Modeling the Effect of Pressure on the Moduli Dispersion in Fluid‐Saturated Rocks

Laboratory experiments of ultrasonic velocities of fluid‐saturated rocks are often much higher than the predictions of the Gassmann theory. This difference is usually attributed to the velocity dispersion caused by fluid pressure relaxation between pores of different shapes and orientation. This paper proposes a simple model to characterize pressure and frequency effects on the elastic moduli of fluid‐saturated rocks in a broad frequency range. The proposed model incorporates micromechanics of wave‐induced fluid pressure relaxation at grain contacts (between crack‐like contacts and stiff pores) into pressure dependency of elastic moduli. Previously, the pressure dependency of the velocities or elastic moduli was ascribed to the progressive closure of cracks with the increasing effective pressure and expressed as an integral of crack compliance over the range of aspect ratios. For isolated cracks, this compliance is a function of crack geometry only. For cracks hydraulically connected to stiff pores, this crack compliance can be replaced by a frequency‐dependent solution of the micromechanical problem of fluid pressure relaxation between a single crack and surrounding pores. The resulting equation expresses the bulk and shear moduli of the fluid‐saturated rock as functions of both pressure and frequency. Furthermore, if pressure‐dependent moduli of both dry and fluid‐saturated moduli are known, the aspect ratio distribution can be obtained from the pressure dependency of the dry moduli, and then the saturated moduli can be computed with no adjustable parameters. The model predictions show reasonable agreement with laboratory data measured using ultrasonic and forced oscillation techniques.