# Water Infusion on the Stability of Coal Specim under Different Static Stress Conditions
## 2 Numerical Model in PFC
### Definition of Saturation Coefficient in Simulation  
In numerous studies, saturation coefficient (\(S_{c}\)) is used to quantify the amount of water in a rock specimen. It is defined as the ratio of the current water content (\(w_{s}\)) of a rock specimen to the maximum water content (\(w_{m}\)):  
\[S_{c}=\
icefrac{{w_{s}}}{{w_{m}}} \tag{1}\]  
The saturation coefficient evidently ranges from 0 for absolutely dry samples to 1 for completely saturated samples. Previous research suggested that the water-induced reduction in the strength of rock materials is primarily attributed to the physical and chemical deterioration of cement material between grains [34; 35; 36; 37]. Therefore, we realized the water-weakening effect on coal behavior under the framework of the particle flow code according to the approach proposed by [PERSON] et al. [38; 39], in which the bond strength and stiffness of the contact between two particles are reduced. In the numerical simulation, we set potential water-weakening contacts (PWCs) randomly distributed in the model. When the sample is in dry state, the PWCs have the same properties as the normal contacts. However, with the increase of saturation coefficient, the PWCs will be activated and converted into weakening contacts increasingly. Herein, the current water content (\(w_{s}\)) and the maximum water content (\(w_{m}\)) are defined as:  
\[\left\{ \begin{array}{l}w_{s}=\
icefrac{{N_{w}}}{{N_{a}}}\\ w_{m}=\
icefrac{{N_{p}}}{{N_{a}}}\end{array} \right. \tag{2}\]  
where \(N_{w}\) is the number of the activated weakening contacts; \(N_{p}\) is the number of the potential water-weakening contacts, and \(N_{a}\) is the total number of contacts in the model. Thus, the saturation coefficient in simulation model can be expressed as:  
\[S_{c}=\
icefrac{{N_{w}}}{{N_{p}}} \tag{3}\]  
[PERSON] et al. [40] studied the water distribution in cylindrical rock sample during soaking process by nuclear magnetic resonance technique. They reported that water permeates from the end surface into the core and the uneven water distribution significantly affects rock behavior. Therefore, the water distribution should be taken into consideration. For rock pillars exposed to underground water or moist atmosphere with low-permeability roof and floor, water mainly infuse into pillars from their side surface. To study water effect in such cases, the top and bottom surfaces of specimens were sealed to make sure water only infuses into the core from the side surfaces in both numerical and laboratory tests. As shown in Equation (4), in the PFC2D model, the saturation coefficient of the model can be determined as:  
\[S_{c}=\int_{0}^{1}s_{i}dD_{i} \tag{4}\]  
where \(s_{i}=\
icefrac{{n_{ni}}}{{n_{pi}}}\) is the saturation coefficient in the \(i\)th section line, \(n_{wi}\) and \(n_{pi}\) is the number of activated weakening contacts and potential water-weakening contacts in the \(i\)th section line, \(D_{i}=d_{i}/R\), where \(D_{i}\) is the relative distance of the \(i\)th section line to the center axis of the model, \(R\) is the radius of the model, \(d_{i}\) is the distance from the section line to the model central axis (Figure 1). According to water distribution characteristics described by [PERSON] et al. [40], we assume that (1) for the dry state (\(S_{c}=0\)), the saturation coefficient in each vertical line is 0. For the fully saturated state (\(S_{c}=1\)), the saturation coefficient in each vertical line is 1; (2) for the partially saturated state, \(s_{i}\) decreases monotonically from the surface to the core; (3) water distribution becomes more and more uniform with the increase of overall water saturation coefficient. Hence, the relation between \(S_{i}\) and \(D_{i}\) can be expressed as:  
\[S_{i}=m-\frac{m(1-m)}{1-m-D_{i}} \tag{5}\]  
where \(m\) is a coefficient changing with the overall water saturation coefficient (\(S_{c}\)) of the model. For a given \(S_{c}\), \(m\) is a constant.