Harmonic Analysis: Elementary Pipe and Collar Models - Measurement While Drilling Signal Analysis, Optimization and Design - page 90

It is interesting to note that the maximum amplitude increase is a factor of
2.0 for this simple model - our six-segment waveguide results indicate that
factors exceeding 2.0 are also possible for more complicated bottomhole
assembly geometries. It is even more important that multiple frequencies exist
which give this maximum constructive interference. These frequencies are
uniformly separated by increments of Zx s /c = S. If frequency shift keying is
used to telemeter results, one or more of these optimal frequencies (each
associated with different amplitudes) can be used to achieve not just 0's and 1's,
but 0's, 1's, 2's, 3's and so on. A simple binary scheme is also possible.
Adjacent to each “good” frequency f good (strong signal) is a “bad” frequency f bad
(weak signal), as our equations show . Importantly , to convey 0's and 1's, it is
not necessary to frequency shift between f good and 0 Hz, that is, bring the siren to
a complete stop, since mechanical inertia demands on the drive motor may be
significant. Instead, one might alternate between f good and f bad .
Case (c) . When the drillbit is an open-ended reflector, we have instead
p 2 (x,t) = {2 cos (Zx s /c)} ½ p s e i Z(-x/c + t) , x > x s . In this case, we have the same
maximum constructive interference factor of 2.0 as in Case (b). Maximum
constructive interference is now obtained at Zx s /c = 0, S, 2S, 3S and so on, and
destructive interference is achieved at Zx s /c = S/2, 3S/2, 5S/2 and so on. Now,
we may justifiably say that all of this is confusing - in practice, how do we
know if we have Case (b) or Case (c)? The answer is interesting - it does not
matter . The important conclusion is that, whatever the model, whether we have
a solid reflector, open reflector or the more complete six-segment waveguide,
multiple frequencies exist that yield good constructive interference and that
these frequencies are close. We do not need a math model at the rigsite to
compute these. Periodically, drilling operations can stop and the MWD pulser
can “sweep” a range of frequencies (that is, slowly change from 1 Hz to 200 Hz,
say). We can “listen” at the standpipe (being careful to subtract out the effects
of surface reflections) to look for good and bad frequencies. Measured signals
on the standpipe will contain the effects of surface reflections, which may also
be good or bad. These FSK frequencies can be used as explained above.
Case (d) . It is easily verified that p 1 = 0 at x = - L and p 2 = 0 at x = +L.
Also, the acoustic pressure solution is antisymmetric with respect to the source
position x = 0, where 'p is maintained. This is an important solution for wind
tunnel determination of 'p. It allows placement of the piezoelectric transducer
anywhere, at any position “x” for measurement of the “p,” which includes the
effects of all reflections needed to set up the standing wave. Then, depending on
whether Equation 3.A.14 or 3.A.15 is used, we can solve for the p s representing
“delta-p” in “p 2 - p 1 = p s e i Zt ” directly.