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The elastic moduli of a solid permeated with an isotropic distribution of flat cracks have been calculated from the energy of a single crack by use of a self‐consistent approximation. The results are applicable for a dense network of cracks and give physically reasonable results up to the point that the shear modulus vanishes. Results for both circular and elliptical cracks are essentially the same if the crack density is characterized by 2N〈A2/P〉/π, where N is the number of cracks per unit volume, A is the area of crack, and P is the perimeter of cracks; for circular cracks of radius a this becomes N〈a3〉. This crack density parameter can be related quantitatively to crack traces observed in thin section. Results for completely dry or saturated cracks, for mixtures of dry and saturated cracks, and for cracks saturated with a compressible fluid are presented. For all cases, both seismic wave velocities decrease with increasing crack density. The velocity ratio VP/VS decreases for dry cracks and increases for saturated cracks. For the analysis of data a plot of VP/VS versus VS uniquely specifies the crack density. Comparison of the theory with wave velocities measured in laboratory rock samples demonstrates its validity for large crack densities. Interpretation of velocity changes before the San Fernando earthquake indicates that the region contained a substantial density of cracks at all times, that the anomalous decrease in VP/VS was due to the vaporization of pore fluid in nearly all of the previously saturated cracks without the introduction of new dry cracks, and that during the period of the recovery of the velocities to previous values the number of cracks in the region away from the epicentral zone decreased as they were resaturated, whereas the crack density increased following resaturation in the epicentral zone. Such use of the theoretical results may be useful in further investigations of preseismic phenomena.

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The effective elastic moduli of a fluid-saturated solid containing thin cracks depend on the degree of interconnection between the cracks. Three separate regimes may be identified: (1) dry (drained), in which fluid in cracks can flow out of bulk regions of compression, (2) saturated isobaric, in which fluid may flow from one crack to another but no bulk flow takes place, and (3) saturated isolated, in which there is no communication of fluid between cracks. Transitions between these cases involve fluid flow, resulting in dissipation of energy. Relaxation of shear stresses in viscous fluid inclusions also results in dissipation. Visco•elastic moduli are derived, by using a self-consistent approximation, that describe the complete ronge of behavior. There are two characteristic frequencies near which dissipation is largest and the moduli change rapidly with frequency. The first corresponds to fluid flow between cracks, and its value can be estimated from the crack geometry or permeability. The second corresponds to the relaxatio•,iof shear stress in an isolated viscous fluid inclusion; its value may also be estimated. Variations of •h• geometry result in a distribution of characteristic frequencies and cause Q to be relatively constant o•!/• many decades of frequency. Fluid flow between cracks accounts for atten, uation of seismic waves in water-saturated rocks and attenuation observed in laboratory measurements on water-saturated rocks and partially molten aggregates. Attenuation in a partially molte, n upper mantle is probably due to fluid flow between cracks, although grain boundary relaxation in an unmelted upper mantle could also account for the seismic low-velocity zone. Grain boundary relaxation in the mantle may cause the long-term shear modulus to be around 20% less than that measured from seismic observations. been interprete d as one in which thin inclusions of fluid exist owing t•o'pa'rtial melting [Anderson and Sammis, 1970]; thus this region may also exhibit a time or frequency dependent elastic response. In this case the elastic properties of the region , Copyright ¸ 1977 by the American Geophysical Union. Paper number 7B0604. inferred from seismic observations may be substantially different from those appropriate for longer-term deformations. In addition, delayed elastic response due to fluid flow may have an influence on the history of deformation before and after large lithospheric earthquakes [Mat•ko and Nur, 1975]. In this paper we analyze the time or frequency dependent elastic properties of a solid permeated with fluid-filled cracks. We find that such a solid exhibits linear viscoelastic behavior and that there are three important relaxation mechanisms. The first is due to the relaxation of a viscous fluid in a single crack in shear, which has previously been analyzed by Walsh [1968, 19•69]. The second involves fluid flow between cracks with different orientations. Although this mechanism has been recognized before [Mat•ko and Nur, 1975], it has up to now not been systematically analyzed. The ...

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