Marine Technology, Vol. 19, No. 1, Jan. 1982, pp. 73-82

Problems in Marine Riser Design

Michael M. Bernitsas 1

Major problems associated with the design of marine risers are identified. Two of these problems, namely, the mathematical modeling of the riser's dynamic behavior and the prediction of the hydrodynamic forces exerted on the riser, are studied. The first problem is studied with the aid of a set of consistent equations describing three-dimensional, nonlinear, large deflections of a tubular beam under tension, and distributed internal and external variable static pressure loads. The analysis of the second problem is based on the Mo- rison formula and experiments. Particular emphasis is given to the identification of flows which cannot be modeled by the Morison equation. For this purpose, experiments have been conducted at the MIT Towing Tank, and the forces exerted on a harmonically oscillating circular cylinder, in any direction 0 with respect to uniform current, have been measured. For the specific values of Reynolds number, reduced velocity, and Keulegan-Carpenter number chosen, significant changes of the hydrodynamic force with angle 0 were ob- served, In addition, force components unpredictable by the Morison equation were measured.

Introduction

THE EVER-INCREASING demand for new energy resources has extended drilling operations to offshore sites. The first offshore drilling facility was installed in 1949, at a 20-ft (6 m) depth. However, it was not until the late 1950's that the marine riser concept was applied. Since then, the expansion of the offshore oil industry has been very fast and it is estimated that by the early 1980's, 30 to 35 percent of the world's crude oil will be produced from beneath the seas [1]. 2

Designs capable of operating in deeper waters and more hostile environments are required by the industry and are attainable with today's technology. However, deep-sea drilling creates a number of new problems which must be resolved before we are able to successfully design a structure capable of such an opera- tion. One of these is the design of marine risers.

The marine riser, the various components of the offshore fa- cility which have some influence on the riser design, and the en- vironmental conditions are described in the first section. In ad- dition, major problems of the design of marine risers are identi- fied.

Forces are exerted on the riser by the drill ship, the tensioning system, the circulating mud, the end systems, the weight of the riser, the hydrostatic pressure, the drill string, the kill and choke lines, the surface waves, and the ocean current. A comprehensive mathematical model of the riser's dynamic behavior is presented in the second section. The issues of the effect of the internal and external static pressure on the riser's rigidity are also ex- plained.

The hydrodynamic force exerted on a riser in a general two- dimensional flow is a function of the body geometry, the fluid properties, the history of the flow and the instantaneous flow characteristics (velocity, acceleration, etc.) [2]. In the third sec- tion, experimental results available in the literature are briefly reviewed and our ability to predict the hydrodynamic force ex- erted on a circular cylinder using the semi-empirical Morison formula is discussed. An effort is made to identify the limits of applicabil ity of this approach, by finding some flows and values of the nondimensional parameters for which this method gives poor results. For this reason, the forces exerted on a harmonically

1 Massachusetts Institute of Technology, Cambridge, Massachusetts; presently at The University of Michigan, Ann Arbor, Michigan.

2 Numbers in brackets designate References at end of paper. Presented at the May 11, 1979 meeting of the New England Section

of THE SOCIETY OF NAVAL ARCHITECTS AND MARINE ENGINEERS.

oscillating circular cylinder, in any direction 0 with respect to a uniform current, have been measured experimentally. The ex- periments have been conducted for specific values of Reynold's number, Re, ratio of amplitude of oscillation to cylinder's di- ameter, So~D, reduced velocity Uo/fo/D, and angle 0 varying between 0 and 90 deg in 15-deg increments.

The marine riser design problem

Description of the riser system. The configuration of a marine riser system varies depending on the type of the sup- porting structure, the site of operation, and the environmental conditions. However, the concept of the riser design hardly changes. The six major components which make up the riser system are as follows (see Fig. 1):

1. The marine riser is a long tubular beam connecting the supporting structure with the wellhead at the seabed and is composed of rigid steel pipes with an average length of 40 ft (12 m) and outer diameter between 16 and 42 in. (40 and 106 cm). These pipes, made out of forged weldable steel, are connected by the riser connectors which are designed to minimize installation time and to provide joints able to withstand high tension loads. In areas of high bending stresses--near the end systems--flexible joints are used instead. The riser diameter, determined by the size of the BOP stack and the wellhead, determines the magni- tude of the external hydrodynamic force and along with the thickness of the pipes it defines the weight of the riser per unit length and the stress level in the material.

Usually, depending on the depth of the water and the size of the riser, additional buoyancy is needed to reduce the required tension at the top of the riser. This buoyancy is provided by buoyancy modules mounted on the riser pipes at the expense of a substantial increase of the hydrodynamic forces [3]:

2. The kill and choke lines are high-pressure pipes needed to control sudden increases of the well pressure. They run along the riser and are mounted directly on the connectors through which they exert concentrated moments and forces on the riser. In recent designs the kill and choke lines are mounted inside the riser.

3. The drilling mud circulates between the riser and the drill string and inside the latter. The mud exerts on the riser static pressure force, Coriolis and centrifugal forces due to the riser's local rotation, and vertical and torsional forces due to its viscosity. Of these forces, only the first is significant.

JANUARY 1982 0025-331618211901-0073500.4510 73

Z DRILL SH IP

TENSIONING SYSTEM

SL IP JO INT

JO INT

K ILL AND CHOKE L INES

MODULE

R ISER CONNECORS

JO INT

BLOWOUT PREVENTER

SEABED WELL HEAD

Fig. 1 Marine riser system

4. The drill string is the instrument that the riser protects from the environmental forces and guides to the wellhead. Its outer diameter is 4 to 6 in. (10 to 15 cm). It may come in contact and wear out the riser, unless the drill collars are properly spaced.

5. The upper-end system consists of a supporting structure, a tensioning system, a slip joint, a ball joint, and a variable-size buoy. The motion of the supporting structure--for example, a drill ship, a semisubmersible, or a submersible--has two com- ponents: first, a fast small-amplitude periodic one due to the surface waves, and second, a slow large-amplitude nonperiodic one due to the wind, the ocean currents, the wave second-order drift forces and the controller action. The second motion is lim- ited when a mooring system is used instead of a dynamic posi- tioning one. In addition, the supporting structure carries the tensioning system, which holds the upper end of the riser through the moonpool.

The slip joint compensates for the vessel heave motion and facilitates the elimination and damping of the vertical motions of the upper end of the riser.

The upper ball joint alleviates excessive bending moments due to rolling and pitching of the supporting structure.

The upper buoy is a variable buoyancy tank, found a few hundred feet below the free surface. It provides additional tension at the top of the riser but is subject to wave forces which may be significant. Consequently, it is designed to be of variable size and, depending on the environmental conditions, its volume is ad- justed to provide maximum tension within the structural limits.

6. The lower-end system consists of a ball joint, a BOP and a marine connector. The ball joint is a stress-alleviation device at the riser lower end, which is an area of high bending stresses.

The BOP, placed on top of the wellhead, has the following four functions [1]: (a) Close in around the drill pipe, circulate a con- ventional well kick, and sustain these conditions over a long pc- riod of time; (b) hang off the drill string and close in the well; (c) reestablish connection to the drill vessel, circulate the well, and sustain the pressure before the reconnection of the drill string; and (d) monitor the pressure relief kill and choke line.

The BOP stack is connected to the riser lower end through the marine connector and to the well through the wellhead con- nector.

Descr ipt ion of the environment. The sources of environ- mental excitations which may influence the riser directly, or in- directly by exerting forces on other components of the riser sys- tem, are the following:

(a) The ocean current. Its speed is usually not greater than 2.5 knots. Timewise the current is slowly varying, while depthwise it may change considerably and even reverse its direction. It ex- erts significant hydrodynamic forces on the riser and the sup- porting structure.

(b) The surface waves. They exert two types of forces on the drilling system: (i) fast small-amplitude oscillatory forces and (ii) slow large-amplitude ones. The wave spectrum is in general a function of the wave frequency w0 and its direction of propa- gation 00 [4].

In the presence of a current the wave properties are changed according to the equations

h = )t o 2 - - - U sinOo (1) 0

/[sin20oll/2 a =aO/[s~n20] (2)

sin0 sin0o/( - oSin0o) where co, )to, 0o, and ao are the original wave characteristics, that is, phase velocity, length, direction of propagation, and ampli- tude, respectively, c, X, O, a are the current-modified wave characteristics, and U is the velocity of the current [5].

Similarly the wave spectrum S(o~,O) is modified:

S(w,O)=I~o)2S(wo, Oo)~ (4)

(c) Strong winds and gusts. They offset the drill ship from its original place and the riser from its vertical position.

Other sources of excitation, not so often encountered, are as follows:

(a) Internal waves. They propagate on the surface between different-density layers [6]. Internal wave currents may be as strong as the ocean surface currents [4].

(b) Microseismic waves. They are surface standing waves created by the vibrations of the bottom of a tank or of the ocean. They generate a second-order pressure term which does not decay exponentially with depth but is constant in the whole domain [71.

(c) Tides. They can influence the riser in two ways: first by varying the water level and second by generating currents of considerable speed. Both effects are significant for shallow-water drilling operations.

(d) Tidal or volcanic waves. Their existence is rather ira- probable. However, in case that a drilling facility is in operation, when such waves appear, its probabil ity of surviving is low. The riser should be disconnected and the drill ship moved away in time.

P re l iminary r i ser design. At first the site of operation is chosen and the required length of the marine riser is determined. The task of the designer is to compute the outer diameter, the thickness, the material of the riser pipes, the required capacity of the tensioning system, the total buoyancy, and the distribution of the buoyancy modules along the riser. The riser design should

74 MARINE TECHNOLOGY

be optimized to keep the maximum stress low. To achieve this task the designer should solve several problems related to the riser behavior. The most important of these are the following:

(a) Modeling of the dynamic behavior of the riser, taking into account all significant forces. This problem is studied in the next section with the aid of the general model developed in reference [81.

(b) Prediction of the hydrodynamic forces exerted on the riser under various flow conditions. We study this problem in final section based on Morison's equation and model tests [9].

(c) Calculation of the resultant stresses in the material due to the dynamic response of the riser [10].

(d) Simulation of the riser dynamic response and statistical analysis of the results [11].

(e) Analysis of the static buckling problem. This is of par- ticular importance for long risers since it has been proven that slender tubular columns may buckle globally, as Euler columns, even if they are in tension along their entire length [12, 13].

(f) Formulation and solution of the static and dynamic design optimization problem. This problem is particularly difficult because of the difficulty in predicting the external hydrodynamic loads. However, the static parametric design optimization problem has been formulated and solved in closed form in ref- erences [14-17].

Each of the foregoing problems has been identified and studied early in the research on marine risers [3]. Many computer pro- grams which calculate the stresses in risers exist [18]. However, none of these problems has been fully resolved and further the- oretical analysis is required.

Dynamic behavior of marine risers

The weight of the riser, the external hydrodynamic loads, the frictional, Corriolis, and centrifugal forces due to the motion of the drilling mud, the concentrated forces exerted on the riser at the points of its occasional contacts with the collars, the con- centrated forces and moments applied on the riser by the kill and choke lines through the riser connectors, and the thermal stresses caused by the variation of the water temperature with the depth, all tend to deform the centerline of the riser to a general three- dimensional curve.

These forces are balanced by the riser's bending rigidity, the inertia forces and moments, and the effective tension, that is, the real tension in the riser pipes modified by the external hydrostatic pressure and the internal mud static pressure.

The end conditions of the problem are defined by the partic-

ular construction of the lower- and upper-end systems, and the motions of the supporting structure.

The first decision that a designer has to make, before mathe- matically modeling his problem, regards the required degree of accuracy of his results. This is dictated by two factors: first, his knowledge of the external loads, and second, the available com- puter funds for the solution of his model. In the particular case of the marine riser, the cost of the structure is such that the sec- ond factor practically sets no limits to the designer. On the other hand, our knowledge of the hydrodynamic loads exerted on the riser is limited. We can find in the literature information about the average hydrodynamic force exerted on circular cylinders [19] but only for a limited number of flows. However, little is known about the pressure distribution on the cylinder's surface. Con- sequently, the riser is studied globally as a beam and not locally as a shell.

By the same argument, the following two assumptions re- garding the motions of the riser contents are justified:

(a) The drill string can either be neglected [18] or assumed to be concentric and in rigid contact with the riser through the collars [20]. The former underestimates the weight of the riser and its contents, while the latter overestimates the riser's bending rigidity. Both approximations are acceptable, since they intro- duce minor errors. In this paper the former approach is adopted. Should the latter be used instead, a simple modification of the equations of motion would be required.

(b) The translational and rotational velocity of the drilling mud is small. Consequently, the centrifugal and Coriolis forces exerted on the mud--and by reaction on the riser--due to the riser's local angular velocity are negligible. For the same reason, the frictional force exerted on the riser due to the mud's viscosity is not taken into account.

The mathematical model. The model used to study the riser dynamic behavior in this section has been developed in reference [8]. Its salient features are the following:

(a) It is a set of consistent equations for large deflections of a beam under tension [21].

(b) It models the bending of the riser in three dimensions. (c) Axial motion effects, which couple the longitudinal and

transverse motions of the riser, are taken into account. (d) The hydrostatic and mud static pressure and their effects

on the riser stiffness are properly modeled. (e) The initial position of the riser may be any three-di-

mensional curve and not necessarily a straight vertical line. (f) The riser material is assumed elastic and linear. (g) It is a Bernoulli-Euler type model which is satisfactory

BOP = blowout preventer /~(s,t) = unit vector in binormal direction D,Do = riser's outer diameter

D~ = riser's inner diameter D(s,t) = force due to hydrostatic pressure

Dm(s,t) = force due to mud static pressure E(s) = Young's modulus

f0 = frequency of cylinder's oscillation fb.fra,fr, = nondimensionalized force per unit

f~,ft,fx,fy length in direction indicated by subscript

hm = z-coordinate of the mud free surface

hw = z-coordinate of the water free surface

Ibb(s), = second moment of cross section l..(s)

Jc = mass inertia tension

Nomenclature

Jbb,Jnn,Jtt = mass inertia moments K(s,t) = curvature

Mb = bending moment in binormal direction

Mn = bending moment in principal normal direction

f~(s,t) = unit vector in principal normal direction

PSD = power spectral density Pe(S,t) = effective tension

Qb = shear force in binormal direction Qm = shear force in principal normal

direction R = torsional moment So = amplitude of cylinder's oscillation t = time

[(s,t) = unit vector in tangential direction T = tension

Uo = velocity, in general u(s,t) = displacement in x-direction v(s,t) = displacement in y-direction w(s,t) = displacement in z-direction Wb(s) = weight of buoyancy modules per

unit length Wr(s) or = riser weight per unit length

WR(s) Wm(s) = drilling mud weight per unit

length Greek symbols

~(s,t) = strain of differential element ds 0 = angle in degrees

pr = density of riser material Pw = water density

r(s,t) = torsion 6% = 27r f0

&(s,t) = angular velocity

JANUARY 1982 75

Z Z

0 (o,o,o)

Fig. 2

ft(s) d s

g fb(S) d s

f (s)ds n

~" y

Freebody diagram for a differential element ds. Local principal directions and hydrodynamic forces

/ ' x

t D(s12 ds)

B(s)ds ~ l -

WR(s)d s

D(s-ds)

; Y

Fig. 4 Freebody diagram for a differential element ds. External hydrostatic pressure and riser's weight

for low frequencies and long wavelengths [22]. (h) Thermal effects are neglected [23]. The quantities involved in this model are depicted in Figs. 2

through 5. Figure 2 shows the three princil~al local directions, tangent

[(s,t), normal ~(s,t), and binormal b(s,t), and the corresponding hydrodynamic forces exerted on the differential element ds.

Figure 3 depicts the structural restoring forces and moments; that is, the tension T(s,t) and torsion R(s,t) in the tangential direction [(s,t), the shear force Q,, (s,t) and the bending moment M~ (s,t) in the principal normal direction r~(s,t), and the shear force Qb (s~t) and the bending moment Mb (s,t) in the binormal direction b(s,t).

Figure 4 shows the weight of the riser element ds, WR (s)ds, and the effect of the hydrostatic pressure force on ds. This force,/~ ....

can be computed by integrating the pressure force on the wetted sides of the element ds. In the case of two-dimensional bending, a formula correct to first order has been derived by Breslin [24, 25].

An alternative method which yields the exact force,/~w, in the case of three-dimensional bending is shown in Fig. 4. If the cy- lindrical element ds was disconnected from the riser, was closed on both ends, and was fully submerged in water, the vector sum of/~w and the hydrostatic pressure forces which would be exerted on the bases of the cylindrical element ds would be equal to the buoyancy B*dsf~. Therefore

B*dsf~ = FIw(s)ds + [ -D(s + ds)[(s + ds)]

- [ -D(s - lds) i (s - ds)] (5)

Fig. 3

Z

Q n(.s - d s)

0(o,o,o)

J

R(s+lds) /

T(s+1ds) /

-~b(S +212 d s)

/ m

/ T(s-}ds)

/ / R(s-lds) __

Freebody diagram for a differential element ds. Structural restoring forces and moments

Z i

f X ?

~%(s + d s)

l Wm(S)d s

D~(s- d s)

Fig. 5 Freebody diagram for a differential element ds. Mud static pressure

76 MARINE TECHNOLOGY

where B* is the buoyancy per unit length of the riser tubes only, without considering the buoyancy tanks.

This method shows that the force exerted on ds due to the hydrostatic pressure is not equal to the buoyancy force. The term

~rD~(s) D(s) i (s) = Pwg ~ [h~ - z(s) l i (s) (6)

which appears in equation (5) has the same effect on the structure as the tension T(s) (see Fig. 3); that is, it increases the riser ef- fective stiffness.

We can use the same method to derive the mud static pressure force exerted on the internal surface of the riser element ds, /~ :

-Wmds]~ = f lmds + [ -Dm(s + ds)i(s + ds)] - [ -Dm(s - ds)[(s - ds)] (7)

Equation (7) shows the riser does not support the total weight of the drilling mud. The effect of the correction term

7r 2 Dm(s)t(s) = Ping ~ Di (s)[hm - z(s)][(s) (8)

which appears in equation (7) is opposite to that of T(s); that is, it decreases the riser effective stiffness.

Since T(s), D(s), and D,n(S) all act in the same direction, [(s), we can introduce a new term Pe (S)

7rD02 Re(s) =- T(s) + Pwg - -~ [hw - z(s)]

7rD] - Pmg~ [h~ - z(s)] (9)

P~(s) has no real meaning but simplifies the notation. Likewise, we introduce We (s)

W~(s) =- -B* (s ) + Win(s) + WR(S)

7r(D~ - D~) + Wb(S) -- Pwg (10)

4

Using the foregoing notation we can derive the equations of equilibrium for the differential element ds subject to the forces and moments shown in Figs. 2 through 5:

Equilibrium of forces in the tangential direction:

0z 1 OPe(S)os K(s)Qn(s) "t- ft(S) -- We(s) -~8 - g [WR(S)

ox + + o woz l + Wb(S)] [0t2 0s ~-~-~s 0t 2 0s] = 0 (11)

Equilibrium of forces in the principal normal direction:

OQAs) - - + K(s)P~(s) + r(S)Qb(s) + In(s)

Os 1 02z 1

- WAs) - - [WR(s) + Wb(S) + Win(s)] K(s) Os 2 g

1 [02u 02x 02v 02y 02w 02z] X ~-~ [~-~ ~s2 + ~-~ ~s 2 +-~-~s 2J = 0 (12)

Equilibrium of forces in the binormal direction:

OQb(s) 7(s)Qn(s) + fb(s) -- We(s)~3(8)

OS

_ 1_ [WR(s) + Wb(S) + Wm(s)] g

[02u , , 02v 02w l X [~7~1[8) -I- ~2(S) -t- -~-"[3(8)] ---- 0 (13)

Equilibrium of moments in the tangential direction:

OR(s) _ K(s)Mn(s) - ;:) [{g~(s)}&(s,t)][(s) = 0 (14) os ~[

Equilibrium of moments in the principal normal direction:

OMn(s) + K(s)R(s) + ~(s)Mb(s) - Qb(s) Os

O - - - [Ig~ (s)}(o(s,t)]ri(s) = 0 (15)

0t

Equilibrium of moments in the binormal direction:

OMb (s) - - -t- Qn(s) - 7-(s)Mn(s)

Os

- ~ [{gc(s)t(o(s,t)] 5(s) = 0 (16) 0t

where K(s) is the local curvature =(o2x/2 (o2,/2

K2(8) (~82 ] ~- 1082 ] "4- [082 ] (17)

r(s) is the local torsion of the riser's centerline

-1 F x/s OylOs Oz/Os -I "r(s)= ,-y-;-v, , l O2xlbs 2 02y/~)s 2 02z/Os21 (18)

^ (s ) L o3x/os~, O'dy/Os 3 03Z/O83_j

and ~'l(S), ~(s), 3'3(s) are the directional cosines of the binormal unit vector b(s)

1 [by 02z 0z 02y] 71(s) = K--~s) ~ss bs 2 bs ~-~s 2] (19a)

1 [Oz 02x Ox b2z] (19b) "y2(s) = K--~s) ~ss 0s 2 0s /)s 2j

1 [0X 02y 0y 02X] 73(s) = K--~s) ~s 0s 2 0s 0s 2] (19c)

To complete the set of equations (11) to (16), which describe the riser dynamic behavior, we should derive the constitutive rela- tions.

Assuming that the planes remain plane [2@ the constitutive relations of bending in the osculating plane (t,ri) is

Mb(s) = E(s) Ibb(S) K(s) (20)

Likewise, since the torsion of the riser centerline, ~(s), also called second curvature [27], is the rate of change of the direction of the binormal, the constitutive relation of bending in the rec- tifying plane (t,~) is

Mn(s) = E(s) Inn(S) ~r(s) (21)

Finally, the constitutive relation of tension is

7r [n02(s ) _ D~(s)] + T(s,to) (22) T(s,t) = e(s)E(s) -~

where e(s) is the strain of the riser element ds(to):

ds(t) - ds(to) c(s,t) -

ds (to)

where to is the time of the initial equilibrium position. Characterist ics of the mathematical model. The mathe-

matical model summarized in the previous section is valid for large three-dimensional deflections, since all nonlinear terms due to squares or products of displacements and slopes are retained. It is understood that the solution of the model is hard. However, it is very useful because it reveals basic properties of the dynamic behavior of the riser and at the same time it gives to the designer the opportunity to appreciate the significance of the errors in- volved in the various simplified riser or cable models which ap- pear in the literature.

Some of the basic properties of the dynamic behavior of the riser are as follows:

(a) The influence of the hydrostatic pressure on the behavior

JANUARY 1982 77

of cables [23] has been a controversial issue for many years [24]. It was modeled correctly to the first order by Goodman [25] and his results have been included in recent riser analyses [28]. In this model the exact answer has been derived by the simple method, shown in Fig. 4, independently of the extension of the riser or the Poisson effect [25]. It is obvious from equations (9) that the hy- drostatic pressure increases the effective tension of the riser and its apparent strength. This actually explains why cables heavier than water may not collapse but stay upright without support.

(b) Opposite is the effect of the static pressure of the drilling mud. It decreases the effective tension. Actually if the riser ends were restrained from longitudinal extension, the internal pressure could cause lateral buckling. In fact this formulation was used in reference [8] to prove that internal pressure may cause global buckling of slender tubular columns even if they are in tension along their entire length. Discrete model analogs and energy methods were used to explain this phenomenon in [13].

(c) The effective tension along the riser is basically deter- mined by its effective weight.

(d) The equations of motion are partial nonlinear differential equations coupled by the slope, curvature, and torsion of the centerline of the riser.

(e) Three-dimensional effects are important for the com- putation of the stresses in the riser, particularly for large de- flections.

(f) The performance of the analysis in the t, rl, and/~- direc- tions has two basic advantages. First it facilitates the prediction of the hydrodynamic forces exerted on the riser and second it makes easy the calculation of the mass of the mud entrained by the riser in each direction [8].

Simplified mathematical models. The models of the riser dynamic response published in the literature are simplified. It is useful to review the assumptions required to derive these models:

(a) The derivative of the angular momentum vector (b/St) ({Jc}w) is Small [20].

(b) The torsional moment R(s,t) is negligible. (These two assumptions are reasonable and simplify the

analysis considerably.) (c) The centerline of the riser is a two-dimensional curve.

This assumption may introduce serious errors for large deflec- tions, since certain neglected terms are of the same order of magnitude as the nonlinear terms retained in some models [29].

(d) All nonlinear terms due to products or squares of slopes are negligible [30]. This assumption makes the analysis valid only for small deflections of the riser and small excursions of the supporting structure and decouples the equations of motion. Consequently, the analysis can be carried out independently in two mutually perpendicular planes. However, the resultant bending moment may not be correct since the results are not additive [31].

Hydrodynamic forces exerted on the marine riser

In the previous section a mathematical model for the dynamic behavior of the marine riser was presented. To solve this set of equations we should first express the hydrodynamic forces ft (s,t), fn (s ,t), and fb (s,t) in a mathematical form.

In early riser designs the kill and choke lines were mounted outside the riser on the connectors. This arrangement made the configuration of the cross section of the riser too complicated for experimental analysis. Very limited experimental data are available in the literature for such configurations. Consequently, the presence of the kill and choke lines was neglected and the riser was assumed to be a circular cylinder. Recent designs make this assumption obsolete since they have these lines mounted inside the riser.

Since the beginning of this century, when Von Karman ob-

served a regular street of vortices behind a circular cylinder in a steady uniform flow, a tremendous experimental and theoret- ical effort has been made to understand this phenomenon [32] and the salient features of viscous fluid flows past circular cyl- inders. However, our knowledge on this subject is still limited.

Components of the hydrodynamic force; the Morison equation. From theoretical analysis and laboratory experiments, it is known that the following hydrodynamic forces are exerted on circular cylinders:

(a) The drag force. It is proportional to Arellfi, rell, where Arel is the instantaneous relative velocity vector of the water with respect to the cylinder ]33]. ,4re1 is normal to the cylinder's axis (empirical).

(b) The lift force. It is proportion~ toArel]Arell and is applied in the direction which is normal to Arel and to the cylinder axis [34] (empirical).

(c) The added-mass force. This is proportional to the added mass and the relative acceleration, tire1, of the water with respect to the cylinder. It is applied in the direction of 5rel [35] (theo- retical).

(d) The pressure gradient force. It is proportional to the absolute fluid acceleration, a fluid, and the displaced fluid volume. Its direction is that of the pressure gradient [36] (theoretical).

Therefore, the total hydrodynamic force, Fhydro, exerted on a circular cylinder can be written in the form

/~hydro ---- PwlDoeDnrellnrel[ "{- PwlDoeLb X 2~lrellArell

~D~I ~D~)l 5 + Pw T (CM -- 1)arel 4- Pw T fluid (23)

where

Pw = density of water Do = outer diameter of cylinder

CD,CL,CM = drag lift and inertia coefficients, respectively

2rel

and d is the unit vector in the direction of the cylinder axis. The Morison equation, which was first used in 1950 [33] to

model the hydrodynamic force exerted by waves on fixed piles, has the form

Fhydro = PwIDoCD Arellffirell .~_1 2 3pwTrDo (CM -- 1) 5re1 (24)

Since 1950, the prediction of forces exerted on circular cylinders has been based, almost exclusively, on the Morison equation. However, its application is still very controversial [37].

Use of equations (23) or (24), for the prediction of the hydro- dynamic forces exerted on marine risers, implies the following assumptions:

(a) The value of length I, along which the flow can be con- sidered as two dimensional, is known or can be estimated. The correlation length is not studied in this paper [38].

(b) Hydroelasticity effects have no influence on the hydro- dynamic forces exerted on the riser [39].

(c) The hydrodynamic torque exerted on the cylinder is ne- glected. This torque is due to the assymmetry of the flow caused by the generated and shed vortices behind the cylinder.

(d) The values of the hydrodynamic coefficients, CD, CL, and CM can be computed theoretically or measured experimentally [19].

The hydrodynamic coefficients. In spite of the efforts of various investigators during the past 50 years to understand and model the properties of viscous fluid flows past circular cylinders, no satisfactory theory for the prediction of the hydrodynamic forces has been established as yet. Consequently, the major effort is concentrated on the experimental calculation of the coefficients CD, CM, and CL.

78 MARINE TECHNOLOGY

The separation point, the rate of growth, the strength, the time of shedding of a vortex, and the motion of the fluid in the wake are functions of the instantaneous velocity and acceleration and of the time history of the flow [2]. All these features of the flow, and in general all our ignorance of the fluid mechanics in the vi- cinity of the cylinder, are lumped in the hydrodynamic coeffi- cients. As a result, CD, CL, and CM strongly depend on the flow pattern and the values of the nondimensional parameters which describe it.

The calculation of the hydrodynamic coefficient is done in the following steps:

(a) A specific flow is modeled experimentally. Thus the input to equation (23) is defined.

(b) The output, that is, the hydrodynamic forces, are mea- sured experimentally.

(c) The coefficients CD, CM, and CL, which describe the system, are computed using the data derived in Steps (a) and (b).

Obviously, this procedure is applicable only to simple flows which can be described by a limited number of nondimensional parameters. More complicated flows are hard to model experi- mentally and in addition a systematic study with respect to all the nondimensional parameters would be extremely time-con- suming. So we can find in the literature experimental and theo- retical studies of the following four simple flows:

(a) Steady flow past a fixed circular cylinder. This case has been studied extensively in the past. The drag coefficient Co has been measured for low and intermediate Reynold's numbers (Re) in 1953 [40], and for high Re in 1961 [41]. The vortex shedding frequency and the lift forces [34] have been measured as well. One nondimensional parameter, namely, Reynold's number, is re- quired to describe the flow.

(b) Cylinder sinusoidally oscillating in calm water [35] or sinusoidally oscillating fluid past a fixed cylinder [36, 42]. Besides Reynold's number, the Keulegan-Carpenter number, KC, is needed to describe this flow:

KC = 2~ S----9- (25) Do

where So is the amplitude of oscillation, and Do the outer diam- eter of the cylinder.

(c) Sinusoidally oscillating cylinder traverse to a uniform current [43].

(d) Sinusoidally oscillating cylinder in line with a uniform current [44, 45].

The last two flows are fully described by three parameters, namely, Reynold's and Keulegan-Carpenter numbers, and the reduced velocity, U*:

U* - Vcur (26) /oDo

where Vcur is the constant current velocity and f0 the frequency of oscillation of the cylinder.

More complicated flows have not been studied (see "Experi- mental work" later in this section).

Application of Morison's equation to marine risers. The analysis of the previous subsection reveals that the hydrodynamic coefficients strongly depend on the pattern of the relative flow of the water with respect to the riser. In general, it is not possible to predict the hydrodynamic forces exerted on circular cylinders for flows which have not been studied experimentally. Conse- quently, the task of the designer is:

(a) To determine the local relative flow patterns along the riser, and distinguish those for which no experimental data are available (treated later).

(b) To identify those flows for which the hydrodynamic forces are not properly modeled by Morison's equation (treated later).

(c) To find those flows which have not been analyzed in the

past, and for which an approximate estimation of the hydrody- namic coefficients from the available experimental data is pos- sible. All other flows should be investigated experimentally. One of these is studied next.

Local flow patterns along the riser. The motion of the drill ship has two components:

(a) A slowly varying, nonperiodic, large offset motion due to the second-order drift forces, the wind, the current, and the action of the controller and thrusters.

(b) A high-frequency small-amplitude periodic motion due to the action of the surface waves.

The relative flow past the riser varies with the water depth because the ocean currents, the influence of the motions of the ship, the wave-induced velocities, and the motion of the riser vary with the water depth. We can identify three zones along the riser where a different flow pattern exists (see Fig. 1).

ZONE A: The water particles have a constant velocity component due to the ocean current and an oscillatory one due to the surface waves. In general the two velocity components are not codirectional. In addition, the riser is moving in some other direction. This flow is as complicated as the most general case of a random relative flow. No experimental data are available [19].

ZONE B: In this zone the wave-induced velocities are at- tenuated but the motion of the riser is still influenced by both components of the ship motion. In general, the riser motion is not sinusoidal. Nevertheless, depending on the exciting force and the excited mode, the motion of the riser can be considered locally as sinusoidal. This approximation simplifies the analysis con- siderably.

ZONE C: At depths greater than 2000 ft (609 m), the cyl- inder is sensitive only to the slowly varying nonperiodic component of the motion of the ship. Consequently, the relative flow can be considered as quasi-steady.

Limitations of the applications of Morison's equation. Earlier in this section it was pointed out that Morison's equation is a combination of empirical and theoretical terms. It is the sum of a drag and an inertia force, expressed in terms of constant coef- ficients and the instantaneous velocity and acceleration, re- spectively.

In ideal fluid flows past circular cylinders, two length scales are involved, namely, that of the nonuniformity of the stream, characterized by ~/I vl [46], and the cylinder diameter. Here- after, for reasons of simplicity, we shall call the former "NU- scale" and the latter "D-scale." The fundamental assumption for the derivation of the two theoretical terms, the added-mass force and pressure gradient force defined earlier in this section, is that the D-scale is very small compared with the NU-scale [46].

In viscous fluid flows, a third length scale is involved, that is, the thickness of the boundary layer, which we shall hereafter call the "5-scale. Viscosity induces two more force components, the skin friction and the form drag. These two terms are lumped in the drag force term of Morison's equation. The fundamental assumption implied is that the 5-scale properties of the flow can be expressed in terms of the NU-scale properties, namely, the instantaneous velocity and acceleration of the undisturbed flOW.

Since Morison's equation expresses the force in terms of the instantaneous velocity, Areb and acceleration, 5rel, of the undis- turbed flow, it is bound to be correct only if there is no flow property independent of Arel and 5re1. This may be the case for other body configurations, but not for circular cylinders. The vorticity generated at the separation point of the boundary layer is accumulated to form large discrete vortices behind the cylinder. That is, another D-scale property is generated. This is a nonun- iformity of the flow which is not in agreement with the first as- sumption. In addition, it is a function of the history of the flow and not its instantaneous properties. Consequently, all effects

JANUARY 1982 79

Fig. 6

ImPVcur,

Direction of uniform current and cylinder's sinusoidal oscillation

related to those large discrete vortices generated and shed behind the cylinder are completely neglected by the Morison equa- tion.

The preceding reveals that: (a) The Morison formula can model properly flows for which

the drag forces are dominant; h)r example, steady flow past a fixed cylinder [32] or pure harmonic flow for Keulegan-Carpenter number greater than 25 [36].

(b) Morison's equation can also model correctly flows for which the inertia forces are dominant and in addition no vortices are ~;enerated; for example, pure harmonic flow for Keulegan- Carpenter number less than 6 [30].

(c) When vortex effects are as significant as inertia and drag effects, Morison's equation should not be applied. Such cases, for example, would be the following: (i) Pure harmonic flow past a fixed circular cylinder for Keulegan-Carpenter number between 6 and 25. As indicated in Sarapkaya's results [42], this is, in fact, the range of maximum differences between measured and pre- dicted forces. (ii) Harmonic oscillations of a circular cylinder transverse to a uniform current and at frequencies close to the vortex shedding one [43]. (iii) A third case is demonstrated by the experiments described next.

Experimental work. In Zone B (defined earlier), the motion of the riser with respect to the water can be approximated with that of a sinusoidally oscillating cylinder in any direction with respect to a uniform current (see Fig. 6). This motion has been modeled experimentally at the MIT Towing Tank.

The technique adopted for the experiments is that of forced oscillations and simultaneous translation of the model in still water. The oscillatory displacement of the model, provided by a dc motor and a scotch-yoke mechanism, was very close to a si- nusoidal curve (see Fig. 7). The model was a 10-in.-long (25.4 cm), 1-in.-diameter (2.54 cm) circular cylinder, with 4-in.-diameter (10.16 cm) thin circular end plates to minimize end effects.

.4

.3

.2

.1

.0

--.1

--.2

--.3

--.4

~So

0

Fig. 7 Oscillatory displacement of the model

t sec

.5

.4

.3

.2

.1

.O

Fig. 8

o

~}, D- - - -

~. ..... T IC

15 30 45 60 75 90

PSDs of fx, fy, fs, and fr for f = 1 Hz versus angle

The values of the nondimensional parameters chosen were dictated by the two purposes of the experiments, namely

(a) To investigate the possibility that small deviations from a steady flow past a fixed circular cylinder could cause significant changes of the hydrodynamic forces.

(b) To demonstrate what was derived by the analysis, given earlier, that Morison's equation cannot model oscillatory force components due to the generation and shedding of vortices.

Thus the experiments were conducted for

Re- Vcu~" D _ 7400, So_ woSo _ D0- 0.375, ands - 0.23 (27)

where

Vcur = constant current velocity So = amplitude of oscillation Do = outer diameter of cylinder

= kinematic viscosity of fresh water ~00 = frequency of oscillation

From equation (27) we can compute the vortex shedding fre- quency, [s = 2.05 sec -1, and the frequency of oscillation of the cylinder, f0 = 1 sec -1.

The power spectral densities (PSD's) of the oscillatory force components due to the cylinder's oscillation and the vortex shedding are shown in Figs. 8 and 9, respectively, as functions of angle 8. The terms fx, fy, fs, and fr are the nondimensionalized force components in the x, y, s, and r-directions, respectively (Fig. 6) [9].

Figures 8 and 9 show strong dependence of the PSD's on angle 8. In addition, comparing the two figures, we see that the force

.8

.7

.6

.5

.4

.3

.2

.1

.0

Fig. 9

/! //

For symbols see Fig. 8 //

/ .A.. jV

/ " .o-- - -'.-..@" %,. . -~ . - -~

15 30 45 60 75 cOO PSD's of fx, fy, fs, and fr for f = 2 Hz versus angle 0

O

80 MARINE TECHNOLOGY

components due to vortex shedding are of the same order of magnitude as those due to the cylinder's oscillation and should not be neglected as in the Morison type of approach.

Conclusions

1. A comprehensive mathematical model for the dynamic behavior of the marine riser is presented and discussed. This model is a set of consistent equations for large three-dimensional deflections of a beam under tension and for internal and external variable static pressure force. Axial motion effects are also in- cluded in the formulation.

2. The external hydrostatic pressure increases the effective tension and consequently the riser rigidity.

3. The normal static pressure diminishes the effective ten- sion. Should the riser ends be restrained from longitudinal ex- tension, the internal pressure could cause lateral buckling.

4. Analysis of the three-dimensional large deflections of the riser, in the local osculating, rectifying, and normal planes, reveals that two-dimensional modeling of riser bending may considerably underestimate the stresses in the riser.

5. Morison's equation, traditionally used for the prediction of the hydrodynamic forces exerted on offshore structures, completely neglects the effects of the large discrete vortices generated and shed behind the cylinder. Consequently, it cannot model properly flows for which the vortex effects are signifi- cant.

6. The limited applicability of Morison's equation was ex- perimentally demonstrated by measuring the hydrodynamic forces exerted on a sinusoidally oscillating circular cylinder in any direction 0 with respect to a uniform current. The vortex effects were found significant and strong dependence of the flow on angle 0 was observed.

7. Morison's equation can model properly two types of flows: (a) those for which the drag forces are dominant, and (b) those for which the inertia forces are dominant and in addition no vortices are shed behind the cylinder.

Acknowledgements

The author would like to thank Professor C. Chryssostomidis for his invaluable assistance. The comments and advice of Pro- fessors J. H. Milgram, F. Noblesse, J. K. Vandiver, and R. J. Vanhouten are deeply appreciated. The experimental work has been supported by the New England Section of The Society of Naval Architects and Marine Engineers; this support is gratefully acknowledged.

References

1 Harris, L. M., Deepwater Floating Drilling Operations, Petroleum Publishing Co., 1972.

2 Sarpkaya, T. and Garrison, C. J., "Vortex Formation and Resis- tance in Unsteady Flow," Journal of Applied Mechanics, Trans. ASME, Vol. 30, Series E, No. 1, 1963, pp. 16-24.

3 "Dynamic Stress Analysis of the Mohole Riser System," Report S.N. 183-2A, National Engineering Science Co., Jan. 1965.

4 Milgram, J. H., "Waves and Wave Forces," Proceedings, Behavior of Offshore Structures Symposium, Vol. 1, 1976, pp. 11-37.

5 Longuet-Higgins, M. S. and Steward, R. W., "The Changes in Amplitude of Short Gravity Waves on Steady non-Uniform Currents," Journal of Fluid Mechanics, Vol. 10, 1965, pp. 529-549.

6 Tritton, D. J., Physical Fluid Dynamics, Van Nostrand Reinhold, New York, 1977.

7 Longuet-Higgins, M. S., "Can Sea Waves Cause Microseisms?" Symposium on Microseisms, Sept. 1952, pp. 74-93.

8 Bernitsas, M. M., "A Three Dimensional, Non-Linear, Large Deflection Model for Dynamic Behavior of Risers, Pipelines and Cables," Journal of Ship Research, to appear in 1982.

9 Bernitsas, M. M.,"Analysis of the Hydrodynamic Forces Exerted on a Harmonically Oscillating Circular Cylinder in any Direction with Respect to a Uniform Current," Proceedings, Behavior of Offshore Structures Symposium, London, Aug. 1979.

10 Morgan, G. W., "Marine Riser Systems," Offshore Drilling and Production Technology, Petroleum Publishing Co., Dallas, Tex., 1976, pp. 44-48.

11 Gardner, T. N. and Kotch, M. A., "Dynamic Analysis of Risers and Caissons by the Finite Element Method," OTC Paper No. 2651, Offshore Technology Conference, Houston, Tex., 1976, pp. 405-421.

12 Bernitsas, M. M., "Riser Top Tension and Riser Buckling Loads," American Society of Mechanical Engineers, Applied Mechanics Division, Vol. 37, Nov. 1980.

13 Bernitsas, M. M. and Taylor, J. E., "Stability of Slender Tubes Under Static Pressure and Tension," Journal of Structural Mechanics, to appear in 1982.

14 Bernitsas, M. M. and Papalambros, P., "Riser Design Optimiza- tion Under Generalized Static Load," Intermaritec '80, Hamburg, Ger- many, Sept. 1980.

15 Bernitsas, M. M., "Static Analysis of Marine Risers," Department of Naval Architecture and Marine Engineering Publication No. 234, The University of Michigan, Ann Arbor, Mich., June 1981.

16 Bernitsas, M. M. and Papalambros, P., "A Study of Global Buckling Effects on Optimum Riser Design," International Conference on Optimum Structural Design, Tucson, Ariz., Oct. 1981.

17 Papalambros, P. and Bernitsas, M. M., "Monotonicity Analysis in Optimum Design of Marine Risers," Progress in Engineering Opti- mization, ASME Book No. I00141 and Journal of Mechanical Design, to appear in 1982.

18 "Comparison of Marine Drilling Riser Analyses," American Pe- troleum Institute Bulletin, Jan. 1977.

19 "A Critical Evaluation of the Data on Wave Force Coefficients," Report W278, British Ship Research Association, Aug. 1976.

20 Botke, J. G., "An Analysis of the Dynamics of Marine Risers," Delco Electronics, Aug. 1975.

21 Jones, N., "Consistent Equations for the Large Deflections of Structures," Bulletin of Mechanical Engineering Education, Wynn Williams, Publishers (U.K.), Vol. 10, 1971, pp. 9-20.

22 Crandal, S. H., Karnopp, D. C., Kurtz Jr., E. F., and Pridmore Brown, D. C., Dynamics of Mechanical and Electromechanical Systems, McGraw-Hill, New York, 1968.

23 Reid, R. O., "Dynamics of Deep-Sea Mooring Lines," Texas A&M University Project 204, Reference No. 68-11F, College Station, Tex., July 1968.

24 Breslin, J. P., "Dynamic Forces Exerted by Oscillating Cables," Journal ofHydronautics, Vol. 8, No. 1, Jan. 1974, pp. 19-31.

25 Goodman, T. R. and Breslin, J. P., "Statics and Dynamics of Anchoring Cables in Waves," Journal ofHydronautics, Vol. 10, No. 4, Oct. 1976, pp. 113-120.

26 Timoshenko, S. P., Strength of Materials, Van Nostrand, New York, 1940.

27 Eisenhart, L. P., An Introduction to Differential Geometry, Princeton University Press, Princeton, N.J., 1947.

28 Young, R. D., Fowler, J. R., Fisher, E. A., and Luke, R. R., "Dy- namic Analysis as an Aid to the Design of Marine Risers," Journal of Pressure Vessel Technology, Trans. ASME, Vol. 100, May 1978, pp. 200-205.

29 Bennett, B. E. and Metcalf, M. F., "Nonlinear Dynamic Analysis of Coupled Axial and Lateral Motions of Marine Risers," OTC Paper No. 2776, Offshore Technology Conference, Houston, Tex., 1977, pp. 403- 412.

30 Burke, B. G., "An Analysis of Marine Risers of Deep Water," OTC Paper No. 1771, Offshore Technology Conference, Houston, Tex., 1973, pp. 449-464.

31 Morgan, G. W., "Dynamic Analysis of Deep Water Risers in Three Dimensions," Petroleum Division, PET 20, American Society of Me- chanical Engineers, Sept. 1975.

32 Roshko, A., "On the Drag and Shedding Frequency of Two- Dimensional Bluff Bodies," NACA Technical Note 3169, National Advisory Committee for Aeronautics, July 1954.

33 Morison, J. R., O'Brien, M. P., Johnson, J. W., and Shaaf, S. A., "The Forces Exerted by Surface Waves on Piles," Petroleum Transac- tions, American Society of Mechanical Engineers, Vol. 189, 1950, pp. 149-154.

34 Bishop, R. E. D. and Hassan, A. V., "The Lift and Drag Forces on a Circular Cylinder in a Flowing Fluid," Proceedings of the Royal Society, Series A, Vol. 227, 1964, pp. 32-50.

35 Keulegan, G. H. and Carpenter, L. H., "Forces on Cylinders and Plates in an Oscillating Fluid," Journal of Research of the National Bureau of Standards, Vol. 60, No. 5, May 1958, pp. 423-440.

36 Sarpkaya, T., "In Line and Transverse Forces on Cylinders in Oscillatory Flow at High Reynold's Numbers," Journal of Ship Research, Vol. 24, No. 4, Dec. 1977, pp. 200-216.

37 Wade, B. G. and Dwyer, M., "On the Application of Morison's Equation to Fixed Offshore Platforms," OTC Paper No. 2723, Offshore Technology Conference, Houston, Tex., 1976, pp. 1181-1189.

38 King, R., "A Review of Vortex Shedding Research and its Appli- cation," Ocean Engineering, Pergamon Press, Vol. 4, 1977, pp. 141- 171.

JANUARY 1982 81

39 Blevins, R. D., Flow-Induced Vibration, Van Nostrand Reinhold, New York, 1977.

40 Roshko, A., "On the Development of Turbulent Wakes from Vortex Sheets," NACA Technical Note 2918, National Advisory Com- mittee for Aeronautics, March 1953.

41 Roshko, A., "Experiments on the Flow Past a Circular Cylinder at Very High Reynold's Number," Journal of Fluid Mechanics, Vol. 10, 1961, pp. 345-356.

42 Sarpkaya, T., "Forces on Cylinders and Spheres in a Sinusoidally Oscillating Fluid," Journal of Applied Mechanics, Trans. ASME, March 1975, pp. 32-37.

43 Bishop, R. E. D. and Hassan, A. V., "The Lift and Drag Forces on a Circular Cylinder Oscillating in a Flowing Fluid," proceedings of the Royal Society, Series A, Vol. 277, 1964, pp. 51-75.

44 Mercier, J. A., "Large Amplitude Oscillation of Circular Cylinder in a Lowspeed Stream," Ph.D. Thesis, Mechanical Engineering, Stevens Institute of Technology, Hoboken, N.J., 1973.

45 Mercier, J. A., "The Forces of Circular Cylinders Due to Combined Action of Waves and Currents," West Gulf Section, SNAME, Feb. 1976.

46 Newman, J. N., Marine Hydrodynamics, MIT Press, Cambridge, Mass., 1977.

47 Bernitsas, M. M., "Contributions Towards the Solution of Marine Riser Design Problem," Ph.D. Dissertation, MIT, Cambridge, Mass., Sept. 1979.

48 Fischer, W. and Ludwig, M., "Design of Floating Vessel Drilling Riser," Journal of Petroleum Technology, March 1966, pp. 272-280.

49 Gnone, E., Signorelli, P., and Giuliano, M., "Three-Dimensional Static and Dynamic Analysis of Deep-Water Sealines and Risers," OTC Paper No. 2326, Offshore Technology Conference, Houston, Tex., 1975, pp. 799-812.

50 Huang, T. and Dareing, D. W., "Buckling and Lateral Vibration of Drill Pipe," Journal of Engineering for Industry, Trans. ASME, Vol. 90, Nov. 1968, pp. 613-619.

51 Osborne, A. R., Brown, J. R., Burch, T. L., and Scarlet, R. I., "The Influence of Internal Waves on Deepwater Drilling Operations," OTC Paper No. 2797, Offshore Technology Conference, Houston, Tex., May 1977, pp. 537-544.

52 Timoshenko, S. P. and Gere, J. M., Theory of Elastic Stability, McGraw-Hill, New York, 1961.

82 MARINE TECHNOLOGY