# Anisotropic Effective Elastic Properties for Multi-Dimensional Fractured Models
## 2 Methodology
### Anisotropic Effective Elastic Properties for 3D Fractured Models  
#### 2.2.1 3D Oscillatory Relaxation Tests  
In this part, we apply [PERSON]'s 2D method to a 3D cracked medium [30]. The governing Equations (1)-(11) would remain the same. As illustrated in Figure 3, we need to conduct six simulation oscillatory relaxation tests for calculating the elements of an equivalent stiffness matrix for a 3D probed model according to the 3D stress-strain correlation. Set \(\Omega=(0,L_{x})\times(0,L_{y})\times(0,L_{z})\) as a domain that represents a 3D interest model. Figure 3a displays the first oscillatory relaxation test in which a uniform vertical displacement along the \(x\)-direction (blue arrow) is applied to the model's red side, with the other sides employing periodic boundary conditions. The opposing side (gray side) is confined, and no shear stress is applied. Thus, the boundary conditions of the \(x\)-, \(y\)-, and \(z\)- directions are imposed as follows:  
\[\left\{ \begin{array}{ll}\mathbf{U}\cdot\mathbf{v}=-\Delta U,&(x,y,z)\in A^{ \prime},\\ \mathbf{U}\cdot\mathbf{v}=0,&(x,y,z)\in A^{\prime},\\ (\sigma v)\cdot\mathbf{x}=0,&(x,y,z)\in A,\\ \mathbf{U}_{dst}=\mathbf{U}_{src},&(x,y,z)\in A^{\prime}\cup A^{k},\\ \mathbf{U}_{dst}=\mathbf{U}_{src},&(x,y,z)\in A^{\prime}\cup A^{b},\end{array} \right. \tag{19}\]  
where \(\sigma\) represents the total stress tensor; \(\mathbf{U}\) denotes the solid displacement; \(\Delta U\) denotes the displacement shift; \(\mathbf{U}_{src}\) denotes the source displacement field; \(\mathbf{U}_{dst}\) denotes the displacement field of the target boundary; \(\mathbf{A}\) is its side boundaries; \(\mathbf{A}^{\prime}\), \(\mathbf{A}^{k}\), \(\mathbf{A}^{l}\), \(\mathbf{A}^{r}\), and \(\mathbf{A}^{b}\) are the left, right, front, back, top, and bottom boundaries, respectively; \(\chi\) is a unit tangent; and \(\
u\) denotes the unit outer normal on \(\mathbf{A}\), so that \(\left[\
u,\chi\right]\) is an orthonormal system. Figure 3b illustrates the second test that is analogous to the first, but with a uniform horizontal displacement applied to the red side from the \(y\)-direction and the opposing side fixed and periodical boundary conditions applied to the other surfaces. So, we have  
\[ \begin{array}{ll}\mathbf{U}\cdot\chi=-\Delta U,&(x,y,z)\in\mathbf{A}^{f},\\ \mathbf{U}\cdot\chi=0,&(x,y,z)\in A^{k},\\ (\sigma v)\cdot\mathbf{v}=0,&(x,y,z)\in\mathbf{A},\\ \mathbf{U}_{dst}=\mathbf{U}_{src},&(x,y,z)\in\mathbf{A}^{l}\cup\mathbf{A}^{r},\\ \mathbf{U}_{dst}=\mathbf{U}_{src},&(x,y,z)\in\mathbf{A}^{t}\cup\mathbf{A}^{b},\end{array}  \tag{20}\]  
Figure 3: Schematic diagram of six oscillatory relaxation simulation tests to determine the equivalent stiffness matrix for a 3D model, including penny-shaped spheroid fractures. Giving every test, let a uniform solid displacement (blue arrow) be applied to the red side, the gray side be confined, and the periodical boundary conditions be imposed on the other sides. (**a**) the first oscillatory relaxation test; (**b**) the second oscillatory relaxation test; (**c**) the third oscillatory relaxation test; (**d**) the fourth oscillatory relaxation test; (**e**) the fifth oscillatory relaxation test; (**f**) the sixth oscillatory relaxation test. Figure 3c-f shows the third to sixth tests whose loaded direction of solid displacement and the relevant boundary conditions are given by  
\[ \begin{array}{ll}\mathbf{U}\cdot\mathbf{v}=-\Delta U,&(x,y,z)\in\mathbf{A}^{t},\\ \mathbf{U}\cdot\mathbf{v}=0,&(x,y,z)\in\mathbf{A}^{b},\\ (\sigma\
u)\cdot\chi=0,&(x,y,z)\in\mathbf{A}_{s},\\ \mathbf{U}_{dst}=\mathbf{U}_{src},&(x,y,z)\in\mathbf{A}^{f}\cup\mathbf{A}^{k},\\ \mathbf{U}_{dst}=\mathbf{U}_{src},&(x,y,z)\in\mathbf{A}^{r}\cup\mathbf{A}^{l},\end{array}  \tag{21}\]  
for the third test,  
\[