The example we present here is for Berea sandstone saturated with water. The parameters used in the calculations are taken from TABLES 1 and 2. Most of the mechanical properties were obtained from measurements made by Coyner (1984). The permeability values for the matrix k(11) and the fractures k(22) are the same as those used by Lewallen and Wang (1998).
The approach described in the preceding text, together with the results obtained in Appendices A and B, has been implemented by writing a Fortran code and computing the eigenvalues for the three compressional modes and their corresponding eigenvectors in the frequency range 10.0 Hz to 1 MHz. The results for the computed velocities and inverse quality factors are then displayed in Figures 1-6. The results for the eigenvectors will be described but not displayed here. We will not present results for the shear modes, but expect them to differ little from results for single-porosity calculations for the same material, since the pore fluid does not significantly affect the shear moduli in these models. (However, shear dispersion effects due to pore-fluid will still be of importance.)
Figures 1 and 2 show that the first compressional wave is dispersive and has its main contributions to attenuation ()centered at about 3 kHz, with significant decrease in the attenuation envelope (by about an order of magnitude) at 100 Hz and 100 kHz. Wave velocity dispersion is localized approximately to the frequency range 1 kHz to 10 kHz, and the total dispersion is less than 1 %. The eigenvector for this mode shows that the storage pore fluid is essentially moving in concert with the solid frame throughout the frequency range considered, with some small but largely negligible deviations above 1 kHz. On the other hand, the fracture pore fluid oscillates out of phase with respect to the solid frame with an amplitude as much as about one half that of the frame amplitude above about 10 kHz. The observed dissipation for this mode is clearly tied to the out of phase motion of the fracture fluid.
Figures 3 and 4 show that the second compressional mode is diffusive at low frequencies, but becomes propagatory with a or greater at about 10 kHz. The wave speed is quite small at these higher frequencies (about 550 m/s), indicating that the wave is probably propagating mostly through some pore fluid along a tortuous path. The eigenvector analysis shows that the storage fluid excitation is again quite small compared to that of the fracture fluid, although it is about two orders of magnitude larger than that observed for the first compressional wave. The main effect observed for the second compressoinal wave is a large oscillation of the fracture fluid relative to the solid frame, so that this mode can be properly characterized in this example as a slow wave (in the sense of single-porosity poroelasticity) through the fracture fluid.
Figures 5 and 6 show that the third compressional mode is diffusive at all frequencies in the range considered. The apparent velocity is much lower (less than 100 m/s) -- even than that of the second compressional mode. The eigenvector analysis for this mode shows that both the storage fluid and the fracture fluid are oscillating with significant and comparable amplitudes, larger than that of the solid frame. The two fluids are also oscillating out of phase with each other. The amplitude of the storage fluid oscillation slightly dominates that of the fracture fluid, which partly explains the increased attenuation for this mode. For this example, we might characterize this mode as a slow wave through the storage fluid, but note that this interpretation is slightly over simplified.
Finally, we note that other examples have been computed in an attempt to verify the nature of the dependence of the mode parameters on the input parameters. It has been observed for example that the second and third compressional wave ``velocities'' at low frequencies are proportional to the square root of the fluid bulk modulus as would be expected for a single-porosity slow wave (Berryman, 1981; Chandler and Johnson, 1981). In the single porosity case, the slow wave velocity is inversely proportional to the square root of the specific storage coefficient (Green and Wang, 1990). If the solid compressibility is small compared with that of the fluid, then the slow wave velocity would be proportional to the square root of the fluid bulk modulus.