Sierra White granite

 

swgdisp_20406080
Figure 7 Showing how the imaginary parts of the dispersion relation for Sierra white granite 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.

 

swg_rhovmu
Figure 8 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 Sierra White granite at  200 kHz. Data are from Murphy. For Sierra White, Gassmann's equation clearly does not apply since the shear modulus $\mu$ must have increased with water saturation. Data and patchy calculation results are therefore compared to the saturation weighted mean of 1/v_s² in analogy to the Gassmann result.

For Sierra White, we have the non-Gassmann-like situation in which the shear wave speed for the drained case is larger than that for the fully saturated case and therefore Re(k_s^*) > Re(k_s). FIG.7 shows how the imaginary parts of the dispersion function change in this case as the real part of k_z² varies from Re(k_s²) to Re((k_s^*)²) (i.e., from water saturated to air saturated). FIG.7 shows four of these curves (S = 0.2 to 0.8). FIG.8 was generated by completing the procedure for 19 equally spaced points in saturation S. FIG.8 shows furthermore that both data and the curve obtained here differ substantially from the simple straightline average that might have been anticipated and, furthermore, that the curve does in fact move in the right direction to agree with the data. This is also a pleasant surprise as it was certainly not known by us what to expect in this situation since our common understanding of poroelasticity does not extend to this rather difficult set of partial saturation problems.