## 4 Numerical Simulation Methods  
### Development of Coal Pillar Softening Model  
To simulate the influence of water accumulation, the first step is to designate the coal pillars as fluid-solid coupling zones. Subsequently, the humidity parameters within the model are accessed using the Fish language embedded in Flac\({}^{3\text{D}}\), and in combination with the relationship between humidity obtained from experiments and the strength of coal pillars, the strength is redefined to simulate the softening of coal pillars. The permeability characteristics of coal pillars and their elastoplastic state are closely related to the stress state of each element within the coal mass, directly determining the distribution and expansion characteristics of the seepage zone. Additionally, the degree of weakening of mechanical parameters within the seepage zone is closely associated with the degree of water saturation. To comprehensively reveal the failure characteristics of coal pillars and the evolution of stability, the development of a Fish language program specifically targeting the permeability characteristics and water-induced softening of coal pillars is crucial. Studies have shown that water immersion significantly promotes the development and expansion of fractures within the coal mass, leading to a notable decrease in the strength of the water-immersed coal mass [34]. The permeability of coal specimens in both elastic and plastic yield states can be calculated using Equation (1). \[\left\{ \begin{array}{l}K_{E}{=375.03-124.32\sigma_{3}-16.07\sigma_{1}{+}10.14 \sigma_{3}^{2}{+}0.15\sigma_{1}^{2}{+}2.15\sigma_{1}\sigma_{3}}\\ K_{P}=142.2316 exp(-2.6349(1-exp(-0.3415\sigma_{n})))\end{array} \right. \tag{1}\]  
In the equation: \(K_{E}\) is the permeability of coal samples under elastic conditions, m\({}^{2}\). \(K_{p}\) is the permeability of coal samples under yield conditions, m\({}^{2}\). \(\sigma_{1}\) is the first principal stress, MPa. \(\sigma_{3}\) is the third principal stress, MPa. \(\sigma_{n}\) is the effective stress experienced by the element, MPa. The permeation of water in coal pillars manifests as low-speed nonlinear flow. Its infiltration process does not occur readily but requires a water pressure gradient to reach a certain threshold known as the initiation pressure gradient of the coal mass. Based on extensive experimental data [35], researchers have investigated the relationship between the initiation pressure gradient and permeability and derived relevant calculation formulas, as shown in Equation (2). \[G=0.03252k^{-1.4155} \tag{2}\]  
Figure 6: Changes of mechanical properties of coal samples. In the equation: \(k\) is the permeability of the coal mass, m\({}^{2}\). \(G\) is the initiation pressure gradient of the coal mass, MPa/cm. From experiments, it is observed that the elastic modulus, cohesion, and internal friction angle of coal samples exhibit an exponential decrease with increasing water saturation. The weakening coefficient of coal sample mechanical parameters is represented as shown in Equation (3). \[ \begin{cases}k_{E}=0.0093+0.9907e^{(-S_{w}/43.1765)}\\ k_{c}=0.6613+0.3387e^{(-S_{w}/31.9171)}\\ k_{\varphi}=0.09068+0.0932\times 0.9305^{S_{w}}\end{cases}  \tag{3}\]  
In the equation: \(k_{E}\), \(k_{c}\), and \(k_{\varphi}\) are the weakening coefficients of the coal sample's elastic modulus, cohesion, and internal friction angle, respectively. \(S_{w}\) is the water saturation of the coal sample, expressed as a percentage. Based on research into the permeation characteristics and weakening effects of water in coal mass, a successful development of a Fish language program for coal pillar softening has been achieved. This program, through secondary development of the model and utilizing a loop calculation mechanism, dynamically adjusts and couples the mechanical and seepage parameters of coal pillars.