Factors Affecting Torque for a Roller Cone Bit
Authors: Warren T.M. 
Published: December 2007

Warren, T.M., SPE, Amoco Production Co.

Summary
Several authors have presented theoretical bit torque relationships derived by equating the energy required to rotate a bit to the energy required to remove a unit volume of rock (energy balance concept). This approach has had little quantitative success in predicting bit torque because of difficulties in quantifying both the drilling efficiency and the specific energy of the rock under a given set of conditions. A torque relationship based on a force balance concept is presented to minimize the problems with the energy-based relationships. The new torque relationship shows that for a given bit, the torque is determined largely by the applied weight on bit (WOB) and the depth of tooth penetration. The model is insensitive to moderate changes in factors such as bit hydraulics, fluid type, and formation type. Laboratory drilling tests and field data were used to confirm the validity of the new torque model.

Introduction
The torque required to rotate a roller cone bit is of interest for several reasons. First, it may give information about the formation being drilled and the condition of the bit. Second, bit torque exerts a significant influence on the "bit walk" experienced in directional wells. Finally, a prediction of bit torque may be useful in matching a bit and mud motor for optimal performance. Previous work relating to bit torque has been based either on a theoretical energy balance at the bit or on an empirical correlation. The following discussion points out the major problems with these techniques and presents an alternative approach.
Energy Balance Concept. The literature has several references that discuss the energy utilized by a roller cone bit in drilling rock . These articles typically relate the mechanical work done by the bit to the energy required to crush a given volume of rock. The specific energy of a rock is the minimum energy required to crush a unit volume of rock and is considered a fundamental property of the rock. The energy required to crush the volume of rock drilled in a unit increment of time is then expressed as
(1)
where E, is specific energy of the rock, d is hole diameter, and R is the rate of penetration (ROP). The total work done by the bit in the same unit time interval is given by
(2)
where W is axial force applied to the bit, N is rotary speed, and M is torque applied to the bit. The bit torque used in Eq. 2 and throughout this paper is the time-averaged torque required to rotate the bit under steady-state conditions. Eq. 2 includes both the axial and rotary work done by the bit. Generally the axial work is much less than the rotary work and therefore is neglected, but for exactness it is retained in the following discussion. Not all the energy applied to a bit is effective in creating new borehole volume. The amount of energy used to crush a fixed volume of rock is strongly related to the size of chips generated. Some energy is dissipated into the unbroken rock around the borehole while the rock stress is being increased sufficiently to form chips. Energy may be used to further reduce the chip size and remove the broken rock from under the bit. Energy is required not only to form and to remove the chips, but also to form a particular hole geometry. The total work done by the bit (Eq. 2) can be equated to the energy required to break the rock drilled in a unit of time (Eq. 1) by adding an efficiency factor, n, to account for nonproductive energy loss. This relationship can be arranged to give an equation for bit torque,
(3)
The usefulness of Eq. 3 depends largely on n having a reasonably constant value or at least a value that can be determined from other known parameters.
Controlled drilling tests indicate that n varies enough to make the energy balance approach suspect. Figs. 1 and 2 show two sets of laboratory drilling data for an 8 3/4-in. [22.2-cm] Intl. Assn. of Drilling Contractors (IADC) Series 1-2-1 bit and an 8 1/2 -in. [21.6-cm] IADC Series 5–1-7 bit. These data were collected while drilling with the full-scale laboratory drilling rig described in Ref. 7. The Series 1–2-1 bit data in Fig. 1 were obtained while drilling Carthage marble using water as the drilling fluid. The Series 5–1-7 bit data shown in Fig. 2 were obtained with a 9.0-lbm/gal [1078-kg/m ] bentonite mud while drilling Indiana limestone.
JPT
P. 1500^