Harmonic Analysis: Six-Segment Downhole Acoustic Waveguide - Measurement While Drilling Signal Analysis, Optimization and Design - page 68

wake behind a bluff body, which does not propagate as sound and does not
possess the required antisymmetries; of course, since such a static drop is
associated with a change in surface hydrostatic pressure, it can be used to
encode information, although at low data rates. Incidentally, delta functions are
used to represent point excitations for jump ropes and violin strings undergoing
transverse motions when similar displacement formulations are used.
We note that the source strength 'p for mud sirens and positive pulsers
used for continuous wave generators depends on flowrate, the mud sound speed,
the mud density, the geometric details of the pulser, and finally, on the manner
in which the pulser is driven. It does not depend on the wave propagation itself,
and in this sense, can be regarded as a known prescribed quantity for acoustic
modeling purposes. The properties of 'p are difficult to assess analytically.
Instead, it is simpler to measure them directly in mud flow loops or in wind
tunnels, an analysis area we develop more fully later in Chapter 9.
As a digression, we emphasize again that a “'p” is not necessary for MWD
signal generation and transmission, as we have discussed for negative pulsers -
in fact, 'p = 0 identically. To model negative pulsers, the forcing function in
Equation 2.1 would take the form “'p” [G(x-x s + H) - G(x-x s - H)] where “'p”
would refer instead to the acoustic pressure drop across the drill collar to the
formation. The difference in delta functions ensures that no net pressure drop is
occurs in the axial direction. More precisely, one models the negative pulser
source as a “couple,” that is, as the derivative of the delta function in a
formulation for u(x,t). To simplify the mathematics for negative pulsers, one
alternatively uses “velocity potentials” for I(x,t) where the axial acoustic
velocity is v = wI/wx and pressure is related to its time derivative. In the MWD
drill collar, the statement U mud I c tt - B mud I c xx = B “'v” G(x-x s ) would now
indicate that a discontinuity in 'v is the natural math model for negative pulsers.
We also observe that a numerical solution to Equation 2.1, say based on
finite differences or finite elements, would lead to very inaccurate results, given
the point force excitation used. Even high-order schemes are complicated by
large truncation errors which lead to dissipation and inaccuracies in modeling
phase effects. The results, for instance, give local sound speeds that may be
different than those characteristic of the actual mud. Inaccurate phase modeling
would render all conclusions for constructive and destructive wave interference
useless. We therefore pursue a completely analytical approach.
2.4.2.2 Harmonic analysis.
We have identified “'p” as the transient acoustic pressure source, which
can take on any functional dependence of time; this dependence, again, is
dictated by pulser geometry, oncoming flow rate, fluid density, and the
telemetry scheme selected. In this chapter, we physically represent “'p” in the
frequency domain as the product of a harmonic function e iZt and a signal