Spiraled boreholes: An expression of 3D directional instability of drilling systems

Borehole spiraling is predicted by analyzing the delay differential equations (DDEs) governing the propagation of a borehole. These evolution equations, expressed in terms of the borehole inclination and azimuth, are obtained from considerations involving a bit/ rock-interaction law that relates the force and moment acting on the bit to its penetration into the rock; kinematic relationships that describe the local borehole geometry in relation to the bit penetration; and a beam model for the bottomhole assembly (BHA) that expresses the force and moment at the bit as functions of the external loads applied on the BHA and the geometrical constraints arising from the stabilizers conforming to the borehole geometry. The analytical nature of the propagation equations makes it possible to conduct a systematic stability analysis in terms of a key dimensionless group that controls the directional stability of the drilling system. This group depends on the downhole weight on bit (WOB), on properties of the BHA, on the bit bluntness, and on parameters characterizing the steering response of the bit. The directional stability of a particular system then is assessed by comparing the magnitude of this group with a critical value, representing a bifurcation of stability, which depends only on the BHA configuration and on the bit walk. If this group is less than the critical value, the system is deemed to be directionally unstable, and borehole spiraling is likely. Stability curves for an idealized BHA with two stabilizers show that the bit walk tends to make drilling systems more prone to spiraling. Applications to field cases are discussed. Simulations conducted by integrating the equations of borehole propagation also are presented. They illustrate that, for unstable systems, the model predicts spiraled boreholes with a pitch comparable with what generally is observed in the field.