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Research Papers

A Semi-Analytical Study of Stick-Slip Oscillations in Drilling Systems

[+] Author and Article Information
B. Besselink

Department of Mechanical Engineering, Eindhoven University of Technology, P.O. Box 513, 5600 MB Eindhoven, The Netherlandsb.besselink@tue.nl

N. van de Wouw

Department of Mechanical Engineering, Eindhoven University of Technology, P.O. Box 513, 5600 MB Eindhoven, The Netherlandsn.v.d.wouw@tue.nl

H. Nijmeijer

Department of Mechanical Engineering, Eindhoven University of Technology, P.O. Box 513, 5600 MB Eindhoven, The Netherlandsh.nijmeijer@tue.nl

J. Comput. Nonlinear Dynam 6(2), 021006 (Oct 22, 2010) (9 pages) doi:10.1115/1.4002386 History: Received October 14, 2009; Revised June 23, 2010; Published October 22, 2010; Online October 22, 2010

Rotary drilling systems are known to exhibit torsional stick-slip vibrations, which decrease drilling efficiency and accelerate the wear of drag bits. The mechanisms leading to these torsional vibrations are analyzed using a model that includes both axial and torsional drill string dynamics, which are coupled via a rate-independent bit-rock interaction law. Earlier work following this approach featured a model that lacked two essential aspects, namely, the axial flexibility of the drill string and dissipation due to friction along the bottom hole assembly. In the current paper, axial stiffness and damping are included, and a more realistic model is obtained. In the dynamic analysis of the drill string model, the separation in time scales between the fast axial dynamics and slow torsional dynamics is exploited. Therefore, the fast axial dynamics, which exhibits a stick-slip limit cycle, is analyzed individually. In the dynamic analysis of a drill string model without axial stiffness and damping, an analytical approach can be taken to obtain an approximation of this limit cycle. Due to the additional complexity of the model caused by the inclusion of axial stiffness and damping, this approach cannot be pursued in this work. Therefore, a semi-analytical approach is developed to calculate the exact axial limit cycle. In this approach, parametrized parts of the axial limit cycle are computed analytically. In order to connect these parts, numerical optimization is used to find the unknown parameters. This semi-analytical approach allows for a fast and accurate computation of the axial limit cycles, leading to insight in the phenomena leading to torsional vibrations. The effect of the (fast) axial limit cycle on the (relatively slow) torsional dynamics is driven by the bit-rock interaction and can thus be obtained by averaging the cutting and wearflat forces acting on the drill bit over one axial limit cycle. Using these results, it is shown that the cutting forces generate an apparent velocity-weakening effect in the torsional dynamics, whereas the wearflat forces yield a velocity-strengthening effect. For a realistic bit geometry, the velocity-weakening effect is dominant, leading to the onset of torsional vibrations.

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References

Figures

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Figure 1

Schematic model of a drill string

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Figure 2

Bottom hole profile between two successive blades (after Ref. 16)

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Figure 3

Stability diagram in (η¯2,τ¯n)-space for γ¯=0 (top) and γ¯=0.5 (bottom). The stable region is depicted in gray, and the unstable region is depicted in white.

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Figure 4

Example of an axial limit cycle in (z1,z2) coordinates (top) and (U,dU/dt) coordinates (bottom)

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Figure 5

Validity of the condition τ¯a≤τ¯b in (γ¯,η¯2)-space. The dash-dotted line shows the line τ¯a=τ¯b for the minimal delay. For increasing delay, the validity region decreases, as shown for τ¯n={5,10,20,40,80,160}. Finally, the dashed line is a numerical approximate of the asymptote for τ¯n→∞.

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Figure 6

Influence of the delay τ¯n on the axial limit cycle for γ¯=0.1 and η¯2=0.1

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Figure 7

Averaged value ⟨z1⟩a and minimum value z1,min for the axial limit cycle with γ¯=0.1 and η¯2=0.1. The black circle denotes the critical delay, at which stability is lost. The bottom graph is a zoomed version of the top figure.

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Figure 8

Value of A in (γ¯,η¯2)-space

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