Spiraled Boreholes: An Expression of 3D Directional Instability of Drilling Systems
Emmanuel Detournay
Civil, Environmental, and Geo- Engineering, University of Minnesota
ABSTRACT: Occurrence of borehole spiraling is predicted by analyzing the delay-differential equations governing the propagation of a borehole. These evolution equations for the borehole inclination and azimuth are obtained from considerations involving: (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 in relation to 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 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 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 the bifurcation value that depends only on the BHA configuration and the bit walk. Stability curves for an ideal BHA with two stabilizers are shown to depend on the bit walk, which tends to enhance conditions for spiraling. An application to a field case is 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 to what is observed in the field. Furthermore, they show that a limit cycle, characterized by bounded magnitude of the oscillations, is attained if a nonlinearity is introduced in the bit response.
