The analytical approximation made in this paper seems to be very effective at capturing the first order behavior of the bulk and shear moduli for cracked porous media in both the dry and saturated cases. The resulting formulas are not rigorous, unlike Gassmann's formulas for low frequency behavior, but these approximate formulas nevertheless have a wider range of validity (considering both porosity and frequency ranges) than either Gassmann (1951) or Mavko and Jizba's (1991) results.
We believe these results will, at the very least, provide a helpful way of understanding the behavior of these complex systems, and may also provide a stepping stone towards more general formulas in the future.