## 2 Simulation of Proppant Particle Arrangement and Micromechanical Analysis
### Physical Modeling and Computer Simulation
When the total number of proppant particles \(n_{p}\) is less than or equal to the maximum number of the first layer in the \(x\)-axis direction \(n_{x}\) (\(n_{p}\leq n_{x}\)), then the temporary dynamic matrix is \(T=Q\) for the circle centers of the highest layer of proppant particles. When the total number of proppant particles \(n_{p}\) is greater than the maximum number \(n_{x}\) of the first layer in the \(x\)-axis direction, arrange the particles one by one along the longitudinal direction (\(y\)-axis direction), ensuring that the distance between the central axis of the particle arrangement and the bottom boundary is minimized. Then, the coordinates \(Q\) of the circle center of all particles can be expressed as:  
\[Q= \begin{bmatrix}x_{1}&x_{2}&\cdots&x_{n_{p}}\\ y_{1}&y_{2}&\cdots&y_{n_{p}}\\ R_{1}&R_{2}&\cdots&R_{n_{p}}\\ \end{bmatrix} . \tag{5}\]  
where \(n_{p}\) is the total number of proppant particles, dimensionless and \(R_{np}\) is the diameter of the circle center, \(R_{np}\in[R_{\min},\,R_{\max}]\), mm. Assuming that the coordinates of the new circle center of the proppant particles are \((x^{\prime}_{i},y_{i}^{\prime})\), and that the particle size of the new circle is \(R_{i}\), we can classify the arrangement of this new center relative to other centers into three main situations: tangential arrangement with the left boundary, tangential arrangement with the right boundary, and middle arrangement. These three situations will be modeled and analyzed separately. 1. When the proppant particles are tangent to the left boundary, as shown in Figure 2b, the coordinates of the new circle center of the particles that may be laid are calculated by the following formula: \[\left\{ \begin{array}{l}x^{\prime}_{i}=R_{i}/2,\\ \sqrt{\left(x^{\prime}_{i}-x_{k}\right)^{2}+\left(y^{\prime}_{i}-y_{k}\right) \right)^{2}}=R_{i}/2+R_{k}/2.\end{array} \right.\] (6)
2. When the proppant particles are not tangential to both the left and right boundaries, they are considered to be in a middle position, as illustrated in Figure 2c. In such cases, where multiple particles occupy the middle position, we identify the particle with the lowest center among them. The coordinates of this lowest central particle can then be calculated using the following formula: \[\left\{ \begin{array}{l}\sqrt{\left(x^{\prime}_{i}-x_{k}\right)^{2}+\left(y^ {\prime}_{i}-y_{k}\right)^{2}}=R_{i}/2+R_{k}/2,\\ \sqrt{\left(x^{\prime}_{i}-x_{l}\right)^{2}+\left(y^{\prime}_{i}-y_{l}\right) ^{2}}=R_{i}/2+R_{l}/2.\end{array} \right.\] (7)
3. When the proppant particles are tangential to the right boundary, as shown in Figure 2d, the coordinates of the new circle center of the particles that may be laid are calculated by the following formula: \[\left\{ \begin{array}{l}x^{\prime}_{i}=w_{f}-R_{i}/2,\\ \sqrt{\left(x^{\prime}_{i}-x_{k}\right)^{2}+\left(y^{\prime}_{i}-y_{k}\right) ^{2}}=R_{i}/2+R_{k}/2.\end{array} \right.\] (8)  
In summary, according to the three cases of micro-arrangements of proppant particles, the objective function is to find the lowest position of the new circle center:  
\[y_{i}=\min\{y^{\prime}_{i}\}. \tag{9}\]  
where \((x^{\prime}_{i},y^{\prime}_{i})\) are the new circle center coordinates of the proppant particle; \((x_{k},\,y_{k})\), \((x_{l},\,y_{l})\) are the existing circle centers in the set of Q; \(R_{i}\) is the new circle diameter, \(R_{i}\in[R_{\min},\,R_{\max}]\), mm; and \(R_{k}\) and \(R_{l}\) are the existing circles diameters in the set of Q, mm. The simulation process for proppant particles is comprised of 7 distinct steps:  
* Given the maximum boundary values along the \(x\)-axis and \(y\)-axis, denoted as \(w_{f}\) and \(L_{f}\), respectively, generate random numbers within a specified radius range, and subsequently create circles randomly based on a distribution range of particle sizes corresponding to the proppant material. * Generate the first circle center (\(x_{1}\), \(y_{1}\)) as the initial circle center (\(R_{1}\), \(R_{1}\)) in the first row (tangent to the \(x\)-axis (0, 0)-(\(w_{f}\), 0)).