# A Dynamic Numerical Simulation on the Grouting Timing in Retained Rib of Pillarless Mining
## 2 The Principle of the Dynamic Numerical Simulation  
### Simulation Flow Chart  
After the coal is mined, caved zones, fractured zones and continuous bending zones tend to be formed above the goaf. In longwall mining, the gangue falls to fill the goaf and subsequently is compressed. In this process, the stress concentration at the coal rib is also gradually weakened. Roof caving is a complex process involving the rock mass deformation, separation and destruction [21]. Therefore, different criteria should be adopted between the gangue falling and the rock yielding in numerical simulation. Rock yielding takes place when the rock stress exceeds its strength, while the roof caving indicates the state of fragmented rock blocks falling into the goaf due to their own weight. [PERSON] [22] simulated the pressure of the hydraulic support on the long wall working face. In this study, two criteria of the gangue falling were adopted including the rock shear strain reaching 0.25 or the roof rock subsiding more than 1 m. The immediate roof caving is mainly caused by the separation of the rock layer, which is ultimately destroyed by the tensile deformation. In the process of numerical simulation, roof subsidence and mining height showed a positive correlation. As a result, maximum plastic tensile strain can be used as the criterion for determining whether the roof surrounding rock is located in the caved zone [23]. The element tensile strain in FLAC3D is defined as Equation (1) [24]:  
\[\epsilon^{t}=\epsilon^{e}+\epsilon^{p} \tag{1}\]  
where \(\epsilon^{t}\) is the element tensile strain, \(\epsilon^{e}\) refers to the element elastic tensile strain and \(\epsilon^{p}\) indicates the element plastic strain.  
The rock experiences tensile failure \(\epsilon^{p}>>\epsilon^{e}\) at the time when \(\epsilon^{t}\approx\epsilon^{p}\). In this study, \(\epsilon^{p}>0.05\) was used as the criterion to evaluate the roof surrounding rock located in the caved zone [25].  
Another important issue about the goaf is the caved zone height. The gangue expansion ratio can be used to determine the height of the caved zone [25]. Studies have shown that the expansion ratio of gangue (\(K_{P}\)) is 1.1-1.5 [26], with a falling gangue porosity of about 0.3 [27; 28].  
Assuming that the gangue fills the mined-out space, the falling gangue porosity is given by:  
\[\frac{h}{h+h_{m}}=0.3 \tag{2}\]  
where \(h\) is mining height, representing the voids in the falling gangue, and \(h+h_{m}\) represents the total volume of the falling gangue.  
So, the expansion ratio can be expressed as Equation (3):  
\[K_{P}=\frac{h+h_{m}}{h_{m}} \tag{3}\]  
where \(h\) is mining height and \(h_{m}\) refers to the height of caved zone. When \(K_{P}\leq 1.43\), the goaf has been filled.  
The calculating process of the numerical simulation model is demonstrated in Figure 1. During the simulation, some routine checks were performed every 100 steps after the calculation initiates. When the equilibrium state is reached, the next 10 m will be excavated. Before the model reaches equilibrium, with the roof and floor contacting, the next 10 m will be excavated also. In other cases, the next 100 steps are expected to be iterated, leading to the next preformation. The calculation will be terminated when all coal seams are mined and an equilibrium is reached. The gangue in the caved zone is assigned to the double-yield constitutive model with the strength of the surrounding rock in the fractured zone reduced.