A multistabilized bottomhole assembly can be taken as a continuous beam-column in the theory of elastic stability. A set of linear algebraic "equations of three moments" will then, be derived to calculate forces at bit and stabilizers as well as the bending deflection at any point.
At first, a two-dimensional analysis is presented. The axis of the borehole is considered as a circular arc in a vertical plane. The lower end of the assembly is the bit which is regarded as simply supported, and the upper end is a point of tangency of the collar with the lower side wall, which has an unknown length from the uppermost stabilizer. All the internal bending moments at stabilizers are also unknowns to be determined at the first stage of theoretical analysis. The conditions of continuity and boundaries are just enough to get the required equations to solve such unknown quantities.
When the axis of the borehole is a space curve. the problem is turned to be three-dimensional and can be easily solved by separating it into two two-dimensionalones: one in the inclination plane and the other in the directional plane. The algorithm is so simple that it can be easily programmed by a pocket computer, even if the number of stabilizers may be quite numerous.
The proper selection of the configuration of bottomhole assembly to get the required side force acting on the bit is of primary importance in controlling of the inclination angle of the bit trajectory. An exact solution with very simple calculation is presented in this article for the analysis of multistabilized bottomhole assembly problems. Another important question of hole deviation forces caused by the interaction between the bit and the anisotropic formation will be discussed later.
Millheim(1) had pointed out that Significant errors may be resulted in analyzing and predicting the ehaviour of a bottomhole assembly if the borehole axis is considered as straight line. In fact, the bit trajectory, especially in directional drilling. is continuously changing both in inclination angle α and in azimuth angle φ. Then, the BHA is lying in a three-dimensionally curvalinear borehole. Suppose a segment of such a borehole AB with a length of L have φA, φA at section A and αB, φB at Section B, its curvature KAB can be determined by means of the following equation:
Equation (1) available in full paper.
where ε is total angular change within length L. The curvature of the arc AB, or the "dog_leg severity" is defined as:
Equation (2) available in full paper
The reciprocal of KAB is the radius of curvature of arc AB
Equation (3) Available in full paper
It should be pointed out here that eq.(l) implied that the space curve between A and B is assumed as a circular arc on an inclined plane. The method of analysis as stated below, contrary to those laborous approximate methods. such as finite elements method(2), difference equation method(3), potential energy method(4), etc, is very simple in calculation, notwithstanding it is an exact solution.