Sierra White granite

 

swgvsq_all2
Figure 1 Square of the velocity data for the Sierra White granite measured by Murphy (1982) at 200 kHz. The dashed lines are the DEM results for compressional and shear when it is assumed that the saturation is homogeneous in each pore. The solid lines are the results for patchy saturation. Clearly, the data all fall closer to the patchy saturation lines at the higher values of liquid saturation. For the very lowest values of liquid saturation, the data seem to mimic the homogeneous saturation curve.

All the data presented here for Sierra White granite are from Murphy (1982). This case is especially simple as the porosity is quite low ($\phi = 0.008$) and therefore the effect of liquid saturation on the density is very small ($< 0.3 \%$ density effect), which we will treat for purposes of hand analysis as negligible. Thus, essentially the entire effect of liquid saturation depends on how the liquid is distributed in the pores and how this affects the bulk and shear moduli only. We will model this simply by considering the effects of gas saturation and liquid saturation separately and then combining the results [Voigt (1928) average] for the patchy saturation effects. For homogeneous saturation, we use DEM with an effective fluid bulk modulus given by the Reuss (1929) average (harmonic mean) of the fluids' moduli.

Murphy describes Sierra White as a granite ``composed of a sparse population of low aspect ratio cracks, embedded in a composite of elastic grains.'' Some preliminary calculations done for the present work indicate that an aspect ratio of $\alpha \simeq 0.005$ to 0.02 should give results very consistent with the measured values for Sierra White. To fit the data at both the fully gas saturated end and the fully water saturated end, we found that K_m = 57.7 GPa and $\mu_m = 31.7$ GPa were good choices. The computation was performed at 21 equally spaced values of saturation for homogeneous saturation. The patchy saturation curve is obtained by connecting the two end points on a plot of velocity squared with a straight line. (For situations with significant density variation, it is preferable to plot the Lamé constants $\lambda$ and $\mu$instead of the squares of the velocities -- but for small porosity this is always a small difference that we choose to ignore here.) The results are shown in Figure 2. Clearly, all the data fall closer to the patchy saturation lines at the higher values of liquid saturation. For the very lowest values of liquid saturation, the data seem to mimic the homogeneous saturation curve, but at these low saturation levels the two curves are very close together anyway.