EQUATIONS SIMPLIFY DYNAMIC ANALYSIS OF DEEPWATER DRILLING RISERS
Li Huagui
Southwest Petroleum Institute
Nanchong, China
A set of nonlinear equations with practical simplifying assumptions on the governing motion equations and operational boundary conditions can improve the dynamic analysis of marine drilling risers.
The dynamic analysis of marine drilling risers includes time domain and frequency domain analyses.1-4
Many dynamic analysis models are very elaborate and inconvenient for practical use.
This article simplifies the four-order nonlinear partial differential equation and boundary conditions used to describe marine drilling riser motion. The nonlinear dynamic analysis is numerically simulated by using a finite difference method.
This dynamic analysis method is simple and accurate, and the numerical simulation takes less time than other comparable methods. The equations are applicable to the design and check of a marine riser system.
The top tension on marine risers was studied using four sea states that correspond to increasingly difficult drilling modes. The results from this numerical method are consistent with those from an analysis of a riser in operation.5
MATHEMATICAL MODEL
The motion equations had to be derived in a convenient form for practical use. The riser configuration and motion were therefore simplified based on the following assumptions:
- The riser's pipe is made of a homogeneous, isotropic, and linearly elastic material.
- The cross section of the riser is a circle.
- The wave, current, and riser motion occur in the same plane. In other words, the riser motion is in two dimensions.
- The riser deflection because of the combined action of wave, current, and other loads is small and finite. The angle between the riser line and true vertical is less than 10.
- The riser pipe is full of drilling mud, and the bending stiffness because of the drillstring may be neglected.
These assumptions make the riser analysis practical and efficient. The riser and coordinate system are shown in Fig. 1. Fig. 2 shows the forces acting on an element of the riser system. The governing motion equation of the riser system is derived from the assumptions and the forces shown in Fig. 2 (Equations 1 and 2).
The lower end of the riser system is connected to the blowout preventer (BOP) stack through the lower ball joint. Because the lateral stiffness of the BOP stack is relatively larger than that of the riser system, the lower end of the riser system and BOP stack are taken as a fixed point with no lateral offset (rotation is allowed however). Equation 3 represents the lower boundary conditions.
If the stiffness of the lower ball joint is Kr, some bending flexure will occur (Equation 4). If Kr = 0, no bending flexure will occur (Equation 5). The angle 0 is a rotational angle.
The upper end of the riser system is connected to the floating rig platform through a flex ball joint. The horizontal motion of the riser system results from the platform's horizontal motion. The heave movement is neglected. Thus, the upper boundary conditions can be represented by Equation 6.
If the stiffness of the upper ball joint is Kr, some bending flexure will occur (Equation 7). If Kr = 0 for the upper ball joint, no bending flexure occurs (Equation 8).
NUMERICAL METHOD
For a discrete model of the governing motion equation, the dynamic response of the riser system has a periodical character in the steady state (Equation 9). Substituting Equation 9 into Equation 2 removes the partial derivative with time.
Then, substituting difference representations for the other derivative items yields the appropriate differential equations. Three-point central difference representations are used for the first and second derivative, and a five-point central difference representation is used for the fourth derivative.
The riser system is divided into n elements. The length of each element is _h. Equation 10 is the set of differential equations obtained.
The upper boundary condition, Equation 6, can be rewritten as Equation 11. The ball joint is considered a hinge joint. When the stiffness of the upper ball joint is zero, the section ABC in Fig. 3 is a straight line (Equation 12).
When the stiffness of the upper ball joint is not zero, the section ABC shown in Fig. 4 is curved by the moment, M = Kr0. The expression y(n - 1) is then given by Equation 13. In this equation, K = -Kr_h/EI.
The lower boundary condition, Equation 3, can be rewritten as Equation 14. The lower ball joint is also considered a hinge joint. When the stiffness of the lower ball joint is zero, the section DOE in Fig. 5 is a straight line (Equation 15).
When the stiffness of the lower ball joint is not zero, the section DOE in Fig. 6 is curved by the moment, M = Kr0. The expression y-1 is then given by Equation 16.
The set of simultaneous difference equations consists of Equations 10, 11, 13, 14, and 16. The set has n + 3 equations; thus, n + 3 unknown values of y1 (i = 1, 0, 1,... n, n + 1) can be determined.
DRILLING LIMITATIONS
Four operation modes in deepwater offshore drilling were considered:
- Normal drilling mode
In this mode, all the routine drilling operations (drilling ahead, tripping pipe, reaming, and circulating mud) can be performed in the local environment and with current hole conditions. Special operations, such as running casing or formation testing, may require more strict limitations.
- Limited drilling mode
In this mode, the drilling operations can be performed only under limited conditions. For example, if the environmental conditions become too harsh for normal drilling operations to proceed, operations may have to be shut down immediately.
- Suspended drilling mode
In this mode, all the drilling operations have been stopped. The riser system is suspended, and it can be moved by the sea current. If necessary, advance preparations for closing the well and disconnecting the riser system with the BOP stack are made.
- Disconnected operation mode
In this mode, the riser system is separated from the BOP stack once the environmental conditions exceed the safe operation limitations of the suspended operation mode.
The important parameters for riser operation include the angle between the lower ball joint and true vertical, the averaged stress, the alternating stress, the angle between the slip flex joint and true vertical, and the proper safety factor for the top tension force. Apparently, the ranges of these parameters are related to the environmental conditions.1-6 Table 1 lists the American Petroleum Institute recommended values of operation limitations.6
The selected top tension force should be larger than the result from the riser analysis. The difference between these two values is the tension loss listed in Table 1. This tension loss is a result of wear, inertia, the compressibility of gas and fluids in the riser, and efficiency. The recommended top tension force should not exceed the adjusted value of the maximum top tension.
EXAMPLE
With input data from an actual riser system in operation, the equations for the numerical simulation method were used to produce the curves of maximum ball joint angle vs. top tension (Fig. 7).5 The original curves for the rig are shown in Fig. 8.
The basic data used in these figures are a water depth of 1,650 ft, a normal drilling mode, a wave height of 10 ft, and a wave period of 8 sec. The sea current is a surface current only with a linear distribution and a 1.5 knot velocity. The static offset of the platform is 2% of the water depth.
Table 2 lists the total top tension; the values from the numerical method presented in this article are consistent with those from the actual operation. The curves in Figs. 7 and 8 are quite similar and therefore support the use of these equations for riser analysis. The numerical simulation method has since been successfully used for dynamic analysis of marine drilling risers on two wells.
REFERENCES
- Burke, B.G., "An Analysis of marine Risers for Deep Water," Journal of Petroleum Technology, April 1974, pp. 455-465.
- Ertas, A., and Kozik, T.J., "Philosophy and Principle of Riser Modeling and Numerical Approaches", presented at the 1986 Offshore Operations Symposium.
- Rajabi, F., Zedan, M.F., and Mangiconachi, A., "Vortex Shedding Induced Dynamic Response of Marine Riser," Journal of Energy Resources Technology, ASME, Vol. 106, June 1984, pp. 214-221.
- Ertas, A., and Kozik, T.J., "Numerical Solution Techniques for Dynamic Analysis of Marine Riser," Journal of Energy Resources Technology, ASME, Vol. 109, March 1987, pp. 1-5.
- Miller, C.A., "Bowdrill 2 Riser Analysis for a Water Depth of 1,650 ft," Stress Engineering Services Inc., March 1987.
- American Petroleum Institute, "Design and Operation of Marine Drilling Riser System," API RP 2Q, April 1984.
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