Drillstring Vibrations: Classic Ideas and Modern Approaches - Wave Propagation in Drilling, Well Logging and Reservoir Applications - page 223

4.4.2.5 Applications to MWD telemetry.
“Mud pulse telemetry,” in which acoustic pulses are transmitted through
the mud column inside the drillstring, represents the most popular form of
Measurement-While-Drilling communications at the present time. However,
other telemetry methods are possible, and are under continual industry
evaluation and research. Bradbury and Wilhoit (1962) importantly considered
the effect of tool joints on the passage of plane longitudinal and torsional waves
along a drillpipe. Approximate solutions to the governing partial differential
equations for an idealized tool joint showed that tool joints had negligible effect
for exciting frequencies of the same order as common rotary speeds, and that the
drillpipe can be taken as a uniform pipe with negligible error.
Barnes and Kirkwood (1972) also considered longitudinal and torsional
vibrations as a means for MWD communication. For their analysis, only zeroth
mode vibrations (i.e., plane wave) were considered, the assumption being that
higher modes would be damped out over short distances. The basic ideas on
modal eigenfunctions and evanescent waves were discussed in Chapters 1 and 3.
In their idealized model, where the drillstring is taken as an infinitely long
cylindrical steel pipe loaded at equal intervals with identical couplings, the
authors show that most of the “passbands” for compressional waves are about
twice the width of those for torsional waves (passbands are frequencies for
which waves propagate without loss). Refer to the paper for further details on
the exact geometry used and the numerical results obtained. Torsional wave
propagation along drillstrings, as a means of MWD acoustic telemetry, was
reconsidered by Squire and Whitehouse (1979). They importantly studied
attenuative effects posed by the drilling mud environment, with and without
gelling, and results of field experiments were reported. The conclusions cited in
the foregoing papers, which have not been independently verified by this author,
are given for completeness only. Many publications related to MWD
transmissions using the drillpipe as the communications medium have appeared
in recent years, and the reader is encouraged to explore the vast literature - at
the present, only mud pulse methods have proven practical operationally.
4.4.2.6 Example Fortran implementation.
Here we will give portions of a Fortran source code listing, showing how
the torsional finite difference equations are introduced by defining the required
A , B , C and W coefficients, and how they are solved using the TRIDI
subroutine call. Because the similarities to the axial case are strong, we will
provide this listing with a minimum of elaboration.