A Geometrical Aperture–width Relationship for Rock Fractures

The relationship between fracture aperture (maximum opening; [Formula: see text]) and fracture width ([Formula: see text]) has been the subject of debate over the past several decades. An empirical power law has been commonly applied to relate these two parameters. Its exponent ([Formula: see text]) is generally determined by fitting the power-law function to experimental observations measured at various scales. Invoking concepts from fractal geometry we theoretically show, as a first-order approximation, that the fracture aperture should be a linear function of its width, meaning that [Formula: see text]. This finding is in agreement with the result of linear elastic fracture mechanics (LEFM) theory. We compare the model predictions with experimental observations available in the literature. This comparison generally supports a linear relationship between fracture aperture and fracture width, although there exists considerable scatter in the data. We also discuss the limitations of the proposed model, and its potential application to the prediction of flow and transport in fractures. Based on more than 170 experimental observations from the literature, we show that such a linear relationship, in combination with the cubic law, is able to scale flow rate with fracture aperture over [Formula: see text]14 orders of magnitude for variations in flow rate and [Formula: see text]5 orders of magnitude for variations in fracture width.