Harmonic Analysis: Six-Segment Downhole Acoustic Waveguide - Measurement While Drilling Signal Analysis, Optimization and Design - page 55

drillbit reflections would not be made; instead, general acoustic matching
conditions would be invoked at impedance mismatches and “the dice would
fall” as they will. However, mathematical complications arise which need to be
addressed. When the source is located within the drilling fluid itself, its
dynamic properties must be modeled. That is, whether the disturbance pressures
(relative to hydrostatic) created by it are symmetric or antisymmetric with
respect to source location are critically important - the dipole or monopole
nature of signal creation must be accounted for in the mathematical model. For
the same telemetry channel and frequency, positive pulsers and mud sirens will
obviously create wave patterns - and hence upgoing MWD signal fields - that
are completely different from those of negative pulsers.
In summary, the least complicated analytical model of signal generation
and wave interference downhole must contain at least the following elements.
(1) A six-segment waveguide is required, the basic elements being the drillpipe;
the MWD collar (containing a pulser and possibly other logging sensors); a
collar beneath it to represent a positive displacement mud-motor (with rubber
stators with different acoustic impedance) or, say, resistivity sub with different
collar dimensions; a drillbit, bit box or bit sub component; an annulus
surrounding the drill collar; and finally, a different annulus surrounding the
drillpipe. (2) The “monopole” or “dipole” nature of the source must be
specified, that is, whether the disturbance pressure field is symmetric or
antisymmetric with respect to the source. And finally, for analysis and modeling
purposes, (3) the source must be located within the MWD drill collar and allow
the propagation of created signals away from it in both directions, as well as the
complete transmission of reflected waves through the source itself - this latter
requirement is necessary because the long waves created (even at frequencies at
high as several hundred Hertz) will effortlessly propagate through valve
openings that never completely close, e.g., siren “rotor-stator gaps” (detailed
experiments show no evidence of reflections from practical MWD pulsers).
The source strength “'p,” which depends on mechanical design, geometry,
drill collar areal cross-section, flow rate, frequency, density and sound speed, is
a quantity that is most accurately determined by laboratory differential pressure
measurements. However, with care, it can also be inferred from field tests when
the only available data is surface data. The value of 'p is not affected by wave
motions and the longitudinal geometry of the waveguide (the cross-sectional
area will affect local signal strength, however). For telemetry modeling, this
strength can be left as “'p,” noting that this will generally depend on frequency.
Only the ratios of upgoing drillpipe pressure to 'p, that is “p pipe /'p,” and
upgoing annular pressure to 'p, that is, “p annulus /'p,” are dynamically
significant. Again, we caution that 'p may vary with frequency in a manner that
depends on the mechanical design of the valve. The objective of the analysis in