Abstract
A phenomenological model for percussive drilling systems is proposed in this paper to explain the experimentally demonstrated existence of an optimal weight-on-bit (WOB), for which the rate of penetration (ROP) is maximized. Several hypotheses have been previously proposed to explain this universal characteristic of percussive drilling, including increased wear of the bit, reduced indexing, and poor cleaning of debris under excessive WOB. Motivated by experimental evidence, we instead consider an increase of the pseudo-stiffness of the bit-rock interface (BRI) with increasing WOB, and investigate its effect on the impact energy transmitted to the rock. The 1D model approximates the dynamics underlying the drilling process by assuming that the impact of the hammer generates a longitudinal wave in the bit. It is shown that the BRI pseudo-stiffness influences the incident wave and associated energy transmitted from the bit to the rock. As a consequence, the drilling efficiency is affected by the dependence of the BRI stiffness on the WOB. The model indicates that there exist optimal conditions for the energy transfer from the bit to the rock in terms of the impedance ratio and the BRI stiffness/WOB. Thus it confirms that there is a sweet spot as seen in practice, which suggests that the root cause of the existence of a sweet spot in the ROP-WOB relationship lies in the nature of the BRI laws, rather than with issues related to bit indexing, bit wear, and/or cleaning of the debris.
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Acknowledgements
This study is a part of the research project INNO-Drill (Technology platform for research-based innovations in deep geothermal drilling) funded by The Research Council of Norway (Grant 254984) and industry partners (Epiroc, Enel Green Power, Lyng Drilling, NOV, Ravel, Robit, Rock Energy, Sandvik Mining and Construction, Tomax and Zaptec).
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A Appendix
A Appendix
1.1 A.1 Hammer-Bit Interface
The equation governing the evolution of the displacement of the HBI following impact is formulated by substituting the following expressions for the velocity and force at the HBI into Eq. (5)
which are obtained by setting \(x=0\) in Eqs. (3) and (4). Thus
After integrating the left term, this equation becomes
whose general solution is
After identifying the constant D using the initial condition \(U(0)=0\), the displacement of the HBI induced by the impact of the rigid hammer is is given by
The displacement increases from \(t=0\) to asympotically reach \(\frac{m_{h}cV_{0}}{EA}\) at large time provided that the elastic bit assembly is unbounded.
1.2 A.2 Bit-Rock Interface
Once the incident wave reaches the BRI, the force balance at the interface is given by
where \(u_{1}\) and \(u_{2}\) represent the displacement at the BRI on the bit and rock side, respectively. Differentiating Eq. (24) with respect to time yields
Given the assumed semi-infinite and homogeneous nature of the rock, no wave reflection in the rock will occur. Hence, the velocity \(v_{1}\) and \(v_{2}\) on both sides of the interface can be expressed as
Substituting Eq. (26) back into Eq. (25) yields the equation governing the evolution of the force applied by the bit on the rock
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Song, X., Aamo, O.M., Kane, PA. et al. Influence of Weight-on-Bit on Percussive Drilling Performance. Rock Mech Rock Eng 54, 3491–3505 (2021). https://doi.org/10.1007/s00603-020-02232-x
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DOI: https://doi.org/10.1007/s00603-020-02232-x