A 3D Analysis of a Bottomhole Assembly Under Large Deflection
- Zifeng Li (Daqing Petroleum Inst., PRC) | Xingrui Ma (Harbin Inst. of Technology, PRC) | Wenhu Huang (Harbin Inst. of Technology, PRC) | Xisheng Liu (U. of Petroleum, PRC)
- Document ID
- Society of Petroleum Engineers
- SPE Drilling & Completion
- Publication Date
- June 1996
- Document Type
- Journal Paper
- 104 - 110
- 1996. Society of Petroleum Engineers
- 4.1.5 Processing Equipment, 1.10.1 Drill string components and drilling tools (tubulars, jars, subs, stabilisers, reamers, etc), 4.1.2 Separation and Treating, 1.6 Drilling Operations, 1.11 Drilling Fluids and Materials, 1.6.1 Drilling Operation Management, 1.6.6 Directional Drilling, 1.5 Drill Bits
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A 3D static mathematical model of a bottomhole assembly (BHA) including steerable downhole motor assembly under large deflection has been established. The forces and deflections of the drillstring have been accurately computed by making use of the method of weighted residuals, the weighted objective function, and the method of optimization to calculate the tangency point and to reduce the effect of the indistinct tangency point conditions. The effects of the weight on bit, the borehole geometry, the geometry and tool face rotation of the BHA on the bit building, and right walking force have been studied separately. There is a significant difference between the bit forces of large deflection analysis and small deflection analysis.
Since the pioneering work by Lubinski,1 the drilling industry has gradually come to accept and appreciate the importance of analysis of BHA, which is now regarded as important in controlling the deviation tendencies of well trajectory, especially in directional and horizontal wells.
Mathematical Methods in BHA Analysis.
There are basically four mathematical methods available to a BHA analysis: classical analytic, finite element, finite difference, and weighted residuals.
In this method, the drillstring displacements and bit forces are expressed in analytic form. The main advantage of this method is that the computation is fast for simple cases where the BHA does not contact the borehole wall. The disadvantage of this method is that it can be used only in a linear analysis and it is very cumbersome in handling complex situations where the BHA contacts the borehole wall.
Finite-Element Method.6-9 This is a well-established numerical method that is widely used in mechanics and structural engineering. The advantage of this method is that it is physically based, adapts to complex material and geometry variations, and generally allows larger element size than the finite-difference method. It is also possible to include nonlinear effects through more complex Lagrangian formulations at a significantly increased cost of establishing the element stiffness matrix. The disadvantage of this method is that it has an inherent difficulty in handling borehole contacts that occur between the predefined nodes. To reduce this problem, either smaller elements or additional internal nodes are needed, resulting in a more complex matrix, thereby reducing the computation speed.
Finite-Difference Method.10 This is a well-established numerical method suitable for solving any types of differential equations. The advantage of this method is that it can account for nonlinear effects and borehole constraints. The disadvantage is that it requires small grid intervals to obtain accurate solutions. This results in a large matrix, while the differential equations and the boundary conditions are complex, and this method will become troublesome.
Weighted-Residuals Method. 2,11-13 This is a well-established semianalytical method in solving differential equations, especially for nonlinear equations. The advantages of this method are that it is suitable for any kind of differential equations and any kind of boundary conditions, and it possesses high accuracy. The disadvantage is that it needs nonlinear optimum processes in solving nonlinear problems.
Linear and Nonlinear Analyses of BHA.
There are linear and nonlinear analyses of BHA, depending upon small or large deflection assumption.
Most analyses of BHA1-4,6-9,11-12 are linear analyses because they make use of small deflection assumption in the derivation process of differential equations or stiffness matrix. These differential equations are linear and easy to solve. On the other hand, because of this small deflection assumption, one cannot account for nonlinear effects that may be important, particularly for large borehole curvatures, oversized holes, and low BHA stiffnesses. Recently, large deflection analyses of BHA containing nonlinear effects have been published.5,10,13 But small deflection assumption is still used in the simplifications of differential equations.
In this paper, a 3D static mathematical model of a BHA, including steerable downhole motor assembly under large deflections, has been established. The forces and deflections of drillstring have been accurately computed by making use of the method of weighted residuals, the weighted objective function, and the method of optimization to calculate the tangency point and to reduce the effect of the indistinct tangency point conditions. The effects of the weight on bit, the borehole geometry, the geometry and tool face rotation of the BHA on the bit building, and right walking force have been studied separately. There is a significant difference between the bit forces of large deflection analysis and small deflection analysis.
For the convenience of establishing a model, we propose the following hypotheses.
1. The components of the BHA, including steerable downhole motor assembly, behave as linear elastic bodies.
2. The bit is centered in the borehole on the hole axis and no moment exists between the bit and the formation.
3. The components of the BHA, including steerable downhole motor assembly, have annular sections and arbitrary properties that remain constant in any segment between stabilizer/bend angle and nearby stabilizer/bend angle.
4. The borehole wall is rigid.
5. The dynamical effects of the drillstring and drilling fluid may be ignored.
6. The drillstring lies on the lower side of the hole from some interval above the upper stabilizer.
The rectangular right-hand coordinate system, shown in Fig. 1, has its origin at the lower end of the BHA (e.g., the bit), the z axis coincides with the tangent line of the wellbore axis and directs to the BHA, the x axis directs to the lower side of the wellbore, and the y axis directs to the right walking direction.
The BHA is a bending beam that is separated into n individual segments by n-1 stabilizers, bend angles, or contact points. The BHA is in 3D bending condition under the actions of its weight, the weight on bit, the torque on bit, and the supports of the well wall.
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