## 2 Simulation of Proppant Particle Arrangement and Micromechanical Analysis
### Mathematical Modeling of the Mechanical Relationships of Proppant  
The microscopic arrangement of two or more proppant particles with different particle sizes is extremely complex. When conducting mechanical analysis and modeling between proppant particles, it is imperative to consider and distinguish the stress conditions not  
Figure 3: Example diagram of 2D simulation of proppant particles ((**a**) before top layer optimization, (**b**) after top layer optimization). only between particles but also between particles and boundaries, subsequently modeling each condition separately to ensure accuracy. 1. Mathematical expression of proppant particle contact Any particle may or may not be in contact with other particles. For the convenience of expression, a contact coefficient \(\delta_{ix}\), \(\delta_{iy}\), \(\delta_{ij}\) is introduced to represent the state of contact. 2. When the proppant particles contact the boundaries along the x-axis (either the left or right side), the contact is noted. \[\delta_{ix}= \begin{cases}1,&x_{i}=R_{i}/2\text{or}x_{i}=w_{f}-R_{i}/2,\\ 0,&x_{i}\
eq R_{i}/2\text{and}x_{i}\
eq w_{f}-R_{i}/2.\end{cases} \] (10)
3. When the proppant particles contact the boundaries along the y-axis (either the upper or lower side), the contact is noted. \[\delta_{iy}= \begin{cases}1,&y_{i}=R_{i}/2\text{or}y_{i}=L_{f}-R_{i}/2,\\ 0,&y_{i}\
eq R_{i}/2\text{and}y_{i}\
eq L_{f}-R_{i}/2.\end{cases} \] (11)
4. When the proppant particles are not in contact on the boundaries along the x-axis and y-axis, the non-contact condition is noted. \[\delta_{ij}= \begin{cases}1,&\sqrt{\left(x_{i}-x_{j}\right)^{2}+\left(y_{i}-y_ {j}\right)^{2}}=R_{i}/2+R_{j}/2,\\ 0,&\sqrt{\left(x_{i}-x_{j}\right)^{2}+\left(y_{i}-y_{j}\right)^{2}}\
eq R_{i}/2 +R_{j}/2.\end{cases} \] (12) where \(i=1,2,3,\ldots,n_{p}\); \(j=1,2,3,\ldots,n_{p}\), and \(i\
eq j\). 5. Mathematical modeling of pressures on proppant particles The mechanical relationship between proppant particles and between particles and boundaries includes three aspects: 1. The pressure on the proppant particle The arrangement of proppants becomes significantly more complex due to their varying particle sizes, resulting in a differing number of contacts between each particle and others. Any proppant particle may or may not come into contact with other particles. Similarly, any proppant particle may or may not be in contact with the upper, lower, left, and right boundaries. Therefore, the mathematical model of mutual contact between proppant particle can be expressed as: \[\left\{ \begin{array}{l}\sum\limits_{j=1}^{n_{p}}\delta_{1j}\textbf{F}_{1}^{ j}+\delta_{1x=0}\textbf{F}_{1}^{x=0}+\delta_{1x=w_{f}}\textbf{F}_{1}^{x=w_{f}}+ \delta_{1y=0}\textbf{F}_{1}^{y=0}+\delta_{1y=L_{f}}\textbf{F}_{1}^{y=L_{f}}=0, \\ \sum\limits_{j=1}^{n_{p}}\delta_{2j}\textbf{F}_{2}^{j}+\delta_{2x=0}\textbf{F }_{2}^{x=0}+\delta_{2x=w_{f}}\textbf{F}_{2}^{x=w_{f}}+\delta_{2y=0}\textbf{F }_{2}^{y=0}+\delta_{2y=L_{f}}\textbf{F}_{2}^{y=L_{f}}=0,\\ \ldots\ldots\ldots\ldots\\ \sum\limits_{j=1}^{n_{p}}\delta_{ij}\textbf{F}_{i}^{j}+\delta_{ix=0}\textbf{F }_{i}^{x=0}+\delta_{ix=w_{f}}\textbf{F}_{i}^{x=w_{f}}+\delta_{iy=0}\textbf{F }_{i}^{y=0}+\delta_{iy=L_{f}}\textbf{F}_{i}^{y=L_{f}}=0,\\ \ldots\ldots\ldots\ldots\\ \sum\limits_{j=1}^{n_{p}}\delta_{(n_{p}-1)j}\textbf{F}_{n_{p}-1}^{j}+\delta_{ (n_{p}-1)x=0}\textbf{F}_{n_{p}-1}^{x=0}+\delta_{(n_{p}-1)x=w_{f}}\textbf{F}_{ n_{p}-1}^{x=w_{f}}+\delta_{(n_{p}-1)y=0}\textbf{F}_{n_{p}-1}^{y=0}+\delta_{(n_{p}-1)y=L _{f}}\textbf{F}_{n_{p}-1}^{y=L_{f}}=0,\\ \sum\limits_{j=1}^{n_{p}}\delta_{ii}\textbf{F}_{n_{p}}^{j}+\delta_{nx=0}\textbf {F}_{n_{p}}^{x=0}+\delta_{nx=w_{f}}\textbf{F}_{n_{p}}^{x=w_{f}}+\delta_{ny=0} \textbf{F}_{n_{p}}^{y=0}+\delta_{ny=L_{f}}\textbf{F}_{n_{p}}^{y=L_{f}}=0.\end{array}