Harmonic Analysis: Six-Segment Downhole Acoustic Waveguide - Measurement While Drilling Signal Analysis, Optimization and Design - page 61

The schematic in Figure 2.4a shows a six-segment waveguide consisting of
(1) a long drillpipe, (2) an MWD drill collar possibly containing other sensors,
(3) the drill collar housing a positive displacement mud motor with possible
impedance mismatches due to the presence of the rubber stator - this waveguide
section can be used to alternatively model, say, a resistivity-at-bit sub, (4) the
drillbit nozzles or the bit sub, (5) a “bottom annulus” surrounding the drillbit, bit
sub and both drill collars, and finally, (6) a long “upper annulus” that extends to
the surface. If the pulser has operated at a given frequency for some time,
standing waves (denoted by double-arrows) are set up at all finite bounded
sections while propagating waves (denoted by single-arrows) traveling away
from the source appear in the semi-infinite drillpipe and upper annulus.
The assumptions implicit in Figure 2.4a are subtle. Consider first the
drillpipe. It is assumed in the model that all waves move upward to the surface
and do not return to the bottom. But at the surface, they will reflect at desurgers
and mudpumps and travel downward - more than likely, though, any waves that
reach the bottomhole assembly will be so weak that our simple upgoing wave
assumption applies (if not, a second, smaller standing-wave pattern is created).
This will not be so for very strong sources, in which case multiple reflections
will be found; when this is so in wind tunnel testing, the “cure” is an appropriate
reduction in signal amplitude, which is easily accomplished by decreasing wind
speed. Similar considerations apply to the annulus and reflections from surface
facilities. Now, both of these upgoing waves will damp as they travel upward,
but since their reflections are assumed to be so weak that their influences on the
standing wave pattern are unimportant, the simple fact that the only waves that
leave the bottomhole assembly are waves that travel upward suffices. This
represents the so-called “radiation” or “outgoing wave” condition used in
physics. Again, for the purposes of modeling wave interactions in the
bottomhole assembly (such as constructive and destructive interference effects
in signal enhancement or destruction), the local effects of irreversible
thermodynamic attenuation are not important.
Now consider the alternative scenarios shown in Figures 2.4b, 2.4c and
2.4d where dark gray caps indicate terminations where strong reflections are
possible. Sections bounded by these terminations will now contain standing
wave patterns as opposed to propagating waves and the complete wave pattern
found in the bottomhole assemblies will differ from that of Figure 2.4a. In
particular, they will depend upon the nature of the reflector. If the termination
acts as a solid reflector, the acoustic pressure locally doubles; if it acts as an
open-end reflector, the acoustic pressure will locally vanish; finally, if it is an
elastic reflector, i.e., a desurger, acoustic reflections will distort in shape and
change in size depending on how wave amplitude and frequency interact with
effective mass-spring-damper parameters.
Again, the same MWD hardware (that is, bottomhole assembly and tool
configuration) operating identically in Figures 2.4a,b,c,d will not produce