## 5 Results and Discussions
### Modelling Insights and Discussions
In Figure 7b, we can see that the Eshelby-Cheng theory predicted that SWS would be higher than the measured results. The shear moduli of S\({}_{1}\) (\(C_{44}\)) and S\({}_{2}\) (\(C_{66}\)) in the Eshelby-Cheng theory are as follows:  
\[C_{44}=\mu[1-\frac{\varphi_{f}(\mu-\mu^{\prime})}{\mu+2(\mu-\mu^{\prime})s_{44}}] \tag{26}\]  
\[C_{66}=\mu[1-\frac{\varphi_{f}(\mu-\mu^{\prime})}{\mu+2(\mu-\mu^{\prime})s_{66}}] \tag{27}\]  
As \(\mu^{\prime}=0\) in our study, SWS can be approximately expressed as follows:  
\[ \begin{array}{l}SWS=\frac{C_{66}-C_{44}}{2C_{43}}\approx\frac{4e}{3e^{2}R}\\ \varepsilon=\frac{C_{11}^{\text{ext}}-C_{33}^{\text{ext}}}{2C_{33}^{\text{ext} }}=\frac{1}{2}(\frac{C_{11}^{\text{dry}}+a_{1}^{2}M}{C_{33}^{\text{dry}}+a_{3 }^{2}M}-1)\\ \alpha_{3}\\ \gamma^{\text{sat}}\approx\frac{4}{3}e\\ SWS\end{array}  \tag{28}\]  
where \(e\) is the fracture density and \(\varphi_{f}=2\pi\alpha\)e. As the value of \(R\) in the four rocks was close (between 0.25 and 0.3), the SWS predictions were similar and not sensitive to changes in porosity. In the theoretical predictions of [1], with increasing porosity, P-wave anisotropy showed a decreasing trend, whereas SWS remained practically constant. This trend was similar to the measured results. However, the theoretical predictions were higher than the measured results. For water-saturated conditions, the theoretically predicted P-wave anisotropy of [1] was as follows:  
\[\varepsilon=\frac{C_{11}^{\text{sat}}-C_{33}^{\text{sat}}}{2C_{33}^{\text{ sat}}}=\frac{1}{2}(\frac{C_{11}^{dry}+a_{1}^{2}M}{C_{33}^{dry}+a_{3}^{2}M}-1) \tag{29}\]  
where direction three is the normal fracture direction. \(C_{11}^{\text{sat}}\) and \(C_{33}^{\text{sat}}\) are the P-wave moduli in the parallel and perpendicular directions, respectively, in water-saturated conditions. \(C_{11}^{dry}\) and \(C_{33}^{dry}\) are the P-wave moduli in the parallel and perpendicular directions, respectively, in dry conditions, and  
\[C_{33}^{dry}=\frac{3\mu(\lambda+\mu)}{3\mu(\lambda+\mu)+4(4\mu^{2}+4\lambda\mu )e}C_{11}^{dry} \tag{30}\]  
where \(\lambda\) and \(\mu\) are Lame parameters of the dry background matrix. Equation (29) can be rewritten as follows:  
\[\varepsilon=\frac{1}{2}(\frac{C_{11}^{dry}+a_{1}^{2}M}{3\mu(\lambda+\mu)+4(4 \mu^{2}+4\lambda\mu)e}C_{11}^{dry}+a_{3}^{2}M}-1) \tag{31}\]  
Firstly, with increasing porosity, the value of \(\frac{3\mu(\lambda+\mu)}{3\mu(\lambda+\mu)+4(4\mu^{2}+4\lambda\mu)e}\) increases from 0.62 to 0.75. According to Equation (6), when the porosity increases, scalar \(M\) decreases greatly. Since \(\alpha_{1}\) is larger than \(\alpha_{3}\) (Equation (5)), \(a_{1}^{2}M\) decreases more than \(a_{3}^{2}M\). As a result, with increasing porosity, these two factors cause the \(C_{11}^{\text{sat}}\) to decrease more than \(C_{33}^{\text{sat}}\), resulting in a decrease in P-wave anisotropy. The predicted P-wave anisotropy was higher than the measured results. Potential explanations for this phenomenon include: (i) Under an experimental frequency of 0.5 MHz, the fluid did not have enough time to flow between the pores and fractures, thus the system did not reach pressure equilibrium in the pore space. This equilibrium is assumed by the Gurevich theory. This can cause a \"stiffening\" effect on the fractures, reducing P-wave anisotropy and causing the measured results to be lower than the theoretical predictions. (ii) Interactions between the fractures were not included in the Gurevich theory, which has been shown to reduce the overall anisotropy in a fractured medium in theoretical studies [5; 6].