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FLEXIBLE AND RIGID RISERS HYDRODYNAMIC COEFFICIENTS BY MODEL TESTING AND FINITE ELEMENT METHOD Antonio Carlos Fernandes (Consultant atPETROBRAS) Marcio Martins Mourelle (PETROBRAS) Otavio Serta (PETROBRAS) Sergio da Silva (IPT) Paulo H. C C Parra (IPT) Abstract — The Design of SCR (Steel Catenary Risers) has been requiring the processing of Finite Element Method Programs both time and frequency domain for extreme and fatigue conditions. These Programs require the knowledge of the drag (Cpj) and mass (C^) coefficients for each riser element. However, not as the vertically span supported conventional risers, the rigid (also the flexible) risers oscillate a greater amount about the static equilibrium configuration. For the former, the Coefficients are taken only as a function of the Reynolds number, but for the latter, the influence of the oscillations, regulated by the Keulegan-Carpenter number, has to be taken into consideration. This normally leads to an interactive process that is suggested by the present work. In order to gain experience with this property, reduced model tests have been performed at the IPT (Institute de Pesquisas Tecnologicas) facilities in Sao Paulo, Brazil. A FEM program, the PETROBRAS' ANFLEX makes it possible to devise a methodology, for full scale application. Further model tests have been analyzed by the present work, evidencing also the influence of the top connection trajectory. INTRODUCTION PETROBRAS isplaning to operate with Steel catenary Risers (SCR) for fluid transport in Campos Basin Offshore Brazil in the near future. This is a very cost effective solution. It is an alternative to the flexible risers whose price increases dramatically with the diameter. See Figure 1 for a typical SCR application Transactions on the Built Environment vol 29, © 1997 WIT Press, www.witpress.com, ISSN 1743-3509

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FLEXIBLE AND RIGID RISERS HYDRODYNAMIC

COEFFICIENTS BY MODEL TESTING AND FINITE

ELEMENT METHOD

Antonio Carlos Fernandes (Consultant at PETROBRAS)Marcio Martins Mourelle (PETROBRAS)Otavio Serta (PETROBRAS)Sergio da Silva (IPT)Paulo H. C C Parra (IPT)

Abstract — The Design of SCR (Steel Catenary Risers) has been requiring theprocessing of Finite Element Method Programs both time and frequencydomain for extreme and fatigue conditions. These Programs require theknowledge of the drag (Cpj) and mass (C ) coefficients for each riser

element. However, not as the vertically span supported conventional risers, therigid (also the flexible) risers oscillate a greater amount about the staticequilibrium configuration. For the former, the Coefficients are taken only as afunction of the Reynolds number, but for the latter, the influence of theoscillations, regulated by the Keulegan-Carpenter number, has to be taken intoconsideration. This normally leads to an interactive process that is suggested bythe present work. In order to gain experience with this property, reduced modeltests have been performed at the IPT (Institute de Pesquisas Tecnologicas)facilities in Sao Paulo, Brazil. A FEM program, the PETROBRAS' ANFLEXmakes it possible to devise a methodology, for full scale application. Furthermodel tests have been analyzed by the present work, evidencing also theinfluence of the top connection trajectory.

INTRODUCTION

PETROBRAS is planing to operate with Steel catenary Risers (SCR) for fluidtransport in Campos Basin Offshore Brazil in the near future. This is a very costeffective solution. It is an alternative to the flexible risers whose price increasesdramatically with the diameter. See Figure 1 for a typical SCR application

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378 Offshore Engineering

For the design of both the SCR and the flexible risers use is made of time andfrequency domain FEM (Finite Element methods) programs. Several of the

programs are available, including the PETROBRAS' ANFLEX . For the useof all of these softwares, it is necessary to specify the hydrodynamiccoefficients C and C (see Equation (1) and (2) below). Despite the

knowledge of the oscillatory effects, which are regulated by the Keulegan-Carpenter (KC) number, the present design practice works with pre-setcoefficients chosen only from the viscous effects.

The present work, with the help of carefully designed model tests and the useof ANFLEX in the model scale, shows that the KC effect should be consideredThis conclusion have been recently exposed in (Fernandes, Mourelle, Serta, da

Silva and Parra, 1997)\ while in the present work some ideas on the modeltesting are discussed more clearly. Besides that, the results of further modeltesting are also presented. Not only the circular trajectory have been imposed atthe top connection, but also the purely vertical and purely horizontaltrajectories. The conclusion is that both the KC effect and the top connectiontrajectory are of fundamental importance for a sound design practice.

THE HYDRODYNAMIC COEFFICIENTS

The Morison Equation * approach is used for slender structures such as therisers under consideration. In this equation the local values of the drag (C^)

and mass (C^ ) coefficients at each section are required. By definition,

^dn-gforce ^

-p|V|VD

inertia force

where D is the transversal diameter, V is the local transversal velocity; theforces are per unit of length; and p is the water density. This work considers

the Reynolds (Key) and the Keulegan-Carpenter (KC) to define the values ofthese coefficients. The KC number is defined as

V TKC = -2L- (3)D ^ '

the Rey number is defined as

V D- _ (4)

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Offshore Engineering 379

where the V^ is the local velocity amplitude, i is the period of oscillation and

v is the cinematic viscosity .

The argument to neglect the KC effect is that the oscillatory effect is usuallysmall for higher KC However, based on Figure 3 and 4, even for KC = 100 theeffect is felt and the coefficients are different from the usual C^ = 1.0 and0 =2.0.

MODEL TESTING DESIGN

During the SCR investigation, PETROBRAS decided to perform model testingfor the complete configuration shown on Figure 1 . The aspects that have beenconsidered are: the Geometry, the Restoring Forces, the Inertia and theDamping. The focus here is on the SCR but the same consideration should beapplied to flexible risers.

GeometryThe main requisite here is the longitudinal geometric similitude. There are twocharacteristics length: one longitudinal, that is related to the depth and theother related to the transversal diameter. For tests in the available facilities suchas at IPT, where the depth is about 4 m, keeping the transversal similitudewould lead to a 1 mm model diameter which is difficult to construct andmanipulate. However, there are good reasons to work with similar depth and asimilar catenary, but no reasons at all to have similar diameter. A similarcatenary shape is important because for oscillatory top trajectories, it will beperformed about the average catenary and the distribution of oscillation shouldbe correlated. On the other hand, the diameter may perfectly be considered anopen variable, used to adjust other characteristics in this case inertial ones as isexplained below.

Restoring ForcesBesides the Geometry, the restoring forces should be correctly represented.However, the restoring effects due to both longitudinal and flexural elasticitymy be neglected. The reason to neglect the former is that its effects is usuallynot important at the prototype scale, where typical first natural period are lessthan 1 s. The reason to neglect the latter is that it has been realized that there isa boundary layer about the extreme connections that concentrates the flexuralvariation* . The boundary layer thickness is given by

where El is the flexural rigidity and H the horizontal tension at the TDP (TouchDown Point). Typically, for a SCR installed in 910 m water depth, A = 10 m

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380 Offshore Engineering

During the research towards model construction it has been observed thatcables with very different materials had the same catenary shape, provided thetop connection tension and linear weight were the same.

Hence, the only restoring effect that matters is the gravitational effect related tothe catenary portion that is suspended above the floor.

InertiaThe other effect that should be considered is the inertia one. There are thesubmerged material mass and the added mass due to the hydrodynamicinteraction. Sectionally, the last one is represented by the C^-l. Since this

coefficient varies with the oscillation, in principle, one is in trouble to adjust themodel parameters. However, it has been realized that small variations in CM do

not influences the response significantly, hence not jeopardizing the inertiacorrelation

The first impression is that the non-dimensional parameter should relate onlythe virtual mass. This is not correct, though. The correct correlation, oncecatenary (gravitational) effects are included relates the MASS RATIO asexplained next. For typical transversal dynamics,

(m + m.,)- —= T(G) — - vv cosG (6)' c%2 ds

where T is the axial tension, 0 is the local angle at each arc position s (s=0corresponds to the TDP), w is the distributed weight, v is the transversaldisplacement, m is the submerged mass and the subscript a indicates the addedmass; t is the time Introducing the length scale

(7)

with the subscript p reversed for the prototype and m for the model (h is thewater depth) and since the water depth for the reference SCR is 910 m and4.08 m for the prototype and model, X^ = 223 .

The time should follow the Froude Law that rules out the proportion betweenthe gravitational to the inertia effects. This Law is traditional in model testingthat includes free-surface effects. These last effects, however, are not relevanthere, since the SCR diffraction is negligible. Here, the gravity comes in by thecatenary restoring capability. Therefore,

(8)

Using the corresponding scale parameters, Equation (6) may be written as

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Offshore Engineering 3 81

(9)

With (8) in mind, the Equation (9) leads to the correct definition of the MASSRATIO O), that should be equal for the prototype and model scales, that is

^ = - ;-- = - — - = Hm (10)*/S W/m/g ^ /

But since

TlD')g (11)

- (12)

the correct mass ratio may be written as

p—H = —*- (13)

7,0:

While in the Literature" one may find for the mass ratio

M,--2_ (14)

and also the specific gravity

s.g = Mr + l (15)

Hence when C^ =2.0, p. equality is equivalent to M,. and s.g. equality But

when CM varies, an interactive process is necessary. However, as said before, a

small variation of C^ will not change the responses. Hence, the feasible

procedure is to find Key and KC operational range; finding the CM range; then

calculate ^. For the present SCR model range CM =2.0 allowing inertiacorrelation. For this case,

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382 Offshore Engineering

h\ = 2.136 (16)

M^ =2.193 (17)

made close enough with a 5 mm distorted diameter model described in Figure 2.

DampingAs in other branches of Ocean Engineering, due the viscosity effects and in thepresent case the oscillatory effects, the perfect correlation is impossible for thedamping. However, the powerful FEM codes available now a days, may beused to calibrate a methodology in model scale. If the correct Physics isconsidered, the same methodology may be applied to the full scale. For thepresent SCR case, the full scale is with the curves shown in Figures 3 and 4.For the model, though the Reynolds number is very small and the correctcoefficients have been obtained by iteration using ANFLEX.

Note that, in principle, the KC number is dependent on the amplitude ofoscillations, that is, using (3),

This value varies throughout the riser length but, since the motion, is about themean catenary it may be regulated by the top amplitude. On the other handusing (4) the Rey number maybe written as

That is, using (18), to correlate KC, one needs only to adjust A at the topconnection. With this fixed KC, only the period i must be modified when using(19) to adjust Rey For the SCR in Figure 1, the typical KC is 23 and the Rey is

2x10^ . With these values, the model period would be .003 s or 348 Hz, a valuevery high that requires special oscillators.

The intention to discuss these ideas and figures here is to leave the door openfor future developments. Due to the cited difficulties, the work left the dampinguncorrelated, assessing its influence via FEM.

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Offshore Engineering 383

MODEL MEASUREMENTS AND FEM

By trial and error, it has been possible to find coefficients to match the FEMcode with model tests measurements within less than 3%. This has been shown

in (Fernandes, Mourelle, Serta, da Silva and Parra, 1997) . The final result isreproduced here in Table 1.

Test

202020302040

Dynamic TensionAmplitudes at the Top

(N)

0.03540.076401246

Error w.r.t.Tests (%)

-2.0-2.70.2

Coefficients used inthe FEM simulation

CD1.91.61.4

02.22.01.5

Table 1 - Dynamic tension amplitudes at the top connection point calculated byFEM with the indicated coefficients which have

been obtained by trial and error.

The trial and error in model scale was helped with auxiliary CFD (Computer

Fluid Dynamic) processing. The data in Figures 3 and 4 goes up to Rey=lo^The CFD code was processed for Key much less than that and the results are

shown in Figures 5 and 6 that used data from (Meneghini and Rosa, 1996)These last figures may be considered extension of Figures 3 and 4.

EFFECT OF TOP TRAJECTORY

As a further investigation, the work decided to verify the effect of the toptrajectory shape. The motivation comes the fact that a circular path is a path asarbitrary as any other. For a riser hanging from a TLP (Tension Leg Platform),the vertical top motion is very small, while the horizontal motions combineslower and waves frequencies. Hence, a circular motion never happens. For a SS(Semi Submersible) platform, the vertical motion may be significant, hence dueto the combination with the horizontal motion, elliptical trajectories areexpected. The same for FPSOs (Floating Production Storage and Offloading)systems.

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384 Offshore Engineering

Conditions

S192020HS192030HS192040HS192080HS193020HS193030HS193040HS193080HS194020HS194030HS194040HS194080H

S192020CSI 9203 OCS192040CS192080CS193020CS193030CS193040CS193080CS194020CS194030CS194040CS194080C

S192020SSI 920308S192040SS192080SS193020SS193030SS193040SS193080SS194020SS194030SS194040SS194080S

DynamicTension

(802,695,749,87

33,614,499,3115,9945,767,3514,1523,3354,68

2,855,9610,3934,494,869,8517,0847,858,0015,3324,4558,56

4,362,494,0012,572,073,735,8618,942,505,017,8624,81

DynamicTension

Filtered(gf)

2,485,689,7433,234,329,2115,8144,207,2313,9823,0451,41

2,605,9110,0534,084,739,7416,8946,487,8415,1124,2054,04

0,462,053,5712,390,623,305,6718,731,844,577,7024,44

PhaseFiltered Sign(degrees)

108,4122,9130,6129,2101,5114,9120,2119,398,0108,2112,2115,9

102,0105,6110,1111,389,495,9102,6102,581,591,195,899,7

45,7136,9127,9141,976,2127,8125,5134,6114,7119,6123,3126,9

sin 9

0.950.840.760.780.980.910.860.870.990.950.930.90

0.980.960.940.931.000.99

0.980.980.99LOO0.990.99

0.720.680.790.620.970.790.810.710.910.870.840.8

averagesimp

0.83

0.91

0.94

0.95

0.99

0.99

0.70

0.82

0.86

Table 2 - Dynamic tension for XX=19° top angle catenary shape, YY averageamplitude, ZZ frequency in RPM and W the type of trajectory at the toptrajectory (C for circular, H for vertical and S for horizontal trajectories)The condition description is given in the first column by SXXYYZZW

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Offshore Engineering 385

Due to the existence of a program like ANFLEX that, whenever calibrated,may be used in the general case, there is no point in trying to investigate allpossible trajectories. Hence, only the circular, the horizontal and the verticalare investigated here. The tests have been made with four frequencies (20, 30,40 and 80 RPM); three amplitudes (20, 30 and 40 mm); three basic catenaries

mean condition (10°, 19°and 28°). The top tension and the phase withrespect to a known acceleration have been registered. The response signs havebeen mostly harmonic for the harmonic excitation. This became clear after

Fourier analysis of the response. Some results are shown in Table 2 for the 19°top angle catenary shape. Some typical results are shown in Figures 7, 8 and 9.The Figure 7 shows for the Near condition that the circular and vertical arevery close and the horizontal do not impose a high dynamic tension (only 12%for the circular motion at 80 RPM). However the horizontal motion becamemore and more important for the Mean (Figure 8) and for the Far conditions(Figure 9). Note that for these last figures as the RPM goes to zero,characterizing a quasi-static situation, the dynamic tension also goes to zero.For higher RPM, though, the dynamic tension increase is practically linear withthe frequency. It is expected that for very high frequency the SCR will freezetransversally, allowing only elastic distention. In the present model tests, thatlimit never showed up even for higher frequencies. This probably is because thethread in Figure 2 is very stiff longitudinally. It is expected that in full scale thatlimit will come up earlier. Finally, it is important to say that other analysis withtables like Table 2, indicate that the dynamic tension varies linearly with theamplitude for a fixed RPM and trajectory.

The phase indicated in Table 2 may be used to assess the Energy dissipation foreach cycle. This quantity may be written as

T I tAE eye,; =JfVdt = jT,V,dt+jTyVydt (20)

o o o

where the f is the top tension, V is the imposed top velocity, and the subscriptindicate horizontal and vertical components.

For the vertical trajectory

y = yo cos(cot + cp) (21)

and the Equation (20) simplifies to

AE cycle = *Trfoy 0 COSOtQSilKf) (22)

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386 Offshore Engineering

where the T^ is the dynamic tension amplitude, <XQ is the mean top angle, and

cp the phase. The Equation (22) is consistent with the fact that there is no

energy dissipation if the mean top angle is 90° .

For the horizontal trajectory

X = XQ COS(cot + (p) (23)

and the Equation (20) simplifies to

(24)

with same as before. The Equation (24) now is consistent with the fact that

there is no energy dissipation if the mean top angle is 0° .

For the circular trajectory

r = TO cos(cot + (p) (25)

and the Equation (20) simplifies to

AE cycle = TdoTQ COS(ao-Cp) (26)

where TQ is the top trajectory radius.

The results (22), (24) and (26) are not frequency dependent . They depend onthe tension and motion amplitudes, and the phase angle. This angle, therefore,is important to quantify the energy dissipation. From Table 2, it is observedwith the sincp column that the frequency dependence is very small leading to

the conclusion that the tests are indeed harmonic. Meanwhile, the average simp

is larger for the vertical motion than for the horizontal motion. A resultsomewhat expect since, for the latter, there are points that hardly move duringthe prescribed oscillations. See Figure 10 for SCR different displacements.

CONCLUSIONS

The experience with the model testing and FEM leads to the followingconclusions:1. Besides the Rey, the KC number must be considered when selecting

hydrodynamic coefficients for oscillatory analysis (typical for the fatigueassessment, for instance).

2. An interactive process see 4. below and information as in Figure 3 and 4may be used to arrive to final coherent coefficients.

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Offshore Engineering 387

3. For the case here studied, the KC distribution along the length is notimportant, but this work is under impression that this conclusion should notbe generalized.

4. The interactive methodology should be as follows: Assume initialcoefficients based on steady state Rey number and process the case; withlocal resultant KC; redefines the coefficients with the help of Figure 3 and 4and reprocess the case; compare the results with the former an iterated againif necessary. It is expected the few interactions are necessary.

5. Besides the Rey and KC number, one should choose carefully the trajectoryfor analysis. It has been shown that a vertical trajectory is very differentfrom the horizontal one and both are different from the circular. It seemsthat a rectilinear trajectory tangent to the catenary at the top will lead to the

higher loads (Andrade, 1997)*. This overestimation may not be sensible forfatigue analysis with tight results. In this case, to associate statistic valuesfor typical trajectories would be the best way to proceed.

ACKNOWLEDGMENTS

The Authors would like to thank PETROBRAS for this great opportunity It isa pleasure to participate in this historical effort to make the SCR a reality inCampos Basin.

REFERENCES

[1] Mourelle,M.M, 1993, "ANFLEX- Analise Nao-Linear de Risers e Linhasde Ancoragem", Manual de Entrada de Dados, 2.0/Rev.3., PETROBRAS,CENPES/DIPREX/SEDEM, May, Brazil

[2] Fernandes,A.C., Mourelle, M.M., Serta,O.B., da Silva, S. and Parra,P.H.C.C., 1997, "Hydrodynamic Coefficients in the Design of SteelCatenary Risers", OMAE97-Yokohama, Offshore Mechanics and ArtieEngineering Conference, Japan.

[3] MorisonJ.R., O'Brien,M.P., Johnson,!.W. and Schaaf,S.A., 1950, "TheForce Exerted by Surface Waves on Piles", Pet. Trans., 189, 149-154,USA.

[4] Martins,C.A., Pesce,C.P., Aranha,J A.P. e Pinto,M 0 , 1996, "Modelo deAnalise da Dinamica do SCR", Escola Politecnica da USP, Monografia 1,Report to PETROBRAS, E&P/GETINP/GECOMP, September, Brazil

[5] Blevins,R.D., 1990, "Flow-Induced Vibration", Van Nostrad Reinhold.New York, NY, USA

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388 Offshore Engineering

[6] Duggal,A.S. And NiedzweckiJ.M., 1993, "Wave Interaction with a Pair ofFlexible Cylinders", Offshore Technology Research Center, Report 01/93-A-42-100, Texas,USA.

[7] Meneghini,J.R. e Rosa,C., 1996, "Utilizacao de Dinamica dos FluidosComputacional para Analise de Escoamento Oscilatorio em torno deCilindros", ", Escola Politecnica da USP, Report to PETROBRAS,E&P/GETINP/GESEM, November, Brazil.

[8] Andrade,B L.R., 1997. Private Communication.

[9] Sarpkaya,T. and Isaacson,M., 1981, "Mechanics of Wave Forces onOffshore Structures", Van Nostrand Reinhold, New York, NY, USA

SpiderDeck \

Flex Joint

Riser WallThickness20.6 mm

(0.812 inches)

Total Riser Length 3310m(Length to Touchdown 1319m)

RiserAnchor

Figure 1 - A SCR recently designed for PETROBRAS to be installedin Campos Basin on the PETROBRAS XVIII Semi Submersible.The free-hanging characteristic is also common for flexible risers,

although with smaller top angle (usually 7 degrees).

Steel Thread00.3 mm

Latex lube

Figure 2 - Longitudinal section of the SCR model. It was assembled externallywith latex tube and internally by a stainless steel thread and lead spheres.

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Offshore Engineering 389

Figure 3 - Drag Coefficient as a function of Keulegan-Carpenter

(K=KC) and Reynolds (Re=Rey) numbers (Sarpkaya and Isaacson, 1981)*

Figure 4 - Mass Coefficient as a function of Keulegan-Carpenter (K=KC)

and Reynolds (Re=Rey) numbers (Sarpkaya and Isaacson, 1981)^

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390 Offshore Engineering

2.0

1.8

1.6

1.4

1.2

1.0

0.8

0.6

0.4

0.2).01 0.10 1.00

Figure 5 - Drag Coefficient as a function of Keulegan-Carpenter (KC) and

Reynolds (Re) numbers based on CFD (Meneghini e Rosa)

2.01.81.61.41.21.00.80.60.40.20.01 0.10 1.00

Figure 6 - Mass Coefficient as a function of Keulegan-Carpenter (KC) andReynolds (Re) numbers based on CFD (Meneghini e Rosa) .

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Offshore Engineering 391

80 —,

40 —

taphtuat 30 mm— CIRCULAR— VERTICAL- HORIZONTAL

40RPM

80

Figure 7 - Dynamic Tension measured during the model tests for indicated

trajectories and amplitudes; catenary top angle 10°

so —,

40 —

40RPM

Figure 8 - Dynamic Tension measured during the model tests for indicated

trajectories and amplitudes; catenary top angle 19 °.

i

40RPM

Figure 9 - Dynamic Tension measured during the model tests for indicated

trajectories and amplitudes; catenary top angle 28 °

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392 Offshore Engineering

Figure 10 - Static SCR shape after horizontal and vertical top displacements.

Transactions on the Built Environment vol 29, © 1997 WIT Press, www.witpress.com, ISSN 1743-3509