Effect of cracks on the pressure dependence of P wave velocities in crystalline rocks

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Abstract
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.