5 Numerical examples Our main result, Eq. (5), shows that the P-wave modulus is complex-valued and frequency dependent regardless of fluid saturation of fractures. This means that waves will have attenuation and dispersion. To explore these effects, we compute the complex phase slowness in the direction normal to the fractures ܸ ି ଵ ൌඥ ߩ ୠ ܥ ଷଷ ⁄ , where ߩ ୠ ൌሺ1െ߶ୠሻ ߩ ߶ୠ ߩ ୠ is mass density of the fluid-saturated background (the effect of fractures on the density can be ignored as their volume fraction is negligibly small). This complex phase slowness can be used to evaluate the frequency dependence of the P-wave phase velocity and attenuation for waves propagating perpendicular to fractures. The P-wave phase speed is the inverse of real part of the complex phase slowness, and the attenuation Q.is given by half the ratio of the real part to the imaginary part of the complex phase slowness. Now, we rewrite Eq. (5) as a function of normalized frequency ߗ [12] ଵయయ ൌౘౘ ଵ ಽ ౘ ൫భ ഃొ షഃొ ൯ାౝ ൮ഀౘ ಾ ౘି ౘ ಽౘ൫భഃొషಳ಼ഃొ ౝ൯శಳ಼ ౝ൲ మ ౘ√ఆ∗ୡ୭୲൬ ಾౘ ౘ ౘ√ఆ൰ାభశ ഃొ ಳ಼ ౝ ಽ ౘ ൫భషഃొ ൯ ಳ಼ ౝ , (16) where ߗ ൌఠுమఎౘெౘ ସౘౘౘ is the normalized frequency and ߜ ൌ ొ ౘ ଵାొ ౘ is a dimensionless (normalized) fracture weakness with values between 0 and 1 [16]. All our calculations are made for a water-saturated sandstone using quartz as the grain material ( ܭ =37GPa, ߩ =2.65×103kgm−3). The dependency of bulk and shear moduli of the background dry on porosity was assumed to follow the empirical model of Krief et al. [20]. To explore the validity of our approximation, we compare the attenuation and dispersion results with the original Norris [14] model, Eq. (1). For the Norris model, we set fracture parameters to satisfy the assumptions of the approximation (݄ ୡ =0.001, ߶ୡ =0.999, ߜ =0.2, ݇ ୡ =200mD, ߟ ୡ =18.1e-6Pa.s). Then, the P-wave speeds and inverse quality factor ܳି ଵ are calculated for different values of B, and the results are shown on Fig. 1.Alternatively, we could have given an input value to ܭ ୡ and then computed B using Eq. (4). However, we prefer to evaluate the results for different values of the dimensionless constant B. Fig. 1 shows diepersion and attenuation of P-waves propagating along the symmetry axis (normal to fracture plane) for different values of parameter B. Symbols show the values obtained by our approximation, Eq. (5), while the curves correspond to the Norris general solution, Eq. (1). We see that for the whole range of parameter B, the approximation matches the general model very accurately. Curves of dispersion and attenuation have a shape typical for a relaxation phenomenon. It is interesting that dispersion and attenuation is significant for both liquid-filled (B=1000) and dry (B=0.001) fractures, but is much lower for intermediate values of the parameter B. This somewhat
RkJQdWJsaXNoZXIy MjM0NDE=