# Anisotropic Effective Elastic Properties for Multi-Dimensional Fractured Models
## 4 Discussion
### Anisotropy Properties for Real Fractures  
Combined with Figures 7 and 8, we see that different fracture densities can induce elastic anisotropy, which can be quantitatively characterized using the equivalent stiffness tensors. We apply 2D anisotropic finite-element modeling to a real crack for anisotropy properties at this part. Figure 10a illustrates an outcrop from the Ordovician carbonate formation in the Tarim Basin, northwest China. Multistage tectonic movements result in a complicated fracture network in the formation [33]. From the Cambrian to the Middle Ordovician, strike-slip faults with numerous fractures were formed by the joint action of north-south and north-east stresses [34]. These secondary fractures are considered as both migration channels and fracture reservoir spaces for hydrocarbon. The outcrop is selected because its fracture characteristics are similar to those of subsurface strata over 7 km, and a complicated fracture network satisfies the simulation scale. Due to the complexity of natural fractures, modeling of irregular geometries and anisotropic features are challenging. The outcrop picture is simplified into a model consisting of background media and superimposed fractures with different shapes, sizes, and orientations to facilitate research on the anisotropy properties of fractures and their stress interaction, as illustrated in Figure 10b. We investigate the cracked model's density and dip-angle. For calculating fracture density, we use the method developed by reference [35; 36], which employs a circular scanning window to calculate the number of total fractures by counting the number of endpoints of complete visible fractures within the window. An equal-area rose diagram instead of the histogram of fracture length is used since it provides an indirect orientation distribution  
Figure 10: 2D real-fracture model and its geometric characteristics: (**a**) rock outcrop of Ordovician carbonates in Tarim Basin, northwest China; (**b**) the model can be composed of the background media and extracted fractures; (**c**) the model’s fracture density distributions where the red and yellow areas in the maps indicate more fractures than others; (**d**) the equal-area rose diagrams of fracture length, which provide dip distribution. (see reference [37] for more details). Figure 10c,d shows the geometric properties of the real model, which clearly exhibits a range of densities and orientations in different areas. We use a numerical approach combined with finite-element modeling to study the anisotropic properties of the real fracture model at different frequencies. The same physical parameters in Table 1 are used, and oscillatory relaxation tests at frequencies of 0.01, 1, 10, 100, and 1000 Hz are applied. The resultant real parts of the effective stiffness coefficients \(M_{11}\) are illustrated in Figure 11. As we see in Figure 11, the real parts of the stiffness matrix of \(M_{11}\) (red circle and red dotted line) show an increasing tendency with increasing frequency, which is not as regular as the simple models in reference [26]. The pictures indicated by the shallow green arrow represent the stress (\(\sigma_{xx}\)) distributions at different frequencies. Additionally, we observe that they change slowly at the beginning, then rapidly, and tend to be flat at high frequency, as the difference between protolith and faults affecting the stiffening of the fluid becomes smaller [38]. The stress (\(\sigma_{xx}\)) occurs an obvious difference between high frequency (100 and 1000 Hz) and low frequency, and the interaction becomes more intense as the frequency increases. The resultant anisotropies of effective elastic properties under the different frequencies can be used to characterize fracture interactions and dynamic stress disturbances. We can characterize the velocity anisotropy and seismic attenuation using these equivalent stiffness tensors. The existence of fractures surely affects the overall elasticity, as shown in Figures 4 and 6.