Bulk modulus for fluid-saturated rocks at high frequency: modification of squirt flow model proposed by Mavko & Jizba

Abstract

SUMMARYThe squirt flow model, proposed by Mavko & Jizba, has been widely used in explaining the frequency-related modulus and velocity dispersion between ultrasonic and seismic measurements. In this model, the saturated bulk modulus at high frequency is obtained by taking the so-called unrelaxed frame bulk modulus into Biot's or Gassmann's formula. When using Gassmann's formula, the mineral bulk modulus is taken as matrix bulk modulus. However, the soft pores (cracks) in rocks have a weakening effect on the matrix bulk modulus. The saturated bulk modulus at high frequency calculated with mineral bulk modulus as matrix bulk modulus is higher than the real values. To overcome this shortcoming we propose a modified matrix bulk modulus based on the Betti–Rayleigh reciprocity theorem and non-interaction approximation. This modification takes the weakening effect of soft pores (cracks) into consideration and allows calculating the correct saturated bulk modulus at high frequency under different soft-pore fractions (the ratio of soft porosity to total porosity) or crack densities. We also propose an alternative expression of the modified matrix bulk modulus, which can be directly obtained from laboratory measurements. The numerical results show that the saturated bulk modulus at high frequency using the original matrix bulk modulus (i.e. mineral bulk modulus) is approximated to that using the modified one only for rocks containing a small amount of soft-pore fraction. However, as the soft-pore fraction becomes substantial, using the original bulk matrix modulus is not applicable, but the modified one is still applicable. Furthermore, the results of the modified squirt flow model show good consistency with published numerical and experimental data. The proposed modification extends the applicable range of soft-pore fraction (crack density) of the previous model, and has potential applications in media having a relatively substantial fraction of soft pores or almost only soft pores, such as granite, basalt and thermally cracked glasses.