## 5 Results and Discussions
### Modelling Insights and Discussions  
The inputs for equivalent medium theories are given in Tables 1 and 3. The Eshelby-Cheng theory (high frequency) predicted P-wave anisotropy to be close to the measured results [3,4]. Although the expressions of the Eshelby-Cheng theory do not contain background porosity, its predicted P-wave anisotropy showed a decreasing trend with increasing background porosity. The Eshelby-Cheng theory is expressed by Equations (1) and (2). Note that in Equation (1), \(\varphi_{f}\) refers to the fracture-occupied porosity, not the background porosity, and in all the fractured samples, the fracture porosities are practically identical. The predicted P-wave anisotropy fluctuations were mainly dominated by parameter \(A_{ij}\) in Equation (2), which relates the applied strain to the \"stress-free\" strain of the inclusion. \(A_{ij}\) is expressed as follows:  
\[A_{ij}=B_{ij}^{-1}L_{kj}^{\prime} \tag{12}\]  
where:  
\[B_{ij}=GS_{j}+2 HS_{ij}+\lambda_{b}+2\mu_{b}\delta_{ij} \tag{13}\]  
Figure 7: Fracture-induced P-wave anisotropy (**a**) and SWS (**b**) variation with porosity. \[L_{ij}=-G-2H\delta_{ij} \tag{14}\]  
\[H=\mu^{\prime}-\mu_{b} \tag{15}\]  
\[G=\lambda^{\prime}-\lambda_{b} \tag{16}\]  
\[S_{j}=\sum_{i=1}^{3}S_{ij} \tag{17}\]  
where \(\lambda_{b}\) and \(\mu_{b}\) are the same constants of the background medium, and \(\lambda^{\prime}\) and \(\mu^{\prime}\) are the lame constants of the inclusion material (water). \(S_{ij}\) is the Eshelby \(S\) tensor, which relates the strain field (due to an ellipsoidal inclusion) to the \"stress-free\" strain in the inclusion. For very thin fractures (with a small aspect ratio), \(S_{ij}\) can be simplified as follows:  
\[S_{11}=S_{22}=\pi\alpha(9-5R)/16 \tag{18}\]  
\[S_{33}=1-\pi\alpha R/2 \tag{19}\]  
\[S_{12}=S_{21}=-\pi\alpha(3-7R)/16 \tag{20}\]  
\[B_{ij}=GS_{j}+2 HS_{ij}+\lambda_{b}+2\mu_{b}\delta_{ij} \tag{13}\]0  
\[B_{ij}=GS_{j}+2 HS_{ij}+\lambda_{b}+2\mu_{b}\delta_{ij} \tag{13}\]1  
\[B_{ij}=GS_{j}+2 HS_{ij}+\lambda_{b}+2\mu_{b}\delta_{ij} \tag{13}\]2  
\[B_{ij}=GS_{j}+2 HS_{ij}+\lambda_{b}+2\mu_{b}\delta_{ij} \tag{13}\]3  
where \(\alpha\) is the fracture aspect ratio, and  
\[B_{ij}=GS_{j}+2 HS_{ij}+\lambda_{b}+2\mu_{b}\delta_{ij} \tag{13}\]4  
In the normal fracture direction (direction three), since \(\alpha\) is so small (\(\alpha=0.018\)), \(\mathrm{S}_{33}\approx 1\) and \(\mathrm{S}_{3}\approx 1\). Because the shear modulus of the inclusion material (water) \(\mu^{\prime}\) is zero, \(B_{33}\approx\lambda^{\prime}\) and \(L_{33}\approx\lambda_{b}+2\mu_{b}-\lambda^{\prime}\). With increasing porosity, \(L_{33}\) decreases, whereas \(B_{33}\) stays nearly constant. This leads to a decrease in \(A_{ij}\). As a result, the fracture-induced perturbation in the normal fracture direction decreases with increasing porosity. However, as the aspect ratios of the fractures were extremely small in this study, the fracture-induced perturbations were nearly negligible in the parallel direction of the fractured samples. As \begin{table}
\begin{tabular}{c c c c c c c c} \hline \multirow{2}{*}{**Sample Number**} & \multicolumn{3}{c}{**Background Matrix Modulus (GPa)**} & \multicolumn{3}{c}{**Grain Modulus (GPa)**} & \multirow{2}{*}{**Fluid Modulus (GPa)**} \\  & \(K_{dny}\) & \(\mu_{dny}\) & \(K_{sat}\) & \(\mu_{sat}\) & \(K_{s}\) & \(\mu_{s}\) & **(GPa)** \\ \hline
1-F & 7.05 & 7.41 & 20.36 & 10.45 & 30.6 & 24.5 & 2.15 \\ \hline
2-F & 5.52 & 5.82 & 15.66 & 7.45 & 30.6 & 24.5 & 2.15 \\ \hline
3-F & 4.69 & 4.14 & 13.33 & 5.19 & 30.6 & 24.5 & 2.15 \\ \hline
4-F & 3.12 & 3.11 & 10.83 & 3.96 & 30.6 & 24.5 & 2.15 \\ \hline \end{tabular}
\end{table} 
Table 3: Main inputs for equivalent medium theories. the fracture-induced perturbations decreased with increasing porosity in the perpendicular direction, the stiffness difference between the parallel and perpendicular directions also decreased. This led to the declining trend observed for the predicted P-wave anisotropy with increasing porosity.