To make numerical predictions of attenuation and dispersion, models must be proposed for the phase 2 (porous grain) parameters.

If the grains are modeled as spheres of radius *R*, the fluid-pressure gradient length
within the grains can be estimated as and the volume to
surface ratio as *V*/*S* = *R*/(3*v _{2}*). The grain porosity is assumed to
be in the form of microcracks and so it is natural to define an effective aperature

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The drained grain modulus *K*^{d}_{2} is necessarily
a function of the crack porosity (and therefore *h*/*R*).
Real crack surfaces have
micron (and smaller) scale asperities present upon them.
If effective stress is applied in order
to make the normalized aperature *h*/*R* smaller (so that, for example,
the peak in squirt attenuation
lies in the seismic band), new contacts are created that
make the crack stronger. In the limit as (large effective stress),
the cracks are no longer present and where *K*_{s} is the mineral modulus of the grain.

Many models for such stiffening could be proposed.
We intentionally make a conservative estimate here in proposing a simple
linear porosity dependence
,where is a fixed constant determined from
fitting ultrasonic attenuation data. Effective medium theories
[see, for example, Berryman *et al.* (2002)] predict
that should be inversely proportional to the aspect ratios of
the cracks present.
As a crack closes and asperities are brought into contact, there is
naturally a decrease in but there should also be a decrease in due to the
fact that the remaining crack porosity becomes more spherical as new
asperities come into contact.
Taking to be constant as crack porosity decreases
is thus a minimalist estimate for how the drained modulus increases.

Thus, the porous-grain elastic properties are taken to be

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10/14/2003