A Theoretical Description of Rotary Drilling for Idealized Down-Hole Bit/Rock Conditions
Authors: Gnirk Paul F. , Cheatham J.B. 
Published: December 2007

Abstract
The results of combined analytical and experimental studies involving simulated multiple bit-tooth penetration into rock are incorporated into a drilling rate equation for roller-cone bits assuming rather idealized downhole conditions. In particular, it is assumed That the rock behaves statically in a ductile fashion during bit-tooth penetration and that the rock chips are instantaneously removed from the bottom of the drill hole. The general analysis demonstrates an application of plasticity theory for the rock/bit-tooth interaction to The formulation of an upper limit on rotary drilling rate.

Introduction
Extensive experimentation involving single and indexed bit-tooth penetration into rock in a confining pressure environment has demonstrated that the pressure environment has demonstrated that the chip formation process is of a ductile, or pseudoplastic, nature at sufficiently low differential pseudoplastic, nature at sufficiently low differential pressures so as to be of interest in rotary drilling. pressures so as to be of interest in rotary drilling. Coincident with the experimentation, analytical consideration has been given to the theoretical problems of single and indexed bit-tooth penetration problems of single and indexed bit-tooth penetration into rock. In general, the analyses have assumed that the rock behaves statically in a rigid-plastic fashion and obeys the Mohr-Coulomb yield criterion. The quantitative comparison between experimental and calculated values of bit-tooth load required for chip formation has been remarkably good for a variety of rocks commonly encountered in drilling and at simulated differential pressures as low as 500 to 1,000 psi.
Results obtained recently for indexed bit-tooth penetration indicate that the work (or energy) penetration indicate that the work (or energy) required to produce a unit volume of rock chip can be minimized by a proper combination of bit-tooth spacing and bit-tooth load for a given rock type and differential pressure. By utilizing this information, it is possible co formulate a drilling rate equation, at least in a preliminary fashion, for a roller-cone bit performing under rather idealized downhole conditions. In particular, through the use of characteristic dimensionless quantities pertinent to a roller-cone bit and to indexed bit-tooth penetration, interrelationships among bit weight, rotary speed, rotary power, bit diameter, rock strength and bit-tooth shape and spacing can be explicitly expressed. In the formulation of the equations, however, it is assumed that the rock chips are instantaneously removed from the bottom of the drill hole and that the rock behaves in a ductile manner during bit-tooth penetration. In addition, the effects of bit-tooth load application And penetration by a yawed tooth at an oblique angle are neglected. Although the analysis is presented in the light of some rather restrictive conditions, it does demonstrate a method of applying fundamental rock/bit-tooth interaction data, obtained by combining the results of analysis and experiment to the formulation of a drilling rate equation for rotary drilling.