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A static drilling riser model using free boundary conditions
Ocean Engng, Vol. 24, No. 5, pp. 431-444, 1997 © 1997 Elsevier Science Ltd Printed in Great Britain. All rights reserved 00294018/97 $17.00 + 0.00
Pergamon S0029-8018(96)00021-2
A STATIC DRILLING RISER MODEL USING FREE BOUNDARY
CONDITIONS
Luc Huyse,*§ Mansa C. Singh# and Marc A. Maes~ *Civil Engineering Department, The University of Calgary, 2500 University Drive N.W., Calgary, Alberta, Canada T2N 1N4 tMechanical Engineering Department, The University of Calgary, 2500 University Drive N.W., Calgary, Alberta, Canada T2N 1N4 $Civil Engineering Department, The University of Calgary, 2500 University Drive N.W., Calgary, Alberta, Canada T2N 1N4
(Received 3 February 1996; accepted in final form 27 March 1996) Abstraet--A static three-dimensional analytical method for drilling risers experiencing large displacements and slip at the top joint is presented. The riser is described in cylindrical coordinates as a three-dimensional tensioned string, without bending or torsional stiffness. The vertical vessel displacement is not taken into account. The equilibrium configuration is obtained from the stationary condition of the total potential energy functional. Kinematic boundary conditions are introduced for the radial displacement only. The riser orientation at the top directly follows from the natural boundary conditions based on the variational treatment of the energy functional. Copyright © 1997 Elsevier Science Ltd
1.
INTRODUCTION: DRILLING RISERS
In recent years, marine risers have attracted considerable attention in offshore engineering since, on the one hand, they are a vital link between the floating platform and the subsea hole, and because of their complex analysis on the other hand. Floating structures are economically attractive for deep water drilling and production. Corresponding challenges arise from the complexity of the interaction of the structure with the environmental loading (De Oliveira et al., 1985). A drilling riser is a conductor pipe, connecting the floating platform and the bore hole (see Fig. 1). At the top, the riser pipe ends at a telescopic joint beneath the vessel. This slip joint allows change in riser length as the vessel heaves or moves laterally. The operability of drilling risers is expressed in terms of maximum angles from the vertical. To increase the riser stiffness, and thus limit these angles, tensioners are installed at the top joint. Buoyancy devices may be added to limit the maximum tension in the riser pipe. An excellent description of drilling risers can be found in Sarpkaya and Isaacson (1981). 2.
PROBLEM FORMULATION
Most static analyses for risers are concerned with the maximum riser response in a vertical plane. However, one must note that the riser may move in various directions at
§Corresponding author. 431
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le
Guide lines
Tensioners Choke and kill lines _ . ~ Riser pipe
Ball joint
.~BOP
L'-"z,. . . . . . . .
•
Fig. 1. Schematic of a drilling riser (Reprinted from Sarpkaya and Isaacson, 1981 with permission from the publisher).
various elevations due to the omnidirectionality of the waves and currents. Such an analysis is extremely complex since it requires information about the waves and the currents as a function of the coordinates x, y and z, in addition to the rig motion (Sarpkaya and Isaacson, 1981). In preliminary design, however, a static response model may be used to perform basic checks to investigate operability criteria (Marsh et al., 1984). For drilling risers, the operability limits are given in terms of the maximum riser angle with the vertical axis (Huyse et al., 1995). In the preliminary design stage, it is common practice to estimate the rig offsets from a set of Response Amplitude Operators (RAO) which relate the rig or vessel response to the significant wave height h~. This is done through the calculation of the mean environmental loading, including the riser and the mooring loads. This offset is then introduced as a boundary condition in the differential equations describing the riser (Garrett, 1993). Since the riser may experience slip at the top joint, the total arc length in the equilibrium
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position is unknown. It is convenient to choose the distance along the vertical axis as the independent variable. The variables r(z) and O(z) then describe the riser horizontal displacement and orientation at a given elevation z in the deformed state (see Fig. 2). In many cases, the main problem which prevents the analysis from being extended into 3 dimensions, is the lack of directionality data for the rig offsets (Cook and Gardner, 1985). Because of the very complex nature of the mooring and the hydrostatic forces, it is very difficult to predict the direction of the rig offset. In the absence of other data, the directionality problem mentioned before may be overcome using the free boundary conditions presented in this formulation. The following assumptions are made in this analysis: 1. The riser has neither bending nor torsional stiffness. 2. Even though the displacements are large, the strains remain small and the material behaves linearly elastic. 3. The wave and current forces have no vertical component; the vessel moves in the horizontal direction only. 4. The angle between the riser and the vertical axis is small. In practice, this assumption is valid for drilling risers only. 3. VARIATIONAL FORMULATION 3.1.
Strain energy in the riser
Since we limit the riser model to a simple tensioned string, only axial deformation is to be included in the internal virtual work formulation. Consider the distance s along the riser to be measured positive up from the bottom of the riser (see Fig. 2). The relative change in arc length, due to the horizontal riser displacement (r, 0) is given by:
U ~ max) y (Zmax) Zmax
- - - - --
- ~ x (Zmax)
. . . . .
~7
Y-__ "~
y (z)
~ ~
x z
r (z)
Fig. 2. Cylindrical coordinate system.
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ds-dz - ~/1 + r '2 + ( r 0 " ) 2 - 1 dz
(1)
where a prime denotes differentiation with respect to the vertical distance z, measured from the bottom of the riser towards the top. Accounting for the small angle approximation, this is simplified into: d s - d z .~ 1 (r,2 + (rO,)2)
dz
(2)
2
This change in length is caused by two different phenomena: strain of the riser without slip and slip of the riser without strain variation (Huang and Rivero, 1986; Huang and Chucheepsakui, 1984; Huan and Saha, 1989). The first mode stores strain energy in the riser, and for the second mode virtual work is required. For both modes the riser tension is driving the process. Since the actual strains are not of primary interest here but only the global deformations, there is no need to uncouple both effects. For small angles the riser tension T is given as: T-TTop + Priserg(Z--Zmax)
(3)
where TTop is the effective tension at the top of the riser, P, ser is the riser mass per unit length, and Zmax denotes the vertical coordinate of the slip joint, i.e. the top of the riser. Consequently, the sum of the strain energy and the internal virtual work per unit vertical length due to the slip at the top joint is given by: 1
f o = ~ (TTop + Priserg(Z--Zmax))(r '2 + (r0J) 2)
(4)
The total 'strain energy' along the riser becomes then: Zmax P
u = ~ Uo dz
(5)
,/
0
3.2. Work o f the external forces Since vertical displacements are not considered, the work of external forces is due to the current force acting over a displacement r in the horizontal plane. The work done by these currents per unit vertical length of the riser is: d V = q.r dz =
Ilqlllldlcos(Oq- O)dz= qr
cos(Oq- O)dz
(6)
where q i s the drag force per unit length of the riser, Oq the direction of the current, and = ~1~ + ~ the horizontal displacement of the riser at depth z. The drag force q per unit vertical length can be expressed as (Sarpkaya and Isaacson, 1981): 1
q = ~ pwfdOrisorvllvll
(7)
where pw is the mass density of the sea water, C~ is the drag coefficient, D the riser drag
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435
diameter, and v the current velocity at the depth z. The total work done by the current forces is then: Zmax
V=
I
Zmax
~ pwGDnserVllvll.rdz =
0
f
qr cos( Oq-O)dz
(8)
0
Constraint equation Due to the slip at the top joint, the total riser length is initially unknown. However, the maximum riser length /max is limited by the stroke capacity of the telescopic joint. This supplementary inequality constraint for the riser length is: 3.3.
Zmax
Zmax
0
J 0
Introducing the positive, dummy variable d 2, this inequality constraint is-transformed into an equality constraint: Zmax
Zmax
fds= f(l+r'2+(rlT)2dz+d2-1max=O 0
(10)
0
This dummy variable d has the following physical interpretation: d=0: the riser length is exactly equal to the maximum allowable length/max, the telescopic joint at the rig is fully stretched. d4=0: the optimal solution lies within the bounded region. The length constraint is not binding in this case, since there is still some stroke capacity left. 3.4.
Functional: the total potential energy
According to the principle of minimum of the total potential energy, the first variation of the total potential energy functional must vanish at the equilibrium position of the riser. Because of the subsidiary condition (10), we make use of the Lagrange multiplier A and construct the following functional I (Mura and Koya, 1992): I = U - V + Ag(d) Zmax
-,-,rO',a,-qr 0
=
fF(r,
r', O, 0", d, h)dz
For I to assume an extremum value, its first variation must vanish:
(ll)
L. Huyse et al.
436 Zmax
31=
~rr 3r + -oO 30 + ~d 6d + o-A 3A + ~r, 3 / + ~
3ff
dz=0
(12)
0
The last two terms in Equation (12) are integrated by parts and substituted for the kinematic boundary conditions for r. Since r is prescribed on the boundaries, its first variation vanishes. Consequently, the following formulas are obtained: Zmax
Znlax
~0r3 / d z = -
(13)
dzz ~r' 6rdz
0
0
Zmax
Zmax
Or' 30'dz =Off
~ 0
0
~ff 60 dz
0
Substituting Equation (13) in Equation (12), the following expression for I is obtained: Zmax
31= f{[00F
OF d (~r'r')]3r + [~F dz
DE 0 d (ff~)]3 dz
(14)
0
OF OF } OF 30 zm"X + ffd M + 0A 6)t d z + 0 f f =0 0
For a first class of admissible functions the equation: OF (m°x 0 ~ 30
= 0
(15)
0
will hold since 30=0 at the top and the bottom of the riser. For this class of functions the riser orientation is prescribed at the top and the bottom. Because of Equation (15), the functional I(r, O, d, A), defined in Equation (11) will assume an extremum if, and only if: OF Or
d (~vFF/ dz \Or ] = 0
(16a)
OF O0
d(OF) dz ~ = 0
(16b)
OF Od
0
OF -0 OA
(16c) (16d)
For another class of admissible functions, the riser orientation 0 is not prescribed on the boundaries. In this case, Equation (15) must still hold because of the Equations (16a)-
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(16d). Since for this class of functions 80 4= 0, Equation (15) is satisfied only if, at both the top and the bottom of the riser: ~F - 0 30"
(17)
The Euler-Equations (16a) and (16b) are the equilibrium equations of the problem in cylindrical coordinates. Solution of these equations, together with the kinematic boundary conditions for r, and the transversality condition (17) for 0 yield the response of the riser. Equation (16c) represents the meaning of the dummy variable d: 3F ad
-
2)td =
(18)
0
In the case d=0, the length constraint (16d) is binding. If, on the contrary, ,~--0, the length constraint is not binding, and may, in fact, be omitted. Let us now concentrate on the transversality condition (17) at the top and at the bottom: ZlTlax
--raO t T+
~1 + r '2 + (rO') 2
=0
(19)
Since at the bottom, z=0, the offset r=0, the natural boundary condition will always be satisfied at the bottom. At the top of the riser, Z=Zmax, this equation has, in the most general case, three solutions: r=O
0'=0
(20)
A = -T~-op,/l~ v + r '2 + (rO') 2 In the first solution there is no rig offset, r=0. For passive mooring systems this is only possible if the significant wave height h~---O. Even if the rig has a dynamical positioning system, a mean offset r=0 is only possible for small hs-values. Since the wave heights are small, the resulting response angles will not be critical. Consequently, we may conclude that r=0 represents the 'trivial' case. When the length constraint is not binding (It--0), the third solution is impossible since TTop4:0. Consequently, when the stroke capacity is not exceeded, the only non-trivial solution is the second one: 19'--0. In the following we will consider only this unconstrained case (2~--0). 4. UNCONSTRAINED PROBLEM: DIFFERENTIAL EQUATIONS For the unconstrained problem, we set up the Euler-equations in terms of the natural coordinate: ~=z/z.... A prime now denotes differentiation with respect to ~: TrO 2 - l * r ' - Tr' - qZ2max c o s ( O q - O) = 0 7~r20" + 2Trr'O" + Tr20 ' + qrZZmax sin(Oq-O) = 0
(21)
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et al.
The following boundary conditions apply to this problem: r(0) = 0
I"(1 ) = OS
(22)
0'(1) = 0 where OS denotes the horizontal displacement at the top of the riser. A convenient method to solve this problem is based on the model formulation by Garrett in Cartesian coordinates (Garrett, 1993). The natural boundary condition is then satisfied using an iterative approach. For a three-dimensional tensioned string (small angle assumption), the equilibrium equations in Cartesian coordinates are given by: d ( dr~] qx(-, r(z), 2 dC T dff] = ~ 0(Z))'Zmax
(23)
d( t
d~ T d ~ / = -q,,(z, r(z), O(z)).z~,x A first estimate of the offset direction is to be assumed and the differential Equation (23) can be solved. A simple iteration scheme acting on this offset direction will then satisfy the natural boundary condition 0"=0. 5. UNCONSTRAINED PROBLEM: GALERKIN SOLUTION In the Galerkin-method, an approximate solution is obtained from the principle of virtual work on the basis of the Euler Equations (16a) and (16b) The Galerkin approach can be stated in the form: I
f (TrO,2 T, r,_ Tr , - qz 2..... cos(O u - O))6r(~)d~ 0 l
+ ~(T'r20 " + 2Trr'O" + TraO ' + qrz~,,a× sin(Oq-O))80(~)d~ = 0
(24)
0 I
I
¢:~ffl(O6r d~ + ff2(~8O d(~ = O 0
0
whereft andfe represent the left-hand sides of the Euler Equations (16a) and (16b) respectively. Now, select a trial function for r and 0, containing some unknown parameters as and bi, but satisfying the boundary conditions. Evaluate the integral expression (24), where the variational operator acts on the unknown parameters as and bj only. Formally, Equation (24) can be regrouped as: 1
I
I
I
0
0
0
0
(25)
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Since each variation of a parameter is completely arbitrary, each integral in Equation (25) must vanish, which yields the solution for the unknown parameters in the trial functions for r and 0. Equation (25) can then be written as a system of m+n+l equations. In the following this method is demonstrated for two families of trial functions.
5.1.
Solution based on a polynomial approximation
For this case, we select a polynomial as the trial function for both r and 0:
r(~) : ~
ai(
(26)
0(0 = ~'~ bj~
(27)
i=1 n
j=O
For these functions, the boundary conditions (22) are no longer satisfied automatically. Two additional conditions apply: in
r(1) = O S ~ ~ ai = OS
(28)
i=l n
0'(1 ) = O ~ ~ bj.j = 0
(29)
j=O
Because of these additional conditions, the variations of ai and bj are no longer arbitrary. The following relation must be satisfied:
i~)a 1 ~-~
- ~
i~ai
(30)
i=2 n
8b, = - ~ j . 6 b j
(3 l)
j=2
Substitute Equations (28)-(31) in Equation (24) and evaluate this integral. After regrouping of the terms, this yields a system of re+n-1 equations in as and b/ 1
f{(TrO'2-7"r'-Tr'-qZ2max c o s ( O q - O ) ) ( ( - O I d ~ = 0, for i = 2...m o 1
f { T'r20" + 2Trr'O" + Tr20 ' + qrZmax 2 sin( Oq-O))(~-j.O }d~ = 0, f o r j = 0, 2, 3 ..... n 0
(32) The m+n+l parameters ai and bj can then be found from Equations (28), (29) and (32).
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5.2.
Solution based on a trigonometric approximation
Select the following class of admissible functions: m
r(~) = OS~ + ~ a i
(33)
sin(iTrO
i=l
m l1
)
0(~) = b0 + ~ bj sin q - ~ )Tr~
(34)
i=1
It is clear that the boundary conditions (22) are satisfied for any values of the parameters ai and bj. Now, substitute Equations (33) and (34) in Equation (24) and evaluate this integral as a function of the unknown parameters ai and b/ 1
f{
( T r f f 2 - T ' r ' - Tr'-qZmax2 cos(0q-0))
sin(ilr~ai i=1
0
}
d~
1
+ f ( (T'r2ff + 2Trr'ff + Tr20 ' + qrZma× 2sin(Oq-O)).
(35)
0
x
(ltt
sin ( j - 2 )~r~ ~bj d~ = 0 j=l
Since each of the variations ~ and ~bj is arbitrary, each integral expression in Equation (35) must vanish. This yields a system of m+n+l equations for each 3a~ and 8bj: 1
f f
{(Trff 2 - T ' r ' - T r ' - q Z ~ a x coS(0q-0)) sin(iTrO}d~= 0 for i = 1..... m
(T'r2ff + 2Tr/O' + rr~O' + qrz~ax sin(0q-0))d~ = 0
0
(36) 6. UNCONSTRAINED PROBLEM: RITZ SOLUTION Another direct method to solve the system of differential Equation (21) is the Ritz method. In this case, a trial function, satisfying the boundary conditions (22), is substituted in the functional (without the length constraint) itself: I=U-V
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441
Zrllax
=
I { ~1 T(/2 + (rO')2)-qr
cos(Oq-O)}dz
(37)
0
= l(al . . . . .
bo..... b,,)
am,
The unknown parameters of the trial function are then estimated by direct minimization of this functional I: =°
8/= 0 ~
:
(38)
31
7.
EXAMPLE
7.1. Comparison of the different solution methods Consider, as an example, the following riser configuration: 0S=40 m, Ca=l.2, TBot=500 KN, Z,,ax=1000 m, 0 , ~ 7 5 kg/m, Dn~r=500 mm, and pw=1030 kg/m 3. Assume the riser to be subjected to a spiral current profile: the current amplitude and the phase angle vary linearly from 0 m/s and 0 ° at the bottom to 2 rods and 180 ° at the top. The different solution methods are compared and the main results are shown in Table 1 and Fig. 3. First, the problem is solved from the differential Equation (23) on the basis of the following directionality assumptions: • 3D: 3D-analysis, accounting for the natural boundary condition: /9' (Zmax)=0. The solution is obtained from the differential Equation (23). • IL: 3D-analysis, assuming that the rig offset is aligned with the top current. In this case, a kinematic boundary condition for the riser orientation 0 at the top of the riser is introduced: 0=-180°. The results in Table 1 show that the in-line assumption leads to somewhat underestimated results for the riser response. Now, the Galerkin method is applied using both polynomial and trigonometric shape functions. The numerical solution of the system of equations is obtained by least squares minimization of the left-hand sides of Equation Table 1. Comparison of the different solution methods Diff. Eqns. IL 3D r,,~[m] [°] ~,op [o] Obo,[°] O,op[°] a t , o,
45.5 5.6 -3.7 144 180
49.2 6.5 -4.6 127 147
Galerkin-polynomial m=2 n=2 m---4n=4
Galerkin-trigonometric m=l n=l m=5 n=5
Ritz-polynomial m=2 n=2 m=4 n=4
45.3 7.0 -2.4 116 140
45.1 6.1 -1.5 113 127
51.5 8.7 -4.1 1 l0 140
46.9 5.9 -3.8 135 143
47.2 5.9 -3.0 130 137
53.3 6.4 -5.7 121 144
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L. Huyse et al.
\
1000 Differential equations - - - Galerkin 800 --Ritz
c-
400
_
\~ ~1 / ' /
,
--
~
]/
Differential equations ]
/
o
200
/1
/
_
600
1000 x
i /
//
800
----
"~
0
e-
-
--
600
-
Galerkin Ritz
z/// z//C
,/Z
//7
,Ill'//
400
~5
J
/ /////~
-
~
Z/"
/ J
'/
200
// I\\
.7 Jl
0 0
10
I
I
I
I
0 /
20
30
40
50
120
Offset r [m]
I \ \1 125
130
135
I
I
140
145
Orientation u [°]
Fig. 3. Comparison of the fourth order polynomial approximations.
(36). An evaluation of the left-hand sides provides for a simple check for the numerical quality of the solution. The integrals are evaluated with the trapezium rule, 100 integration steps are used. Finally, the Ritz solution is calculated for polynomial trial functions in Equations (26) and (27). From the plots in Fig. 3, we may conclude that for the displacement r, the higher order terms in the various trial functions are not very important. It is to be noted, however, that the numerical evaluation of the coefficients ai and bj becomes rather difficult for higher order m or n; appropriate scaling and starting values are required. The Galerkin method yields responses which are too small. The underestimation of the maximum riser displacement is about 5%. It is interesting to note that the Ritz and Galerkin solution are slightly different in this example: the Ritz solution is based on an energy method, while the Galerkin solution is based on the principle of virtual work (see Table 2). It is also observed that even though the currents rotate over 180 ° , the riser itself stays within a small sector. This suggests that, in this case, a further simplification of the analysis into 2D may be justified. When the exponent in the current velocity shape function increases, and the top currents become more and more important, this 2D-plane will approach the plane of the top current direction. 7.2. Error estimation An appropriate definition of the measure of error is necessarily application dependent. An extensive discussion of the possible sources of error and their estimation is given in (Mikhlin, 1964, 1991). In this discussion it is assumed that algorithm, rounding and perturbation errors are negligible compared to the approximation error introduced by the direct solution method (Galerkin or Ritz). For the Galerkin method this can be checked to some extent since the sum of square errors for the Equation (36) gives an indication about the
443
A static drilling riser m o d e l T a b l e 2. Trial function coefficients for the G a l e r k i n and Ritz solutions
Galerkin Polynomial
Order
Variable
a01b0
a~lb,
a2[b2
a,lb3
2
r(z) 0(z) r(z) 0(z) r(z) 0(z) r(z) 0(z) r(z) 0(z) r(z) 0(z)
0.000 2.021 0.000 2.352 0.000 1.969 0.000 2.233 0.000 1.915 0.000 2.119
121.8 0.849 102.4 - 1.029 21.10 0.241 20.45 0.088 151.7 1.043 112.4 0.563
- 81.76 -0.425 -47.09 2.498 . . -2.084 -0.048 - 111.7 -0.522 - 1.948 -2.143
12.58 - 1.345 . . . . 1.017 0.011
4 Galerkin Trigonometric
1 5
Ritz
2
Polynomial
4
a,lb,
a4[b4 -27.90 0.017 . . -0.119 -0.002
-
0.219 0.002
-
-
-
-73.31 4.160
2.831 -2.190
-
Table 3. Error estimation for the different solution methods Galerkin-polynomial m=2 n=2 m=4 n--4
r [m] c~ [°] 0 [o] r,.a~ [m] a .... [°]
3.8 2.2 11 3.9 0.5
Galerkin-trigonometric m=l n=l m=5 n=5
U n i f o r m closeness 3.3 4.5 3.0 0.9 3.1 1.6 8 20 11 Error on design variables r,,o~ and c~,,o~ 2.3 4.1 2.0 0.6 0.4 0.6
Ritz-polynomial m=2 n=2 m=4 n=4
6.4 2.0 17
4.6 1.1 5
2.3 2.2
4.2 0.1
numerical quality of the solution. For the Ritz method the value of the total potential energy can be used for this purpose. Consequently, the solution obtained from the differential equations is considered 'exact'. Only a basic error estimation is performed here. If we restrict ourselves in this example to operability criteria for the drilling riser, the error on the orientation 0 is not important. In Table 3 the uniform closeness, i.e. the maximum difference between the exact function value and its approximation, for r and r'=a are listed, together with the error on the design variables Fmax and Otmax. 8.
CONCLUSIONS
A natural boundary condition for the 3D-analysis of a drilling riser is derived from variational principles assuming the vessel is free to move along a circle. The formulation of this problem in terms of a direct solution method causes numerical difficulties. The problem is highly non-linear and requires extremely adequate scaling. The integrands in each of the equations derived from Equation (24) may differ by several orders of magnitude for each coefficient to be estimated. Substitution of this natural boundary condition in the differential equations formulated in Cartesian coordinates is convenient. A simple iteration scheme on the natural boundary condition 0'=0 can then account for the natural boundary condition.
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Acknowledgements This research was made possible through the financial support of the Natural Science and Engineering Council of Canada (NSERC operating grant to the third author and support to the first author) and Mobil Exploration and New Ventures. REFERENCES Cook, M.F. and Gardner, T.N. 1985. Riser and Vessel Motion Data From Deepwater Drilling Operation, 17th Annual Offshore Technology Conference, OTC 5004, Houston, TX. De Oliveira, J.G., Goto, Y. and Okamoto, T. 1985. Theoretical and Methodological Approaches to Flexible Pipe Design and Application, 17th Annual Offshore Technology Conference, OTC 5021, Houston, TX. Garrett, D.L. 1993. Riser Development----Characterization of Current Loads on a Drilling Riser, File 054-1-. 1993. TN1, Dallas E&P Engineering, Mobil RDC, Dallas, TX. Huang, T. and Chucheepsakui, S. 1984. Large Displacement Analysis of a Marine Riser, 3rd Offshore Mechanics and Arctic Engineering Conference, New Orleans, LA. Huang, T. and Rivero, C.E. 1986. On the Functional in a Marine Riser Analysis, 5th Offshore Mechanics and Arctic Engineering Conference, Tokyo, Japan. Huang, T. and Saha, K.G. 1989. Polar Coordinates and Riser Analysis, 8th Offshore Mechanics and Arctic Engineering Conference, The Hague, The Netherlands. Huyse, L., Maes, M.A., Gu, G.Z. and Johnson, R.C. 1995. Response Based Design Criteria for Marine Risers, ! 5th International Conference on Offshore Mechanics and Arctic Engineering, Copenhagen, DK Conference, 317-339. Marsh, G.L., Denison, E.B. and Pekera, S.J. 1984. Marine Riser System for 7500-Ft. Water Depth, 16th Annual Offshore Technology Conference, OTC 4750, Houston, TX. Mikhlin, S.G. 1964. Variational Methods in Mathematical Physics, Pergamon Press, New York, NY. Mikhlin, S.G. 1991. Error Analysis in Numerical Processes, John Wiley & Sons, New York, NY. Mura, T. and Koya, T. 1992. Variational Methods in Mechanics, Oxford, University Press, NY. Sarpkaya, T. and Isaacson, M. 1981. Mechanics of Wave Forces on Offshore Structures, Van Nostrand Reinhold Company, New York, NY.
Pergamon S0029-8018(96)00021-2
A STATIC DRILLING RISER MODEL USING FREE BOUNDARY
CONDITIONS
Luc Huyse,*§ Mansa C. Singh# and Marc A. Maes~ *Civil Engineering Department, The University of Calgary, 2500 University Drive N.W., Calgary, Alberta, Canada T2N 1N4 tMechanical Engineering Department, The University of Calgary, 2500 University Drive N.W., Calgary, Alberta, Canada T2N 1N4 $Civil Engineering Department, The University of Calgary, 2500 University Drive N.W., Calgary, Alberta, Canada T2N 1N4
(Received 3 February 1996; accepted in final form 27 March 1996) Abstraet--A static three-dimensional analytical method for drilling risers experiencing large displacements and slip at the top joint is presented. The riser is described in cylindrical coordinates as a three-dimensional tensioned string, without bending or torsional stiffness. The vertical vessel displacement is not taken into account. The equilibrium configuration is obtained from the stationary condition of the total potential energy functional. Kinematic boundary conditions are introduced for the radial displacement only. The riser orientation at the top directly follows from the natural boundary conditions based on the variational treatment of the energy functional. Copyright © 1997 Elsevier Science Ltd
1.
INTRODUCTION: DRILLING RISERS
In recent years, marine risers have attracted considerable attention in offshore engineering since, on the one hand, they are a vital link between the floating platform and the subsea hole, and because of their complex analysis on the other hand. Floating structures are economically attractive for deep water drilling and production. Corresponding challenges arise from the complexity of the interaction of the structure with the environmental loading (De Oliveira et al., 1985). A drilling riser is a conductor pipe, connecting the floating platform and the bore hole (see Fig. 1). At the top, the riser pipe ends at a telescopic joint beneath the vessel. This slip joint allows change in riser length as the vessel heaves or moves laterally. The operability of drilling risers is expressed in terms of maximum angles from the vertical. To increase the riser stiffness, and thus limit these angles, tensioners are installed at the top joint. Buoyancy devices may be added to limit the maximum tension in the riser pipe. An excellent description of drilling risers can be found in Sarpkaya and Isaacson (1981). 2.
PROBLEM FORMULATION
Most static analyses for risers are concerned with the maximum riser response in a vertical plane. However, one must note that the riser may move in various directions at
§Corresponding author. 431
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le
Guide lines
Tensioners Choke and kill lines _ . ~ Riser pipe
Ball joint
.~BOP
L'-"z,. . . . . . . .
•
Fig. 1. Schematic of a drilling riser (Reprinted from Sarpkaya and Isaacson, 1981 with permission from the publisher).
various elevations due to the omnidirectionality of the waves and currents. Such an analysis is extremely complex since it requires information about the waves and the currents as a function of the coordinates x, y and z, in addition to the rig motion (Sarpkaya and Isaacson, 1981). In preliminary design, however, a static response model may be used to perform basic checks to investigate operability criteria (Marsh et al., 1984). For drilling risers, the operability limits are given in terms of the maximum riser angle with the vertical axis (Huyse et al., 1995). In the preliminary design stage, it is common practice to estimate the rig offsets from a set of Response Amplitude Operators (RAO) which relate the rig or vessel response to the significant wave height h~. This is done through the calculation of the mean environmental loading, including the riser and the mooring loads. This offset is then introduced as a boundary condition in the differential equations describing the riser (Garrett, 1993). Since the riser may experience slip at the top joint, the total arc length in the equilibrium
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position is unknown. It is convenient to choose the distance along the vertical axis as the independent variable. The variables r(z) and O(z) then describe the riser horizontal displacement and orientation at a given elevation z in the deformed state (see Fig. 2). In many cases, the main problem which prevents the analysis from being extended into 3 dimensions, is the lack of directionality data for the rig offsets (Cook and Gardner, 1985). Because of the very complex nature of the mooring and the hydrostatic forces, it is very difficult to predict the direction of the rig offset. In the absence of other data, the directionality problem mentioned before may be overcome using the free boundary conditions presented in this formulation. The following assumptions are made in this analysis: 1. The riser has neither bending nor torsional stiffness. 2. Even though the displacements are large, the strains remain small and the material behaves linearly elastic. 3. The wave and current forces have no vertical component; the vessel moves in the horizontal direction only. 4. The angle between the riser and the vertical axis is small. In practice, this assumption is valid for drilling risers only. 3. VARIATIONAL FORMULATION 3.1.
Strain energy in the riser
Since we limit the riser model to a simple tensioned string, only axial deformation is to be included in the internal virtual work formulation. Consider the distance s along the riser to be measured positive up from the bottom of the riser (see Fig. 2). The relative change in arc length, due to the horizontal riser displacement (r, 0) is given by:
U ~ max) y (Zmax) Zmax
- - - - --
- ~ x (Zmax)
. . . . .
~7
Y-__ "~
y (z)
~ ~
x z
r (z)
Fig. 2. Cylindrical coordinate system.
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ds-dz - ~/1 + r '2 + ( r 0 " ) 2 - 1 dz
(1)
where a prime denotes differentiation with respect to the vertical distance z, measured from the bottom of the riser towards the top. Accounting for the small angle approximation, this is simplified into: d s - d z .~ 1 (r,2 + (rO,)2)
dz
(2)
2
This change in length is caused by two different phenomena: strain of the riser without slip and slip of the riser without strain variation (Huang and Rivero, 1986; Huang and Chucheepsakui, 1984; Huan and Saha, 1989). The first mode stores strain energy in the riser, and for the second mode virtual work is required. For both modes the riser tension is driving the process. Since the actual strains are not of primary interest here but only the global deformations, there is no need to uncouple both effects. For small angles the riser tension T is given as: T-TTop + Priserg(Z--Zmax)
(3)
where TTop is the effective tension at the top of the riser, P, ser is the riser mass per unit length, and Zmax denotes the vertical coordinate of the slip joint, i.e. the top of the riser. Consequently, the sum of the strain energy and the internal virtual work per unit vertical length due to the slip at the top joint is given by: 1
f o = ~ (TTop + Priserg(Z--Zmax))(r '2 + (r0J) 2)
(4)
The total 'strain energy' along the riser becomes then: Zmax P
u = ~ Uo dz
(5)
,/
0
3.2. Work o f the external forces Since vertical displacements are not considered, the work of external forces is due to the current force acting over a displacement r in the horizontal plane. The work done by these currents per unit vertical length of the riser is: d V = q.r dz =
Ilqlllldlcos(Oq- O)dz= qr
cos(Oq- O)dz
(6)
where q i s the drag force per unit length of the riser, Oq the direction of the current, and = ~1~ + ~ the horizontal displacement of the riser at depth z. The drag force q per unit vertical length can be expressed as (Sarpkaya and Isaacson, 1981): 1
q = ~ pwfdOrisorvllvll
(7)
where pw is the mass density of the sea water, C~ is the drag coefficient, D the riser drag
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diameter, and v the current velocity at the depth z. The total work done by the current forces is then: Zmax
V=
I
Zmax
~ pwGDnserVllvll.rdz =
0
f
qr cos( Oq-O)dz
(8)
0
Constraint equation Due to the slip at the top joint, the total riser length is initially unknown. However, the maximum riser length /max is limited by the stroke capacity of the telescopic joint. This supplementary inequality constraint for the riser length is: 3.3.
Zmax
Zmax
0
J 0
Introducing the positive, dummy variable d 2, this inequality constraint is-transformed into an equality constraint: Zmax
Zmax
fds= f(l+r'2+(rlT)2dz+d2-1max=O 0
(10)
0
This dummy variable d has the following physical interpretation: d=0: the riser length is exactly equal to the maximum allowable length/max, the telescopic joint at the rig is fully stretched. d4=0: the optimal solution lies within the bounded region. The length constraint is not binding in this case, since there is still some stroke capacity left. 3.4.
Functional: the total potential energy
According to the principle of minimum of the total potential energy, the first variation of the total potential energy functional must vanish at the equilibrium position of the riser. Because of the subsidiary condition (10), we make use of the Lagrange multiplier A and construct the following functional I (Mura and Koya, 1992): I = U - V + Ag(d) Zmax
-,-,rO',a,-qr 0
=
fF(r,
r', O, 0", d, h)dz
For I to assume an extremum value, its first variation must vanish:
(ll)
L. Huyse et al.
436 Zmax
31=
~rr 3r + -oO 30 + ~d 6d + o-A 3A + ~r, 3 / + ~
3ff
dz=0
(12)
0
The last two terms in Equation (12) are integrated by parts and substituted for the kinematic boundary conditions for r. Since r is prescribed on the boundaries, its first variation vanishes. Consequently, the following formulas are obtained: Zmax
Znlax
~0r3 / d z = -
(13)
dzz ~r' 6rdz
0
0
Zmax
Zmax
Or' 30'dz =Off
~ 0
0
~ff 60 dz
0
Substituting Equation (13) in Equation (12), the following expression for I is obtained: Zmax
31= f{[00F
OF d (~r'r')]3r + [~F dz
DE 0 d (ff~)]3 dz
(14)
0
OF OF } OF 30 zm"X + ffd M + 0A 6)t d z + 0 f f =0 0
For a first class of admissible functions the equation: OF (m°x 0 ~ 30
= 0
(15)
0
will hold since 30=0 at the top and the bottom of the riser. For this class of functions the riser orientation is prescribed at the top and the bottom. Because of Equation (15), the functional I(r, O, d, A), defined in Equation (11) will assume an extremum if, and only if: OF Or
d (~vFF/ dz \Or ] = 0
(16a)
OF O0
d(OF) dz ~ = 0
(16b)
OF Od
0
OF -0 OA
(16c) (16d)
For another class of admissible functions, the riser orientation 0 is not prescribed on the boundaries. In this case, Equation (15) must still hold because of the Equations (16a)-
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(16d). Since for this class of functions 80 4= 0, Equation (15) is satisfied only if, at both the top and the bottom of the riser: ~F - 0 30"
(17)
The Euler-Equations (16a) and (16b) are the equilibrium equations of the problem in cylindrical coordinates. Solution of these equations, together with the kinematic boundary conditions for r, and the transversality condition (17) for 0 yield the response of the riser. Equation (16c) represents the meaning of the dummy variable d: 3F ad
-
2)td =
(18)
0
In the case d=0, the length constraint (16d) is binding. If, on the contrary, ,~--0, the length constraint is not binding, and may, in fact, be omitted. Let us now concentrate on the transversality condition (17) at the top and at the bottom: ZlTlax
--raO t T+
~1 + r '2 + (rO') 2
=0
(19)
Since at the bottom, z=0, the offset r=0, the natural boundary condition will always be satisfied at the bottom. At the top of the riser, Z=Zmax, this equation has, in the most general case, three solutions: r=O
0'=0
(20)
A = -T~-op,/l~ v + r '2 + (rO') 2 In the first solution there is no rig offset, r=0. For passive mooring systems this is only possible if the significant wave height h~---O. Even if the rig has a dynamical positioning system, a mean offset r=0 is only possible for small hs-values. Since the wave heights are small, the resulting response angles will not be critical. Consequently, we may conclude that r=0 represents the 'trivial' case. When the length constraint is not binding (It--0), the third solution is impossible since TTop4:0. Consequently, when the stroke capacity is not exceeded, the only non-trivial solution is the second one: 19'--0. In the following we will consider only this unconstrained case (2~--0). 4. UNCONSTRAINED PROBLEM: DIFFERENTIAL EQUATIONS For the unconstrained problem, we set up the Euler-equations in terms of the natural coordinate: ~=z/z.... A prime now denotes differentiation with respect to ~: TrO 2 - l * r ' - Tr' - qZ2max c o s ( O q - O) = 0 7~r20" + 2Trr'O" + Tr20 ' + qrZZmax sin(Oq-O) = 0
(21)
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The following boundary conditions apply to this problem: r(0) = 0
I"(1 ) = OS
(22)
0'(1) = 0 where OS denotes the horizontal displacement at the top of the riser. A convenient method to solve this problem is based on the model formulation by Garrett in Cartesian coordinates (Garrett, 1993). The natural boundary condition is then satisfied using an iterative approach. For a three-dimensional tensioned string (small angle assumption), the equilibrium equations in Cartesian coordinates are given by: d ( dr~] qx(-, r(z), 2 dC T dff] = ~ 0(Z))'Zmax
(23)
d( t
d~ T d ~ / = -q,,(z, r(z), O(z)).z~,x A first estimate of the offset direction is to be assumed and the differential Equation (23) can be solved. A simple iteration scheme acting on this offset direction will then satisfy the natural boundary condition 0"=0. 5. UNCONSTRAINED PROBLEM: GALERKIN SOLUTION In the Galerkin-method, an approximate solution is obtained from the principle of virtual work on the basis of the Euler Equations (16a) and (16b) The Galerkin approach can be stated in the form: I
f (TrO,2 T, r,_ Tr , - qz 2..... cos(O u - O))6r(~)d~ 0 l
+ ~(T'r20 " + 2Trr'O" + TraO ' + qrz~,,a× sin(Oq-O))80(~)d~ = 0
(24)
0 I
I
¢:~ffl(O6r d~ + ff2(~8O d(~ = O 0
0
whereft andfe represent the left-hand sides of the Euler Equations (16a) and (16b) respectively. Now, select a trial function for r and 0, containing some unknown parameters as and bi, but satisfying the boundary conditions. Evaluate the integral expression (24), where the variational operator acts on the unknown parameters as and bj only. Formally, Equation (24) can be regrouped as: 1
I
I
I
0
0
0
0
(25)
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Since each variation of a parameter is completely arbitrary, each integral in Equation (25) must vanish, which yields the solution for the unknown parameters in the trial functions for r and 0. Equation (25) can then be written as a system of m+n+l equations. In the following this method is demonstrated for two families of trial functions.
5.1.
Solution based on a polynomial approximation
For this case, we select a polynomial as the trial function for both r and 0:
r(~) : ~
ai(
(26)
0(0 = ~'~ bj~
(27)
i=1 n
j=O
For these functions, the boundary conditions (22) are no longer satisfied automatically. Two additional conditions apply: in
r(1) = O S ~ ~ ai = OS
(28)
i=l n
0'(1 ) = O ~ ~ bj.j = 0
(29)
j=O
Because of these additional conditions, the variations of ai and bj are no longer arbitrary. The following relation must be satisfied:
i~)a 1 ~-~
- ~
i~ai
(30)
i=2 n
8b, = - ~ j . 6 b j
(3 l)
j=2
Substitute Equations (28)-(31) in Equation (24) and evaluate this integral. After regrouping of the terms, this yields a system of re+n-1 equations in as and b/ 1
f{(TrO'2-7"r'-Tr'-qZ2max c o s ( O q - O ) ) ( ( - O I d ~ = 0, for i = 2...m o 1
f { T'r20" + 2Trr'O" + Tr20 ' + qrZmax 2 sin( Oq-O))(~-j.O }d~ = 0, f o r j = 0, 2, 3 ..... n 0
(32) The m+n+l parameters ai and bj can then be found from Equations (28), (29) and (32).
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5.2.
Solution based on a trigonometric approximation
Select the following class of admissible functions: m
r(~) = OS~ + ~ a i
(33)
sin(iTrO
i=l
m l1
)
0(~) = b0 + ~ bj sin q - ~ )Tr~
(34)
i=1
It is clear that the boundary conditions (22) are satisfied for any values of the parameters ai and bj. Now, substitute Equations (33) and (34) in Equation (24) and evaluate this integral as a function of the unknown parameters ai and b/ 1
f{
( T r f f 2 - T ' r ' - Tr'-qZmax2 cos(0q-0))
sin(ilr~ai i=1
0
}
d~
1
+ f ( (T'r2ff + 2Trr'ff + Tr20 ' + qrZma× 2sin(Oq-O)).
(35)
0
x
(ltt
sin ( j - 2 )~r~ ~bj d~ = 0 j=l
Since each of the variations ~ and ~bj is arbitrary, each integral expression in Equation (35) must vanish. This yields a system of m+n+l equations for each 3a~ and 8bj: 1
f f
{(Trff 2 - T ' r ' - T r ' - q Z ~ a x coS(0q-0)) sin(iTrO}d~= 0 for i = 1..... m
(T'r2ff + 2Tr/O' + rr~O' + qrz~ax sin(0q-0))d~ = 0
0
(36) 6. UNCONSTRAINED PROBLEM: RITZ SOLUTION Another direct method to solve the system of differential Equation (21) is the Ritz method. In this case, a trial function, satisfying the boundary conditions (22), is substituted in the functional (without the length constraint) itself: I=U-V
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Zrllax
=
I { ~1 T(/2 + (rO')2)-qr
cos(Oq-O)}dz
(37)
0
= l(al . . . . .
bo..... b,,)
am,
The unknown parameters of the trial function are then estimated by direct minimization of this functional I: =°
8/= 0 ~
:
(38)
31
7.
EXAMPLE
7.1. Comparison of the different solution methods Consider, as an example, the following riser configuration: 0S=40 m, Ca=l.2, TBot=500 KN, Z,,ax=1000 m, 0 , ~ 7 5 kg/m, Dn~r=500 mm, and pw=1030 kg/m 3. Assume the riser to be subjected to a spiral current profile: the current amplitude and the phase angle vary linearly from 0 m/s and 0 ° at the bottom to 2 rods and 180 ° at the top. The different solution methods are compared and the main results are shown in Table 1 and Fig. 3. First, the problem is solved from the differential Equation (23) on the basis of the following directionality assumptions: • 3D: 3D-analysis, accounting for the natural boundary condition: /9' (Zmax)=0. The solution is obtained from the differential Equation (23). • IL: 3D-analysis, assuming that the rig offset is aligned with the top current. In this case, a kinematic boundary condition for the riser orientation 0 at the top of the riser is introduced: 0=-180°. The results in Table 1 show that the in-line assumption leads to somewhat underestimated results for the riser response. Now, the Galerkin method is applied using both polynomial and trigonometric shape functions. The numerical solution of the system of equations is obtained by least squares minimization of the left-hand sides of Equation Table 1. Comparison of the different solution methods Diff. Eqns. IL 3D r,,~[m] [°] ~,op [o] Obo,[°] O,op[°] a t , o,
45.5 5.6 -3.7 144 180
49.2 6.5 -4.6 127 147
Galerkin-polynomial m=2 n=2 m---4n=4
Galerkin-trigonometric m=l n=l m=5 n=5
Ritz-polynomial m=2 n=2 m=4 n=4
45.3 7.0 -2.4 116 140
45.1 6.1 -1.5 113 127
51.5 8.7 -4.1 1 l0 140
46.9 5.9 -3.8 135 143
47.2 5.9 -3.0 130 137
53.3 6.4 -5.7 121 144
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\
1000 Differential equations - - - Galerkin 800 --Ritz
c-
400
_
\~ ~1 / ' /
,
--
~
]/
Differential equations ]
/
o
200
/1
/
_
600
1000 x
i /
//
800
----
"~
0
e-
-
--
600
-
Galerkin Ritz
z/// z//C
,/Z
//7
,Ill'//
400
~5
J
/ /////~
-
~
Z/"
/ J
'/
200
// I\\
.7 Jl
0 0
10
I
I
I
I
0 /
20
30
40
50
120
Offset r [m]
I \ \1 125
130
135
I
I
140
145
Orientation u [°]
Fig. 3. Comparison of the fourth order polynomial approximations.
(36). An evaluation of the left-hand sides provides for a simple check for the numerical quality of the solution. The integrals are evaluated with the trapezium rule, 100 integration steps are used. Finally, the Ritz solution is calculated for polynomial trial functions in Equations (26) and (27). From the plots in Fig. 3, we may conclude that for the displacement r, the higher order terms in the various trial functions are not very important. It is to be noted, however, that the numerical evaluation of the coefficients ai and bj becomes rather difficult for higher order m or n; appropriate scaling and starting values are required. The Galerkin method yields responses which are too small. The underestimation of the maximum riser displacement is about 5%. It is interesting to note that the Ritz and Galerkin solution are slightly different in this example: the Ritz solution is based on an energy method, while the Galerkin solution is based on the principle of virtual work (see Table 2). It is also observed that even though the currents rotate over 180 ° , the riser itself stays within a small sector. This suggests that, in this case, a further simplification of the analysis into 2D may be justified. When the exponent in the current velocity shape function increases, and the top currents become more and more important, this 2D-plane will approach the plane of the top current direction. 7.2. Error estimation An appropriate definition of the measure of error is necessarily application dependent. An extensive discussion of the possible sources of error and their estimation is given in (Mikhlin, 1964, 1991). In this discussion it is assumed that algorithm, rounding and perturbation errors are negligible compared to the approximation error introduced by the direct solution method (Galerkin or Ritz). For the Galerkin method this can be checked to some extent since the sum of square errors for the Equation (36) gives an indication about the
443
A static drilling riser m o d e l T a b l e 2. Trial function coefficients for the G a l e r k i n and Ritz solutions
Galerkin Polynomial
Order
Variable
a01b0
a~lb,
a2[b2
a,lb3
2
r(z) 0(z) r(z) 0(z) r(z) 0(z) r(z) 0(z) r(z) 0(z) r(z) 0(z)
0.000 2.021 0.000 2.352 0.000 1.969 0.000 2.233 0.000 1.915 0.000 2.119
121.8 0.849 102.4 - 1.029 21.10 0.241 20.45 0.088 151.7 1.043 112.4 0.563
- 81.76 -0.425 -47.09 2.498 . . -2.084 -0.048 - 111.7 -0.522 - 1.948 -2.143
12.58 - 1.345 . . . . 1.017 0.011
4 Galerkin Trigonometric
1 5
Ritz
2
Polynomial
4
a,lb,
a4[b4 -27.90 0.017 . . -0.119 -0.002
-
0.219 0.002
-
-
-
-73.31 4.160
2.831 -2.190
-
Table 3. Error estimation for the different solution methods Galerkin-polynomial m=2 n=2 m=4 n--4
r [m] c~ [°] 0 [o] r,.a~ [m] a .... [°]
3.8 2.2 11 3.9 0.5
Galerkin-trigonometric m=l n=l m=5 n=5
U n i f o r m closeness 3.3 4.5 3.0 0.9 3.1 1.6 8 20 11 Error on design variables r,,o~ and c~,,o~ 2.3 4.1 2.0 0.6 0.4 0.6
Ritz-polynomial m=2 n=2 m=4 n=4
6.4 2.0 17
4.6 1.1 5
2.3 2.2
4.2 0.1
numerical quality of the solution. For the Ritz method the value of the total potential energy can be used for this purpose. Consequently, the solution obtained from the differential equations is considered 'exact'. Only a basic error estimation is performed here. If we restrict ourselves in this example to operability criteria for the drilling riser, the error on the orientation 0 is not important. In Table 3 the uniform closeness, i.e. the maximum difference between the exact function value and its approximation, for r and r'=a are listed, together with the error on the design variables Fmax and Otmax. 8.
CONCLUSIONS
A natural boundary condition for the 3D-analysis of a drilling riser is derived from variational principles assuming the vessel is free to move along a circle. The formulation of this problem in terms of a direct solution method causes numerical difficulties. The problem is highly non-linear and requires extremely adequate scaling. The integrands in each of the equations derived from Equation (24) may differ by several orders of magnitude for each coefficient to be estimated. Substitution of this natural boundary condition in the differential equations formulated in Cartesian coordinates is convenient. A simple iteration scheme on the natural boundary condition 0'=0 can then account for the natural boundary condition.
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Acknowledgements This research was made possible through the financial support of the Natural Science and Engineering Council of Canada (NSERC operating grant to the third author and support to the first author) and Mobil Exploration and New Ventures. REFERENCES Cook, M.F. and Gardner, T.N. 1985. Riser and Vessel Motion Data From Deepwater Drilling Operation, 17th Annual Offshore Technology Conference, OTC 5004, Houston, TX. De Oliveira, J.G., Goto, Y. and Okamoto, T. 1985. Theoretical and Methodological Approaches to Flexible Pipe Design and Application, 17th Annual Offshore Technology Conference, OTC 5021, Houston, TX. Garrett, D.L. 1993. Riser Development----Characterization of Current Loads on a Drilling Riser, File 054-1-. 1993. TN1, Dallas E&P Engineering, Mobil RDC, Dallas, TX. Huang, T. and Chucheepsakui, S. 1984. Large Displacement Analysis of a Marine Riser, 3rd Offshore Mechanics and Arctic Engineering Conference, New Orleans, LA. Huang, T. and Rivero, C.E. 1986. On the Functional in a Marine Riser Analysis, 5th Offshore Mechanics and Arctic Engineering Conference, Tokyo, Japan. Huang, T. and Saha, K.G. 1989. Polar Coordinates and Riser Analysis, 8th Offshore Mechanics and Arctic Engineering Conference, The Hague, The Netherlands. Huyse, L., Maes, M.A., Gu, G.Z. and Johnson, R.C. 1995. Response Based Design Criteria for Marine Risers, ! 5th International Conference on Offshore Mechanics and Arctic Engineering, Copenhagen, DK Conference, 317-339. Marsh, G.L., Denison, E.B. and Pekera, S.J. 1984. Marine Riser System for 7500-Ft. Water Depth, 16th Annual Offshore Technology Conference, OTC 4750, Houston, TX. Mikhlin, S.G. 1964. Variational Methods in Mathematical Physics, Pergamon Press, New York, NY. Mikhlin, S.G. 1991. Error Analysis in Numerical Processes, John Wiley & Sons, New York, NY. Mura, T. and Koya, T. 1992. Variational Methods in Mechanics, Oxford, University Press, NY. Sarpkaya, T. and Isaacson, M. 1981. Mechanics of Wave Forces on Offshore Structures, Van Nostrand Reinhold Company, New York, NY.