Geology Reference
In-Depth Information
We return to our inverse problem, and obtain measurements for the
displacement function u(x,t) = g(ct+x) + g(ct-x) at some point x = x
*
. Thus,
we can write
g(ct+x
*
) + g(ct-x
*
) = u(x
*
,t) (4.2.168)
where the right side can be regarded as a known function of time. If we denote
g(ct+x
*
) = g{c(
t + x
*
/c
)} = Y(t), it follows that we can express g(ct-x
*
) =
g(ct+x
*
-2x
*
) = g{c(
t + x
*
/c
- 2x
*
/c)}= Y(t - 2x
*
/c). That is, Equation 4.2.168
becomes
Y(t) + Y(t - 2x
*
/c) = u(x
*
,t) (4.2.169)
If we compare Equation 4.2.169 with Y(t) - Y(t-h) = Z(t) in Equation 4.2.164,
we have = - , h = 2x
*
/c > 0, and Z(t) = u(x
*
,t). Thus, the solution to Equation
4.2.169 becomes
(- )
n
u{x
*
,t-2nx
*
/c}
g(ct+x
*
) =
(4.2.170)
n=0
Since, at any given instant in time t, the incident wave g(ct+x
*
) and the total
response u{x
*
,t-2nx
*
/c} are both available, Equation 4.2.170 becomes a
polynomial equation in that can be solved by algebraic methods. This
concludes our discussion on axial vibrations. The author hopes that this section
will provide a focus for constructive development and discussion.
4.3 Lateral Bending Vibrations
High-amplitude bending, transverse, or lateral
downhole
vibrations are
widely acknowledged to be responsible for “twist-offs” and expensive drillstring
failures. The axial and torsional oscillations discussed in this topic, which can
be dangerous, play only secondary roles in this severe failure mode. Yet, it is
well known that even large, lateral bending vibrations cannot be detected uphole
at the surface and corrected, even in vertical wells where borehole wall contact
and damping are known to be weak - thus, no safety warnings are available for
this significant downhole event. This paradox has eluded practitioners and
researchers alike for years, and a deterministic model of this mysterious
phenomenon has remained high on research and development priorities.
4.3.1 Why explain this drilling paradox?
For one, the requirement for a predictive model is importantly needed in
drilling operations, where cost and safety issues are forever looming. Second,
the subject of drillstring vibrations cannot be recognized as useful and credible
unless the physical explanations for such fundamentally significant observations
are identified and understood.
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