Borehole Acoustics –New Approaches to Old Problems - Wave Propagation in Drilling, Well Logging and Reservoir Applications - page 370

that all of the qualitative Stoneley wave features obtained numerically by
previous authors follow straightforwardly from the closed form expressions
derived here. Also, extremely good consistency and agreement with all prior
results is obtained - exact agreement with Biot's complete theory was expected
for frequencies in the 100 to 300 Hz range, but good qualitative agreement for
frequencies extending into the kilohertz range was obtained from direct
calculations. What are some of these properties? We will prove them first, and
then illustrate their consistency with calculations published by other authors.
10.1.2 Properties of Stoneley waves from KWT analysis.
In this section, will analytically prove a number of properties uncovered by
previous investigators in other contexts, using methods similar to those leading
to Equation 10.1 or other entirely numerical approaches. In doing so, we
emphasize the simplicity and physical correctness of the kinematic wave
approach introduced in Chapter 2 - of course, the methodology applies to a wide
variety of problems in engineering and not just Stoneley waves. Unless
otherwise noted, V and are assumed to be constant.

10.1.2.1 Dissipation due to permeability.
The exponential convention used in deriving the foregoing results assumed
an e - i t dependence in all wave properties, so that “exp ( i t)” growth or
damping is found accordingly as i is positive or negative. Because Equation
10.4 shows that i is always negative, waves will always damp as they
propagate in space. This is expected because i is proportional to (the square
root of) permeability; but as we showed in Chapter 2, other growth and damping
mechanisms occur in transient and heterogeneous media and i < 0 alone does
not guarantee damping. Now, combining Equations 10.2a and 10.4 yields
i (k) = - ( mud V 5/2 /R well ) {

oil )} k 1/2 < 0

/(2 oil (10.7)

This result is new and shows that the damping rate is proportional to the square
root of permeability; however, other parameters related to the borehole and pore
fluid also enter in the manner given above. Equation 10.7 also shows that
permeability based attenuation decreases at higher frequencies, in agreement
with a result of Chang et al (1988).
10.1.2.2 Phase velocity and attenuation decrement.
The present author has observed that detailed numerical solutions for
“phase velocity versus frequency” and “attenuation decrement versus
frequency” by many independent investigators show phase velocity and
attenuation as closely related “mirror images,” a frequent occurrence not likely
the result of mere coincidence. These results are summarized schematially in
our Figure 10.2 below.