These observations show that there is a significant problem with
up-scaling Biot's theory, *i.e.*, that the resulting system of
equations is no longer of the same form as Biot's theory. This is
certainly no failing of Biot's theory, but rather a failing of any
attempted application of Biot's theory directly to the
up-scaled macro-system. Biot's theory predicts correctly tha
compressional and shear wave attenuation both depend on the integral
of the permeability along the path of each wave. But the
permeability itself along the same path averages as the inverse of
the permeability (harmonic mean). Thus, the overall permeability
depends most strongly on the smallest permeabilities present in the
system, while the wave attenuation depends most strongly on the
largest permeabilities in the system (Berryman, 1988). When we try
to up-scale under these circumstances, we have an inherent problem
due to the fact that Biot's theory contains only one permeability;
yet, for heterogeneous systems, there are two very distinct measures
of permeability (the mean and the harmonic mean) that play significant roles.

10/14/2003