Self-Similar and Self-Affine Properties of Two-Dimensional Fracture Patterns in Rocks

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In this paper, we investigate the fractal properties of binary maps of rock fractures at different scales and different geological types, as well as different families of fracture patterns obtained from a two-dimensional Laplacian growth model (LGM). From these analyses we figure out which families of the LGM patterns match the structural properties of the fracture binary maps. The LGM is defined in terms of a nonlinear map that depends on two parameters, λ and \(\mathfrak{a}\), that respectively define the area and shape of the elements of the aggregate that conforms the patterns. The fractal dimension and roughness exponent of the LGM patterns are found to depend on \(\mathfrak{a}\), with \(0<\mathfrak {a}<1\). From a detailed statistical analysis of these patterns we found that the fractal dimensions of capacity, correlation and information decrease monotonically as \(\mathfrak{a}\) increases. We also found that the values of these three fractal dimensions tend to collapse on top of each other as \(\mathfrak{a}\lessapprox1\). Remarkably, the fractal properties of rock fractures in the scales from millimeters up to a few meters appear to be well represented by the fractal structure of the LGM families of patterns with \(\mathfrak{a}=0.15\) and 0.30, while the fractal properties of rock fractures in the scale of kilometers seems to be well represented by the LGM family with \(\mathfrak{a}=0.90\). In addition, the three fractal dimension values of fracture binary maps in the scales from millimeters up to meters were found to be different between them. Nonetheless, for fractures in the scale of kilometers, the values of the three fractal dimensions are very close to each other as an indication of self-similar behavior. Analysis of the corrections to the scaling of the roughness exponent, ζ, suggests that they are negligible for the LGM family of fracture patterns with \(\mathfrak{a}=0.9\). This finding points to a self-affine structure for this family of patterns. In fact, the calculated roughness exponent results are in the range of values characteristic of rock fractures.