# Grouting Process Simulation Based on 3D Fracture Network Considering Fluid-Structure Interaction
## 2 Methodology
### Modeling of 3D Fracture Network
#### 2.1.2 Statistical Analysis of Fracture Geometric Characteristic Parameters  
In this study, the fracture was simulated by the Baecher disc model [30] which assumed every fracture as a thin disc (Figure 2). The fracture disc model can be defined by the following formula:  
\[C=c(O,\ V,\ R,\ A) \tag{1}\]  
the formula defines a fracture disc with center point \(O\), occurrence \(V\), radius \(R\) and aperture \(A\). Where \(O=(x_{0},\ y_{0},\ z_{0})\), \(V=(\alpha,\ \beta)\), \(\alpha\) and \(\beta\) are the dip direction and dip angle of the fracture disc respectively, and \(n\) is the normal vector of the fracture disc.  
The relationship between the various parameters can be expressed as follows:  
\[\left\{\begin{array}{l}a(x-x_{0})+b(y-y_{0})+c(z-z_{0})=0\\ (x-x_{0})^{2}+(y-y_{0})^{2}+(z-z_{0})^{2}\leq R^{2}\\ a=sin\beta\acute{s}ina,\ b=sin\beta cos\alpha,\ c=cos\beta\\ A=f(R)\end{array}\right. \tag{2}\]  
According to the exposed surface fracture catalog data and digital borehole data, the distributions of fracture geometric characteristic parameters can be determined.  
1. Fracture space location The Poisson process [31] is widely used to describe fracture location. The fractures are mutually independent and the uniform distribution function are adopted to obtain the coordinates \((x_{0},y_{0},\ z_{0})\) of the fracture center point.  
Figure 1: Modeling process of 3D fracture network.  
Figure 2: Baecher disc model.  
2. Fracture density The Mauldon method [32] is adopted to estimate the fracture volume density. The following equation is used to estimate the trace area density: \[\lambda_{a}=\frac{n_{1}{+}2n_{2}}{2 WH}\] (3) where \(\lambda_{a}\) is the trace area density, \(n_{1}\) is the number of traces which one end can be observed, \(n_{2}\) is the number of traces which both end can be observed, \(W\) is the width of rectangular window, \(H\) is the height of rectangular window. Then, Equation (4) is adopted to obtain the fracture volume density: \[E(\lambda_{v})=\frac{E(\lambda_{a})}{E(D)E|sin\
u|}\] (4) where \(\lambda_{v}\) is the fracture volume density, \(D\) is the fracture diameter, \(sin\
u\) is the sine value of the dip.
3. Fracture size To simulate the size of the fracture surface, statistical analysis of the fracture trace length is needed first. [PERSON] et al. [33] put forward the estimation formula of trace length: \[l= \frac{n_{1}{+}2n_{0}}{2N}\frac{\pi WH}{W+H}\] (5) where \(l\) is the fracture trace length in the window, \(n_{0}\) is the number of traces which neither end can be observed, \(n_{1}\) is the number of traces which one end can be observed, \(N\) is the total number of fracture traces in the window, \(W\) is the width of rectangular window, \(H\) is the height of rectangular window. When the disc model is used to simulate the fracture, the fracture size is expressed by its diameter. the fracture diameter distribution can be confirmed based on the distribution of trace length.
4. Fracture occurrence According to [PERSON] and [PERSON] [34], the fisher distribution can be used to fit fracture occurrence and obtained relatively better results.
5. Fracture aperture [PERSON] et al. [17; 18] conducted a lot of statistical studies and obtained the expression of the relation between fracture aperture and fracture trace length: \[A= \beta l^{n}\] (6) where \(A\) is the fracture aperture, \(l\) is the fracture trace length, \(\beta\) and \(n\) are constants related to the properties of fractured rock mass.  
In this study, the value of \(n=1\) is chosen to reflect the self-similarity and fractal of the fracture network [35; 36; 37], and then the relationship between fracture aperture and fracture trace length is linear; based on this relationship, the existing survey data are linearly fitted to get the value of \(\gamma\). Finally, the fracture aperture can be obtained by Equation (6) based on the data of trace length.