## 4 Result of Numerical Modeling  
### Determination of the Hydraulic Fracture  
Assume that the direction of the maximum principal stress is horizontal. Predetermine that the new fracture in DFN has connected its downstream vertex with its upstream vertex. For example, vector \((u,v)\) represents the start vertex (indicated by \(u(u_{x},u_{y})\)) and the final point (indicated by \(v(v_{x},v_{y})\)) of a branch. If \(u_{x}\) is greater than \(v_{x}\), vertex \(u\) is marked as an upstream vertex. In addition, vertex \(v\) is made as a downstream vertex. On the contrary, vertex \(v\) is marked as upstream vertex. All the vertices in Figure 4 and intersections in Figure 5 are the elements of set \(V\).  
Select \(N\) vertices from the set \(V\) as source points. If the abscissa of these \(N\) vertices is smaller than other vertices', these \(N\) vertices are denoted as initial vertices. In addition, found \(N\) vertices are also regarded as terminal points, if the abscissa of these \(N\) vertices are larger than other vertices'. Assume \(N=12\). The initial vertices and terminal points can be obtained in this DFN by comparing their abscissa values, which are shown in Figure 7. The vertices marked by red '*' are the start vertices (indicated by \(v_{1},v_{2},...,v_{12}\)), and the terminal points are marked by a blue '*' (indicated by \(u_{1},u_{2},...,u_{12}\)).  
Basing on Definition 3 of Section 2, the DFN is a collection of three sets, indicated by \(G(V,E,W)\). All endpoints of the fracture in Figure 3 and intersection points in Figure 4 are the element of set \(V=v_{1},v_{2},..,v_{n}\), where \(v_{j+12}=u_{j}(j=1,2,...,12)\), and \(V_{1}=v_{2}5,v_{2}6,..,vn\) describe the intersection points. The element of set \(E\) is composed of the fracture in DFN. Set \(W\) is computed by Equations (6) and (7) in Section 3.3. Take in situ stress \(S[\sigma_{xx},\sigma_{xy},\sigma_{yy}]=[6.0\times 10^{7}Pa,1.0\times 10^{7}Pa,5.0 \times 10^{7}Pa]\) in Equation (2), [PERSON]'s ratio \(\
u=0.35\) in Equations (5) and (6), \(\gamma=1.0\times 10^{6}J/m\) in Equation (7). Based on Definition 2.2-2.3 in Section 2, \(p_{ij}=(v_{i},u_{j})\) presents the weight from \(v_{i}\) to \(u_{j}\) (\(i,j=1,2,...,12\)), and \(\delta(u,v)\) represents the shortest path with the minimum sum of the fracture weights. Using [PERSON]'s algorithm with energy, the hydraulic fracture of DFN can be obtained. The main steps of algorithm are as follows:  
Step 1. Set \(x=1\).  
Step 2. For the initial vertex \(v_{x}\), initialize \(S=\{v_{x}\}\), \(T=V\setminus S,dist[v_{i}]\) and \(ver[x]\):  
\[S[v_{i}]=\begin{cases}0,v_{i}\
otin S\\ 1,v_{i}\in S\end{cases},\]  
\[dist[v_{i}]=\begin{cases}U_{M},i\
eq x\&(v_{0},v_{i})\in E\&naturrefracture\\ U_{M}+U_{S},i\
eq x\&(v_{0},v_{i})\in E\&newfracture\\ \infty,(v,v_{i})\
otin E\\ 0,i=x\&(v_{x},v_{i})\in E\end{cases}\quad i=1,2,...,n,\]  
and  
\[ver[x]=0,x=1,2,...,12.\]  
Step 3. Select the vertex \(v_{i}\) in \(T\), such that \(dist[v_{i}]=\min\limits_{v_{j}\in T}dist[v_{j}]\). If \(dist[v_{i}]=\infty\), stop, else go to Step 4.  
Step 4. Set \(S=S\cup v_{i}\), \(T=T\setminus v_{i}\), and \(S[v_{i}]=1\). If \(T=\varnothing\), stop, else go to Step 5.  
Step 5. Update \(dist[v_{j}]=\min\{\omega_{0j},dist[v_{i}]+\omega_{ij}\}\). Go to Step 3.  
Step 6. Find \(p_{xj}\), and let \(P_{x}=\min\limits_{j=1,2,...,12}\{p_{xj}\}\).  
Step 7. Set \(ver[x]=1\), \(x=x+1\) and go to Step 2 until \(x=12\).  
Figure 7: The start vertex and terminal vertex of discrete fracture network.  
Step 8. \(\delta(u,v)=\min_{j=1,2,\ldots,12}\{P_{j}\}\).  
In addition, the algorithm flow chart is shown in Figure 8. From the above steps, it can be shown that the potential distances are now recorded as updated edge weights so that no additional memory is required. Through combining the vector method and geological parameter with [PERSON]'s algorithm, the hydraulic fracture of DFN is obtained, which is marked by a red color in Figure 9.