Browsing by Subject "Delay differential equations"

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    A Nonlinear Dynamical Model of Borehole Spiraling
    (2015-12) Marck, Julien
    With the emergence of new measurement devices, the non-smooth nature of the borehole geometry has been comprehended more accurately. In many happenstances, the borehole has been seen taking the shape of a corkscrew or helix. Referred to as micro-tortuosity, or more commonly spiraling, this drilling dysfunction correlates with lower-than-expected rates of penetration, increased shocks and vibrations, damage to components and tools, and smaller drift diameters. In this dissertation, the relevant mechanism and parameters leading to borehole spiraling are identified and studied. Spiraling is predicted by conducting a stability analysis of the linearized delay differential equations governing the borehole propagation. These evolution equations, expressed in terms of the borehole inclination and azimuth, are obtained from considerations involving: (a) a bit/rock interaction law that relates the force and moment acting on the bit to its penetration into the rock; (b) kinematic relationships that describe the local borehole geometry in relation to the bit penetration; and (c) a beam model for the bottom-hole 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 to a critical value representing a Hopf bifurcation of stability, which depends only on the BHA configuration and the bit walk. If this group is less than the critical value, the system is referred to as being 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. The influence of the design of push-the-bit rotary steerable systems (RSS) on the onset of spiraling is also discussed, as well as the ability of the stabilizers to tilt freely or not. For directionally unstable systems, the resulting limit cycle, corresponding to the spiral, has been captured by the introduction of the relevant nonlinearity: a saturation of the bit tilt. This nonlinearity enables to characterize further the influence of the directional stability on the borehole geometry and illustrates how the amplitude of the spiral also depends on the borehole inclination and the RSS force. Applications to field cases are discussed and model predictions are tested against actual spiraled-hole data. Simulations conducted by integrating the equations of borehole propagation also are presented. For unstable systems, the model predicts spiraled boreholes with a pitch comparable to what generally is observed in the field. The general good agreement between the model predictions and the field data suggests potential direct implementations. It could lead to model-based control algorithms limiting micro-tortuosity or to better bit and BHA designs, if complemented with field campaigns aiming at refining the model parameters.