Drillstring Vibrations: Classic Ideas and Modern Approaches - Wave Propagation in Drilling, Well Logging and Reservoir Applications - page 203

is an algebraic quantity that is variable versus length. Since N(x) is a known
variable, the cross-sectional geometry and materials should be selected so that
the group velocity remains more or less constant and never vanishing. Clearly,
it is impractical to redesign the entire drillstring; most of the danger, perhaps,
may be removed just by improving the transmission properties of the drill collar.

4.3.4.5 Viscous dissipation.
We developed our model for wave trapping in attenuation-free media in
order to obtain simple closed form solutions. This strategy also allowed us to
circumvent the problems which arise in dealing with dimensionless groups in
addition to those of Equations 4.3.14 to 4.3.16. The more complete beam
equation accounting for internal viscous dissipation, which we do solve
numerically later, takes the form
EI 4 v/ x 4 + (N v/ x)/ x + v/ t + A 2 v/ t 2 = 0 (4.3.34)
where the damping factor satisfies > 0. If now we substitute Equations 4.3.2
and 4.3.4, we obtain on taking real parts,
A(

r 2 - i 2 ) - i = EI k 4 - Nk 2

(4.3.35)

whereas by taking imaginary parts, we have
i = kN x /(2 r A) - /2 A (4.3.36)
From Equation 4.3.36 above, we observe that bending vibrations dissipate like
exp {kN x /(2 r A) - /2 A}t. Not only does the inclusion of the v/ t term
introduce an attenuative - t/2 A in the exponential factor, but viscous
dissipation imparts the phase change evident from Equation 4.3.35. Such phase
effects are discussed in detail in Chapter 1. However, since | r | >> | i |,
modifications for dissipation should not affect our results qualitatively. Later in
this section, the qualitative ideas developed in this section will be quantified,
and combined with the results of our axial vibrations study to produce a
numerical and analytical model that simulates coupled axial and bending
vibrations only. Again, torsion is introduced much later, while coupled axial,
torsional and bending vibrations are considered in concluding work.

4.3.5 Surface detection of downhole bending disturbances.
That the violent lateral vibrations occurring downhole cannot be detected
from the surface as high amplitude bending waves, of course, does not mean that
such catastrophic events cannot be monitored using other related measurements.
They can be monitored and we will demonstrate how.

4.3.5.1 Detecting lateral vibrations.
Vandiver, Nicholson and Shyu (1989) note that unlike longitudinal and
torsional vibrations, lateral vibrations are not transmitted to the surface by the