Frequency Dependence of Hertz Exponent of Velocity-pressure Power Law in Petroelastic Models and Practical Consequences

A good knowledge of the simultaneous effect of fluid substitution and pressure variations on seismic properties is a key point for the seismic monitoring of reservoir exploitation. The fluid substitution is well described by Biot-Gassmann's poroelastic equations. The simplest model describing the pressure dependence of the seismic velocities is Hertz power law, characterized by a power exponent commonly called Hertz exponent. This study is focussed on the pressure dependence of the velocities and emphasizes the frequency dependence of Hertz exponent, which is not commonly appreciated. A simplified theory is proposed and experimentally checked. Furthermore, pressure dependence of the velocities is stronger at low frequencies (typically in the seismic frequency band) than at high frequencies (typically in the ultrasonic frequency band or in the sonic log frequency band). As a consequence, common procedures using ultrasonic laboratory measurements or sonic log data to calibrate the petroelastic models for the pressure dependence of the seismic velocities in rocks, neglecting in passing the pressure dependence of Hertz exponent, tend to underestimate the effect of pressure on the seismic velocities.