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Stochastic torsional stability of an oil drill-string

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Abstract

The aim of this paper is to investigate the influence of uncertainties on the torsional vibration of drill-strings, in order to find out which uncertainty affects most significantly the torsional stability. The unstable torsional behavior is commonly associated to polycrystalline diamond compact bits, and manifests itself in the form of stick-slip oscillations. The stick-slip is a severe type of self-excited vibration characterized by large fluctuations in the rotation of the bit. It not only increases the bit wear, but also can cause drill-string failures. The analysis were done using a mathematical model of the drill-string based on classical torsion theory discretized by means of the finite element method. The bit-rock torque was included in the model as a nonlinear boundary condition at the bottom end of the drill-string. The values of the model parameters are typical values of a real drilling situation, which are subject to a high degree of uncertainty, what justifies a stochastic analysis. We have built probability distributions for the uncertain parameters and used Monte Carlo method to obtain the stochastic stability maps.

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Notes

  1. Bit rate of penetration: the downward speed of the bit. The bit rate of penetration is a measure of how fast the rig is drilling the hole. Higher penetration rates imply quicker drilling progress and less time being spent in the process [12].

  2. When one uses linear basis functions and chooses the elements such that \(\rho (z)\), G(z), J(z) do not vary inside them, the finite element mass and stiffness matrices are given by [22]:

    $$\begin{aligned} {\mathbf {M}}^{(e)} = \frac{\rho J \, l}{6} \left[ \begin{array}{cc} 2 &{} 1\\ 1 &{} 2 \end{array}\right] \phantom {aaaaaa} {\mathbf {K}}^{(e)} = \frac{GJ}{l} \left[ \begin{array}{cc} 1 &{} -1\\ -1 &{} 1 \end{array}\right] \end{aligned}$$
    (3)

    where l is the finite element length.

  3. Hysteresis: is the history dependence of a physical system outputs. The system outputs depend not only on the present inputs, but also on the past inputs.

  4. The vertical load per unit horizontal area is given by \((f_{Bit})_{z} = \frac{(F_{Bit})_{z}}{\pi (R_{Bit})^{2}}\), where \((F_{Bit})_{z}\) is the weight-on-bit, \(R_{Bit}\) is the bit radius.

  5. The sign function extracts the sign of its argument. For all \(x \ne 0\) we define [27]: \({{{\mathrm{sgn}}}}(x) := \frac{x}{|x|}\). If \(x=0\) we define \({{{\mathrm{sgn}}}}(x) := 0\).

  6. Cutter wearflat: horizontal flat surface below the cutter.

  7. The coefficient of variation function measures the dispersion of a random variable, \({{\mathrm{cv}}}:= {{{\mathrm{\upsigma }}}}/{{{\mathrm{\upmu }}}}\), where \({{\mathrm{\upmu }}}\) is the mean function and \({{\mathrm{\upsigma }}}^{2}\) is the variance function.

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Acknowledgements

The authors acknowledge the financial support of the Brazilian agencies CAPES, CNPq and FAPERJ.

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Authors and Affiliations

  1. Department of Mechanical Engineering, Centro de Tecnologia, Federal University of Rio de Janeiro, Cidade Universitária, Rio de Janeiro, 21945-970, Brazil

    B. F. Nogueira & T. G. Ritto

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  1. B. F. Nogueira
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  2. T. G. Ritto
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Correspondence to B. F. Nogueira.

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Appendix A: data used in simulations

Appendix A: data used in simulations

See Table 2.

Table 2 Values of the exact variables
Full size table

The values of the material properties \(\rho _{DS}\) and G are referred to ASTM-A36 structural steel. The values of the geological properties \(\mu \) and \(\sigma _{C}\) are referred to sandstone wet [37, 38], the values of the bit-rock interaction constants \(\alpha _{1}\), \(\alpha _{2}\) and \(\alpha _{3}\) were obtained by least squares fit from field data [20].

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Nogueira, B.F., Ritto, T.G. Stochastic torsional stability of an oil drill-string. Meccanica 53, 3047–3060 (2018). https://doi.org/10.1007/s11012-018-0859-6

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