## 1 Introduction
When Equation (9) is substituted into Equation (7), the following expression of the water permeability of rock strata is obtained:  
\[\omega=\frac{Q}{PL}, \tag{2}\]1  
The expression of the karst fracture opening is obtained from the transformation of Equation (10):  
\[\omega=\frac{Q}{PL}, \tag{2}\]2  
In the karst curtain grouting project, the scale of the karst crack structure is much larger than that of the roughness, and the viscosity time variation of the slurry is often ignored in the actual grouting process. Therefore, the cubic law can be directly used to describe the flow of slurry in karst fissures, without considering the influence of the roughness of karst fissures and the change of slurry viscosity. The expression is as follows:  
\[\omega=\frac{Q}{PL}, \tag{2}\]3  
where \(Q_{v}\) is the flow rate of the slurry (m\({}^{3}\)/s); \(\rho\) is the density of the slurry (kg/m\({}^{3}\)); \(g\) is the acceleration of gravity (N/kg); \(w\) is the width of the karst fracture (m); \(h\) is the water head height converted by pressure (m); \(r\) is the diffusion distance of the slurry (m); and \(\eta\) is the dynamic viscosity of the slurry (Pa\(\cdot\)s). When Equation (11) is substituted into Equation (12), the flow model of slurry diffusion in karst fissures in karst curtain grouting is obtained:  
\[\omega=\frac{Q}{PL}, \tag{2}\]4  
where \(\gamma\) is the weight of the slurry (N/m\({}^{3}\)). Because the unit grouting quantity represents the quality of the slurry injected into the grouting hole per meter, the theoretical equation of the unit grouting quantity in karst curtain grouting is obtained by further transformation of Equation (13):  
\[\omega=\frac{Q}{PL}, \tag{2}\]5  
where \(q_{w}\) is the unit grouting quantity (kg/m), and \(t\) is time used after grouting in grouting section (s). In karst curtain grouting, the unit grouting quantity is affected by many factors, and the theoretical relationship equation between the unit grouting quantity and the water permeability of rock strata (Equation (14)) is more complicated. In order to simplify the form of the mathematical model, a new coefficient, \(K\), is introduced. It is defined as the comprehensive coefficient of karst curtain grouting, so that its expression is as follows:\[\omega=\frac{Q}{PL}, \tag{2}\]6  
It can be seen that the physical quantities contained in the comprehensive coefficient of karst curtain grouting defined in this paper basically reflect the three controlling factors of slurry diffusion, namely, the characteristics of slurry fluid, the characteristics of grouted rock strata, and the technical factors of grouting. Therefore, if the degree of karst development is different, \(K\) will be different values. In view of the fact that \(K\) is a complex coefficient, it is difficult to solve the value theoretically, so the field test and nonlinear function fitting are used to determine the value of \(K\) in this paper. Therefore, the theoretical relationship between the unit grouting quantity and the water permeability of rock strata in karst curtain grouting can be expressed by the following formula:  
\[\omega=\frac{Q}{PL}, \tag{2}\]7  
The theoretical relationship curve between the unit grouting quantity and the water permeability of rock strata in karst curtain grouting can be expressed in the form of \[\omega=\frac{Q}{PL}, \tag{2}\]8  
In engineering practice, the process of predicting the unit grouting quantity in karst curtain grouting is as follows: (i) According to the hydrogeological conditions of karst curtain grouting construction area, the suitable curtain grouting test section is selected for the field test. The unit grouting quantity and the water permeability of rock strata obtained in the test are recorded; (ii) using the test data and nonlinear function fitting, the comprehensive coefficient of karst curtain grouting is determined; (iii) Equation (16) is used to predict the unit grouting quantity by using the water permeability of rock strata obtained in the main project.