A Preliminary Fractal Interpretation of Effects of Grain Size and Grain Shape on Rock Strength

Rock strength is an important consideration to indicate the integrity and stability of wellbore drilling, tunnels, and other underground excavations. In this research, by assuming the rock rain as a 3D spheroid with a 2D ellipse projection area, a fractal model is developed to quantify the rock strength based on the fractal dimensions of grain size and grain aspect ratio. The validity of the modeling is verified using available experimental data. It can well explain the rock strength changes along with the equivalent diameter of grains found in the common igneous rocks, sedimentary rocks and metamorphic rocks, i.e., granite, marble, limestone, and sandstones. A sensitivity analysis indicates a high fractal dimension of grain size leads to a low value of grain size and a fine-grained rock matrix while a high fractal dimension of grain aspect ratio indicates a low value of grain aspect ratio and a high-sphericity rock matrix. The difference between a 2D equivalent diameter and a 3D equivalent diameter is also discussed and is found to diminish under a low fractal dimension of grain size and a low fractal dimension of grain aspect ratio. Those rocks exhibiting high rock strengths tend to be a fine-grained rock matrix with a low value of grain size (i.e., the major length), a small equivalent diameter, a low sphericity, but a high aspect ratio.