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example, at mud-motor and MWD collar interfaces, as well as at entrances and
exits of drillbit nozzles and bit subs or bit boxes and in the annulus. All of these
effects must be modeled correctly, but unfortunately, none of the numerous
models with which the author is familiar addresses even the simplest reflection
problem properly. We will review some fundamental issues before developing a
comprehensive acoustic waveguide model encompassing all of these effects.
2.2.4 Drillbit as a reflector
.
In many studies, the ratio of total nozzle cross-sectional area to the cross-
sectional area of the bit or bit sub is seen to be small and the drillbit is
accordingly modeled as a solid reflector. This appears to be reasonable from an
engineering perspective but it is completely wrong. This is obvious from the
following “thought experiment.” It is worthwhile noting that, for a 12 Hz carrier
wave in water with a sound speed of 5,000 ft/sec, the wavelength is 5,000/12 or
about 500 feet - at lower frequencies, the wavelengths are much longer. Let us
consider a positive pulser or a siren, located about 100 feet from the drillbit, in
the process of obstructing the oncoming mud flow. A positive signal (that is, an
over-pressure) is created that propagates uphole. At the same time, a negative
signal is created which travels downhole. If the drillbit is a solid reflector, this
negative signal will reflect as a negative signal and travel uphole past the source
to combine with the long positive signal already traveling uphole. The net result
is a wave, by virtue of destructive interference, with mostly zero disturbance
pressure amplitude.
It is known from operational experience, of course, that signals created
using positive pulsers and mud sirens
are
measurable at the surface - and, in
fact, that they have been observed even in the surface borehole annulus -
therefore, the solid reflector model must be wrong. Let us, then, reconsider the
downgoing negative wave on this basis. If the drillbit nozzles represent,
alternatively, an effective open-ended reflector, negative pressures will reflect as
positive pressures - these positive pressures will add to the positive pressures
that initially travel uphole and hence increase the possibility of detection. An
open-end reflector will therefore double the propagating pressure associated
with an infinite uniform pipe - the “transmission efficiency” should approach
1.0, or nearly twice the 0.5 considered previously, only because the end is not
entirely opened in an acoustic sense. An open-end model also allows wave
transmission up the annulus and is consistent with field observation.
These simple explanations on drillbit reflections are in fact supported by
results of our detailed waveguide calculations which assume very general
acoustic impedance matching conditions. Again, we note that the transmission
efficiency for a uniform drillpipe infinite in both directions is 0.5. For the
simple open reflector model above, this increases to 1.0, and interestingly,
values exceeding 1.0 are possible, a phenomenon not surprising to designers of
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