Drillstring Vibrations: Classic Ideas and Modern Approaches - Wave Propagation in Drilling, Well Logging and Reservoir Applications - page 217

C = GJ (4.4.2)

where G is the “shear modulus” and J is the “polar moment of inertia” (again,
for circular cross-sections, J = a 4 /2, a being the radius). For such problems,
Equation 4.4.1 becomes

T = GJ / x (4.4.3)

In the simplest case, angular momentum considerations show that the angle of
twist satisfies the classical wave equation

2 / t 2 - (C/ J) 2 / x 2 = 0

(4.4.4)

where we have assumed a nondissipative system without external excitation. In
general, Equations 4.4.2 and 4.4.4 suggest the definition

c s 2 = G/ (4.4.5)

At the same time, following previous work on attenuation, the obvious extension
of Equation 4.4.4 is

2 / t 2 -

/ t - c s 2

2 / x 2 =

e

(4.4.6)

Note that c s is the “shear wave propagation velocity,” and that it is
different from the “axial speed of sound” c = (E/ ) 1/2 for longitudinal
disturbances. In Equation 4.4.6, we have introduced a right side e term to
represent the effect of external excitation due to borehole friction and contact.
The fact that different distinct characteristic speeds exist along a drillstring is
important to future MWD telemetry methods, especially with respect to
multiplexing , that is, transmitting multiple signals along the same
communications channel. For example, downhole information may be
conveyed to the surface using axial pipe waves propagating at a speed (E/ ) 1/2 ,
torsional pipe waves at a speed (G/ ) 1/2 , and acoustic mud pulse signals at a
speed (B mud / mud ) 1/2 , where B mud and mud represent the bulk modulus and
density of the mud. Lateral bending waves, as discussed previously, cannot be
used for MWD telemetry because they are associated with low signal-to-noise
ratios at the surface and also frequency-dispersive.
We have also introduced an attenuation term / t along the lines noted
for axial disturbances. The damping factor may in general be different from
the appearing in 2 u/ t 2 + u/ t - E 2 u/ x 2 = 0 for axial vibrations. For
detailed expositions on torsional mechanics, the reader is referred to classic
topics by Timoshenko and Goodier (1934), Love (1944), Den Hartog (1952),
and Timoshenko (1958), and the more detailed vibrations textbook by Graff
(1975). Although Equation 4.4.6 is suggestive of transient torsional waves, it is
important to understand that it applies as well to the combined static and