# Computer 3D Simulation of Proppant Particles
## 3 Contact Model of Proppant Particles
### Determination Phenomenon and Treatment of Proppant Particle Arrangement  
The phenomenon of determining the three-dimensional simulation of proppant particles needs to be described and expressed through three parts: bottom layer, even layer in the middle, and odd layer in the middle. (1) Proppant particles at the bottom layer (\(k=1\)):  
a. A new sphere is generated between two adjacent spheres in matrix \(T_{0}\), as shown in subgraph (c) in Figure 5.  
b. A new sphere is generated between two non-adjacent spheres in matrix \(T_{0}\), as shown in subgraph (d) in Figure 5.  
c. When the distance between the centers of any two spheres in matrix \(T_{0}\) is greater than \(2R\), no new spheres are generated between the two spheres. A new sphere center is generated based on two or more existing sphere centers. Therefore, the new sphere center can be represented as  
\[\left\{ \begin{array}{l}\sqrt{\left(x_{ij1}-x_{(i1)(j1)1}\right)^{2}+\left(y_ {ij1}-y_{(i1)(j1)1}\right)^{2}+\left(z_{ij1}-z_{(i1)(j1)1}\right)^{2}}=R\\ \sqrt{\left(x_{ij1}-x_{(i2)(j2)1}\right)^{2}+\left(y_{ij1}-y_{(i2)(j2)1}\right) ^{2}+\left(z_{ij1}-z_{(i2)(j2)1}\right)^{2}}=R\\ z_{ij1}=z_{(i1)(j1)1}=z_{(i2)(j2)1}=\frac{R}{2}\end{array} \right., \tag{13}\]  
\[\left\{ \begin{array}{l}\sqrt{\left(x_{ij1}-x_{(i1)(1)1}\right)^{2}+\left(y_ {ij1}-y_{(i1)(j1)1}\right)^{2}+\left(z_{ij1}-z_{(i1)(j1)1}\right)^{2}}=R\\ \sqrt{\left(x_{ij1}-x_{(i2)(j2)1}\right)^{2}+\left(y_{ij1}-y_{(i2)(j2)1} \right)^{2}+\left(z_{ij1}-z_{(i2)(j2)1}\right)^{2}}=R\\ \sqrt{\left(x_{ij1}-x_{(i3)(j3)1}\right)^{2}+\left(y_{ij1}-y_{(i3)(j3)1} \right)^{2}+\left(z_{ij1}-z_{(i3)(j3)1}\right)^{2}}=R\\ z_{ij1}=z_{(i1)(j1)1}=z_{(i2)(j2)1}=z_{(i3)(j1)1}=\frac{R}{2}\end{array} \right. \tag{14}\]  
where (\(x_{ijk}\), \(y_{ijk}\), \(z_{ijk}\)) is the coordinate of the new spherical center of the proppant particle. (\(x_{(i1)(j1)1}\), \(y_{(i1)(j1)1}\), \(z_{(i1)(j1)1}\)), (\(x_{(i2)(j2)1}\), \(y_{(i2)(j2)1}\), \(z_{(i2)(j2)1}\)), and (\(x_{(i3)(j3)1}\), \(y_{(i3)(j31)}\), \(z_{(i3)(j31)}\)) all represent the existing sphere center of the first layer. According to the above three conditions, search for the location of the new spherical center with the shortest distance from the four boundaries of (0, 0)-(0, \(w_{j}\)), (0, 0)-(0, \(H_{f}\)), (0, \(H_{f}\))-(\(w_{f}\), \(H_{f}\)), and (\(w_{f}\), 0)-(\(w_{f}\), \(H_{f}\)) through the traversal search method  
\[\min\left(\min_{j}x_{j},\min_{j}y_{j},\min_{j}(w_{f}-x_{j}),\min_{j}(H_{f}-y_{j })\right) \tag{15}\]  
(2) Proppant particles in the intermediate layer (\(1<k<n\)):  
a. There are two situations when the proppant particles are tangent to the boundary (0, 0)-(\(w_{f}\), 0) and not tangent to both the boundary (0, 0)-(0, \(H_{f}\)) and the boundary (\(w_{f}\), 0)-(\(w_{f}\), \(H_{f}\)). A proppant particle is located on top of one sphere, or on top of two spheres. The new spherical center coordinates of the proppant particles that may be placed are calculated by the following formula:  
\[\left\{ \begin{array}{l}\sqrt{\left(x_{ijk}-x_{(i1)(j1)(z1)}\right)^{2}+\left(y_{ ijk}-y_{(i1)(j1)(z1)}\right)^{2}+\left(z_{ijk}-z_{(i1)(j1)(z1)}\right)^{2}}=R\\ y_{ijk}=\frac{R}{2},\,z_{ijk}=z_{(i1)(j1)(z1)}+R\end{array} \right. \tag{16}\]  
\[\left\{ \begin{array}{l}\sqrt{\left(x_{ijk}-x_{(i1)(j1)(z1)}\right)^{2}+\left( y_{ijk}-y_{(i1)(j1)(z1)}\right)^{2}+\left(z_{ijk}-z_{(i1)(j1)(z1)}\right)^{2}}=R\\ \sqrt{\left(x_{ijk}-x_{(i2)(j2)(z2)}\right)^{2}+\left(y_{ijk}-y_{(i2)(j2)(z2)} \right)^{2}+\left(z_{ijk}-z_{(i2)(j2)(z2)}\right)^{2}}=R\\ y_{ijk}=\frac{R}{2}\end{array} \right. \tag{17}\]  
b. When the proppant particles are tangent to the boundary (0, 0)-(\(w_{f}\), 0) and tangent to the boundary (0, 0)-(\(0_{f}\), \(H_{f}\)), the new spherical center coordinates of the proppant particles that may be placed are  
\[\left\{ \begin{array}{l}\sqrt{\left(x_{ijk}-x_{(i1)(j1)(z1)}\right)^{2}+\left( y_{ijk}-y_{(i1)(j1)(z1)}\right)^{2}+\left(z_{ijk}-z_{(i1)(j1)(z1)}\right)^{2}}=R \\ x_{ijk}=y_{ijk}=\frac{R}{2}\end{array} \right.