## 2 Establishment of a Nonlinear Discrete Creep Model
### Model Parameter Identification and Verification  
Before proceeding with model parameter identification, it is necessary to conduct uniaxial creep tests on the sampled return airway from the 11,000 working faces in the 11 mining areas of Quandian Coal Mine. Due to the low strength, softness, and fragility of the sampled coal seam, efforts were made to improve the testing process. Following the guidelines provided by the ISRM standard [45], a limited number of standard cylindrical specimens with dimensions of \(\varphi\)50 mm \(\times\) 100 mm were manufactured and tested in the laboratory. The average uniaxial compressive strength of the three specimens was measured at 3.44 MPa, while the average yield strength was 2.92 MPa, approximately 85% of the ultimate failure strength of the specimens. The CVISC creep model requires the determination of four parameters: \(E_{m}\), \(\eta_{m}\), \(E_{k}\), and \(\eta_{k}\). To establish these parameters, the uniaxial creep experimental program from a previous study [46] was consulted. Uniaxial creep tests were performed on three specimens, designated as #1, #2, and #3, using the graded loading method. Stresses were incrementally applied at a rate of 5 N/s during the test, and each stress level was sustained for a duration of 12 h. This testing protocol was replicated for all three specimens. The ratio of each stress level to the peak strength was calculated under the ultimate destructive strength of the specimen, resulting in stress levels of 10%, 20%, 40%, 60%, and 80%. These five stress levels were established for analysis. In cases where the loading load failed to induce accelerated creep even after more than 12 h at the last stress level, the next level of loading was increased by 5% of the peak strength difference. This process was repeated until accelerated-state creep was observed in the specimen. Based on the uniaxial strength test results of the soft coal specimens, the initial deformation parameters and strength parameters of the blocks were determined. By comparing the initial values with the simulation results, the short-term deformation and mechanical parameters of the soft coal specimens were obtained through parameter adjustment, as presented in Table 1.  
Figure 4: Nonlinear discrete creep models: (**a**) Coulomb slip unbonded contact model; (**b**) Numerical specimen model. The nonlinear least squares method is used to identify the parameters of the loading data for each creep stage between the accelerated creep stages. The parameters of the block variables at all levels are obtained, and the joint model is modified based on the short-term parameters. Taking the example of the sample with a yield stress of 2.92 MPa entering the accelerated creep, the parameters of the modified joint model show minimal changes, and the initial values are still used. The calibration of the block parameters in the accelerated creep stage after correction is presented in Table 2.  
The uniaxial creep test of soft coal was numerically simulated using the established particle discrete creep model as the calculation criterion. The actual experimental results were compared with the numerical simulation results, and a comparison diagram is presented in Figure 5 to verify the validity of the model. Illustrated in the figure are strain-time plots for soft coal specimens subjected to varying axial pressures in laboratory tests. The points of corresponding colors on the graph depict strain-time plots generated by the established creep model at different axial pressures. Despite some divergence between the numerical simulation results and the test results during the accelerated-state creep stage, the overall error falls within acceptable limits. Notably, during the decay-state creep stage and steady-state creep stage, the two trends exhibit close alignment, providing affirmation of the model's rationality. Additionally, both the test and simulation curves exhibit the characteristic three-stage pattern of decay-state creep, steady-state creep, and accelerated-state creep.