# Anisotropic Effective Elastic Properties for Multi-Dimensional Fractured Models
## 1 Introduction  
Cracks are the key factor in determining the deformability of rocks, properties of fluid movement, and tectonic history. We need to enhance ways for detecting and characterizing fractures. Due to the fact that natural fractures in geological formations frequently have a complicated shape and distribution, their interactions cause the complex anisotropic features of effective elastic properties. Thus, knowing the natural fractures' elastic behavior is essential for interpreting and predicting seismic responses in terms of rock properties and enables us to bridge the gaps in the spatial and temporal scale [1; 2]. The overall effective elastic coefficient of a fractured rock is frequently composed of the stiffness tensor of the background host rock and fractures [3], which attract considerable attention because of their contribution to the elastic property. The theoretical medium models (TMM) have been widely applied to calculate the effective elastic properties of fractured rocks analytically [4; 5; 6; 7; 8]. However, due to the dilute assumption of fracture densities, these models ignore fracture interactions. The second-order [PERSON]'s model takes into account the pairwise interaction among cracks [9]. [PERSON] [10] proposed an equivalent-media theorythat was sensitive to fracture length and studied the frequency-dependent anisotropy of fractured porous rocks using the squirt flow mechanism. [PERSON] et al. [11] united linear-slip theory and anisotropic Gassmann equations to analyze equivalent stiffness coefficients. The self-consistent approximation (SCA) and differential effective medium (DEM) were applied to control the concentration of fractures to investigate the consequences of fracture interactions [12; 13; 14]. [PERSON] et al. [15] applied the T-matrix method to characterize the pores-fracture interaction. Meanwhile, the idealization and oversimplification of real rock severely limit the utility of theoretical or semi-analytical models. In an attempt to circumvent certain assumptions inherent in effective medium theories [16; 17; 18; 19; 20; 21; 22; 23], numerical simulations have been extensively applied for studying the elastic properties of complex rocks. Numerical methods can be used to characterize real fractures with different shapes, sizes, partial interconnections, and intersections, as well as provide independent validation of theoretical methods and laboratory measurements as an efficient means of determining their general applicability. [PERSON] et al. [24] used numerical experiments to analyze the correlation between material structure and seismic attenuation anisotropy in porous media. [PERSON] et al. [25; 26] employed 2D numerical oscillatory relaxation experiments to study the frequency-dependent effective stiffness matrix using the least-squares optimal approach. The method is used to characterize the frequency- and incidence-angle- dependent attenuation and velocity dispersion in cracked porous media for the P wave [27; 28; 29] and P-SV wave [30]. [PERSON] et al. [31] expanded [PERSON]'s relaxation tests to a three-dimensional (3D) case for computing the equivalent stiffness matrix. We apply [PERSON]'s numerical method to explore the elastic and seismic anisotropy characteristics of real fractures in this paper. Particular attention should be paid to the anisotropy variation of fractures generated by different frequencies and crack densities. The following is the outline. In the first part of the technique, we demonstrate the novel form of [PERSON]'s 2D algorithm using three oscillatory relaxation tests with different fracture densities. Additionally, six oscillatory relaxation tests are used to spread [PERSON]'s algorithm to three-dimensional fractured media. Following that, the effective stiffness of the 2D numerical method is compared with that of the classical Hudson's model and the Eshelby-Cheng model with different fracture densities. Then, 3D fractured media are compared with TMM in elastic responses to different fracture densities.