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Compressional and shear waves have almost the same asymptotic behavior
at low frequencies, but the analysis for shear waves is much shorter,
so we will present only the shear wave analysis here.

The wavenumber *k*_{s} for shear wave propagation is determined by
(14), and when we have
, so

| |
(20) |

Thus, when the loss tangent is a small number, we find the shear
wave quality factor is
| |
(21) |

Total attenuation along the path of a shear wave is then determined
by the integral along the path of the wave. We assume for the sake of argument that
the fluid is the same throughout the reservoir. So all fluid factors
as well as frequency are constant. The solid material parameters
and and also the porosity (which is hidden
in ) may vary in the reservoir, but these variations will be
treated here as negligible compared the variations in the permeability
. Thus, we find that the total attenuation along a path of
length is approximately proportional to . The average attenuation per unit length of the travel path
is therefore proportional to , which is just the
mean of the permeability along the wave's path. This result is also
true for the compressional waves, but the other multiplicative factors
are a bit more complicated in that case.

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** Up:** Low Frequency Asymptotics for
** Previous:** Low Frequency Asymptotics for
Stanford Exploration Project

10/14/2003