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/(3v2). The grain porosity is assumed to be in the form of microcracks and so it is natural to define an effective aperature h for these cracks. If the cracks have an average effective radius of R/NR where NR is roughly 2 or 3 and if there are on average Nc cracks per grain where Nc is also roughly 2 or 3 then the permeability and porsity of the grains is reasonably modeled as
The drained grain modulus Kd2 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 Ks 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