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identical acoustic wave patterns because of boundary condition differences.
Also, changing the mud (or, “c”) will affect each differently. In some cases, the
upgoing wave is strong; in others, it is not. In MWD field tool testing, the
effects of the complete acoustic channel must be understood. Simply having
pressure transducers in the same location along the standpipe means very little,
since the upgoing signals can vary widely depending on borehole length,
frequency and mud sound speed, and the locations of surface components like
mudpumps and desurgers. With these physical arguments explained clearly, we
now turn to a mathematical expression of these ideas and provide exact
analytical solutions. We emphasize that computational finite difference and
finite element solutions to the general formulation are the worst way to proceed,
since numerical dispersion and diffusion errors lead to effective local sound
speeds that differ significantly from actual ones, thus implying large phase
errors. Great effort was expended to obtain analytical solutions; owing to the
complexity of the linear system derived below, recourse to computerized
algebraic manipulation methods was necessary, as will be described in detail.
2.4 Six-Segment Downhole Waveguide Model
In order to develop a realistic and accurate acoustic waveguide simulator
for the bottomhole assembly, the source and the annulus, it is important to
understand every component that affects the combined wave that eventually
leaves the MWD drill collar. We have already discussed pressure symmetries
and antisymmetries for source-field modeling, appropriate acoustic impedance
treatment at areal and material changes, and the proper application of radiation
conditions. In this chapter we neglect the effects of repeating drillpipe joints
because they are not significant - these have not proven to be detrimental to
field operation even in long wells where the initial signals are weak. One final
modeling assumption is discussed, related to the events near the drillbit where
the directions of the waveguide and the local mud flow both reverse.
We specifically refer to the lowest portion of the borehole occupied by the
drill collar, drill bit and the lower annulus. What happens when a downward
traveling wave passes through the bit nozzles and turns up into the annulus?
One should not confuse hydraulic viscous losses with acoustic losses. For
instance, the flow of fluid through a sharp bend will lead to large pressure
losses, requiring higher pump power; however, the transmission of sound,
particularly sound associated with long waves, through that same fluid and bend
will be nearly lossless. How efficient is sound transmission through a bend?
Fortunately, this question can be answered because, for frequencies several
hundred Hertz or below, our waves are in fact long compared to the cross-
sectional dimensions of the pipe or annulus. For this, we draw upon an exact
solution of Lippert (1954, 1955) obtained in his well known studies of long
wave reflection from ninety-degree bends.
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