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<jats:p> 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. </jats:p>

<jats:p> Coal is a complex viscoelastic porous medium with fractal characteristics at different scales. To model the macroscale structure of coal, a fractal viscoelastic model is established, and the P-wave velocity dispersion and attenuation characteristics are discussed based on the complex modulus derived from this model. The numerical simulation results indicate that the fractional order [Formula: see text] and relaxation time [Formula: see text] greatly affect the P-wave velocity dispersion and attenuation. The fractal viscoelastic model indicates a full-band velocity dispersion between 1 Hz and 10<jats:sup>4</jats:sup> Hz. Meanwhile, the P-wave velocity has a weaker dispersion with the fractal viscoelastic model than with the Kelvin-Voigt model and Zener model between 1 Hz and 10<jats:sup>4</jats:sup> Hz for the same relaxation time and elastic modulus, but the velocity at 1 Hz based on the fractal viscoelastic model is higher with the Kelvin-Voigt model and Zener model. Simultaneously, the velocities of five coal samples are tested, and the attenuation factor is calculated using a low-frequency system. The experimental results indicate a strong dispersion in coal in the range of 10–250 Hz. The classic Kelvin-Voigt model and Zener model cannot describe the dispersion characteristics of coal, but the fractal viscoelastic model can describe them well by using the appropriate fractional order and relaxation time. </jats:p>

<jats:p> Shear wave velocity (Vs) is a vital parameter for various petrophysical, geophysical, and geomechanical applications in subsurface characterization. However, obtaining shear sonic log is often challenging since it often costs extra budget and time to acquire. Conventional methods for predicting Vs often rely on empirical relationships and rock physics models. However, these models often fall short in accuracy due to their inability to account for the complex nonlinear factors affecting the relationship between Vs and other parameters. We propose a physics-guided machine learning approach to predict shear sonic log using the various physical parameters (e.g. natural gamma ray, P-wave velocity, density, resistivity) that can be routinely obtained from standard logging suites. Three types of rock physical constraints including the mudrock line, empirical P- and S- wave velocity relationship and multi-parameter regression from the logging data, are combined with three physical guidance strategies including constructing physics-guided pseudo labels, physics-guided loss function and transfer learning, to blind test four wells based on one training well in a clastic reservoir. Compared to pure supervised ML, all the model that incorporates physical constraints significantly improves prediction accuracy and generalization performance, demonstrating the importance of incorporating first-order physical laws into data-driven network training. The multi-parameter regression relationship combined with the strategy of constructing physics-guided pseudo labels gives the best prediction performance, with the average root mean square error (RMSE) of the blind test dropping by 47%. </jats:p>