Geophysical Ray Tracing - Wave Propagation in Drilling, Well Logging and Reservoir Applications - page 315

r 2 - i 2 = c 2 (k x 2 + k y 2 +k z 2 ) + i

(6.48)

and the imaginary frequency
i = - /2 (6.49)
The damping rate, as in Equation 6.42, is again a constant independent of k, and
therefore, the wave frequency.

6.4.4 The quality factor Q.
In geophysical problems, the dimensionless measure of attenuation Q is
commonly used (Aki and Richards, 1980; Toksoz and Johnston, 1981; Kennett,
1983; Bourbie, Coussy, and Zinszner, 1987). If a volume of material is cycled
in stress at a fixed frequency, Q can be defined as
1/Q = - E/2 E (6.50)
where E is the peak strain energy stored in the volume, and - E is the energy
lost per cycle due to material imperfections. A detailed discussion on
attenuation modeling is given in Chapter 2 dealing with kinematic wave
formalisms. There, this laboratory Q is recast into a form useful for standing
and propagating wave analyses. The propagation result for wave amplitude (as
opposed to energy ) was obtained as
A(x) = A 0 exp (- r x/2cQ) (6.51)
Standard geophysical conventions (Clay, 1990) often take propagating
amplitude in the form
A(x) = A 0 exp (-

x) (6.52)

with the definition
= f/cQ (6.53)
where f is the frequency in Hertz and Q may additionally depend on frequency
(since = 2 f, an alternative to Equation 6.53 is = /2cQ, which is consistent
with the conclusion drawn by directly comparing Equations 6.51 and 6.52). The
kinematic wave model developed in Chapter 2 renders Q' s determined from
standard experimental procedures for uniform media (Toksoz and Johnston,
1981) usable in general wave propagation modeling studies over heterogeneous
formations covering large space and time scales.

6.4.5 A simple example.
Wave dissipation occurs because of viscous fluid losses in rock and
frictional losses in solids. The “damping” or “quality factor” Q depends on the
type of rock; it is nearly constant in the seismic frequency range for most rocks
(its value varies from ten to several hundred for common rocks). Geophysicists
are interested in the dependence of Q on rock structure, pore fluid and external
forces such as overburden pressure.