Borehole Acoustics –New Approaches to Old Problems - Wave Propagation in Drilling, Well Logging and Reservoir Applications - page 369

10.1.1 Analytical simplifications and new “lumped” parameters.
In order to obtain practical, usable results, it is desirable to re-consider
Equation 10.1 in a frequency range consistent with modern logging tools, say
100 to 500 Hz, and approximate it for larger values of . To do this, we draw
on the asymptotic Bessel expansion K n (z) exp(-z) ( /(2z) for large z (note
that this is independent of the order n). If we now write V T = ( r + i i )/k, our
expansions lead to two surprisingly simple formulas that depend on two
“lumped” dimensional constants V and only and not seven, namely,
V = 1/ { mud (1/B mud +1 / G shear )} > 0 (10.2a)
= ( mud V 5/2 /R well ) { /(2 oil oil )} > 0 (10.2b)
in our notation. Thus, the “independent” parameters in Equation 10.1 only act in
specific combinations with others to affect the physical problem. In the above,
R well is the well radius, is the rock porosity, is the formation permeability,
G shear is the solid shear modulus, B mud is the bulk modulus of the borehole mud,
mud is the mud density, oil is the pore fluid bulk modulus and oil is the pore
fluid viscosity. If our complex wave frequency is denoted by = r (k) + i
i (k), where k is the wavenumber, the real and imaginary frequencies become
r (k) = Vk - k 1/2 (10.3)
i (k) = - k 1/2 < 0 (10.4)
The form taken by Equation 10.3 indicates that V is a velocity; moreover, from
Equation 10.2a, it is sound speed in the borehole due to mud modified by the
effects of wall elasticity. (Note the similarity between Equations 10.3 and
Equation 7.16 for ocean waves - wave tank analogies can be derived for
experimental work, e.g., see Chin (2001).) The meaning of , at this point, is
not clear; we will later show that it is the single fundamental parameter
controlling both dispersion and dissipation. These results are easily obtained
using modern algebraic manipulation software, e.g., Maple TM , Maczyma TM or
Mathe mati c a TM - however, the author cautions that he has on occasion
uncovered mistakes and that due diligence is the rule.
Two velocities can be constructed from Equation 10.3, namely, a “group
velocity” and a “phase velocity” satisfying, respectively,

r (k)/ k = r k (k) = V - ½ k -1/2

c g = (10.5)

r (k)/k = V - k -1/2

c p = (10.6)

As is well known from classical physics, the group velocity is the speed with
which wave energy propagates, while the phase velocity, aside from its
application to tracking wave phase, is not physically significant. It is important