## 2 Characterization of the Rough Discrete Network Model
### RDFN Model Based on Fourier Transform  
[PERSON] et al. [30] proposed RDFN models and modelling methods based on three different geometric types, including curves, triangular polylines, and fractal fracture network models. However, each RDFN model only enables the construction of a separate geometric representation function. To unify the model representation method, several geometric joint trace representation models based on the Fourier transform principle were proposed to represent different geometric joint models. The Fourier expansion of multi-stage trapezoidal wave is selected as Equation (2). Figure 3 shows the trapezoidal wave geometry corresponding to Equation (2)  
\[f(x)=\frac{4A_{\text{max}}}{\pi\omega d}\sum_{n=1}^{\infty}\frac{\sin((2n-1) \omega d)}{(2n-1)^{2}}\sin((2n-1)\omega x),n=1,2,3,\ldots \tag{2}\]  
where \(T\) is the waveform period length, \(A_{\text{max}}\) is the amplitude height, \(d\) is the rising width, which determines the width of the trapezoidal platform and the geometry of the Fourier curve, \(\omega\) determines the frequency of the curve, and \(n\) is the Fourier series of the curve.  
The joint trace that conforms with the actual shape can be obtained by changing the above parameters. Table 1 shows the Fourier parameters of different curves and Figure 3 displays the established curve conditions. The trapezoidal wave shape can be changed by modifying the trapezoidal wave d value. For example, the trapezoidal wave can evolve into a triangular wave when the d value is modified to T/4 and a square wave when the value of \(d\) is small. Figure 3 shows the trapezoidal waves under different parameters. The unified joint RDFN modelling of complex geometric shapes, such as straight, triangular, and rectangular steps can be obtained by modifying the Fourier series.  
Appropriate parameters are selected to simulate the actual joint model based on the above three sets of linear DFN joint information to establish three sets of Fourier joint models with different trace lengths and a certain density on the MATLAB platform. Thereafter, the RDFN joint models (as shown in Figure 4c) with crossing network characteristics are generated based on the above steps to generate line joint sets in the DFN model. The Fourier curve information is shown in Table 2. Following several on-site investigations, the structural plane was not found to be a simple form of the same type. There are many different forms of structural planes, such as straight lines, curves, or waving lines. Therefore, a numerical joint model containing linear, fractal, and Fourier types based on the MATLAB program was established to more accurately simulate real rock mass joints. The geometric forms of each type of joint can be modified according to the actual joint distribution of the site survey. For example, amplitude, frequency, trace length, and other parameters can be appropriately modified and adjusted according to the actual distribution and fluctuations. The mixed RDFN models are shown in Figure 4 and their related mixed joint information is presented in Table 2.  
Figure 3: Principle of the Fourier-based RDFN models.