Box-counting methods to directly estimate the fractal dimension of a rock surface

Surfaces of rocks are usually not perfectly “smooth”, and two box-counting methods, i.e. the conventional cubic covering method (CCM) and improved cubic covering method (ICCM), can directly describe the irregularities of a rock fracture surface without any approximate calculations. Our investigation showed that if the scale δ of covering cubes is greater than the sampling interval S0, the CCM and ICCM cannot completely cover the object rough surface. Considering this, we presented two new cubic covering methods, namely the differential cubic covering method (DCCM) and relative differential cubic covering method (RDCCM) to directly evaluate the fractal dimension of a rough surface according to the definition of box-counting dimension. Experimentally, a 3D laser profilometer was used to measure the topography of a natural surface of sandstone. With the CCM, ICCM, DCCM and RDCCM, direct estimations of the fractal dimension of the rock surface were performed. It was found the DCCM and RDCCM usually need more cubes to cover the whole fracture surface than the CCM and ICCM do. However, the estimated fractal dimensions by the four methods were quite close. Hence, three Takagi surfaces with known fractal dimensions of 2.10, 2.50 and 2.90 were adopted to further examine the four box-counting algorithms. Results showed that for a low fractal dimension Takagi surface, the DCCM and RDCCM gave accurate results within the ranges determined by small covering scales, whereas the CCM and ICCM always overestimate the fractal dimension for all the potential scale ranges investigated in current work; for high fractal dimension surfaces, the CCM and ICCM provided very good results within the ranges determined by small covering scales, and oppositely, the DCCM and RDCCM cannot provide a good estimation of the fractal dimension within such scale ranges but can determine approximate results at large scales.