nonlinear drag. External excitation consists of time independent current, Steady wind, and second order mean wave drift forces.

INTRODUCTION

In offshore operations, several types of mooring and anchoring systems are being designed and deployed, depending on the type and projected time of operation, and the environmental conditions. The search for more efficient means of production and recovery, has led to innovative ideas in the area of mooring system design. One such concept is the Differentiated Compliance Anchoring System (DICAS), a type of Spread Mooring System (SMS) that can

be effectively used, if properly designed, in relatively moderate weather conditions, such as those encountered offshore Brazil. The first DICAS was installed in the Caravela field, Santos basin, Brazil, in 1993 at water depth of 195 meters, and is still in operation. DICAS is a type of spread mooring

system with different stiffnesses in the mooring lines at the bow and stern of the vessel. Such differences in stiffnesses, somewhat restrict the bow of the vessel from moving due to their higher pretensions while allowing the stern of the vessel to move relatively freely. This property allows the vessel to partially weathervane as the environmental conditions vaiy. Due to the characteristics of this system, the DICAS design must take into account the slowly changing directions of the environmental excitations as well as the best layout for the production risers.

The design of a DICAS, however, is tedious and time consuming due to the high number of parameters that must be considered, such as the number, length, orientation and pretension of the catenaries, fairlead position, etc., as well as the environmental conditions in which the system is to operate. A design methodology for SMS based on nonlinear dynamics and bifurcation theory1, can be used to analyze the dynamics of the system while eliminating intense computations in SMS design. A previous introductory study of the slow motions of DICAS for the Campos Basin2 based on these principles, shows the richness of the system nonlinear dynamics Under vaiying current

directions, fairlead positions of the mooring lines, and mooring line

pretensions. In this paper, the horizontal plane Slow motion dynamics of a DICAS under varying directions of the external excitations are analyzed using nonlinear dynamics and bifurcation theory. The effect of water depth on the dynamical behavior of the system is considered as well.

MATHEMATICAL MODEL

The slow motion dynamics of the DICAS in the horizontal plane (surge, sway and yaw) are formulated in terms of the vessel equations of motion, mooring line model, and external excitation. The mathematical model is based on the large drift angle, slow horizontal plane motion, fifthorder maneuvering equations3'4.

Equations of Motion

The horizontal plane geometry of a DICAS is shown in Figure 1, with two reference frames: (x,y) = inertial reference frame with its origin located at mooring terminal 1; (X,Y,Z) = body fixed reference frame with its origin located at the center of gravity of the vessel(CG). In addition, n is the number

V

of mooring lines;

ate the moO'rdinates of the ith mooring line

in the (x,y) frame; (x,y) arethe body-fixed coordin tes of the ithmooring

line fairleath is positive forward of the CG, and Yp s positive on the port

side of the vessel; 1'

is the horizontal distance of the uk mooring line between the attachment point on the vessel and the mooring point at the ocean

floor; and

r

is the drift angle.

The mooring lines are numbered

counterclockwise starting fiorn the line moed connected to terminal l

Figure 1. Geometiy of DICAS

The equations of motion in the horizontal plane are given by2:

(m+mx)â(m+my)rv=XH+X{7e l}+Fsurgei

(1)

(m+y)+(m+mx)ru=YH+{TyF2}+F3.way.

(2)

(172 +J72)i= NH +

{xP(7?ay - i%

_{Y(7rge F2)}+ N,}

(3)

where m is the mass of the vessel; 1 is the moment of inertia about the Z-axis; mx, my and '72 are the added masses and moment of inertia in surge, sway and yaw, respectively; u, v, and r are the relative vessel velocities with respect to

(1)

In addition, in EquationS (l)-(3), 7%e and

represent the tension components in the horizontal plane of the ith mooring line in the surge and sway directions;

F? and F? correspond to the mooring line nonlinear

damping components in surge and sway, respectively; and F, F, and

N are the external forces and moment acting on the vessel due to steady wind and second order wave drift forces.

The kinematics of the system axe governed by Equations (8)-( 10):

x=ucosWvsiflV+Ucosa,

(8)

$P=vcosW+usin!y+Usna,

(9)

(10)

where U Uj is the absolute value of the relative velocity of the vessel with respect to Water, and a is the current angle measured with respect to the (x,y) frame as defined in Figure 1.

Mooring Line Model

The mooring lines of the DICAS are modeled quasistatically by open-water catenary chains, which include touchdown and nonlineardrag8. Mooring lines are classified as Single (S) or Double (D). Single mooring lines have an Average Breaking Strength (ABS) of 5159 KN; Double lines have an ABS Qf 9715 KN9.

The total tension of th catenary, T,

j8

water in surge, sway, and yaw, respectively; XH, _H and NH are the velocity dependent horizontal plane hull hydrodynamic forces and moment expressed in terms of the large drift angle, slow motion derivatives as56'7

XH = Xu + + +XvrVT, (4)

= YV +

Yv3 +

+ }r + Yju,irI +

(5)

NH

+ NuvI,1 + N,,v2r.

(6)

The first three terms in (4) represent the third order approximation of the vessel resistance, R7:

T=j7,2+7,2 =Tocosh[,J.

where 7, is the horizontal tension component in the catenazy, T, is the vertical tension compOnent, P is the weight of the catenary per unit length (1510 kg/s2

for Single lines; 3094 kg/s2 for Double lines)9, and £ is the horizontally

projected length of the suspended portion of the catenary.

Mooring Line Drag Forces: The drag forces in the surge and sway

directions F and F,d on each of the mooring lines are of the form8:

F=FAcos$FLsinfl,

(12)

P=FAsinfl+FLcos$,

(13)

where

_fi

is the angle from the X-axis to the direction of the mooring line measured counterclockwise, and FA and FL are the mooring Line drag forces in the directions parallel and perpendicular to the catenary motion, respectively.

External Excitation

External excitation corresponds to time independtnt current, wind, and mean second-order wave drift forces, with direction of excitation with respect to the (xy) reference frame as shown inFigure 1. The current force is formulated in the maneuvering equations by introducing the relative velocities of the vessel with respect to water. Thus,

Wind forces and moment corresponding to steady wind action exerted on the ship can be modeled as'°:

where p0 is the density; Ui,, is wind velocity at a standard height of 10 m above water ar is the relative angle of attack between the wind direction and the Ship heading; Ar and AL are the transverse and longitudinal projected areas

of the vessel, respectively; L is the length of the vessel; C,(ar), C(ar),

FpjrgeFxwind+Fxwavei (14)

Fvi,,4+Fyw#jye,

M wind +M wave (15) (16) wini __pcUw2Cxw(ar)AT, (17) 'ywind

=PaUw2cw(t2r)j4L,

(18) vind = PaUw2Czw(u1r)L14L, (19)

and C(a,) are Wind longitudinal and lateral force and moment coefficients, expressed in Fourier Series as follows:

C(a,) =

+ cos(na,.), (20)

n=1

Cyw(ar)=ônsin(nar),

(21)

C(a)=

n(?2(2r)

(22)

n=1

Coefficients ,

, t,

and in Equations (20X22) depend on the type of vessel, superstructure location, and loading conditions. These coefficients can be obtained from a suitable set of data.

The mean wave drift excitation in the horizontal plane isl 1:

F'xwaye=Pw&WAdCOS3(0oV), (23)

wave

= pgLCd

sin3(00- V,), (24)

M p =pgL2 Cj sin 2(9

-

yf), (25)

where Pw is water density; O

is the absolute angle of attack; g is the

gravitational constant; and Cj,

and Cj are the drift excitation

coefficients in surge, sway and yaw, respectively:

=JS(wo){ FxD(WO) 1

Lo.5pga2°'

(26) Ca=JC,=JS(w0I F,.,(w0)

_L

(27) o

L0.5Pwga2f0

S(wo)[_F(w0)

1w,, (28) o

O.5pLga2f 0

In the expressions above, the quantities in square brackets are the drift excitation operators; a is the wave amplitude; o.k, is the wave frequency. The two parameter Bretschnider spectrum is used to relate S(w0) in terms of the significant wave height IfJ/3 and significant wave period T113 as:

where A and B are nonlinear functions of H1 3äEid T1

PARTICULARS OF THE SYSTEM

DICAS Geometry

The DICAS under consideration consists of a converted FPSO system, whose geometiy is shown in Table 12. The system consists of nine mooring lines (six

moOred forward of the CG; three moored aft of the CG), with nominal mooring line length tT= 1910 meters each. The system is geometrically

symmetric with respect to its X-axis, With three different points of attachment, with three mooring lines attached to each of them as shown in Figure 1. The types of catenaries (single, double) in the system, their orientation with respect to the X-axis, the body-fixed coordinates of the mooring line fairlead, and the mooring line pretensions as percentages of ABS, are shown on Table 2. This configuration has been shown to provide highly stable slow motion dynamics and adequate displacement properties for operation in the Campos Basin, Brazil, in the absence of wind and wave excitation under a current speed of 2 knots2.

The production risers in the system are placed near the bow of the vessel, where the system has the least horizontal displacement as the environmental excitations change direction, due to the relatively high pretensions in the forward mooring lines. These are flexible risers, with a maximum allowable horizontal displacement of60-75 meters (20-25% of the water depth), with respect to a prescribed initial orientation12.

Environmental Conditions

The DICAS configuration described in the prcvious sub-section is to be

deployed in the Marlin field, Campos basin, Brazil, under a water depth of 300 meters. The predominant direction of each environmental force at that site, along with the percentage of occurrences for each, are as follows'3:

Table 1: Geometric properties of the FPSO

Length overall (WA) 272.80 rn

Length between perpendiculars (LWL, L) 259.40 m

Beam(B)

43.10m

Draft(D)

16.15m Block Coefficient (C8) 0.83 Displacement(4) 1.5374x 10 ton 9.11Ox 106 kg my 1.360x 108 kg 7.180x 1011 kg.m2

j

5.430x 10 kg.m2

predominant direction oeeurrence

surface current: S-N 54.56%

wind: NE-SW 33.50%

waves: NE-SW 38.60%

The values for the magnitudes of surface current, wind, and mean second order wave drift forces acting on the system, are based on a. 10-year return period,

and are shown on Table 3.

Table 2: Mooring line arrangement Of DICAS

Table 3: 10-year period return values, Marlin field13 Force Predominant Direction Magnitude

surface current S current speed: 1.58 rn/s

wind NE wind speed: 25.06 rn/s

waves NE max. wave height: 8.6 m

max.period: .

11.7s

significant wave height: 4.7 m

significant period: 9.2 s

peak period 12.9 s

DYNAMIC ANALYSIS OF DICAS

Analyses of nonlinear dynamical. systems undergoing

changes in the

enviro mental excitatiOn over long periods of time require, in general, trial and error with numerous and lengthy nonlinear tune simulations in each trial The

methodology for the design of SMS based on nonlinear

dynamics and

bifurcation theory as several parameters vary over the life of the moored vessel fairlead co rdinates (m) Pretension

(%ABS) 1 S 2680 64.85

-15.24

23.71 2 D 2710 64.85

-1524

24.48 .3 D 2840 64.85

-15.24

24.48 4 D 66° 64.85. 15.24. 4.48 5 D 89° 64.85 15.24 24.48 6 S 92° 64.85 15.24 23.71 7 D 175° -129.70 0.00 14.51 8 S 180° -129.70 0.00 13.74 9

0

185° -129.70 0.00 14.51

(in this case direction Ofcurrent, wave, andwind)1'7 can be used to eliminate such a tedious process, and is adopted for DICAS.

Theoretical Background: Nonlinear DynamicsfBifurcatlon Theory

The slow motion dynamics of DICAS are studied based on nonlinear dynamics and bifurcation theory. First, the system is modeled as a Set of six first-order nonlinear differential equations by combining equations of motion (1)-(3) and kinematic relations (8)-(iO). Then, the stability properties of a specific PICAS

configuration are obtained around an equilibrium position by performing eigenvalue analysis for a given set of environmental forces. If all eigenvalues have negative real parts, such equilibrium is locally stable and all trajectories initiated near that equilibrium position will converge to it in forward time. If at least one eigenvalue has a positive real part, such equilibrium is unstable, and a small disturbance from equilibrium will cause the system trajectories to diverge from it'd.

Once the local stability properties of DICAS have been established, bifurcation sequences can be studied to find qualitative changes in the dynamic behavior of the system as one or several parameters (direction of current, wind, waves) vaiy with respect toaprescribed DICAS orientation. These sequences are used

to construct stability charts in the two or three-dimensional parametric design space that separate regions of qualitatively different dynamical behavior (stable,

unstable, periodic, quasiperiodic, chaotic)7.

These charts, known as "catastrophe sets," can be used toy eliminate the number and length of

nonlinear time simulations in the DICAS design process. Nonlinear simulations are used to calculate the dynamical tensions, maximum displacement of the vessel, relative motion of the vessel endpoints, etc.

Analysis of the System Under. Variable External Excitation

To study the dynamics of DICAS as the environmental excitation changes direction over the lifetime of the system, catastrophe sets are constructed around the principal equilibrium position of the system in the design space of: the current direction (a), and the wave direction

_(0d)

The wind direction () is

inroduced as a parameter as shown in Figure 2. The excitation directions are measured with respect to the (x,y) reference frame according to the convention shown in Figure 1. The values for the magnitudes of the external excitations correspond Wa 10-year return period (Table 3).

Figure 2 shows a series of catastrophe sets in the (a, Od) parametric space for the following ranges in the current and wave directions:

0°a.360°, OOd36O,

for four different values of the wind direction O ranging from 0° (or 360°) to 270° in intervals of 90°. These sets present two regions that denote qualitatively different dynamical behavior of the system. These regions, numbered I and 11, and denoted as R-I and R-fl, have the following characteristics:

(1) R-I: Stable equilibrium. All eigenvalues in the system have negative real parts. A small random disturbance from this equilibrium position initiates

trajectories converging toward it in forward time.

(ii) R-J1: Unstable equilibrium with a two-dimensional unstable manifold (i.e. a

complex conjugate pair of eigenvalues with positive

real part). A small

disturbance from equilibrium results in a limit cycle around it15. The system loses its stability dynamically when crossing from R-I toR-fl. At the boundaiy between these tWo regions, a Hopf bifurcation occurs7, resulting in oscillations around the equilibrium position. The limit cycle exhibited by the system may be stable or unstable, depending on the interaction between the mooring lines as

the system oscillates1. A DICAS may fall in R-U as

the environmental conditions change. If operating in this region is deemed necessary, the limits of the amplitude of oscillations of the system in terms of mooring line tensions and vessel relative displacements must be studied. In such a casó, further analyses based On the center manifold theorem'4 and nonlinear time simulations are required.

Figure 2. Effect of external excitation on catasimphe set o DICAS

The catastrophe sets in Figure 2 show that the principal equilibrium of the

system is stable under a large rangeof current, wave, and wind directions. The system falls into oscillatory regionR-ll, undergoing a limit cycle behavior, for current directions at or near right angles with respect to the vessel orientation. Waves destabilize the system in the range of following to beam seas, and

stabilize the system in or near head seas. This is partly due to the fact that this

pretensioned forward mooring lines, thus limiting the motions of the system. The same principles apply for wind fOrces, as shôv nm the figure.

The nonlinear time simulations in Figure 3 show the richness of the DICAS dynamics depicted in the catastrophe sets in Figure 2 as functions of the drift angle . Simulation (1) in Figure 3 corresponds to a stable principal

equilibrium for the parameter values (a= 100°, 8d= 165°, O=O°). A small

decrease in the current angle (15°) results in a stable limit cycle around the

principal equilibrium, as shown by simulation (2) for the parameter values

(a=85°, 0d=165°, _O=0°). As previously mentioned, if the DICAS is to be designed such that the environmental conditions of scenario (2) apply, further analyses would be required to determine tbe plausibility of such a design.

nondinensiona1 time -0.3

0 25 50 75 100 125 150 175

Figure 3.Driftangle, DICAS: stable and oscillatoiyprincipal equilibrium

DICAS Odentatlon

As shown in Figure.2, there are ranges of current directions with respect to the (x,y) reference frame (0°

60°; 100°

260°; 290° 60°) for which the system yields a stable principal equilibrium irrespective of the direction of wind and waves A DICAS may be designed with a suitable choice of vessel orientation based on the catastrophe sets of Figure 2. An appropriate orientation for the system would be that in which the predominant current direction (coming from the south, see Table 3) is aligned to the (x,y) and

(X,Y) frames of the vessel (i.e. the relative angle between the current and the vesselis 180°). This would be achieved by orienting the system with its bow pointing into the principal direction of the current (i.e. sOuth).

I

-0.05-

-0.1--0.15

₍₁₎ -02-,

a)

-0.25-Figure 4 shows the relative horizontal displacement of the vessel's bow at equilibrium with respect to the position of the bow in the absence of external excitation, as a function of the current angle in the predominant range SW-SE. lEt this application, the water depth is 300 m; wind and waves are co-linear with

directions ranging from E-N. The vessel is oriented with its bow pointing south.

Figure 4. Relative horizontalbow displacement; principal orientation: bow south

As shown in Figure 4, the maximum horizontal displacement of the bow in these excitation ranges is under 30 meters (approx. 27.8 m), which is less than the allowable limits of motion for operation with flexible risers. Additional analyses are needed, however, to determine the dynamical tensions iii the

mooring lines and the maximum displacement of the vessel.

An alternative DICAS orientation that renders a highly stable system according

to the catastrophe sets in Figure 2, would be to place the vessel at 00 with respect to the current principal direction (i.e. the bow of the vessel facing north). In such a case, the counterpart of Figure 4 is shown on Figure 5. From this figure, it is shown that the maximum relative bow horizontal displacement for the directions of wind and waves shown (ápprox. 23.2 m), is less than the allowable limit of motion This orientation also yields smaller displacements than those shown in Figure 4, and it is therefore a better design configuration in terms of vessel displacements.

It is important to point out, however, that in this particular case, such alternative orientation yields smaller relative displacements due to the fact that the principal

direction of the surfacecurrent is opposite tothe direction of wind and waves. If all forces were to act in the same principal direction, the alternative of placing the vessel in following seas would result in an increase in the tensions of the aft catenanes. This could possibly restrain the aft part of the vessel from moving, while allowing the fore part of the vessel to move relatively freely.

5-0-

cuflent dhectiOn

SW SE

FIgure 5. Relative horizontal bow displacement; principal orientation: bow north

EFFECT OF WATER DEPTH

The water depth at the Marlin field/Campos basin is not constant, and it is

therefore important to consider the dynamics of the system under different water depths.

Figure 6 shows a series of catastrophe sets in the (a, Od) parametric space (wind direction Os,, is fixed at 180°), for three different values of the water depth, ranging from 250 meters to 350 meters. The qualitative behavior of the system about its pnncipal equihbrium position in regions R-I and R-U in this figure, is the same as that in Figure 2.

As shown in Figure 6, the stable region R-I tends to increase with increasing water depth., This is due to higher resistance (drag) on the system as a result of

increasing the length of the suspended catenaries. Such an increase in the

catenary length, however, causes the horizontal displacement of the vessel, and the tensions in the mooring lines, to increase as well.

50

Din ction of Wind and waves 45 -E 40- N NE 35 -NNE 30- N 25-20 I-._.

--

-

--15

--:

_--

10-Figure 6. Effect of water depth on catastrophe set of DICAS

CONCLUDING REMARKS

Preliminary design of a DICAS based on its slow motion dynamics in the horizontal plane can be developed using the methodology presented in this paper. To illustrate some of theapplications of the design methodology, it has been shown that, for a prescribed DICAS configuration, it is possible to select an appropriate system orientation that yields a stable configuration under a wide range of external excitations without resorting to trial and error, and lengthy nonlinear time simulations. This orientation was seiected based oncatastrophe Sets with the predominant directions of theexternal excitations as parameters. An increase in water depth increases the stable domaln of the system due to longer suspended portions of the catenaries, and thereby induced higher drag. It rósults, however, in higher mooring line tensions and increased horizontal displacement of the system. The DICAS concept in deeper waters may

therefore not be feasible, unless the horizontal pretensions in the mooring lines are increased, and buOys are placed along the catenaries to reduce the vertical tensions on the vessel exerted by their weight.

ACKNOWLEDGMENTS

This work was sponsored by the University of Michigan/Sea Grant/Industry Consortium in Offshore Engineering under Michigan Sea Grant College Program, project number RIF-35 under grant number DQC-NA36RG0506 from the Office of Sea Grant, National (eanic and Atmospheric Administration

(NOAA), U.S. Department; of Commerce, and funds from the state of

Michigan. Industiy participants include Amoco, Inc.; Conoco, Inc.; Exxon Production Research; Mobil Research and Development; and Shell Companies

Foundation. The U.S. Government is authorized to produce and distribute reprints for governmental purposes notwithstanding any copyright notation appearing hereon.

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