compliant layers and thus considered Eq. (1) in the limit ݄ ୡ →0, ܭ ୡ →0 and ߤ ୡ →0 such that both ܭ ୡ and ߤ ୡ (and hence ܮ ୡ →0) are O(݄ ୡ). By assuming that both pores and fractures are saturated with the same fluid with the viscosity ߟ ୠ ൌ ߟ ୡ ൌ ߟ , Brajanovski et al [12] obtained the equation େ ଵయయ ൌେ ଵౘ ൬ ಉౘ ౘି ౘి ଵ൰మ ටಡಏౘౡౘైౘ ౘి ౄమୡ୭୲൬ටౡ ౘ ಡ ಏి ౘై ౘ ౘhౘ మ ౄ൰ାభౖొ , (2) Where ܼ ൌ limౙ→ ౙౙ is the normal excess fracture stiffness of the dry frame given by Schoenberg and Douma [16]. Implicit in the derivation of Eq. (2) was an assumption ܭ ≫݄ ୡ/ܼ , (3) When both pores and fractures are saturated with a liquid, Eq. (2) exhibits significant attenuation and velocity dispersion. However the model is limited to the case where the fluid is the same in both matrix pores and fractures, and there is an upper limit on the fluid compressibility (Eq. (3)). Below we develop a model that overcomes these limitations. 3 Arbitrary fluid in the fractures The analysis in the previous section suggests that the effect of fracture fill on the overall modulus of the porous and fractured medium depends on how the fluid bulk modulus scales with ݄ ୡ as ݄ ୡ →0. To analyze this effect, we use the following parameterization ܭ ୡ ܭ ⁄ ൌ ݄ܤ ୡ , (4) where B is a dimensionless nonzero constant that defines the type of fluid in fractures, liquid, gas or intermediate. If B is large enough (e.g., if the fluid is a liquid), ܭ ୡ may satisfy the condition (3). In this case, taking the limit ݄ ୡ →0 in Eq. (1), we obtain Eq. (2) with fluid properties ܭ and ߟ replaced by the corresponding values for the fluid in the pores ܭ ୠ and ߟ ୠ. Conversely, ܤ ൌ0 corresponds to dry fractures. When ݄ ୡ →0, we have ܭ ୡ →0, and ߶ →1, and thus ߙ ୡ →1, ܯ ୡ → ܭ ୡ, and ܥ ୡ → ܮ ୡ ܭ ୡ. Combining these results with the parameterizations (4), and considering cotሺ ݔ ሻൎ1 ݔ ⁄ for any complex ݔ with | ݔ |≪1 and cot൫√݅ ݔ ൯→݅ for any real ݔ ≫1, Eq. (1) in the limit ݄ ୡ →0 can be simplified as ଵయయ ൌ ଵౘ ଵାొ ొ ౝ ൬ഀౘ ಾ ౘି ౘ ೋొ ಳ಼ ౝ భశೋొ ಳ಼ ౝ൰మ ටഘആౘౘಾೖౘౘ ಽౘ ಹమୡ୭୲൬ටೖ ഘౘ ಾആ ౘౘ ಽ ౘౘ ಹమ൰ାభశೋొ ಳ಼ ಳ಼ ౝ ౝ, (5) Eq. (5) is the approximation of Eq. (1) for a porous and fractured medium with an arbitrary fracture fill. Here, we have introduced a dimensionless constant B to define the type of fracture fluids, so that we can attain gas and liquid limiting cases. Additionally, we can define low and high frequencies with respect to fluid flow between fractures and background. Therefore, in the following section, we derive and analyze some limiting cases of fluids and frequencies.
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