Influence of random parameter joint length on rock electrical conductivity

Rock joint hollowness coefficient is an important parameter when resolving practical mining problems. Geophysical methods used to resolve this problem are indirect. Thus the interpretation of their results may cause certain difficulties as a result of the uncertainty of the physical relationships between the parameters of joints and the measurement results. One of the ways to resolve this problem is to combine experimental research methods with analytical and numerical simulation. The studies were aimed at investigating the electrical conductivity of a two-dimensional medium in the presence of thin insulating (non-conducting) joints. This paper proposes an analytical method for assessing the dependence of the specific conductivity of a medium with inclusions in the form of elliptical joints on their half-length. This dependence is show to have the form of an exponent depending on the square of the length of the maximum semi-axis as an argument. The simulation method is based on the assumption of the elliptical shape of a joint when the length of the minor semi-axis of the ellipses tends to zero. A review of publications and their results presented in this paper showed that this method for determining the specific conductivity of the medium with thin joints is one of the best in terms of compliance with experimental data. Its predictions are close to those of the Effective Media Approximation (EMA). However, the proposed method is distinguished by the simplicity of the formulas and their physical visibility essential for the use in interpreting the data of a physical experiment. In two-dimensional formulation, numerical simulation of the specific electrical conductivity of a sample of a medium measuring 1×1 m with elliptical joints of conductivity less than that of the matrix was carried out in the COMSOL Multiphysics environment. A square sample of unit sizes with unit conductivity was considered in which 25 joints with uniform distribution along the length occurred. 40 models were built wherein the maximum length of the joints varied from 0.01 to 0.4 sample size in increments of 0.01 m. The satisfactory concordance of the results of numerical and analytical models, both visual and confirmed by statistical estimates, has been shown. It was noted that when the size of the joints changes to achieve the value of the maximum semi-axis a = 0.15 m, the influence of single joints that do not extend beyond the boundaries of the sample prevails. Above this value, at a > of 0.15 m, the influence of joint coalescence, as well as their extension to and beyond the sample boundaries begins to affect. Comparison of the proposed theoretical model of electrical conductivity, depending on the square of the length of the maximum semi-axis of a joint, with a similar model in the form of an exponent with a linear dependence showed a better concordance of the proposed model with observations at the stage of the lack of joint coalescence and their extension to the sample boundaries at a a > 0.15 m. The proposed model has a lower coefficient of determination compared to the full range including both intervals, but higher than that of the model with a linear dependence in the exponent argument. This indicates the universal nature of the proposed model.