Drillstring Vibrations: Classic Ideas and Modern Approaches - Wave Propagation in Drilling, Well Logging and Reservoir Applications - page 137

tests with hard formation bits under rock and borehole fluid pressures simulating
a deep hole in a impermeable dolomite. Bit weight, torque and rotary speed
were carefully controlled on a hydraulically-driven drilling machine that was
shock and vibration isolated. Individual parameter effects on penetration rate
were obtained and fitted to polynomial relations that relate our u, u t and u x .
Warren (1981) similarly gave laboratory derived drilling models for full-
scale soft formation bits, relating weight-on-bit, rotary speed, bit size, bit type,
and rock strength to penetration rate. In his study, the laboratory rig WOB and
rotary speed were servo-controlled so that they were constant during the test.
Warren (1983) explored efficiency factors affecting tricone bit performance,
introducing torque as another variable into his model, while Warren (1984),
dealing with roller cone bits, studied penetration rate performance as influenced
by hole cleaning. Gray, Armstrong and Gatlin (1962) studied 2D rock breakage
in drag-bit drilling at atmospheric pressure. They varied cutting tip rake angles,
depths of cuts and cutting speeds in a comprehensive test matrix, and obtained
motion pictures of the cutting process at camera speeds of 5,000 to 8,000 frames
per second, shedding insight into the brittle failure mechanism in rocks. Again,
detailed functional relationships connecting u, u t and u x were obtained.
Single-tooth impact results. Podio and Gray (1965) and Yang and Gray
(1967) gave analogous relationships for idealized single-blow bit-tooth impact
tests on saturated rocks under confining pressure. These papers offered
excellent reproductions of oscilloscope traces showing simultaneous force,
velocity, and displacement at different initial instances of time. Figure 4.2.4
provides a crude sketch, typical of the authors' experimental results.
In these papers, brittle, transitional and plastic collisions were considered
through a range of pore pressures and fluid saturations. Results were obtained
for impacts parallel and perpendicular to bedding planes. Observed
relationships connecting u, u t and u x were typically nonlinear and not exactly
simple; however, similarities in shape demonstrated repeatedly over a range of
conditions indicate that well-defined rock-bit interaction models do exist,
possibly taking forms not unlike Equation 4.2.22 under limited displacement
ranges. Others dealing with similar subjects include Outmans (1960), who
provided a theoretical framework for instantaneous rate of penetration; and
Maurer (1962), who gave rate versus weight-on-bit formulas derived from rock-
cratering mechanisms under “perfect cleaning” conditions. Also, Maurer (1965)
described experimental findings on crater formation, and Eronini (1982)
characterized the dynamic interaction between an impacting tool and rock with
models derived from fracture mechanics.