Geology Reference
In-Depth Information
Again, positive values of N* represent compression near the drillbit, while
negative values denote tension at the surface. It is seen from Figure 4.3.1 that
C
g
*
, for all practical purposes, vanishes for zero and negative N*, that is, for
those portions of the drillstring uphole of the neutral point. Those small
amplitude waves that do “leak” past this “focal point” should dampen
eventually, because of drillstring length and borehole wall and mud contact. On
this basis, disturbance waves generated downhole near the bit will accumulate at
the neutral point and should not be detectable uphole.
Note that regions of low nondimensional group velocity are defined by a
continuous spectrum of values
(consistently with the speculation in Aldred and
Sheppard (1992)), and
not
by a discrete set of natural frequencies, as obtained in
simple eigenvalue-type, resonant vibration analyses. What happens to the lateral
vibration amplitude
a
? Since the total energy flux (proportional to
a
2
C
g
*
) is
expected to be locally conserved, bending amplitudes should grow indefinitely
at the focus; they are, in fact, singular like 1/(C
g
*
)
1/2
as C
g
*
goes to zero
(Landahl, 1972). Figure 4.3.2 shows how the wavenumber decreases
significantly for those waves that do move uphole to the surface; they stretch
beyond recognition, owing to refractive (that is, variable N(x)) effects, so that
they will be undetectable, even if they have not fully dissipated.
The quantities plotted in Figures 4.3.1 and 4.3.2 are tabulated, respectively,
in Tables 4.3.1 and 4.3.2; these general, nondimensional results are provided to
furnish numbers for comparative use along the drillstring in practical
application. Bending vibrations maintain their observable character only for
positive N*, that is, only along the section of the drill collar beneath the neutral
point. Above the neutral point, observable motions are not possible because
energy densities and wavenumbers are both small. Those disturbances
originating at the bit will travel upward to the neutral point, where they may
remain; as discussed, they may not reflect downward toward the bit from the
neutral point. Bending vibrations are therefore confined to this lowest part of
the bottomhole assembly. And because material drill collar damping is weak,
these strong vibrations persist, providing a potential source of damaging high-
cycle fatigue. This prediction is substantiated by recent MWD measurements,
which track the frequency of high amplitude stresses; by and large, these are
greatest in the lowest fifty feet of the drill collar beneath the neutral point.
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