Geology Reference
In-Depth Information
d{log
e
(MC
g
)} = [3kN
x
/(2 A
0
)] dt
(4.3.28)
log
e
(MC
g
)
[3kN
x
/(2 A
0
)] dt
(4.3.29)
and finally,
the closed form spatial distribution for the wave momentum
M(x) = {1/C
g
(x)} exp { [3kN
x
/(2 A
0
)] dt }
(4.3.30)
Now , s i nc e
0
> 0 by requirement, and because N'(x) < 0 as the wave makes its
way uphole to the surface, it follows that the second factor above vanishes
exponentially far away at the surface, even without explicitly considering
internal viscous dissipation. But near the neutral point, the effects of this factor
are locally weak; the dominant term is the near-zero group velocity C
g
.
Similar considerations apply to energy density using Equation 4.3.20.
Since explicit dependencies are disallowed because N is (by assumption) solely
dependent upon x and not t, Equation 4.3.20 reduces to
E/ t + (C
g
E)/ x = 2
i
E (4.3.31)
because the term
r,t
(k,x,t) vanishes identically. Thus, the integral
corresponding to this simplified energy equation, following a derivation similar
to that for the momentum derivation above, is
E(x) = {1/C
g
(x)} exp { [2kN
x
/(2 A
0
)] dt }
(4.3.32)
and physical interpretations analogous to those for M(x) apply. The functions
C
g
(x) and k(x) are given in our earlier graphs and tables.
Close to the neutral point, the wave momentum density M(x) and wave
energy density E(x) are both singular like 1/C
g
(x) as C
g
(x) approaches zero.
This indicates that the singularity is locally strong. In fact, as earlier tabulated
and plotted formulas for the group velocity show, the singularity is strongly
algebraic. Note that N(x) varies linearly with x along the drillstring, so that any
explicit N dependence can be replaced by a constant multiple of x. Our model
explains why high concentrations of bending amplitude are more the rule than
the exception, and why their mysterious absence at the surface is really quite
natural when axial load variations are accounted for along the length of the
drillstring. Analyses for axial and torsional vibrations can be similarly
performed. But because analogous sources of spatial inhomogeneity do not
exist in the governing equations, their motions are not dynamically restricted or
physically altered by
kinematic barriers
such as those introduced here. Thus,
instabilities associated with axial and torsional vibrations can and should be
modeled using conventional eigenfunction, natural frequency, or resonant modal
techniques. Incidentally, surface measurements of these non-lateral modes, as
we shall see, properly deconvolved, should provide useful information on
downhole vibrations and lithology. This possibility will be pursued later.
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