Figure 2. Dispersion magnitude (difference between high- and low-frequency velocities) as a function for parameter B. As B increases, the dispersion first decreases, reaches zero, and then increases again. Fig. 2 shows the dispersion magnitude (difference between high- and low-frequency velocities) as a function for parameter B. As B increases, the dispersion first decreases, reaches zero, and then increases again. We also see that the dispersion is almost insensitive to B both for very small and very large values of B (corresponding to highly compressible gases and liquids, respectively), but quite sensitive to B for values of B in a range around the critical value where dispersion reaches zero. This critical value can be obtained by equating low- and high-frequency limits, Eqs (8) and (13). This gives ܤ ∗ ൌ ఈౘெౘ ౘି ఈౘெౘ ଵౝ ొ ଵ , (17) The value ܤ ∗ given by Eq. (17) corresponds to a critical fracture fluid modulus case, where there will be zero dispersion and attenuation in the general porous and fractured model. For the parameters used in the numerical example of Fig. 1, Eq. (5) gives ܤ ∗ ൎ0.59. This value is quite close to 1.0, and thus we see very small dispersion and attenuation for ܤ ൌ1. 6 Conclusions We have developed a model for wave propagation in a porous medium with aligned fractures such that pores and fractures can be filled with different fluids. The model considers the fractured medium as a periodic system of alternating layers of two types: thick porous layers representing the background, and very thin and highly compliant porous layers representing fractures. The results show that in the low-frequency limit the elastic properties of such a medium can be described by Gassmann equation with a composite fluid, whose bulk modulus is a harmonic (Wood) average of the moduli of the two fluids. At higher frequencies, the model predicts significant dispersion and attenuation. The dispersion and attenuation are the highest when both pores and fractures are saturated with liquids. The dispersion and attenuation are also significant (but somewhat weaker) when the pores are filled with a liquid but fractures are dry or filled with a highly compressible gas. However, there is an intermediate case where no dispersion is observed. This can be explained by observing that when the medium is uniformly saturated with a liquid, wave-induced compression causes flow from fractures into pores due to high compliance of the fractures. Conversely, when 10 -3 10 -1 10 1 10 3 0 100 200 300 400 B P-wave speed difference (m/s)
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