In order to describe the law of IP secondary field of water-bearing rocks changing with time from a certain quantitative relationship, many mathematical models have been proposed, such as single exponential model, multi-exponential model and hyperbolic model. Professor Kimi of China Geo University, based on nearly 1000 IP discharge curves measured on more than 400 water-bearing sand samples composed of different factors, summed up a mathematical model that can approximately describe most IP secondary fields in the time range of 0. 18 ~ 15s, that is, a linear equation with logarithmic time axis. On this basis, a new parameter-deviation degree, which can reflect the aquifer condition, is proposed.

Mathematical model of (1) induced secondary field

The experimental results show that during the observation time of 0. 18 ~ 15s, when the time axis is logarithmic abscissa and △V2 is arithmetic ordinate, most curves are straight lines, and the equation is:

Geophysical methods, techniques and instruments for water exploration

Where: k is the slope of the attenuation curve; B is a constant, the attenuation voltage per unit time after power failure (t = 1).

Figure 2-2-4 is the result of drawing the experimental curve of water-bearing sand sample (humidity w = 5%) on two coordinates. The calculated result according to formula (2-2-5) is compared with the measured single logarithmic straight line, and they are completely consistent.

Figure 2-2-5 is the measurement result of a known water source well.

Fig. 2-2-4 Experimental results of δV2 attenuation curve of natural sand samples with water.

Figure 2-2-5 Measured Delta V2 Flow Curve of Known Wells

Firstly, three indexes are used to fit the △V2(t) curve, and good results are obtained. Then, using the observation data of △V2(t), it is found that the attenuation curve also satisfies the linear equation of formula (2-2-5) in the time range of t = 0.25 ~ 100 s, and there are:

Geophysical methods, techniques and instruments for water exploration

Therefore, equation (2-2-5) has certain universal significance.

When the two ends of equation (2-2-5) are divided by the primary field △V 1, that is, it is expressed by the polarizability η(t), the equation still has the properties of a linear equation.

(2) Deviation parameter

The so-called deviation degree, that is, the fitting degree between the measured attenuation curve and the linear equation (2-2-5), is expressed by the mean square relative deviation r, which is called "deviation degree".

Geophysical methods, techniques and instruments for water exploration

Where: n is the number of sampling points; Is the average of the polarizabilities of the sampling points in the observation period.

R can be used to measure the "linearity" of the measured attenuation curve. The greater the R, the worse the linearity, and the less satisfying the mathematical model given by Equation (2-2-5), so R is called deviation degree, that is, the degree of deviation from an ideal straight line. In other words, the R curve on the aquifer should have obvious minima, that is, low-value anomalies.

To sum up, in the underground water-bearing geological body, the apparent polarizability ηS and St have high value anomalies at half attenuation, and the deviation r has low value anomalies. Therefore, it is necessary to comprehensively analyze the characteristics of various IP anomalies in order to achieve good geological results of IP water exploration.