A simulation study of a dynamic positioned semi-submersible conduction offshore drilling is carried out to demonstrate the effect of intoducing criteria related to the riser response. The floater is a semi-submersible which is equipped with 4 azimuthing thrusters each able to produce 1000 kN located at the four corners at the two pontoons.

The operational draught is equal to 24 m, vessel mass at operational draught is 45000 ton. the length is 110 m. and the breadth is 75 m. Radius of gyration in roll is 30 m, in pitch equal to 33 m and in yaw 38 m. The undamped resonance periods in roll and pitch are found to be equal to 55 s and 60 s. respectively.

The riser properties are the same as applied throughout the present example study. The current profile and the current surface velocity as a function of time are assumed to be known. The latter is shown in Figure 6. A time window of 4000 seconds is applied in the simulation. Here, the mean values and variances (and possibly higher moments if relevant) of the top and bottom angle response processes are taken as known based on the measurements. However, the estimation of these values based on measurements is a topic on its own. This applies in particular to cases where prediction of the mean value and variance into the near future is relevant, see e.g. Leira (1998).

The following variance is assumed to apply for the top angle: It is constant and equal to 0.2 (degrees)² until 500 seconds. and is then increased linearly to 0.55 (degrees)² at 4000 seconds. The variance of the bottom angle is taken as one fourth of the variance for the top angle.

　

The simplest dynamic position scheme is based on a fixed set point which is typically just above the wellhead. For (his case, the reliability indices for the top and bottom angles can be computed as a function of time. The reliability index is defined by the relationship Pf = Φ(-β) where Pf is the probability of failure for a given time interval, and Φ() is the standard normal cumulative distribution function. The probability of failure is computed based on a Gumbel extreme value distribution as referred to a 20 minute stationary period. The probability is obtained from the corresponding cumulative distribution function by inserting the critical values of the respective angles. The reliability index is subsequently obtained from the inverse standard normal distribution function. Presently, the critical value for the top angle is set to 5 degrees. and to 3 degrees for the bottom angle.

The reliability indices as functions of time are shown in Figure 7. It is seen that the reliability index for the bottom angle becomes zero and negative in the time interval from 1100 to 2000 seconds. This implies that the failure probability exceeds 50%. Typically, a much smaller probability would be permitted. in particular for the bottom angle. If a "reliability index monitoring" is performed, some kind of action should hence be undertaken. Such an action could e.g. correspond to moving the surface floater.

The most direct way of taking the riser angle criteria into account in the dynamic positioning algorithm is by application of equations (12) to (15). This algorithm is based on the instantaneous mean values of the riser angles. and its application is illustrated in Sorensen et al.(2000a). For such a positioning algorithm, the dynamic response components are excluded from the control loop. However, "reliability index monitoring" (or monitoring of maximum dynamic measured angles) can still be performed to decide when counteracting measures need to be applied.

Application of regression curves was investigated for this control scheme, and was found to give identical results to an algorithm based on a complete Finite Element Model.

　

Considering the variation in reliability indices as a function of time for both the lower and upper angle, it seems natural to consider dynamic position control schemes that are more or less based on safety index target levels. It is possible to establish a control scheme completely based on reliability-index criteria. However, here we consider criteria based on a linear combination of fixed weights and running weights. The latter are computed on-line as functions of the instantaneous values of the reliability indices.

The basic version of this scheme is based on making the weight factors for the top and bottom angles become linear combinations of a fixed term plus a function of the instantaneous reliability index. Leira et al. (2001):

where βT is the estimated reliability index for the top angle, βB is the estimated reliability index for the bottom angle, βCRt andβCR_b are the critical reliability indices for the top and bottom angles. and C1, C2 are constants. The critical index is taken as 2.0 for the bottom angle and 1.0 for the top angle. The incremental index Δβ is a "smoothing increment" for the reliability index to avoid singularities in the expression.

The reliability indices corresponding to this strategy are shown in Figure 8.

Generalization of the algorithm for optimal positioning of the floater can easily be extended to cover other types of riser configurations. Examples are risers which are restricted at the lower end. which is frequently achieved by introducing a taper-joint to make the rotation angle zero. The design criteria entering the control loop will then be based on allowable stresses rather than allowable angles.

　

ABB Industry AS is gratefully acknowledged for fruitful cooperation.