Numerical results and discussion

This equation can be rewritten as F P s y3 = P W t f Eu Eu Eu x y z q F q q ž E x E y E z Fg 1 y 3 = P u. 18 Ž . f We notice that the expression between paren- thesis is a divergence term, therefore, the aver- age pore fluid pressure in 3-D wave motion for an isotropic porous medium saturated with a fluid can be rewritten as F F Fg 1 P s y = P Wq = P uy = P u, 19 Ž . av f 3 f where F P s y = P Wqa =P u , 20 Ž . Ž . av f or simply a q 2f a s . 3 This expression of pore fluid pressure has been used in the modelling of acoustic wave velocity and attenuation dispersion vs. fre- quency. Formation and saturant parameters used Ž . are those of Dvorkin and Nur 1993 as reported Ž . by Parra 1997 . We avoid the lengthy mathe- matical derivation as it has been clearly and extensively treated in the two papers cited above and give directly expressions used to determine velocity and attenuation: 1 V s , a s v Im Y 21 Ž . Ž . p Re Y Ž . where for our model, 2 r 1 y f q r f q vr u Ž . Ž . a f f Y s , 2 a M q F f y1 f r rr q f hf a f u s y q j . 22 Ž . 2 v r f k r v f f Ž . Ž . The notation Re and Im indicate the real and complex part of a complex number.

3. Numerical results and discussion

We analyse the effect of frequency on the attenuation and phase velocity curves, by mod- elling six cases with permeability equal to 5, 20, 50, 100, 500 and 1000 md. The formation con- sidered is a water saturated porous rock. Forma- tion and saturant parameters are given in Table 1. We first present the velocity–frequency and attenuation–frequency dispersion curves ob- Ž . tained using the original BISQ model Fig. 1 and it is extended version to transversally Ž . Ž isotropic porous media after Parra 1997 Fig. . 2 . In the latter, we considered the case of normal isotropic medium; the transversally isotropic case can be easily dealt with by find- ing wavenumber roots to the dispersion equa- Ž . tion given in Parra 1997 and using them to determine the velocities and attenuation of Ž . quasi-compressional waves fast and slow and quasi-SV waves. The velocity and attenuation dispersion curves obtained using our reformu- Ž . lated BISQ model Fig. 3 show significant dispersion of the same order as that predicted by the original BISQ theory. The main difference observed is the reversed behaviour with respect to the change in permeability. In the BISQ Ž . model Dvorkin and Nur, 1993; Parra, 1997 , the transition zone between the low-frequency limit and the high-frequency limit of acoustic wave velocity and attenuation peak shifts to- ward higher frequency when permeability in- Ž . creases Figs. 1 and 2 . In our model, the re- Ž . verse trend is observed Fig. 3 . The phase velocity at low frequency in the new model Ž . Fig. 4a is nearly the same as that obtained Ž . using Parra’s 1997 version of the BISQ-model Ž . Fig. 3a , but slightly greater than that given in Ž . the paper of Dvorkin and Nur 1993 . Despite this difference the low-frequency phase velocity obtained is found to be consistent with the Fig. 1. The effect of frequency for permeability 5, 20, 50, Ž . 100, 500, and 1000 md. a Compressional velocity vs. log Ž . frequency and b inverse attenuation quality factor 1r Q vs. log frequency as predicted by BISQ theory after Ž . Dvorkin and Nur 1993 . Ž . theory Parra, 1997 . The attenuation curves Ž . Fig. 3b depict the same trend with respect to permeability as observed in the velocity disper- Ž . sion curves Fig. 3a . Here, again, attenuation values are of same order of magnitude as those predicted by the BISQ model of Dvorkin and Ž . Nur 1993 in Fig. 1b. Curves of compressional velocity and attenuation dispersion predicted by Ž . the Biot’s 1956a,b theory are presented for Ž . comparison Fig. 4 . Fig. 5 displays velocity and attenuation dispersion curves for a perme- ability of 20 md along with the velocity of the saturated rock in the limit of zero frequency Ž . calculated with the help of Gassman’s 1951 equations. These equations applies at low fre- quency where scattering, solid–fluid inertial ef- fect, local flow effect are negligible so that the estimated velocity reflect the effect of fluid Ž saturation at seismic or sonic frequency Winkler Fig. 2. The effect of frequency for permeability 5, 20, 50, Ž . 100, 500, and 1000 md. a Compressional velocity vs. log Ž . frequency and b inverse attenuation quality factor 1r Q vs. log frequency as predicted by BISQ theory after Parra Ž . 1997 . Fig. 3. The effect of frequency for permeability 5, 20, 50, Ž . 100, 500, and 1000 md. a Compressional velocity vs. log Ž . frequency and b inverse attenuation quality factor 1r Q vs. log frequency as predicted by our reformulated BISQ theory. . 1986, Nolen-Hoeksema et al., 1995 . As one may expect, the calculated velocity, using Ž . Gassman’s 1951 equations, fits very well to the low frequency velocity limit of the Biot’s Ž . 1956a,b model curve because Gassmann equa- tions are also the low frequency limit of the Ž . Biot theory Biot, 1956a . The low-frequency Ž velocity limit from BISQ theory Dvorkin and . Nur, 1993; Parra, 1997 curves are identical, but still lower than that predicted by Biot and Gassmann. The same remark applies to the re- formulated BISQ curve. The difference in low frequency velocity limit between the BISQ the- Ž . ory Dvorkin and Nur, 1993; Parra, 1997 and our model on the one hand, and the Biot–Gas- saman theory in the other hand can be explained by assuming that the rock may still contain some highly compressible residual gases, which have little effect on rock elastic moduli, but can Fig. 4. The effect of frequency for permeability 5, 20, 50, Ž . 100, 500, and 1000 md. a Compressional velocity vs. log Ž . frequency and b inverse attenuation quality factor 1r Q vs. log frequency as predicted by Biot theory after Biot Ž . 1956a,b . Fig. 5. Comparison of the different models for a perme- Ž . ability of 20 md. a Compressional velocity vs. log fre- Ž . quency and b inverse attenuation quality factor 1r Q vs. log frequency as predicted by the different models. induce a noticeable change in density so that the difference in low-frequency velocity limits is Ž mainly due to the density effect Dvorkin et al., . 1995 .

4. Conclusion