Use of Single-Cutter Data in the Analysis of PDC Bit Designs: Part 2Development and Use of the PDCWEAR Computer Code
Authors: Glowka D.A. 
Published: December 2007

Summary
Algorithms are developed for use in the PDCWEAR computer code, which analyzes the performance and wear of PDC bits in drilling rock. A cutter-interaction model is generalized to permit prediction of individual cutter forces across the bit face in an arbitrary design. These forces are used to predict cutter wear-flat temperatures and wear rates as well as integrated performance parameters, such as the weight on bit (WOB), drilling torque, bit side forces, and bending moments. Use of PDCWEAR is demonstrated by analyzing a typical PDC bit design. A parametric study of bit design and operating variables is conducted to provide general conclusions related to the effects of bit profile, number of cutters, rotary speed, water-jet assistance, and cutter-wear mode.
"Within limits, the WOB and torque required to drill at a specified penetration rate are relatively Independent of the number of cutters used on the bit."

Introduction
Part 2 of this paper describes the Part 2 of this paper describes the development of PDCWEAR,* a computer code that uses the single-cutter test results presented in Part 1 (Ref. 1) as a basis for predicting bit performance. The code also uses PDC cutter-wear models developed in our earlier work to predict bit wear and the effects of that wear on subsequent bit performance. General trends related to the effects of bit design and operation, as predicted by the code, are also identified and discussed. The origin of the algorithm used for cutter-geometry calculations in PDCWEAR is STRATAPAX, a computer code that computes cross-sectional areas and volumes of cut for each cutter in an arbitrary bit design. STRATAPAX has an optimization routine that adjusts cutter radial placement to equalize either cutting volumes or crosssectional areas, as specified by the user. Extensive modifications to the algorithm were made in developing PDCWEAR to account for the effects of wear and to calculate quantities needed to compute cutter forces, temperatures, and wear rates.

Bit-Analysis Algorithms
This section briefly describes the algorithms used to calculate individual cutter forces, temperatures, and wear rates as well as integrated bit-performance parameters. A more detailed description, including derivation of the equations, is provided in Ref. 4. Fig. 1 shows a simplified schematic of an arbitrary PDC bit body in a borehole. A single, worn cutter with backrake angle B is shown mounted at an arbitrary location on the bit body. Four coordinate systems are used to describe cutter geometries. The borehole coordinate system (x, y, z,) is a nonrotating, Cartesian coordinate system with a z axis parallel to the longitudinal axis of the borehole. The bit coordinate system (x', y', z') is a Cartesian coordinate system that rotates and advances with the bit will that has a z' axis parallel to the longitudinal axis of the bit and parallel to the z axis of the borehole coordinate system. The cutter coordinate system (xo', yo', zo') is a two-dimensional, rectangular coordinate system that lies in the plane of the diamond face of each cutter. The cutter profile coordinate system (xo, yo, zo) is the projection of the cutter coordinate system onto a radial plane running through the longitudinal axis of the bit.
Fig. 2 shows parameters used to describe cutter locations. The cutter compact radius, r, is a measure of the cutter size. The radial position of the center of the compact face is R. The longitudinal position of the cutter, h', is measured from an arbitrary reference plane. The circumferential location of the cutter, 0, is measured counterclockwise from the x' axis. Finally, the inclination angle, c, defines the tilt of the cutter's longitudinal axis with respect to the bit's longitudinal axis. Fig. 3 illustrates the forces acting on individual cutters and on the bit as a whole. A penetrating force, F, develops normal to the cutter wear-flat, and a drag force, Fd, develops perpendicular to F. As shown in Ref. 1, the magnitudes of these forces depend on the rock type being drilled, the rate of penetration (ROP) of the bit, the cutterwear state, and the degree of interaction among the adjacent cutters. The penetrating and drag forces of multiple cutters may be integrated to determine the vector sum of the forces and the resultant bending moments. The sum of the vertical (z' direction) force components is the WOB. The sum of moments produced by the drag forces about the z' axis is the drilling torque, . The vector sum of forces and bending moments in the x' and y' directions are the bit side forces, Fx' and Fy', and the bit bending moments, Mx' and My'. These side forces and bending moments ideally are negligible in any bit design; otherwise the bit will drill in an unbalanced manner, rotating about an axis other than the bit centerfine, drilling an overgauge hole, and deviating from its intended path. To determine the cutter forces and thus the integrated bit performance parameters, an algorithm in PDCWEAR determines the cutting profile of each cutter, based on geometry considerations.
JPT
P. 850