Signal Processor Design and Additional Noise Models - Measurement While Drilling Signal Analysis, Optimization and Design - page 163

studied in the literature due to their complexity. Typically, in conventional
models, individual partial differential equations for axial, torsional and lateral
vibrations are solved subject to standard boundary conditions, e.g., sinusoidal
displacements are prescribed and resonant conditions are obtained.
There are serious limitations associated with such approaches. Assuming
that bit displacement is sinusoidal does not imply that a Fourier component of
the general transient problem is being considered - in fact, by assuming that bit
motion is always sinusoidal, one importantly precludes the modeling of highly
nonlinear events like “bit bounce.” In the author's topic, a general boundary
condition related to rock-bit interactions is used, and the transient “up and
down” displacement effect of the rotating bit is in fact modeled using
“accordion-like, displacement sources” like those in earthquake engineering.
When the general initial value problem is solved subject to specified starting
conditions and bottomhole geometrical constraints, the resulting bit motion will
be sinusoidal - or, erratically bouncing, if the rock-bit interaction dictates.
Other common fallacies exist that the topic addresses which may prove
important to MWD signal processing. For example, one can state that all
resonances are dangerous; however, not all dangerous events arise from
resonances. The well known fact that many drillstring failures occur at the
neutral point in the drill collar while undergoing strong lateral vibrations is one
consequence that cannot be predicted by traditional resonance-based models.
The neutral point is simply the location within the drillstring where axia l forces
change from tension to compression, that is, axial stresses vanish at the neutral
point. But what do axial forces have to do with lateral transverse vibrations?
It turns out that axial and transverse vibrations are dynamically coupled.
An exact, closed form solution based on “group velocity” methods in theoretical
physics shows that axial vibration energy tends to coalesce and trap near the
neutral point and instigates high-cycle bending fatigue, e.g., see Chin (1988).
Mathematically, the solution represents a “singularity” in the governing partial
differential equations, popularly termed a “black hole.” This implies serious
consequences in developing MWD drillpipe telemetry technology based on
transmissions through metal itself. Severe lateral vibrations at the drillbit may
affect the rock-bit interaction boundary condition enforced for axial vibrations,
resulting in axial bit bounce. Because lower frequencies are typically involved,
direct bit bounce effects can be removed using high-pass filters; unfortunately,
bit bounce is associated with significant fluctuations in mud flow rate through
the nozzles, and hence, with large changes to siren torque, control system
response, signal strength and phase, and so on, that will require both better
designed tools and very robust signal processing.
It is well known that free vibrations of axial, torsional and lateral bending
modes are each associated with different characteristic frequencies. Signals
correlated with these frequencies, which are typically different from those of the
MWD signal, can be removed by low-pass, high-pass or notch filters (see