frequency limit, we can also get the same expression as given by Eq. (8). To clarify the physical meaning of the low-frequency Eq. (8) we again consider liquid and dry (or gas) cases. For liquid-filled fractures (large B), we have ଵయయ ൌ ଵౘ ൬ ഀౘ ಾ ౘି ౘ ଵ൰మ ಾ ౘ ౘ ಽ ౘାೋొ భ , (10) while for ܤ ≪1 ଵయయ ൌ ଵౘ ܼ , (11) The liquid limit, Eq. (10), corresponds exactly to the low frequency limit of the result of Brajanovski et al [12], with only the bulk modulus of the liquid in the pores affecting the overall modulus. This result may be understood from Eq. (10), which shows that when the bulk moduli of the two fluids are comparable, the effect of the fracture fluid is negligible for its relatively small saturation. In turn, Eq. (11) is exactly the modulus of the dry medium [12, 16]. This is because when ܭ ୡ is very small (much smaller than ݄ ୡ ܭ ୠ), Wood Eq. (10) for the effective fluid modulus reduces to ଵ ∗ ൌௌౙ ౙ, (12) and thus ܭ ∗ →0, which means that the whole porous and fractured model can be considered as dry or nearly dry medium. Physically, this is the result of the fact that at low frequencies, the pore pressure is equilibrated between pores and fractures, so when the pressure in fractures is zero, thus is also zero in the pores. This is the drained – or dry – limit. 4.2.2 High frequencies In the high-frequency limit ߱ →∞, the cotangent function in Eq. (5) can be replaced by . So, we can get the expression of P-wave modulus at high frequencies ଵయయ ൌ ଵౘ ଵ ೋొ భ ାౝ, (13) In liquid and gas cases we have ଵయయ ൌ ଵౘ, (14) and ଵయయ ൌ ଵౘ ܼ , (15) respectively. Note that at high frequencies, the fluid pressure does not have time to equilibrate between pores and fractures, and thus they can be considered ‘hydraulically isolated’ [12, 17]. Thus the P-wave modulus in this limit corresponds to the modulus of a porous medium with isolated fractures. In the liquid case, the modulus given by Eq. (14) is the same as if there were no fractures. This is because liquid can stiffen the otherwise very compliant fractures so that P-wave velocities for waves propagating parallel and perpendicular to layering are both approximately equal to the modulus of the background medium [12, 16, 19]. Conversely, when fractures are dry, the P-wave modulus (15) is the same as for a medium with dry isolated fractures (cf Eq. (11)). .
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