4 Limiting cases 4.1 Fluid limits 4.1.1 Liquid in fractures If the fluid in the fractures is liquid, then ܤ ≫1 and Eq. (5) gives ଵయయ ൌ ଵౘ ൬ ഀౘ ಾ ౘି ౘ ଵ൰మ ටഘആౘౘಾೖౘౘ ಽౘ ಹమ௧൬ටೖ ഘౘ ಾആ ౘౘ ಽ ౘౘ ಹమ൰ାೋొ భ , (6) Note that Eq. (6) is exactly the same as Eq. (2). This shows that the result of Brajanovski et al [12] is valid not only if both pores and fractures are saturated with the same liquid, but also when the two liquids are different. 4.1.2 Dry or nearly dry fractures When fractures are dry or nearly dry, Eq. (5) cannot, strictly speaking, be used because it was derived by assuming that B is nonzero. Instead, we take the limit ܤ →0, directly in Eq (1). This gives ଵయయ ൌ ଵౘ ܼ ൬ ഀౘ ಾ ౘ ౘ൰మ ටഘആౘౘಾೖౘౘ ಽౘ ಹమୡ୭୲൬ටೖ ഘౘ ಾആ ౘౘ ಽ ౘౘ ಹమ൰, (7) Incidentally, exactly the same result is obtained by taking the limit ܤ →0 in Eq. (5). This means that Eq. (5) is valid even in the limit of small B. Eq. (7) gives the P-wave modulus for a porous medium with dry or gas-filled fractures, and it is quite different from Eq. (5) for liquid case. To further analyze the reason for the difference, we derive the limiting cases of low and high frequencies next. 4.2. Frequency limits 4.2.1 Low frequencies In the low-frequency limit ߱ →0, the cotangent function in Eq. (5) can be replaced by the inverse of its argument. Thus, Eq. (5) is reduced to ଵయయ ൌ ଵౘ ଵ ೋొ భ ାౝ ቌഀౘ ಾ ౘି ౘ భ ಳ಼ ౝ ೋొ శಳ಼ ౝቍ మ ಾ ౘ ౘ ಽ ౘାభశೋొ ಳ಼ ಳ಼ ౝ ౝ , (8) In the low frequency limit, the fluid pressure should be fully equilibrated between pores and fractures. Thus in this limit the result must be consistent with the anisotropic Gassmann equations for a fractured medium saturated with a single composite fluid [17] with a bulk modulus defined by ଵ ∗ ൌௌౘౘ ௌౙ ౙ, (9) Eq. (9) is known as the Wood equation, and corresponds to so-called uniform saturation of the partial saturation theory [7, 8, 18]. So, if we replace ܭ ୠ with ܭ ∗ in Eq. (5), and then take the low
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