## 5 Control Technology of Roof Fall
### Grouting Reinforcement Technology  
#### 5.2.1 Grouting Fluid Permeation Diffusion Equation  
The chamber dome is a typical pore-fracture dual medium that can be assumed to be a continuous medium. Slurry diffusion is an unsteady Darcy seepage motion with grouting pressure as the main driving force, which satisfies the following osmotic continuous differential equation:  
\[-\left[\frac{\partial(\rho v_{x})}{\partial x}+\frac{\partial(\rho v_{y})}{ \partial y}+\frac{\partial(\rho v_{z})}{\partial z}\right]=\rho(\alpha+n \beta)\frac{\partial p}{\partial t} \tag{17}\]  
where \(\rho\) is the density of the grouting fluid, \(\alpha\) and \(\beta\) represent the compressibility of the chamber dome and slurry, respectively. \(n\) is the porosity of the chamber dome and \(v_{\rm i}\) (\(i\)=\(x\), \(y\), \(z\)) is the equivalent  
Figure 15: Effect of bolt spacing on the crack development in the chamber dome. permeation velocity of the slurry in the chamber dome. \(p\) is the grouting pressure. Since the flow of the slurry in the chamber dome cracks belongs to the Darcy flow, the equivalent permeation velocity of the slurry can be expressed as the differential form of the grouting pressure \(p\)  
\[v_{x}=-\frac{1}{\gamma}\frac{\partial(K_{\mathrm{e}}p)}{\partial x},v_{y}=- \frac{1}{\gamma}\frac{\partial(K_{\mathrm{e}}p)}{\partial y},v_{x}=-\frac{1}{ \gamma}\frac{\partial(K_{\mathrm{e}}p)}{\partial z} \tag{18}\]  
where \(K_{\mathrm{e}}\) is the equivalent slurry permeability coefficient, indicating the permeability of the slurry in the chamber dome cracks. \(\gamma\)=\(\rho g\) is the bulk density of the slurry where \(g\) is the acceleration of gravity. Assuming that the grouting slurry is an incompressible fluid, that is, \(\rho\) is a constant, the basic differential equation for the penetration and diffusion of the grouting slurry in the chamber dome can be obtained by substituting the Equation (17) into the Equation (18):  
\[-\left[\frac{\partial^{2}(K_{\mathrm{e}}p)}{\partial x^{2}}+\frac{\partial^{2 }(K_{\mathrm{e}}p)}{\partial y^{2}}+\frac{\partial^{2}(K_{\mathrm{e}}p)}{ \partial z^{2}}\right]=S\frac{\partial p}{\partial t} \tag{19}\]  
In the above equation, \(S\)=\(\rho g(\alpha\)+\(\mu\beta)\) is the chamber dome crack slurry storage coefficient, which represents the amount of stored slurry released due to the compression of chamber dome cracks and slurry expansion when the slurry hydraulic head is lowered by 1 unit per unit volume of saturated surrounding rock. Equation (19) requires the determination of the equivalent slurry permeability coefficient \(K_{\mathrm{e}}\) and the equivalent slurry storage coefficient \(S\) of the chamber dome. Then it can be solved by combining appropriate boundary conditions and initial conditions. The equivalent slurry storage coefficient \(S\) is a constant obtained in the test, while the chamber dome stress and the grouting pressure affect the equivalent slurry permeability coefficient \(K_{\mathrm{e}}\) of the chamber dome. A large number of engineering experiments have shown that disturbances, such as chamber excavation, cause the chamber dome to form a stress relaxation zone, a stress increase zone and an initial stress zone from the shallow to the deep. These zones greatly change the permeability characteristics of the chamber dome and influence the penetration and diffusion of slurry in the chamber dome. Therefore, in order to consider the influence of the stress redistribution and the grouting pressure on the slurry diffusion law, [PERSON] et al. [25] (2015) proposed the negative exponential relationship between the material stress and the permeability coefficient based on a large number of experimental results. This relationship can express the equivalent slurry permeability coefficient of the grouting slurry in the chamber dome as:  
\[K_{\mathrm{e}}=K_{0}e^{-\lambda(\alpha_{\mathrm{m}}-\alpha_{\mathrm{m}}-p)} \tag{20}\]  
where \(\alpha_{\mathrm{m}}\)=(\(\sigma_{1}\)+\(\sigma_{2}\)+\(\sigma_{3}\))/3 is the average stress of the chamber dome after excavation.