## 1 Introduction  
With the increasing number of deep and long tunnels being constructed, water and mud burst disasters in front of these tunnels have become more frequent due to surrounding water-bearing structures, and this has become an important problem that needs to be solved in the construction of tunnels [1, 2]. The water permeability of a water-bearing structure (that is, its permeability characteristics) directly determines the magnitude and scale of water inrush. It is essential to estimate the magnitude and scale of water inrush through the proper evaluation of the permeability characteristics of water-bearing structures [3]. In general, laboratory measurements of borehole samples, as well as field methods such as pumping or pressure experiments, are used to understand the permeability characteristics of water-bearing structures in front of tunnel faces [4]. However, these methods are expensive, and due to a shortage of samples and in situ experiments, their results often suffer from hysteresis and one-sidedness. The induced polarization method has the advantage of being sensitive to the porous media of formations and not being subject to topographical factors, and it is being used more and more frequently to predict the hydraulic characteristics of the formation [5, 6]. The induced polarization method is a geophysical method based on the induced polarization effect in porous media, which is dependent on the difference between the induced polarization parameters of different rock and soil media. By observing the complex conductivity and polarizability of the rock-soil medium, we can derive information such as the \"real part\", characterizing the charge conduction characteristics, and the \"imaginary part\", characterizing the charge storage characteristics in complex conductivity [7; 8]. Permeability determines the resistance of porous materials to fluid flow [9; 10]. In the detection of disaster-causing, water-bearing structures in tunnels, the fluid is usually groundwater, and its properties are relatively stable. Therefore, the permeability (\(k\)) can be used to evaluate the permeability of the entire formation. In 1957, the induced polarization method was first proved to be applicable to groundwater detection, which showed that the permeability of shallow aquifers can be evaluated by the relevant parameters of the induced polarization decay curve [11]. [PERSON] et al. (1996) used the constant phase angle model to quantitatively interpret the results of field spectral induced polarization measurements, which could separate the electrical volume and interface effect. By using a Kozeny-Carman-like equation, the hydraulic permeability can be estimated [12]. Combined with the Kozeny-Carman formula, the relationship between permeability, porosity and specific surface area can be obtained [13; 14; 15; 16]. In addition, some scholars have found that the correlation between the imaginary part of complex conductivity and the specific surface area is weak when using British sandstone for experiments and that it is difficult to satisfy the accuracy requirements for predicting permeability [17]. [PERSON] et al. (2007) verified the permeability coefficient of different field data using the empirical equations by [PERSON] et al. (1996) and [PERSON] and [PERSON] (2002) and showed the limitations of their method; they suggested that using the relaxation time for estimating the permeability [18]. In the study of relaxation time, [PERSON] and [PERSON] (2001) proposed to use the relaxation time distribution to assess the grain size distribution [19]. [PERSON] and [PERSON] (2003) found a correlation between the distributions of pore throat diameters and relaxation time through the sandstone test. After that, increasing literature showed that relaxation time was positively correlated with the square of pore throat diameter or grain size [20; 21; 22]. In terms of the prediction of permeability by relaxation time, [PERSON] et al.