Improved projective covering method for fractal dimensions of rock discontinuities based on stochastic analysis

The strength, deformability, and flow properties of rock discontinuities are strongly affected by the surface characteristics. Therefore, a quantitative description of the topography of the discontinuities is very important. The projective covering method (PCM) is useful in calculating the fractal dimensions to measure the irregularity and roughness of fracture surfaces. However, there is a defect in the division of a grid cell into two triangles, which is, for every grid cell, only one dividing scheme is used to calculate the fractal dimensions with the projective covering method, despite the availability of two schemes. Moreover, it has been confirmed that when a small grid cell is divided by a different triangulation division scheme, differing fractal dimensions are calculated. To obtain a grid cell division method whose result is consistent with the surface morphology of the studied fracture surface, which comprises thousands of grid cells, improved projective covering method (IPCM) was propose based on stochastic analysis. In this method, a random number was generated by the random function and its parity was judged. If the number was odd, the small grid cell was divided using one scheme. Otherwise, it was divided by the other scheme. With this method, the fractal dimensions of the discontinuity of a redsandstone was calculated and 120 fractal dimensions were obtained, which formed a sample space. Secondly, the distribution characteristics of the sample space was determined, and the average of the sample was regarded as the accurate fractal dimensions of the redsandstone discontinuity. The analysis shows that the sample of fractal dimensions follows a normal distribution, the calculated results by the projective covering method are the maximum or minimum values of the fractal dimensions estimation, and because the result of the dividing scheme using stochastic analysis method is more consistent with the surface morphology, the fractal dimensions obtained by improved projective covering method are more accurate.