Steady-state solutions of a propagating borehole

This paper analyzes a general class of stationary trajectories for deep boreholes drilled using rotary systems. These solutions correspond to helical wells twisting around a vertical axis, which can degenerate into straight or circular boreholes; they arise when the forces acting on the bit, and thus the penetrations of the bit into the rock, are invariant in a basis attached to the bit. Under these stationary conditions, the deformed configuration of the bottomhole assembly (the lower part of the drillstring) is also invariant. The paper formulates the equations governing these equilibrium solutions from considerations involving the interaction between the bit and the rock, but also between the bottomhole assembly and the borehole through the contact points at the stabilizers and at the rotary steerable system (the tool used to steer the bit). It is shown that the stationary solutions are completely defined by four parameters characterizing the geometry of the wellbore: two at the scale of the bottomhole assembly (the curvature and inclination of the helical axis), and two at the scale of the bit (the bit tilts, proxies for the borehole diameter). The key dimensionless parameters that control the directional response of the drilling system are finally identified, as well as the critical values of some parameters at which a pathological change in the response takes place.