Setdown of a Catenary-moored Gravity Platform - Science Direct

Marine Structures 9 (1996) 721-742

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095 I -8339(95)000

© 1996 Elsevier Science Limited Printed in Great Britain• All rights reserved 0951-8339/96/$15.00 19-4

Setdown of a Catenary-moored Gravity Platform B. P a d m a n a b h a n & R. C. Ertekin Department of Ocean Engineering, School of Ocean and Earth Science & Technology, University of Hawaii at Manoa, 2540 Dole St., Holmes Hall 402, Honolulu, HI 96822, USA (Revised 10 January 1995; accepted 23 January 1995)

ABSTRACT The' setdown phase of the installation of a gravity platform requires a careful analysis of each step involved to prevent any damage to the structure. To do this, it is important to include all the significant forces, including the tensions in the mooring lines, in a setdown simulation. In this article, the setdown of a conical gravity platform is simulated when temporary mooring lines that assist the setdown are fully accounted for. The setdown is achieved by bah'asting the tanks in the platform according to a sequence controlled basically by the stability of the platform, as well as by its final location. A theory is developed to account for all the exact hydrostatic and ballasting forces, as well as the tensions in the mooring lines, which are catenary. Based on this theory, an interactive microcomputer program is developed to instantly monitor, report, and take action to correct any undesirable movements of the plalform that may lead to negative metacentric height. The numerical scheme devised is applied to a conical platform to analyze its real-time behavior during the setdown process. The effect of catenary mooring lines on the setdown process is shown by comparing the stability parameters with and witJ~out the mooring lines. Also, different ballasting sequences are shown to have a significant effect on the safety of the platform. Key words: setdown, gravity platform, stability, catenary mooring.

1 INTRODUCTION

Conical gravity platforms are commonly considered for oil drilling in the arctic environment. These platforms are superior to other platforms 721

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B. Padmanabhan, R. C. Ertekin

because of their high resistance to ice loads and the fact that they can be easily relocated for exploratory or production drilling at other sites. The installation of an arctic gravity structure involves the towing of the platform to site, positioning of the structure over the installation site and setdown of the platform over a previously prepared berm. The setdown of the platform is usually achieved by ballasting the platform, sometimes assisted by mooring lines. Some of the engineering requirements are that; 1 (i) the ballasting must lead to a set of equilibrium positions, (ii) the structure must stay clear of the sea floor at each stage of the operation to reduce the risk of collision, and (iii) one must be aware of contingencies at each stage of the ballasting operation, such as overflooding of tanks, instability of the structure, sinking of the structure or breaking of the mooring lines. A rigorous analysis of setdown would involve solving the equations of motion in time domain taking into account the forces acting on the platform - - hydrodynamic, hydrostatic, mooring, current, ice and viscous forces. Because of the complexity of the problem and the importance of static forces, it is common practice to simplify the problem to a multiposition static equilibrium problem. Transient phenomenon from one equilibrium to another is neglected which means that the procedure of finding equilibrium is reduced to an iterative one, wherein forces and moments on the platform at various attitudes are resolved until equilibrium is reached. The whole setdown process is then a set of equilibrium configurations corresponding to a particular ballasting scheme. There are potentially an infinite number of solutions to a setdown problem, each corresponding to a particular ballasting scheme. This makes the setdown problem an ideal candidate for interactive analysis by means of computer graphics for visualization, wherein a variety of ballast schemes can be tried and tested. A static equilibrium problem can be solved on the basis of the principle of virtual work, the energy approach; or based on the Newtonian principle that the sum of the forces and moments should be zero. Such approaches were used by Metcalf2 and Zurru & Chillemi I in the context of upending and setdown analysis of platforms. The energy approach was used by Zurru & Chillemi.l They solved the problem by minimizing the potential energy of the platform. The potential energy equation of the system was set up and the problem solved as an unconstrained nonlinear optimization problem. Metcalf2 used the Newtonian approach in his upending simulation model, OPUS (Offshore Platform Upending Simulation). In this model, the equilibrium configuration is attained by iteratively resolving the forces and moments, computing the resulting force system, and repositioning the platform until the forces and

Setdown of a gravity platform

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moments vanish for a particular configuration; however, the stability of the platform in OPUS is not ascertained explicitly, but is known only when infinitesimal changes to the platform configuration are made. Recently, Campanile 3 proposed a method for solving the equilibrium of floating bodies by potential energy minimization which can also be adapted to the upending or setdown problem. The main objective of this article is to simulate the setdown process as a multi-position static equilibrium problem using computer graphics for visualization. We first discuss the concept of equilibrium. The Newtonian approach used in this paper to solve for the equilibrium is then outlined. This solution is followed by the development of the equations of equilibrium, including the effect of ballast and mooring lines oll the setdown of a platform. The lost-buoyancy method is used to include the effect of ballast. The present method is then applied to a conical gravity platform to simulate the setdown process at different water depths.

2 THEORY The vir~Lualwork done by an equilibrium system of forces when the system undergoes a virtual rigid body displacement pattern is zero according to the principle of virtual work. For a conservative system of forces, the equilibrium is then represented by the stationary points of the potential energy function, which is equivalent to stating that the sum of the forces and moments should be equal to zero. If the system is given small displacements from the equilibrium, then the work done governs the stability of the system. If the work done is negative, then the equilibrium is stable, if it is positive, the equilibrium is unstable. A positive work for some displacements; and negative for others then indicates a saddle point in the equilibrium, and therefore, the stability depends on the direction of the impelling forces (see, e.g., Goldstein4). 2.1 Present method

In this paper, the Newtonian approach to solving the equilibrium is used. The forces and moments are expressed as a function of configuration coordinates, {X°}, of the platform. For example, the configuration coordinates for a floating platform are the draft of the platform and its angles of inclination about a defined coordinate axis system. If the platform is displaced to a new set of configuration {X ~}, then the forces on the platform are expressed by a Taylor series expansion as

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B. Padmanabhan, R. C. Ertekin

{F(X')} = {F(X°)} + [K] {X ~ - X ° } r + {E},

(1)

where {F} is the vector of forces and moments; [K] is the total (hydrostatic plus any mooring) stiffness matrix and is related to the stability of the system; and {E} is the vector of residual forces and moments. If {X l} represents the equilibrium configuration coordinates, i.e., {F(Xl)} = 0, and {X °} is such that the difference {X l - X °} is small, then we can rewrite eqn (1) as

{ x ' ) = ( x °) - [K]-'

(2)

Equation (2) can be iterated until the forces and moments vanish. The iterative process is described next. An initial estimate of the equilibrium coordinates {X °} is assumed. Based on {X°}, {F(X°)} is calculated. {X l } is then calculated from eqn (2). If [[{F(X1)} II (divided by the weight of the platform for forces, and weight times a characteristic length, such as the base diameter of the platform for moments) is less than a given tolerance, the iteration is ended since convergence is achieved. It should be noted that in eqn (2), the residual forces have been assumed to be negligible, and convergence to the equilibrium solution will be slow if the impelling forces are large and if the initial approximation, {X°}, is not good. Therefore, to arrive at a good initial estimate of the equilibrium, the method devised by Metcalf2 is used. In Metcalf, 2 the forces and moments at the current configuration are evaluated. If there are imbalances, changes to the configuration coordinates are made based on the sign of the forces and moments. For example, if there is a net force acting upwards, then an upward displacement to the platform is given. Initially, the changes made are constant. But when there is a change in the sign of the forces and moments, a bisection scheme is used to speed up the algorithm. The calculation of the components of the stiffness matrix, [K], is described next.

2.2 Coordinate systems The forces and moments are expressed with respect to a body-fixed coordinate system. The body-fixed coordinate system, referred to by Oxl x2x3 (see Fig. 1) is at the center of the base of the platform. The unit base vectors in this coordinate system are {e~,e2,e3}. The global coordinate system, referred to by OXo~Xo2Xo3, is situated at the still water surface with the Xo3 axis pointing vertically upwards and earth bound. The unit base vectors in the global coordinate system are {eol, eo2, eo3}. The forces acting on a freely floating body in quiescent water are the

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O

"~1

Fig. 1. Coordinate system.

weight of the body and the hydrostatic pressure forces. In view of the symmetry of the conical gravity platform, it has been assumed here that the forces acting are symmetric with respect to the OXol Xo3 plane. This assumption requires that the platform be ballasted symmetrically with respect to the oxol Xo3 plane and the arrangement of the mooring lines is also symmetric with respect to this plane. In view of this assumption, the configuration coordinates chosen are the draft at the centerline, T, and the trim angle ~b (rotation about the x2 axis, clockwise positive) of the platform. The equation of the waterplane is (see Fig. 1). x3 = xl tan ~ + T.

(3)

The transformation between the global and body-fixed coordinate systems can then be written as Xo,

[