Harmonic Analysis: Six-Segment Downhole Acoustic Waveguide - Measurement While Drilling Signal Analysis, Optimization and Design - page 66

source, which arise from interference and re-reflections, will depend on the
geometry and boundary conditions associated with the waveguide assumed.
In summary, the mathematical formulation must permit convenient
implementation of farfield boundary conditions, but it must allow free passage
of pressure waves through the source point, as well as permitting us to represent
the MWD source by a “jump in pressure” without specifying the exact local
pressure (the exact source pressure will, of course, be available from the solution
of the boundary value problem). Before proceeding, we again emphasize that a
jump in pressure, or delta-p, is not necessary for signal creation. For example,
two oppositely-facing speakers placed in close contact so that their outer rims
touch, would model a negative pulser. These will obviously generate pressures
that are symmetric with respect to the source point. However, they are
associated with a zero delta-p, and instead execute monopole-type “breathing
volume oscillations.” The dipole sources considered in this topic instead
generate propagating signals associated with displacement changes and
antisymmetric pressures produced without simultaneous volume creation.
2.4.2 Mathematical formulation.
In this section, we will derive the boundary value problem formulation
describing the coupled acoustical interactions discussed above. This
formulation consists of governing partial differential equations, dipole source
model, matching conditions at impedance junctions, and farfield radiation
conditions. Again, the objectives of the model are two-fold. We wish to design
a model that allows us to study the conditions under which the drillpipe signal is
good or bad and the extent to which improvements in MWD telemetry can be
made. Second, we desire to obtain predictive values for strong annular MWD
signal because their correlation with drillpipe signals can be useful in
monitoring gas influx or in providing improved signal detection.
2.4.2.1 Dipole source, drill collar modeling.
A number of physical quantities were cited earlier for potential use as
candidate dependent variables; however, none of these fulfill the requirements
discussed. It turns out that the “Lagrangian displacement” variable u(x,t)
familiar to seismologists provides the flexibility needed to simultaneously model
impedance, farfield, and source “delta-p” boundary conditions. The quantity
“u” is a length representing the lineal displacement of a fluid element from its
equilibrium position, which arises as a result of propagating compression and
expansion waves that momentarily displace it from its initial position. This
length, which varies with space and time, satisfies the one-dimensional wave
equation “U mud u c tt - B mud u c xx = 0,” where U mud and B mud represent, respectively,
the mud density and bulk modulus. When desirable, in this topic, subscripts will
be used to denote partial derivatives, in order to simplify the presentation.