A Fractal Model of Fracture Permeability Considering Morphology and Spatial Distribution

In fractured reservoirs, the fracture system is considered to be the main channel for fluid flow. To better investigate the impacts of fracture morphology (tortuosity and roughness) and spatial distribution on the flow capacity, a fractal model of fracture permeability was developed. Based on micro-computed tomography (CT) images, the 3D structure of the fracture was reconstructed, and the fractal characteristics were systematically analyzed. Finally, the control of permeability by fracture morphology and spatial distribution in different fractured reservoirs was identified. The results demonstrate that the complexity of the fracture distribution in 2D slices can represent the nature of the fracture distribution in 3D space. The permeability fractal prediction model was developed based on porosity (φ), spatial distribution fractal dimension (Df), tortuosity fractal dimension (DT), and opening fractal dimension of the maximum width fracture (Db). The permeability prediction results of the fractal model for Samples L-01 (limestone), BD-01 (coal), BD-02 (coal), S-01 (sandstone), M-01 (mudstone), and C-01 (coal) are 0.011 md, 0.239 md, 0.134 md, 0.119 md, 1.429 md, and 27.444 md, respectively. For different types of rocks, the results predicted by the model show good agreement with numerical simulations (with an average relative error of 2.51%). The factors controlling the permeability of fractured reservoirs were analyzed through the application of the mathematical model. The permeability is positively exponentially correlated with the fractal dimension of spatial distribution and negatively exponentially correlated with the fractal dimension of morphology. When Df < 2.25, the fracture spatial structure is simple, and the morphology and spatial distribution jointly control the seepage capacity of fractured reservoirs. When Df > 2.25, the fracture spatial structure is complex, and the impact of morphology on seepage capacity can be disregarded. This work can effectively lay the foundation for the study of fluid permeability in fractured reservoirs by investigating the effects of fracture morphology (tortuosity and roughness) and spatial distribution on flow capacity.