# Research on Mechanical Properties and Failure Mode of Conglomerate Based on Discrete Element Method
## 4 Discussion  
A conglomerate reservoir shows a strong heterogeneity due to its complex sedimentary environment and tectonic setting, and the mechanical properties of the conglomerate reservoir can only be studied on a macroscopic scale by laboratory mechanical experiments. Different from the traditional indoor test, the numerical simulation based on the discrete element method can easily extract the detailed information of basic particle position, contact force and other parameters [20], and conduct an in-depth analysis. The interaction between two adjacent particles in a particle aggregate can be described by the normal contact force, the normal contact force perpendicular to the contact plane, and the tangential contact force parallel to the contact surface. [PERSON] [28] put forward a cross-scale correlation for analysis, pointing out that the mechanical parameters of particles determine the force chain and its stability of materials, while the complex dynamic response of the force chain network determines the macroscopic mechanical properties of materials. Therefore, it is of practical significance to study the fabric characteristics of particles to reveal their macroscopic mechanical characteristics based on the basic components of the discrete element. The geometric anisotropy of conglomerate discrete element particles is usually composed of its particle contact normal and branch vector. In this paper, the fabric tensor is used to quantify the contact force between conglomerate particles. \[\varnothing_{ij}=\frac{1}{N_{c}}\sum_{C=1}^{N_{c}}n_{i}^{c}n_{j}^{c} \tag{2}\]  
In Formula (2), \(\varnothing_{ij}\), \(N_{c}\), \(n_{i}^{c}\) are the normal fabric tensor, unit vector and total contact number, respectively. The second order anisotropic tensor of normal contact is:  
\[\alpha_{ij}^{c}=\frac{15}{2}\varnothing_{ij}^{{}^{\prime}} \tag{3}\]  
In Formula (3), \(\varnothing_{ij}^{{}^{\prime}}\) is the partial tensor of \(\varnothing_{ij}\). The mechanical anisotropy mainly depends on the contact force related to the direction of the contact plane. It can be divided into the anisotropy of the normal and tangential contact forces, as shown below:  
\[\varnothing_{ij}=\frac{1}{N_{c}}\sum_{C=1}^{N_{c}}n_{i}^{c}n_{j}^{c} \tag{4}\]  
\[X_{ij}^{n}=\frac{1}{N_{c}}\sum_{C=1}^{N_{c}}\frac{f^{n}n_{i}n_{j}}{1+\alpha_{ kl}^{c}n_{k}n_{l}} \tag{5}\]  
\[X_{ij}^{t}=\frac{1}{N_{c}}\sum_{C=1}^{N_{c}}\frac{f^{t}t_{i}n_{j}}{1+\alpha_{ kl}^{c}n_{k}n_{l}} \tag{6}\]\[\alpha_{kl}^{n}=\frac{15}{2}\frac{X_{ij}^{n}}{\overline{f}^{0}} \tag{7}\]  
\[\alpha_{kl}^{t}=\frac{15}{3}\frac{X_{ij}^{t}}{\overline{f}^{0}} \tag{8}\]  
In Formula (8), \(X_{ij}^{n}\) is the normal fabric tensor of particle contact force; \(X_{ij}^{t}\) is the tangential fabric tensor of particle contact force, \(f^{n}\), \(f^{t}\) are the values of normal and tangential contact force respectively; \(\overline{f}^{0}\) and \(X_{ij}^{n}\) are the average normal contact force; \(X_{ij}^{n}\) and \(X_{ij}^{t}\) are the partial tensors of \(X_{ij}^{n}\) and \(X_{ij}^{t}\). The anisotropy of particle contact force caused by different factors can be characterized by the above four factors. [PERSON] [29] expressed the degree of anisotropy by the invariant of the partial tensor, as shown in the formula:  
\[\alpha^{*}=sign(S_{r})\sqrt{\frac{3}{2}}\alpha_{ij}^{*}\alpha_{ij}^{*} \tag{9}\]  
In Formula (9), \(\alpha^{*}\) is the above four fabric tensors, and the double shrinkage normalized quantities of \(\alpha_{ij}^{*}\) and \(\alpha_{ij}^{\prime}\) are defined by \(S_{r}\):  
\[S_{r}=\frac{\alpha_{ij}^{*}\alpha_{ij}^{\prime}}{\sqrt{\alpha_{kl}^{*}\alpha_ {kl}^{*}\sqrt{\alpha_{mm}^{*}\alpha_{mm}^{*}}}} \tag{10}\]  
In Formula (10), \(sign(S_{r})\) represents the relative direction of the main direction of \(\alpha_{ij}^{*}\) to the stress tensor; A positive \(\alpha_{ij}^{*}\) indicates that the principal direction is close to the principal direction of the stress tensor, and a negative sign is the opposite.