A Geometrical Aperture–width Relationship for Rock Fractures

Abstract

The relationship between fracture aperture (maximum opening; dmax) and fracture width (w) 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 (n) 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 firstorder approximation, that the fracture aperture should be a linear function of its width, meaning that n = 1. 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, 2 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 ~14 orders of magnitude for variations in flow rate and ~5 orders of magnitude for variations in fracture width.

Predicting Electrokinetic Coupling and Electrical Conductivity in Fractured Media Using a Fractal Distribution of Tortuous Capillary Fractures

“…We found β = 0.009, τ = 1.5 and D f = 1.8 to give the minimum RMSE for fractured samples. Note that Ghanbarian et al (2019) found the values of β between 0.1 and 0.001 by fitting their model to experimental data of tensile fractures in the Krafla fissure swarm of northeast Iceland [76]. Besides the 1:1 line (the solid line), the 1:10 and the 10:1 lines are also shown in Figure 8 (the dashed lines).…”

“…where φ is the porosity of porous media and α is the ratio of the minimum width to the maximum width of the fractures (α = w min /w max ). It has been shown that the aperture a (m) is related to the width w (m) of the fracture by the following linear scaling law, e.g., [71,72,75,76]:…”

Predicting Electrokinetic Coupling and Electrical Conductivity in Fractured Media Using a Bundle of Tortuous Capillary Fractures

“…We found β = 0.009, τ = 1.5 and D f = 1.8 to give the minimum RMSE for fractured samples. Note that Ghanbarian et al (2019) found the values of β between 0.1 and 0.001 by fitting their model to experimental data of tensile fractures in the Krafla fissure swarm of northeast Iceland [77]. Besides the 1:1 line (the solid line), the 1:10 and the 10:1 lines are also shown in Fig.…”

“…where φ is the porosity of porous media and α is the ratio of the minimum width to the maximum width of the fractures (α = w min /w max ). It has been shown that the aperture a (m) is related to the width w (m) of the fracture by the following linear scaling law [e.g., 72,73,76,77]:…”

Optimal fluid stretching for mixing-limited reactions in rough channel flows

“…In many natural and engineering applications, rough channel flows are common. We consider self-affine rough walls, as rough surfaces in nature are often found to be statistically self-affine (Mandelbrot 1983; Kertesz, Horvath & Weber 1993; Ponson, Bonamy & Bouchaud 2006; Ghanbarian, Perfect & Liu 2019). Self-affine surfaces are scale invariant in that the standard deviation of the height difference between two points separated by lateral distance can be expressed as , where is the Hurst exponent that characterizes the surface roughness (Drazer et al.…”