The effect of voids on the seismic velocities of rocks and composites has been studied by many researchers (e.g., Eshelby, 1957; Mal and Knopoff, 1967; Mori and Tanaka, 1973; Kuster and Toksöz, 1974; Gangi, 1978, 1981; Carlson and Gangi, 1985; Wilkens et al., 1991; Berge et al., 1992; Ju and Chen, 1994; Berryman and Berge, 1996). In general, these studies show that when porosity is held constant, thinner voids (cracks) lower the seismic velocities significantly more than spherical pores, or vesicles (e.g., fig. 1 of Toksöz et al., 1976). This effect is caused by the more substantial lowering of both the bulk and shear moduli of the rock (the effective moduli) when cracks, rather than vesicles, are present. For example, given a hypothetical basalt composed of 50% plagioclase and 50% augite with a porosity of 5%, the effect of spherical voids (e.g., vesicles) is to lower the P- and S-wave velocities by 4% and 3%, respectively, whereas the effect of voids with an aspect ratio of 0.1 (i.e., the length of the semi-minor axis is one-tenth that of the semi-major axis) is to lower the P- and S-wave velocities by 13% and 15%, respectively (e.g., Toksöz et al., 1976). Hence, the shapes of the voids and cracks that populate rocks have a substantial influence on their seismic velocities.

Kuster and Toksöz (1974) proposed a theoretical model in which cracks and voids are approximated by spheroids having different aspect ratios, and the pressure dependence of P- and S-wave velocities is accounted for by a distribution or spectrum of spheroids with different aspect ratios. Cheng and Toksöz (1979) developed an inverse procedure for estimating the aspect ratio spectrum of a rock from its measured velocities. The effective elastic moduli of a medium populated by cracks that have m discrete aspect ratios are given by

and

 ,

where and are, respectively, the effective bulk and shear moduli of the material at confining pressure P_n, K and µ are the bulk and shear moduli of the grains, K' and µ' are the bulk and shear moduli of the pore fluid, _m and c(_m) are the m-th aspect ratio and corresponding concentration, dc(_m, P_n) is the change in concentration from the initial concentration (at P = 0) of cracks with aspect ratio _m at confining pressure P_n, and T₁(_mn) and T₂(_mn) are expressions calculated from the moduli and porosity of the rock (see Kuster and Toksöz, 1974).

The Kuster-Toksöz model does not account for crack-crack interaction. The approximation is therefore valid only for low concentrations of cracks. To have that condition satisfied, Kuster and Toksöz (1974) imposed a limit on the concentrations such that

  for any .

We also imposed this constraint, although Berryman (1980) and Berryman and Berge (1996) demonstrated that this strict constraint is not necessary for small _m and low porosity samples (e.g., _m < 0.01, and porosity < 20%). Another limitation of the Kuster-Toksöz model is that the modeling results are nonunique and model dependent. Berge et al. (1992) and Berge (1996) have compared various schemes for estimating elastic properties of multiphase composites and have shown the Kuster-Toksöz model to be useful for materials with porosities <20%.

We used the theory of Kuster and Toksöz (1974) to model void-shape distributions using the seismic velocities in samples from Hole 990A. The data used in this study are bulk densities and porosities, and P- and S-wave velocities measured at confining pressures ranging from 10-200 MPa in 10 basalt samples from Hole 990A. Porosities were calculated from the measured bulk and grain densities of the samples. The data were inverted for pore aspect ratio distributions and apparent grain moduli using a slightly modified form of the method of Cheng and Toksöz (1979).