Brown and Korringa’s expression for the saturated bulk modulus at high frequencies: Modification of Mavko and Jizba’s squirt flow model

Abstract

The squirt flow model developed by Mavko and Jizba has been widely applied to quantify elastic moduli/velocities of fluid-saturated rocks at ultrasonic frequencies and the related modulus/velocity dispersion between ultrasonic and seismic frequencies. In the model, the high-frequency saturated bulk modulus is obtained by taking the unrelaxed frame bulk modulus as the drained/dry one as the input to Biot’s or Gassmann’s formula. However, when using Gassmann’s formula, the “new” rock matrix contains rock mineral matrix and fluid-saturated soft pores, which is heterogeneous at the microscopic scale and thus breaks the fundamental assumption of microhomogeneity for Gassmann’s formula. Therefore, the high-frequency saturated bulk modulus computed by Gassmann’s formula is inaccurate, especially when the soft-pore fraction (SPF) (the ratio of soft porosity to total porosity) or crack density is large. To this end, we have derived the Brown and Korringa’s expression (Mavko-Jizba-Gurevich [MJG]-BK model) for the high-frequency saturated bulk modulus in Mavko and Jizba’s model based on Berryman and Milton’s generalized Gassmann’s equations for the composite porous media, which correctly characterizes the microheterogeneity of new rock matrix. The parameters in the MJG-BK model are totally determined by measured quantities in the laboratory. Numerical example indicates that the MJG model is consistent with the MJG-BK model at small SPFs or crack densities. When the SPF/crack density becomes large, the difference between them becomes large, the MJG model loses its accuracy, and the MJG-BK model is preferred. Furthermore, experiment data from the laboratory validate the effectiveness of the proposed MJG-BK model. In summary, the model develops Brown and Korringa’s expression for Mavko and Jizba’s squirt flow model and can be used to calculate the high-frequency saturated bulk modulus at different SPFs or crack densities more accurately.