Drillstring Vibrations: Classic Ideas and Modern Approaches - Wave Propagation in Drilling, Well Logging and Reservoir Applications - page 133

where the damping factor is generally small. As in Equation 4.2.1 for u(x,t),
our sign convention for gravity is chosen so that in the free-fall limit, the
reduced equation M 2 u/ t 2 = - Mg leads to the familiar

u(t) = -1/2 gt 2 (4.2.18)

from particle mechanics.
Clayer, Vandiver and Lee (1990) reevaluated this surface model in light of
field data and numerical simulations. The authors conclude that the mass-
spring-damper model is sufficiently accurate for surface modeling, and probably
suffices for most engineering analyses. Equation 4.2.17 appears to be
satisfactory, and its use is retained in this topic.

4.2.4.2 Conventional bit boundary conditions.
In their early paper, Dareing and Livesay (1968) introduced the “obvious”
drill bit boundary condition

u d (0,t) = u 0 sin t (4.2.19)

at x = 0 , where is related to the “3 rpm” frequency typical of tricone bits,
and u 0 is the peak-to-peak cone vertical displacement (alternatively, we might
have taken u d (0,t) = u 0 cos t). The frequency may be nonzero or zero. If

nonzero, the displacement u 0 sin t (or, u 0 cos t) is marked by a zero time

average, that is,

period u d (0,t) dt = u 0 period sin

t dt = 0 (4.2.20)

Thus, the drillstring never penetrates the formation on the average ! If zero, we
have at best a fixed nonzero bit displacement, yielding uninteresting solutions.
The use of u d (0,t) = u 0 sin t, of course, only partially addresses problems
satisfying transient boundary conditions of the form u d (0,t) = f(t). Also,
noticeably absent from it is any rock-bit interaction: the most important part of
the physical problem - rock-bit interaction - is absent. Thus, the boundary
condition in Equation 4.2.19 implicitly assumes hard rock with zero mean rate-
of-penetration. But the very fact that something has been assumed at x = 0
raises the most concern. If we wish to specify rock-bit boundary conditions
instead of Equation 4.2.19, then how do we additionally introduce “sin t” to
characterize bit motion? It seems that only one, but not both models, is possible;
fortunately, it is possible to fix this shortcoming using ideas from Chapter 1.
Other inconsistencies are found with Equation 4.2.19, particularly with
respect to predicted downhole resonances. Researchers have attempted to drill
at the resonances predicted by the simple theory, in order to increase rate-of-
penetration, but without success. Dareing (1972) explained why drillstrings do
not experience resonant vibrations frequently, and gave three possible reasons.