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The equations of motion and stress/strain relations for the linear dynamics of a two-phase, fluid/solid, isotropic, porous material have been derived by a direct volume averaging of the equations of motion and stress/strain relations known to apply in each phase. The equations thus obtained are shown to be consistent with Biot’s equations of motion and stress/strain relations; however, the effective fluid density in the equation of relative flow has an unambiguous definition in terms of the tractions acting on the pore walls. The stress/strain relations of the theory correspond to ‘‘quasistatic’’ stressing (i.e., inertial effects are ignored). It is demonstrated that using such quasistatic stress/strain relations in the equations of motion is justified whenever the wavelengths are greater than a length characteristic of the averaging volume size.

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[1] Permeability of illite-rich shale recovered from the Wilcox formation and saturated with 1 M NaCl solution varies from 3 Â 10 À22 to 3 Â 10 À19 m 2 , depending on flow direction relative to bedding, clay content (40-65%), and effective pressure P e (2-12 MPa). Permeability k is anisotropic at low P e ; measured k values for flow parallel to bedding at P e = 3 MPa exceed those for flow perpendicular to bedding by a factor of 10, both for low clay content (LC) and high clay content (HC) samples. With increasing P e , k becomes increasingly isotropic, showing little directional dependence at 10-12 MPa. Permeability depends on clay content; k measured for LC samples exceed those of HC samples by a factor of 5. Permeability decreases irreversibly with the application of P e , following a cubic law of the form k = k 0 [1 À (P e /P 1 ) m ] 3 , where k 0 varies over 3 orders of magnitude, depending on orientation and clay content, m is dependent only on orientation (equal to 0.166 for bedding-parallel flow and 0.52 for flow across bedding), and P 1 (18-27 MPa) appears to be similar for all orientations and clay contents. Anisotropy and reductions in permeability with P e are attributed to the presence of crack-like voids parallel to bedding and their closure upon loading, respectively.

Numerous correlations of one-way only if the variation of seismic velocity with reflection travel times and depths to reflecting depth is known. outline our statistical models and methods. tant correlation criterion. Other reliable criteria (all of which imply changes of physical

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To test the “bed of nails” model, we have made detailed measurements of P wave velocities in five low‐porosity, crystalline rocks at effective pressures to 500 MPa and fit two equations based on the model to the laboratory data. The first equation, V(P) = V0(1 + P/ Pi)(1 ‐ m)/2, applies at relatively low pressures because it assumes that the grain modulus is very much larger than the crack modulus. It can be fit to four of the five data sets. The fit to the data for a monomineralic epidote yields values for V0, Piand m of 8.02±0.02 km/s, 1.2±0.5 MPa, and 0.9845±0.0004, respectively, with a rms error of 6.28 m/s. The second equation, 1/V2 (P) = (1/Vc2 ‐ Vg2)/(1 + P/Pi)1 ‐ m + 1/Vg2 assigns a constant velocity to the grains and applies when the modulus of the cracks is of the order of the grain modulus at high pressures. This equation can be fit to three of the data sets; the fit to data for a diopside pyroxenite yields values of Vc, Vg, Pi, and m of 6.20±0.04 km/s, 8.28±0.02 km/s, 7±1 MPa, and 0.20±0.05, with a rms error of 17.9 m/s. For all seven fits to the laboratory data the rms errors range from 0.1 to 0.3% and are of the order of the limits of precision of the measurements. The “bed of nails” model explains the pressure dependence of P wave velocities in the samples remarkably well, as evidenced by the small rms errors. The variation with pressure of P wave velocities in these rocks clearly reflects the increasing stiffness of cracks. The fact that the first equation fits four of five data sets is one of several indications that cracks significantly affect the mechanical properties of the rocks even at 500 MPa. Finally, we note that different kinds of cracks have markedly different mechanical properties; the best fitting model parameters reflect the nature of the cracks which populate the samples.

The reciprocity relationship describing the relations among the fields resulting from the interchange of point sources and receivers may be extended to the seismic case. Seismic reciprocity can be described either in terms of the scalar product of the vectors representing the excitation of the source and the field at the receiver, or in terms of a Green’s tensor describing these two quantities. Theoretical reciprocity relations give no information concerning reciprocity in the cases for which the scalar product vanishes. A simple experiment in the vector case demonstrates that reciprocity is not obtained when the scalar product of the two vectors vanishes.