# Computer 3D Simulation of Proppant Particles
## 3 Contact Model of Proppant Particles  
As shown in Figure 5, the three-dimensional coordinates of the underlying particles are established and discretized for three-dimensional processing. The spherical center coordinate of any proppant particle is recorded as (\(x_{i}\), \(y_{i}\), \(z_{i}\)). The maximum number of particles laid in various directions in three-dimensional space can be represented by \(n_{x}\), \(n_{y}\), \(n_{z}\), where  
\[ \begin{split} n_{x}&=\text{int}\Big{(}\frac{w_{f}}{ R}\Big{)}\\ n_{y}&=\text{int}\Big{(}\frac{H_{f}}{R}\Big{)}\\ n_{z}&=\text{int}\Big{(}\frac{L_{f}}{R}\Big{)} \end{split}  \tag{1}\]  
In the formula, \(n_{x}\) is the maximum number of proppant particles laid in the \(x\)-axis direction, which is also the width direction of the fracture, dimensionless. \(n_{y}\) is the maximum number of proppant particles laid in the \(y\)-axis direction, which is also the height direction of the fracture, dimensionless. \(n_{z}\) is the maximum number of proppant particles laid in the \(z\)-axis direction, dimensionless. int (\(x_{0}\)) represents the downward rounding of any real number \(x_{0}\). \(w_{f}\) is the fracture width, mm. \(H_{f}\) is the fracture height, mm. \(L_{f}\) is the fracture length, mm. \(R\) is the diameter of the proppant particle, mm. According to the arrangement rule of proppant particles, the three-dimensional simulation of proppant particles is processed according to random phenomena and deterministic phenomena. Figure 5: Arrangement of proppant particles in the bottom layer (\(k=1\)) of three-dimensional space. Random Phenomenon and Treatment of PropWhen (0, \(H_{f}\))-(\(w_{f}\), \(H_{f}\)) boundary contacts, the new spherical center coordinates (\(x_{ijk}\), \(y_{ijk}\), \(z_{ijk}\)) of the proppant particles are calculated by the following formula:  
\[\left\{ \begin{array}{l}x_{iny1}=\frac{R}{2}(\lambda_{m}+2(m-1)+1)\\ y_{iny1}=H_{f}-\frac{R}{2}\\ z_{iny1}=\frac{R}{2}\end{array} \right. \tag{7}\]  
(4) Last, a random phenomenon is caused by the contact of proppant particles with the (\(w_{f}\), 0)-(\(w_{f}\), \(H_{f}\)) boundary. Random phenomena generate random numbers \(\xi_{n}\) (\(n=1\), 2, \(\ldots\), \(n_{y}\)-1). The remaining space is allocated according to the weighted average method of random numbers; that is, the random weighted distribution coefficient is introduced as  
\[\lambda_{n}=\frac{\xi_{n-1}}{n_{y}-1}\left(\frac{H_{f}}{R}-n_{y}\right),\left( n=1,2,\cdots,n_{y}\right) \tag{8}\]  
When (\(w_{f}\), 0)-(\(w_{f}\), \(H_{f}\)) boundary contacts, the new spherical center coordinates (\(x_{i}\), \(y_{i}\), \(z_{i}\)) of the proppant particles are calculated by the following formula:  
\[\left\{ \begin{array}{l}x_{nx_{i}1}=w_{f}-\frac{R}{2}\\ y_{nx_{i}1}=\frac{R}{2}(\lambda_{n}+2(n-1)+1)\\ z_{nx_{i}1}=\frac{R}{2}\end{array} \right. \tag{9}\]  
The new spherical center coordinate set \(Q\) for the bottom layer (\(k=1\)) is  
\[Q= \begin{bmatrix}x_{1}&x_{2}&\cdots&x_{n}\\ y_{1}&y_{2}&\cdots&y_{n}\\ z_{1}&z_{2}&\cdots&z_{n}\end{bmatrix} . \tag{10}\]  
In Figure 5b, a temporary dynamic matrix T0 is set as the set of the innermost spherical centers of the bottom layer (\(k=1\)) in three-dimensional space. When the total number of proppant particles np is less than or equal to \(2n_{x}+2n_{y}-4\), the temporary dynamic matrix \(T_{0}\) equals \(Q\). When the total number of proppant particles \(n_{p}\) is greater than \(2n_{x}+2n_{y}-4\), the new sphere may be generated by two or more spheres together, and it is not possible to express the new sphere centers using a certain mathematical formula. Therefore, consider using arrays to express the set of new sphere centers. Take the shortest distance between the new sphere centers and the four boundaries (0, 0)-(0, \(w\)), (0, 0)-(0,\(H\)), (0, H)-(\(w_{f}\), \(H\)), and (\(w_{f}\), 0)-(\(w_{f}\), \(H_{f}\)) as the target, and recursively arrange the spheres one by one in a continuous manner towards the middle.