## 5 Discussion  
We tested our experimental results against predictions of the Galvin model from an experimental point of view. Predicted P- and S-wave velocities in water-saturated conditions were well below the measured data. The Galvin model uses the Gassmann theory to calculate the fluid-saturated background matrix modulus. However, the Gassmann theory is valid only at sufficiently low frequencies because it assumes that the induced pore pressure is equilibrated throughout the pore space (i.e., there is sufficient time for the pore fluid to flow and eliminate wave-induced pore pressure gradients). The experimental frequency (0.5 MHz) was higher than the low-frequency range assumed by the Gassmann theory, so pore pressure was not equilibrated throughout the pore space. This  
Figure 6: Measured anisotropy and theoretical predictions of the Galvin model. (**a**) P-wave anisotropy; (**b**) S-wave anisotropy. acts to'stiffen' the rock and make P- and S-wave velocities higher than in the theoretical prediction. Similar observations can be found in ultrasonic experimental studies of tight stones, and [PERSON] (2018) indicated that the dispersion effect caused by local fluid flow between pores and microcracks makes measured velocity higher than in Gassmann-theory-based prediction [39]. Considering the differences between the measured data and theoretical predictions, several modifications were made to the Galvin model. (i) The [PERSON] (1995) squirt flow model was used to replace the [PERSON] model to calculate the fluid-saturated background matrix modulus [40]. The parameter Z in the [PERSON] model was found by matching the measured P-wave velocity and theoretical prediction. (ii) The elastic moduli of the fractured rocks in the low- and high-frequency limits were obtained by the Thomsen model instead of the linear-slip model or the [PERSON] (2003) model. Figures 7-9 show comparisons of the measured P-, SH-, and SV-wave velocities with the prediction of the Galvin model and the modified Galvin model. In water saturation, predicted P-, SH-, and SV-wave velocities in the modified model were significantly higher than those in the Galvin model, indicating that squirt flow can increase both bulk and shear moduli. Predicted P-wave velocities of the modified model were very close to measured data. In the modified model prediction, SH- and SV-wave velocities in water saturation were higher than in dry conditions; this is in agreement with measured data but contradicts the Galvin model. However, the modified model predicted that SH- and SV-wave velocities were still lower than the measured results in water-saturated conditions. For dry conditions, since the measured results of sample 1# were used as inputs in the Galvin model and modified Galvin model, both models yielded the same predictions of dry velocities of three modes of waves in 1#. In samples 2# and  
Figure 7: Measured velocity and theoretical predictions of the Galvin model (dashed curves) and the modified model (solid curves) for sample 1# (fracture density 0). (a) P-wave (P-wave velocity measurement error is smaller than the size of labels); (b) SH-wave; and (c) SV-wave. Figure 8: _Cont._  
Figure 8: Measured velocity and theoretical predictions of the Galvin model (dashed curves) and the modified model (solid curves) for sample 2# (fracture density 0.0312). (**a**) P-wave (P-wave velocity measurement error is smaller than the size of labels); (**b**)-SH-wave; and (**c**) SV-wave. Figure 9: _Cont._  
Figure 10 shows comparisons of the measured P-wave anisotropy with the prediction of the Galvin model and the modified Galvin model. In water-saturated conditions, the predicted P-wave anisotropy of the modified model was higher than that of the Galvin model and close to the measured results; Meanwhile, S-wave anisotropy prediction in the modified model was slightly lower than in the Galvin model. However, in dry conditions, the modified model yielded nearly the same P-wave anisotropy and S-wave anisotropy as the Galvin model.