Summary of results

The fitting parameters used to match the data in all three of the examples shown are listed in Table 2. For comparison, the values of bulk and shear moduli for quartz are often quoted as K_m = 37.0 GPa and $\mu_m = 44.0$ GPa, respectively. The fitting parameters obtained here lie in the ranges $30.0 \le K_m \le 57.7$ GPa and $31.7 \le \mu_m \le 69.0$ GPa. According to Walls (1982), the mineralogy of some other Schuler-Cotton Valley sandstones had about 72% quartz with the next most common mineral being quartz overgrowth ranging from 7-12%. The specific sample of Spirit River sandstone used by both Walls (192) and Knight and Nolen-Hoeksema (1990) had only about 34% quartz and 30% siderite, with the next most abundant components being chert, dolomite, and quartz overgrowth - each being in the range 7-10%. Siderite and dolomite both have significantly different, stronger bulk and shear moduli compared to quartz. So we conclude that the range of values observed in our fitting parameters are all quite possible, physically reasonable values but hard to check otherwise.

The observed grain sizes for Schuler-Cotton Valley sandstone and Spirit River sandstone were 100 $\mu$m and 125-150 $\mu$m, respectively. At the frequencies used in the experiments, the wavelengths for Schuler-Cotton Valley were $\lambda_p \simeq 18$-24mm, and $\lambda_s \simeq 18$mm. For Spirit River, the wavelengths were $\lambda_p \simeq 3$-4mm and $\lambda_s \simeq 4$mm. So, for Schuler-Cotton Valley, the grain size over wavelength is about 0.005, while, for Spirit River, it was about 0.035. We expect that the effective medium theory approach should be valid whenever these ratios are less than about 0.3, and certainly for an order of magnitude or more as is the case for these samples.