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

The occurrence of borehole spiraling is predicted by analyzing the delay differential equations that govern the propagation of a borehole. These evolution equations for borehole inclination and azimuth are obtained by considering the following three model components: (i) a bit/rock interaction law that relates the force and moment acting on the bit to its penetration into the rock; (ii) kinematic relationships that describe the local borehole geometry from the bit penetration; and (iii) a beam model for the bottom-hole assembly (BHA) that can be used to express the force and moment at the bit from the external loads applied on the BHA and the geometrical constraints arising from the stabilizers conforming to the borehole geometry. The analytical nature of borehole propagation equations makes it possible to conduct a 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 bluntness of the bit, and on parameters characterizing its response. The directional stability of a particular drilling system can be assessed by comparing the magnitude of this group with a critical value that depends only on the BHA configuration and on the bit walk. If this dimensionless group, which depends on the actual drilling conditions, is less than the bifurcation value, the system is deemed to be directionally unstable, and borehole spiraling is likely. Stability curves for an ideal BHA with two stabilizers are shown to depend on the bit walk. An application to a field example is discussed. Borehole simulations, based on solving the equations of propagation, also are presented to illustrate that, for unstable systems, the model reproduces spiraled boreholes with a pitch, comparable to what is observed in the field.