Nowadays, coal resources are still the world main energy source. However, with the continuous increase in mining depth, the problem of geological disasters in production is becoming more and more serious. Among them, coal and gas outburst accidents are called the “number one killer” of coal mine safety production, which seriously restricts coal mines security [1]. The complex resistivity method is an emerging electrical exploration technology, which uses the induced polarization effect of the rock under the excitation of an external electric field to achieve the purpose of geophysical exploration [2]. The most important significance is that the complex resistivity parameters are closely related to the lithology and micro-fracture structure of the rock. Therefore, the complex resistivity method is widely used in the evaluation of fluids in rock micro-fractures, such as soil pollutant monitoring [3–7], groundwater remediation Monitoring [8], and bacterial activity monitoring [9], etc., show the broad application prospects of the complex resistivity method. In recent years, many researchers have used complex resistivity to study the electrical characteristics of coal and rock, and analyzed the relationship between complex resistivity characteristics and parameters, gas content and coalbed methane content [10–12].

Pelton et al. proposed the Cole-Cole model, and proposed that different minerals can be distinguished by induced polarization effects at multiple frequencies, and then ex-tended it to the second-order Cole-Cole model to describe the low-frequency information of electromagnetic induction [13]. Then Zhu et al. studied the Cole-Cole model’s ex-tended model Dias model, combined with the damped least square method and the Monte Carlo method based simulated annealing method to reduce the uncertainty of the solution [14]. The research of Ke shizhen believes that the dispersion phenomenon of complex resistance is mainly caused by the polarization of the medium, and improved the Cole-Cole model to include multiple polarization points, and proposed a new interpretation method [15]. Fan focused on the relationship between rock polarizability and current size, cation exchange capacity, salinity of saturated solution, membrane potential and rock particle size under laboratory conditions [16]. Jiang et al. deduced the induced polarization potential response equation, gave a theoretical model of induced polarization potential, and proposed a model and interpretation method for quantitatively solving oil saturation and formation water resistivity [17]. Yin et al. studied the mechanism of rock dispersion and demonstrated that the rate-frequency dispersion char-acteristics of rock complex resistivity are affected by measurement frequency, properties of pore fluids, electrochemical properties of the surface of framework particles, and clay content [18]. Guo et al. conducted testing and research on four types of structural coals with different metamorphic degrees, and the results showed that: with the increasing frequency, the complex resistivity modulus of different metamorphic coals showed a decreasing trend, and the complex resistivity modulus at the low frequency stage showed obvious all directions [19]. The opposite sex, the reason for this is related to the length of time that the polarizable dielectric in the coal is polarized [20].

In this paper, a forward modeling study of Cole-Cole is carried out to study the difference in induced polarization properties of coal and rock, which is reflected in the model parameters ρ₀, m, τ, c and the real part and imaginary part calculated from them. Change the values of different parameters of the model to study the changes of the spectrum curve. In the inversion process, amplitude individual inversion, phase individual inversion, and amplitude and phase combined inversion were carried out respectively. The effect of joint inversion is better than that of individual inversion.

Pelton et al. [13] analyzed and summarized the data through a large number of rock and ore sample experiments and field exploration experiments, and finally obtained the complex resistivity spectrum expression of the excitation effect of rock and ore in the frequency domain:(1) ρ(iω)=ρ0{1−m[1−11+(iωτ)c]}

Charge rate (polarizability) m, a parameter describing the strength of the polarization effect of the rock (ore); zero-frequency impedance ρ₀ is the resistivity in a DC electric field; relaxation time τ, describing the amount of delay in the process of induced polarization, unit: s; frequency correlation coefficient c, dimensionless, indicating the shape of the attenuation curve, related to the size of the particles, and the value range is 0 < c < 1. The effects of different parameters on the complex resistivity of rocks are studied separately.

The least square method (also known as the least square method) is a mathematical optimization technique. It finds the best function match of the data by minimizing the sum of squares of errors. In this paper, the damped least squares method is selected for inversion. The method was proposed by Marquardt in 1963 [21]. He perfected it on the basis of K. Levenbevg, adding the steepest descent method and the linearization method (Taylor series) comprehensive.(2) △m=[(WdJ)τ(WdJ)+αI]−1(WdJ)(d−F(mk))

Among them: △m is the model modification; W_d is the covariance matrix of the observation data; J is the Jacobian matrix; α is the damping factor; d is the measured data; m_k is the initial model of the kth iteration; F(m_k) is the forward model data.

Given the initial model m_k, calculate the model modification △m from the above equation, and mk+1= mk+△m as the new initial model for the next iteration until the fit is less than the preset threshold So far, m_k+1 is the final inversion result at this time.

The zero-frequency resistivity ρ₀ affects the amplitude of the complex resistivity, but has no effect on its shape and does not affect the phase curve; the polarizability m mainly affects the amplitude trend and the phase minimum; the time constant τ mainly affects the phase minimum Corresponding frequency; frequency correlation coefficient c mainly affects the speed of amplitude attenuation and the size and position of the phase mini-mum.