CONCLUSIONS

The main point of this exercise has been to show that the DEM equations are appropriate to use in this context (since the wavelengths are sufficiently long compared to the grain sizes) and that the DEM equations do in fact predict that right kinds of behavior in the high frequency range (> 200 kHz). Gassmann's equations have clearly failed in this region (i.e., the shear modulus is not independent of the fluid content), as would be expected. The quasi-static assumptions explicitly used in Gassmann's derivation are not satisfied in this frequency regime, and particularly so in rocks having low permeabilities ($\mu$D), as is the case for all the samples considered here. We can understand both qualitatively and semi-quantitatively what is happening in these samples by making use of DEM as a modeling tool. To do more detailed modeling requires much more detailed information about the constituents, their spatial distribution, their bonding characteristics, and the distribution and character of voids and cracks. We are still some ways from being able to determine all of these parameters in real rocks, but nevertheless can conclude that the methods described and used for modeling here do correctly capture the physics of these complicated high-frequency acoustics problems.