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_Journal of Structural Geology 32 (2010) 1444e1449_ Contents lists available at ScienceDirect Journal of Structural Geology journal homepage: www.elsevier.com locate jsg Lacunarity analysis of fracture networks: Evidence for scale-dependent clustering Ankur Roy a, Edmund Perfect a,*, William M. Dunne a, Noelle Odling b, Jung-Woo Kim c,d a Department of Earth and Planetary Sciences, University of Tennessee, Knoxville, TN 37996-1410, USA b School of Earth and Environment, University of Leeds, Leeds LS2 9JT, UK c USDA-ARS Environmental Microbial Safety Laboratory, 173 Powder Mill Road, BARC-EAST, Beltsville, MD 20705, USA d Korea Atomic Energy Research Institute, Radioactive Waste Technology Development Division, 1045 Daedeok-daero Yuseong-gu Daejeon, 305-353, Korea Article info Article history: Received 3 September 2009 Received in revised form 12 August 2010 Accepted 22 August 2010 Available online 24 September 2010 Keywords: Fractures Joints Lacunarity Clustering Gliding-box algorithm Abstract Previous studies on fracture networks have shown that fractures contained within distinct mechanical units ("stratabound") are regularly spaced while those that terminate within the rock mass are clustered ("non-stratabound"). Lacunarity is a parameter which can quantify the distribution of spaces between rock fractures. When normalized to account for differences in fracture abundance, lacunarity characterizes the distribution of spaces as the degree of clustering in the fracture network. Normalized lacunarity curves, L*(r), computed using the gliding-box algorithm and plotted as a function of box-size, r, were constructed for natural fracture patterns from Telpyn Point, Wales and the Hornelen basin, Norway. The results from analysis of the Telpyn Point fractures indicate that such curves are sensitive to differences in the clustering of different fracture sets at the same scale. For fracture networks mapped at different scales from the Hornelen basin, our analysis shows that clustering increases with decreasing spatial scale. This trend is attributed to the transition from a "stratabound" system at the scale of sedimentary cycles (100e200 m) that act as distinct mechanical units to a "non-stratabound" fracture system geometry at the finer 10’s of meters thick bedding scale. © 2010 Elsevier Ltd. All rights reserved. 1. Introduction Fractures control or influence important behaviors in geological systems such as fluid storage, contaminant transport, seismicity, and rock strength. In the context of joints, a key attribute that influences these characteristics is the geometry of the fracture network. To better understand joint geometry it is necessary to consider fractures from the perspective of mechanical stratigraphy. Joints in sedimentary rocks fall in two categories, those that terminate randomly within the rock mass and those that terminate at distinct mechanical layer boundaries (Gross et al., 1995). Lithologic contacts, as well as pre-existing fractures, can serve as mechanical layer boundaries, thereby dividing the rock mass into discreet mechanical units (Gross, 1993). For our study, only lithologic contacts are considered as mechanical layer boundaries. Fractures that terminate at lithologic contacts are termed as "stratabound" while the ones that randomly terminate within the rock mass are "non-stratabound" (Odling et al., 1999; Gillespie et al., 1999). The former often display a log-normal distribution for length or other non-power law type distributions and appear to be regularly spaced as seen in the siliceous layers of the Monterey Formation (Gross et al., 1995). The "non-stratabound" fractures, however, have a wide range of length distributions (e.g. joint patterns at the Oliana anticline, Shackleton et al., 2005), sometimes yielding a power law, and are typically clustered (Odling et al., 1999; Gillespie et al., 1999). Interface strength and the contrast between the rheology of layers control the ability of joints to propagate through lithologic contacts. Analog and numerical experiments suggest that weak interfaces inhibit joint propagation by sliding or opening, and similarly cracks terminate at contacts with soft and ductile layers (Shackleton et al., 2005 and references therein). In this case, the joints developed are "stratabound" and their spacing is proportional to the bed thickness (Narr and Suppe, 1991; Wu and Pollard, 1995 and references therein; Gross et al., 1995; Gillespie et al., 1999; Odling et al., 1999; Cooke et al., 2006). The driving condition for such joint formation is the result of either remote extension or possibly thermal relaxation (Hobbs, 1967; Engelder and Fischer, 1996; Bai and Pollard, 2000). A. Roy et al. Journal of Structural Geology 32 (2010) 1444e1449 1445 Joint spacing distributions can be measured from 1D scanlines (La Pointe and Hudson, 1985). Semi-variograms constructed from such measurements have been independently employed by La Pointe and Hudson (1985) and Chiles (1988) for quantifying the spatial heterogeneity of fracture networks. The ratio of the standard deviation to the mean of the spaces along a scanline has also been used by Gillespie et al. (1999) to discern between clustered and anticlustered veins. Given that rock properties can vary with direction, if possible it is more useful, although certainly more time consuming, to characterize joint spacing distribution in two dimensions using an area or map approach (Wu and Pollard, 1995; Rohrbaugh et al., 2002). In this paper, we present a technique modified from Plotnick et al. (1996) for analyzing clustering of joint populations in a two-dimensional representation. To quantify the clustering of fractures, we use the concept of lacunarity (Mandelbrot, 1983). This approach is based on a multiscale analysis of spatial or temporal dispersion (Plotnick et al., 1996). Stated simply, lacunarity characterizes the distribution of spaces or gaps in a pattern as a function of scale. For a fracture pattern, therefore, it can be employed to quantify the degree of fracture clustering at a given spatial resolution. To implement lacunarity as a tool for our purpose, we have introduced a new normalization of this parameter. It is distinct from that of Plotnick et al. (1996) and completely removes the effect of fracture abundance on the lacunarity values. We use a set of three maps from Wales, U.K (Rohrbaugh et al., 2002) to demonstrate the usefulness of our normalized lacunarity measure over that proposed by Plotnick et al. (1996) and show its effectiveness in discerning between different sets of fractures within the same network. We then use normalized lacunarity to analyze a set of four maps from the Devonian sandstones of Hornelen basin, Norway (Odling, 1997) to investigate clustering of fractures at different scales. Finally, we interpret our observations from this sedimentary package in terms of mechanical stratigraphy as a function of scale. 2. Lacunarity and its quantification A useful conceptual perspective for understanding lacunarity is to evoke the idea of translational invariance. Consider a uniform sequence of alternating 0’s and 1’s like 101010101... This sequence will map onto itself if a copy is made and moved over by two digits so that the original cannot be distinguished from the translated copy. This property is called translational invariance. In terms of lacunarity, a translationally invariant pattern exhibits no clustering, because all of the gap sizes (denoted by zeroes in our example) are the same. This behavior is not observed in the case of a slightly more heterogeneous sequence, such as 101000101... where the gaps have a range of sizes, including a cluster of three gaps in the middle. The greater the degree of gap clustering, the greater the lacunarity. Lacunarity is a scale-dependent parameter because sets that are uniform at a coarse scale might be heterogeneous at a finer scale, and vice-versa. Lacunarity can thus be considered as a scale-dependent measure of textural heterogeneity (Allain and Cloitre, 1991; Plotnick et al., 1993). Quantifying lacunarity as a function of scale can be achieved by using the gliding-box algorithm (Allain and Cloitre, 1991; Plotnick et al., 1996). This algorithm slides a window or box of a given length, r, translated in increments of a chosen unit length across the pattern. In the case of all our analyses, this unit length is chosen to be at the pixel scale (size of the smallest dot that can be drawn on a computer screen). The box-size, r, is generally a multiple of this assigned unit length. The interrogator box searches for occupied sites in the pattern at each step and counts them as s(r). The total number of steps, N(r), required to cover the entire pattern is given by: N(r) = rt / (r - 1) Here, E is the Euclidean dimension of the pattern (for fracture maps, E = 2) and rt is the total length of the set. The first and second moments of the distribution of the number of occupied sites at each step, Z1(r), and Z2(r) respectively, are given by: Z1(r) = s(r) Z2(r) = s^2 - [s(r)]^2 Here s(r) and ss^2(r) are the arithmetic mean and variance of s(r), respectively. The lacunarity is then defined as a function of box-size, L(r), by: L(r) = Z2(r) / [Z1(r)^2] In terms of the mean and variance of s(r) the lacunarity can also be expressed as: L(r) = (s^2 - [s(r)]^2) / [s(r)]^2 Lacunarity is thus the dimensionless ratio of the dispersion (variance) to the square of the central tendency (mean) at a given scale. 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