Mathematical Evaluation of Hydraulic Fracture Complexity Based on Digital Rock Modeling and Fractal Geometry

The fractal natural microstructure of shale reservoirs significantly influences hydraulic fracture propagation and reservoir stimulation. However, there is a lack of quantitative mathematical descriptions for the coupled regulation of micropores, natural fractures, and injection rates. This study develops a mathematical evaluation method for hydraulic fracture evolution in complex microstructured reservoirs using digital core technology, fractal geometry and a hydraulic–mechanical–damage coupling algorithm. High-resolution SEM images were used to reconstruct the microscopic fractal features. Integrated digital image processing and fractal analysis, along with geometric indices such as fractal dimension, fracture coverage, and stimulated area, and statistical measures including directional entropy, variance, and the Pearson correlation coefficient, were employed to systematically quantify fracture network evolution and complexity under different injection rates. Results show that fracture morphology, spatial complexity, and mineral damage mechanisms are jointly controlled by microstructure and injection rate. In particular, the directional distribution of pores and natural fractures is found to exert a dominant control on the propagation paths and branching behavior of hydraulic fractures, revealing a strong coupling between microstructural anisotropy and fracture directionality. Increased injection rates enhance fracture complexity and stimulation range, with varying effects from different microstructures. At low rates, fracture propagation is mainly determined by the initial microstructure, whereas at high rates, fractures tend to develop multiple pathways. Natural fracture structures contribute more to fracture complexity at high rates. The proposed comprehensive fracturability index (FI)-based fracturability evaluation model provides a systematic, quantitative approach to optimizing fracturing processes.