## 1 Introduction
However, because the membership degree and the weight of the index of the prediction method of fuzzy theory are not easy to determine, and the evaluation results obtained by using different evaluation models are different, the prediction method of fuzzy theory has many shortcomings. It is not very practical in engineering practice. [PERSON] et al. [23] used the grey prediction model to predict the unit grouting quantity. However, the prediction method of grey theory is mainly based on a large number of grouting data, and it does not consider the grouting mechanism, so its science and feasibility have yet to be verified. [PERSON] et al. [24] proposed and established the genetic neural network model of grouting quantity, and used the genetic neural network method to predict the unit grouting quantity. [PERSON] et al. [25] introduced the back-propagation neural network and information diffusion method into grouting practice to predict the unit grouting quantity. However, the prediction method of artificial neural network theory lacks a unified mathematical basis. For its structure selection and the initial value setting of weight, it is necessary to rely on experience, and the model obtained is usually a local optimal solution rather than a global optimal solution, so its generalization ability is poor. At the same time, this method is a learning method based on large samples. Only enough learning samples can train the prediction model with high accuracy, but the actual situation is that there cannot be enough learning samples in the process of grouting test and construction. In most cases, it is faced with small sample data, and the learning speed of this method is slow and the efficiency is low, so the prediction method of artificial neural network theory also has many unreasonable points. From the above analysis, it can be seen that there are many irrationalities in the prediction methods of the unit grouting quantity in karst curtain grouting at present. The theoretical research in this aspect is lacking, and the grouting construction is lack of scientific guidance. Therefore, it is necessary to carry out the theoretical study on the prediction of the unit grouting quantity in karst curtain grouting and to find a reasonable and applicable method for predicting the unit grouting quantity. Study on the Relationship between the Unit Grouting Quantity and the Water Permeability of Rock Strata in Karst Curtain Grouting  
Some scholars [18; 20; 21] collated and summarized the grouting data of some typical karst curtain grouting areas and made a regression analysis of these grouting data by means of mathematical statistics. It is pointed out that there is no linear ratio between the unit grouting quantity and the water permeability of rock strata in karst curtain grouting, but these two physical quantities are not two unrelated independent variables, and there is a certain correlation between them. This shows that there is a nonlinear relationship between the unit grouting quantity and the water permeability of rock strata in karst curtain grouting. In general, both the unit grouting quantity and the water permeability of rock strata can reflect the size of rock cracks. In fact, the permeable paths in rock mass are called water paths, and the water paths which slurry can pass through are called slurry paths. Different rock masses have their own unique water paths and slurry paths [26]. Therefore, there is the following quantitative relationship between water paths and slurry paths of rock mass:  
\[M\geq m, \tag{1}\]  
where \(M\) is the number of water paths of rock mass, and \(m\) is the number of slurry paths of rock mass. In grouting engineering, the unit grouting quantity mainly depends on the condition of slurry paths of rock mass, and the water permeability of rock strata mainly depends on the condition of water paths of rock mass.