Signal Processor Design and Additional Noise Models - Measurement While Drilling Signal Analysis, Optimization and Design - page 171

end, for which we already have a solution algorithm. Now we ask, “What
problem is physically solved if we simply replace 'u' by 'I' in the formulation?”
Thus, we wish to interpret the formulation w 2 I/wt 2 - c 2 w 2 I/wx 2 = 0, with the
jump [wI/wx] specified through the source position and wI/wx = 0 at the drillbit.
At the drillbit,wI/wx = 0 implies that wu/wt = 0, that is, a solid reflector. From
our definitions, a jump in [wI/wx] is simply a jump in the velocity wu/wt. In other
words, we have a discontinuity in axial velocity, as one might envision for
accordion motion or for a pulsating balloon, which models a negative pulser.
Once the solution for I(x,t) is available, local pressures would be calculated by
evaluating wI/wt and then re-expressing the result in terms of wu/wx, which is, of
course, proportional to the acoustic pressure. That is, our displacement dipole
formulation for open drillbits is identical to that for monopoles with solid
reflectors - the latter solution is a “free” byproduct of the first !
To understand this, in perspective, recall that a dipole source (that is,
positive pulser or mud siren) is associated with antisymmetric disturbance
pressure fields and velocities continuous through the source point, while a
monopole source (or negative pulser) is associated with a symmetric disturbance
pressure fields and velocities discontinuous through the source. Contrary to
popular engineering notions, it is not necessary to have a nonzero “delta-p” in
order to have MWD signal generation; in fact, the “delta-p” associated with
negative pulser applications is identically zero whatever the valve motion.
The slightly different formulation w 2 u/wt 2 - c 2 w 2 u/wx 2 = 0, with the jump
[wu/wx] specified through the source position (again, a “delta-p” function of time
prescribed at the pulser) and wu/wt = 0 at the drillbit (that is, a solid reflector
assumption) has also been addressed previously and a numerical solution
algorithm is already available. If we replace 'u' by 'I,' what does the resulting
boundary value problem solve? The only difference from the foregoing
formulation is 'wI/wt = 0' at the bit. Now, wI/wt = 0 implies, per our definitions,
that wu/wx = 0. In other words, the formulation for I solves for the pressures
associated with negative pulsers when the drillbit satisfies a zero acoustic
pressure, open-ended assumption. Pressures are obtained as before.
The above paragraphs demonstrate how solutions to our dipole source
formulations (for positive pulsers and sirens) under a Lagrangian displacement
description provide “free” solutions for monopole formulations for negative
pulser problems without additional work other than minor re-interpretation of
the pertinent math symbols. However, the extension can be interpreted much
more broadly even outside the context of mud pulse telemetry. The above
approaches and results also apply directly to MWD telemetry applications where
the transmission mechanism involves axial elastic wave propagation through
drillpipe steel. A dipole source would model, say, piezoelectric plates
“oscillating back and forth,” while a monopole source might model piezoelectric
transducers stacked so that they “breathe in and out, much like pulsating