Massillon sandstone

 

reimdisp_mss560_20406080_ray3
Figure 5 Showing how the imaginary parts of the dispersion relation for Massillon sandstone change in the complex k_z² plane as k_z varies from k_sa to k_sw. The real part of the dispersion relation is either zero or very close to zero along this line and therefore the desired points are those where the imaginary part crosses the zero line.

 

mss_rhovmu
Figure 6 Comparison between the points that solve the dispersion relation for the patchy cylinder, plotted as $\rho/\mu = 1/v_s^2$versus water saturation S, for Massillon sandstone at 560 Hz. Data are from Murphy. The Gassmann curve is computed assuming that the shear modulus $\mu$ is constant and that the only quantity chaning is therefore the density $\rho$.

For Massillon, we have the Gassmann-like situation in which the shear wave speed for the drained case is smaller than that for the fully saturated case and therefore Re(k_s^*) < Re(k_s). FIG.5 shows how the imaginary parts of the dispersion function change in this case as the real part of k_z² varies from Re((k_s^*)²) to Re(k_s²) (i.e., from air saturated to water saturated). FIG.5 shows four of these curves (S = 0.2 to 0.8). FIG.6 was generated by completing the procedure for 19 equally spaced points in saturation S. FIG.6 shows furthermore that the curve obtained actually fits the data for Massillon better than Gassmann does (the straight line between the end points). This is a bit of a surprise as virtually everyone (including the present authors) have often considered these data to be the best known proof of the accuracy of Gassmann's equations for partial saturation problems.