## 4 Results and Discussion
### Discussion  
For the 20/40 mesh LZ sand, the multi-particle size calculation model presented in this paper was compared with the single particle size model developed by [PERSON] et al. [30], as shown in Figures 10 and 11. Figure 11: Comparison of the fracture conductivity of LZ sand [30]. Figure 10: Comparison of the crushing rate of LZ sand [30]. Figure 9: Fracture conductivity of XY ceramic. The average error between the model values and the experimental values of the single particle-size LZ sand crushing rate was 20.16%. The average error between the model values and the experiment values of the multi-particle LZ sand crushing rate was 13.68%. The average error between the model values and the experimental values of the single particle-size LZ sand fracture conductivity was 21.97%. The average error between the model values and the experimental values of the multi-particle LZ sand fracture conductivity was 11.39%. The particle size distribution of the multi-particle LZ sand was more in-line with the actual particle size distribution of the LZ sand, and the error between the model values and the average experimental values was smaller. For the 20/40 mesh XY ceramic, the multi-particle-size calculation model presented in this paper was compared with the single particle-size model developed by [PERSON] et al. [30], as shown in Figures 12 and 13. The average error between the model values and the experiment values of the single particle-size XY ceramic crushing rate was 12.61%. The average error between the model values and the experimental values of the multi-particle XY ceramic crushing rate was 9.26%. The average error between the model values and the experimental values of the single particle-size XY ceramic fracture conductivity was 13.90%. The average error between the model values and the experimental values of the multi-particle XY ceramic fracture conductivity was 9.08%. The particle size distribution of the multi-particle XY ceramic was more in-line with the actual particle size distribution of the XY ceramic, and the error between the model values and the average experimental values was smaller. Figure 12: Comparison of the crushing rate of XY ceramic [30]. Figure 13: Comparison of the fracture conductivity of XY ceramic [30]. Each calculation of fracture conductivity requires several computer simulations and calculations, and it is not known how many times the computer simulation converges to a stable result, which consumes a lot of time and effort. The convergence of the number of computer simulations and calculations was investigated by studying the error between the cumulative average and the total average. In Figure 14, the relationship between the cumulative average and total average of the LZ sand fracture conductivity is presented. When the number of simulations was small, the error of the LZ sand calculation results was large. When the number of simulations reached 10 times, the error between the cumulative average and the total average was less than 3%. When the number of simulations reached 27 times, the error between the cumulative average and the total average was less than 1%. Furthermore, as the number of simulations increased, the error tended towards zero. Therefore, setting the number of times for the simulations and calculations for the LZ sand at 27 was able to meet the accuracy requirements. In Figure 15, the relationship between the cumulative average and total average of the XY ceramic fracture conductivity is presented. The more computer simulations conducted, the more accurate the calculated results. When the number of simulations reached eight times, the error between the cumulative average and the total average was less than 3%. When the number of simulations reached 25 times, the error between the cumulative average and the total average was less than 1%. Furthermore, as the number of simulations increased, the error tended towards zero. Therefore, setting the number of times for simulations and calculations for the LZ sand at 25 was able to meet the accuracy requirements.