preventing fluid mixing). In this paper we consider the simplest situation of this kind: a porous rock saturated with one fluid and permeated by a single set of aligned planar fractures filled with another fluid. For such a medium, we derive a dispersion equation following a method originally proposed by Brajanovski et al. [12] for a porous fractured medium saturated with a single fluid. The paper is organized as follows. First, we review the theory in case of a medium saturated with a single fluid. Then we extend the method to the situation where the porous background and fractures are saturated with different fluids and derive the corresponding dispersion equation, which yields expressions for dispersion and attenuation due to wave induced flow between pores and fractures. To explore the behavior of attenuation and dispersion, we explore various limiting cases and present several numerical examples. Finally, we discuss the physical nature of the results obtained. 2 Liquid saturated porous and fractured medium Brajanovski et al. [12] developed a model for a porous medium with aligned fractures. The medium comprises a periodic (with spatial period ܪ ) stratified system of alternating layers: relatively thick layers of a background material (with a finite porosity ߶) and relatively thin layers of a high-porosity material representing the fractures. This double porosity model is a limiting case of a periodically layered poroelastic medium studied by White et al. [1] and Norris [14]. Norris showed that for frequencies much smaller than Biot’s characteristic frequency ߱ ൌ ߟ ߶݇ ߩ ⁄ , and also much smaller than the resonant frequency of the layering ߱ୖ ൌܸ ୮ ܪ ⁄ , the compressional wave modulus of a periodically layered fluid-saturated porous medium composed of two constituents, b and c, can be written in the form: ଵయయ ൌౘౘ ౙౙ ൬ ഀౘಾౘି ౘ ഀౙಾౙ ౙ൰మ ටഘആౘౘಾೖౘౘ ಽౘ ಹమୡ୭୲൬ටೖ ഘౘ ಾആ ౘౘಽౘౘ ౘమ ಹ൰ାටഘആೖౙౙಾౙౙಽౙಹమୡ୭୲ቆටೖഘౙ ಾആౙౙಽౙౙ ౙమಹቇ, (1) In Eq. (1), both constituents are assumed to be made of the same isotropic grain material with bulk modulus ܭ , shear modulus ߤ and density ߩ , but they have different solid frame parameters: porosity߶, permeability , dry bulk modulus ܭ , shear modulus ߤ and thickness fraction ݄ . The layers b and c may be saturated with different fluids with bulk modulus ܭ , density ߩ and dynamic viscosity ߟ , as indicated by adding ‘b’ and ‘c’ in the subscript. Parameter ܥ ൌ ܮ ߙ ଶ ܯ denotes the fluid-saturated P-wave modulus of layer given by Gassmann’s equation [15], where ߙ ൌ1െೕౝ is Biot’s effective stress coefficient, ܯ is pore space modulus defined by ெ ଵೕ ൌ ఈೕି థೕ ౝ థೕೕ and ܮ ൌ ܭ 4 ߤ /3 is the dry P-wave modulus of the layer ݆ . To construct a model for a porous medium permeated by parallel fractures, Brajanovski et al. [12] considered parameters with subscript b to represent the porous background, and parameters with subscript c to represent fractures (cracks). They then assumed fractures to be very thin and very
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