Drillstring Vibrations: Classic Ideas and Modern Approaches - Wave Propagation in Drilling, Well Logging and Reservoir Applications - page 145

follows that the frequency satisfies f = 3 N rpm /60 Hz. But since = 2 f, the

accordion-like length variation at the bit may be written as

[u] x=xbit = u 0 sin t, = 2 (3N rpm )/60 (4.2.42)

where u 0 is the maximum axial displacement due to the cones. Higher harmonic
motions may be introduced as necessary. PDC bits rely on the shearing action
of the cutters to make hole, and differ significantly from roller cone bits in form
and in cutting mode. Equation 4.2.42 still applies, with the adjustment factor
(instead of 3) varying anywhere from 9 to 20 depending on bit design. The
amplitude u 0 is also likely to change; it may be necessary to include higher order
harmonics, taking the displacement source function in the form f(t) = u 0 sin t +
u 1 sin 2 t + ... Note that PDC axial vibrations, excited by flatter bit surfaces, are
characterized by stronger DC modes than are “3 rpm” roller-cone-driven
drillstring vibrations.

4.2.6 Simple solution for rate-of-penetration.
Clayer, Vandiver and Lee (1990) studied the effects of surface and
downhole boundary conditions on drillstring vibration. The authors'
experimental and numerical work shows that simple displacement boundary
conditions (not to be confused with our more accurate “displacement source”
approach) do not describe observed events well, but suitably chosen linear
models will capture the essential physics . These conclusions were based on
multi-channel surface data for a drillstring (with a 17.5 inch tricone drilling in
hard limestone) acquired with an instrumented sub inserted just below the power
swivel. This sub collected high-bandwidth force, torque, acceleration and rotary
speed data, and direct impedance measurements of the drilling rig in axial and
torsional motion.

4.2.6.1 Field motivation.
It turns out that the real-time bottom boundary condition is a transient
function of several parameters. In brief, the effective stiffness and damping of
the rock depends strongly on the dynamic WOB and related parameters such as
rpm, mud flow rate and bit type. Comparisons between experiment and
numerical simulations suggest that damping at the drillbit due to formation
interaction, closely related to rock fracture mechanisms, significantly affects
drillstring dynamical response. The authors suggest that a simple massless
spring-dashpot model might suffice. The spring emulates the elasticity of the
rock as the bit pushes against it, while the damper represents energy loss
mechanisms related to fracture, rock and cuttings displacement. All of these
observations lend support to Equation 4.2.22, first proposed in Chin (1988a,b) as
a simple rock-bit interaction model.