# Coupled Numerical Method for Modeling Propped Fracture Behavior
## 3 Results  
As introduced above, DEM simulation was used to generate fractures filled by the particles. The first step is generating the particles in the model space randomly. Although a regular layout is easy to create, this particle generation procedure can distort the behavior of the particle assembly [40]. Due to this reason, YADE software applies a random particle generation technique. At the beginning of the simulation, proppant particles were randomly generated in a closed box placed above the fracture. The box was closed until the summarized kinetic energy of elements decreased almost to zero (the system reached a quasi-static state). Then the bottom of the box was opened, and the particles could fell into the fracture under gravity. This random particle placement procedure allows the representation of proppant transport and placement by the fracturing fluid, neglecting particle settlement by gravity. The filled fracture can be seen in Figure 8.  
The resulting geometry was pushed by an artificial assembly from the top of the fracture with 500 psi to enable proppant compaction and slippage, which occurs during actual fracturing and is considered in API standard proppant conductivity measurements [48], and then converted into  
Figure 7: The flow chart of the method. The first step in the FEA analysis was to specify the material properties. As described in Section 2.1.2, the proppant Young's modulus and Poisson ratio were 41,306 MPa and 0.22. The Young's modulus of the formation was systematically varied in a range of 1000 MPa to 30,000 MPa. The Poisson ratio was 0.22 in each case. The second step was to create a numerical mesh; then, the parts were meshed using second-order tetrahedron elements. Finally, mesh refinements were applied to the stress concentration areas, such as the particles and surfaces of the fracture. Mesh of the assembly can be seen in Figure 9, while Table 2 summarizes the nodes and elements in each case. One constraint was applied during the simulations to model fracture behavior: fixed support was used at the bottom of the formation (Figure 10b). In contrast, negative pressure was applied on the inside surfaces of the fracture to demonstrate the closure pressure acting as the initiator of deformation and embedment (Figure 10a). \begin{table}
\begin{tabular}{c c c} \hline \hline
**Proppant Diameter [mm]** & **Nodes** & **Elements** \\ \hline \(1\) & 379,043 & 187,117 \\ 1.37 & 195,523 & 100,071 \\ 1.74 & 173,006 & 93,518 \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Number of the nodes and elements. Figure 8: Fracture filled by proppant. Figure 9: Numerical mesh of the formation (**a**) and proppant particles (**b**). Contact behavior was applied and modeled on connecting surfaces. There were only bonded contacts in the model, entailing no sliding or separation between faces or edges allowed. Automatic contact detection was utilized in the FEA software. The default contact formulation was adjusted Multi-Point Constraint (MPC) to bonded contacts. In Figure 11 undeformed and deformed assembly can be seen colored by the extent of total deformation. The deformed body is represented at two times magnification to get the results perceived. Comparing and analyzing the assemblies, one can obtain that the contacts worked well according to the rock and proppants' visible displacement and deformation. After FEM simulated deformation, static structural deformed geometry was exported into ANSYS CFX environment. To reduce the number of nodes and the computational time to achieve reasonable computational efficiency, the fracture was halved along a symmetry plane of the deformed assembly, assuming similar behavior for the rest of the fracture. Figure 12. shows the analyzed cross-section of the formation and proppants. Figure 11. The undeformed and deformed assembly (2\(\times\) deformation). Figure 12. The analyzed cross-section. Figure 10. Surface pressure inside the fracture (**a**) and fixed support at the bottom of the formation (**b**).