Percolation conditions in binary fractal fracture networks: Applications to rock fractures and active and seismogenic faults

Abstract

[1] The fracture network patterns in crystalline rock are treated using a binary fractal model described by three fractal geometric parameters: the fractal dimension D of the spatial distribution of fractures, the fractal dimension a of the fracture length distribution, and the maximum fracture length (l max ) normalized by the domain length (L), l max /L. We present a percolation threshold relationship describing the boundary between percolating and nonpercolating conditions in random binary fractal fracture networks (RBFFNs) and apply this to observations of natural fractures. Our results indicate that the three fractal parameters identified indeed control the connectivity of fracture networks, and hence the migration of fluids through rock fractures, and active and seismogenic faults. Moreover, our numerical approach may be applied in the future to investigating fluid migration associated with seismicity based on fractal parameters determined from earthquake catalog data.

Fractal properties of seismicity in regions affected by large, shallow earthquakes in western Japan: Implications for fault formation processes based on a binary fractal fracture network model

“…A cumulative analysis was executed in order to evaluate the growth of the fault network pattern (i.e., the fault formation process) from the behavior of fractal dimensions D 2 and a on the basis of the binary fractal fracture network (BFFN) model [ Nakaya et al , 2003] and another fracture network model similar to BFFN [e.g., Darcel et al , 2003]. Darcel et al [2003] and Nakaya et al [2003] have reported that a fractal‐based fracture network model such as BFFN can describe the geometry and connectivity of fracture network patterns using two kinds of fractal parameters for the size distribution and the spatial distribution of rock fractures and active and seismogenic faults. The four fundamental modes (M1, M2, M3 and M4; Figure 9) of the relationship between D 2 and a are explained as follows under the condition that the number of fractures increases and the fracture network size expands with time.…”

Percolation conditions in fractured hard rocks: A numerical approach using the three‐dimensional binary fractal fracture network (3D‐BFFN) model

“…The procedure for generating a two‐dimensional RBFFN model (2D‐RBFFN) has been reported previously [e.g., Darcel et al , 2003; Nakaya et al , 2003]. In the present study, we develop a 3D‐RBFFN model based on the 2D‐RBFFN model.…”

“…For example, fracture connectivity determines the pathway of fluid flow in a fracture network system. On the basis of evidence presented in many previous studies, it is natural to assume that properties of fracture geometry such as size distribution, spatial distribution, and the distributions of aperture, throw, and spacing must be fractal or partial fractal [e.g., Okubo and Aki , 1987; Hirata et al , 1987; Hirata , 1989; Ohno and Kojima , 1992; Marrett and Allmendinger , 1992; Gillespie et al , 1993; Bloomfield , 1996; Cladouhos and Marrett , 1996; Odling , 1997; Bour and Davy , 1997; Van Dijk et al , 2000; Gillespie et al , 2001; Wilson , 2001; Nakaya et al , 2003]. Many phenomena in geology and geophysics, such as fault systems and seismic activity, behave as fractals.…”

“…Theoretical and numerical approaches have expressed phase changes, percolation, and connectivity within fracture systems as a function of the system's geometric parameters, especially fractal dimensions [e.g., Robinson , 1983, 1984; La Pointe , 1998; Balberg et al , 1991; Acuna and Yortsos , 1995; Berkowitz et al , 2000; de Dreuzy et al , 2000, 2001a, 2001b; Bour et al , 2002; Darcel et al , 2003; Nakaya et al , 2003; Goto and Otsuki , 2004]. These results indicate that the two‐dimensional fracture network connectivity threshold is critically dependent on the fractal dimensions of length distribution, spatial distribution, and minimum fracture length ( l min ) normalized by the domain length (L), l min /L; however, few studies have sought to clarify the effect of fracture orientation on two‐ and three‐dimensional fracture connectivity [e.g., Robinson , 1983, 1984].…”

“… Nakaya et al [2003] considered two‐dimensional binary fractal fracture network (2D‐BFFN) models described by three fractal geometric parameters: the fractal dimension (D) of the spatial distribution of fractures, the exponent ( a ) of the power‐law cumulative fracture length distribution, and the maximum fracture length ( l max ) normalized by the domain length ( L ), l max /L, and discussed percolation conditions for 2D‐BFFN patterns as well as that for the model of Darcel et al [2003]. Despite the fact that their research focused on a model relevant to natural fracture network patterns, it is not clear whether their conclusive results related to percolation conditions based on two‐dimensional networks are entirely satisfactory for three‐dimensional networks.…”