Example: Predictions for fluid-saturated porous glass

Plona (1980) observed two distinct compressional waves in a water-saturated, porous structure made from sintered glass beads. The speeds predicted by Biot's equations of poroelasticity are compared to the values observed by Plona shown in Figure 1.

The input parameters to the model are K_s = 40.7 GPa, $\Gm_s = 29.7$ GPa, $\Gr_s = 2.48$ g/cc, K_f = 2.2 GPa, $\Gr_f = 1.00$ g/cc, $\Gn = 1.00$centistoke, and $\Gw = 2\Gp\times 500$ kHz. The drained frame moduli K_d and were calculated assuming spherically shaped glass particles and needle-shaped inclusions of voids, following Berryman (1980c). We use $\Gt = \Gv^{-{1\over2}}$ for the tortuosity. The permeability variation with porosity was taken to obey the Kozeny-Carman relation

= const^3/(1-)^2,   which has been shown empirically to provide a reasonable estimate of the porosity variation of permeability. We choose $\Gk_0 = 9.1 \times 10^{-8}$ cm² ($\simeq 9.1$ D) at $\Gv_0 = 0.283$ and then use (GkGv) to compute the value of for all other porosities considered. No entirely satisfactory model for the characteristic length h has been found. However, dimensional analysis suggests that h² must be comparable to , so we have taken

h^2/= h_0^2/_0 = const.   At $\Gv_0 = 0.283$, we choose h₀ = 0.02 mm corresponding to an average pore radius ${1\over5}$ to ${1\over7}$ of the grain radius (the glass beads in Plona's samples were 0.21-0.29 mm in diameter before sintering).

The theoretical results for the fast compressional wave and the shear wave agree with Plona's measurements within the experimental error ($\pm 3 \%$ relative error in measured speeds and and an absolute error of $\pm 0.005$ in measured porosity), showing that the Biot-Gassmann equations agree very well with data for these simple synthetic rocks.