Fractal Geometry of Rocks

Abstract

The analysis of small-and ultra-small-angle neutron scattering data for sedimentary rocks shows that the pore-rock fabric interface is a surface fractal ͑D s 2.82͒ over 3 orders of magnitude of the length scale and 10 orders of magnitude in intensity. The fractal dimension and scatterer size obtained from scanning electron microscopy image processing are consistent with neutron scattering data.[S0031-9007(99)08945-0] PACS numbers: 61.43.Hv, 61.12.Ex, Owing to the limited size range over which the fractal properties are usually observed, the issue of the apparent fractal geometry of various natural objects is a contentious one. In their critique of 96 recent reports on the fractality of a wide range of physical systems, Avnir et al. pointed out the contradiction between the narrow range of the appropriate scaling properties for declared fractal objects (centered around 1.3 orders of magnitude) and the public image of the status of experimental fractals [1], which for rocks has previously been based on limited experimental evidence (about 1.5 decades in length scale). A notable exception is the x-ray study of Bale and Schmidt on coals (Ref. [2], 2 decades in length scale, 7.5 decades in intensity). In this study we extend the range of length scales studied for rocks to over 3 decades (10 decades in intensity) and show that sedimentary rocks are in fact one of the most extensive fractal systems found in nature.Sedimentary rocks are formed from a mixture of organic and inorganic debris deposited in an aqueous environment, buried and compacted at elevated temperatures over geological periods of time. Remarkably, there is no percolation threshold observed in sedimentary rocks, which indicates a microstructure more complex than one originating from just a collection of compacted grains. According to the antisintering hypothesis of Cohen, the rock/pore interface evolves by maximizing the internal surface area in response to the secular equilibrium between the rock matrix and the formation brine [3]. Various studies performed on rocks of different origin and lithology over length scales in the range 20 Å to 100 mm have shown that sedimentary rocks are often effective fractals [4]. Experimental tools used in these studies include molecular adsorption [5], microscopic techniques [6,7], and small-angle scattering (SAS) methods. SAS methods are particularly well suited for testing the porematrix interface: They are noninvasive, average over the entire sample volume, and include correlation information. Previous small-angle neutron and x-ray scattering (SANS and SAXS) studies on shales [8-10] and sand-stones [7,9,11] demonstrated the surface fractal geometry of the pore-matrix interface in the scale range 20 Å to about 2000 Å.Recent progress in neutron scattering instrumentation enables one to access the microstructure of rocks well beyond the conventional SANS Q limit of Q min 3 3 10 23 Å 21 . The 80-m SANS instrument D11 at ILL [12] has resolution Q min 8 3 10 24 Å 21 and the Bonse-Hart geometry USANS facility at ORNL [13] c...

Surface and bulk properties of deposits grown with a bidisperse ballistic deposition model

“…Moreover, the apparently universal surface fractal dimension of ballisticlike deposits is another important point for further investigation and suggests possible applications of this class of growth model. Indeed, the estimate D F = 2.9 is very close to the value D F ≈ 2.85 obtained experimentally in sandstones [38,39], and not very far from D F = 2.7 − 2.8 obtained in gold vapor deposition [40]. Another interesting question is what happens in ballisticlike models with varying angles of deposition, which have been recently applied to growth of silicon or silicon compounds [41,42,43].…”

Frequency-dependent viscous flow in channels with fractal rough surfaces

“…Experimental observations, however, indicate that pore roughness may exhibit self-similar characteristics over a wide range of length scales. Experimental studies based on different experimental techniques [23][24][25] (e.g., scanning electron microscopy, small-angle neutron scattering, nitrogen adsorption isotherm) have demonstrated that, for instance, the pore space of sandstones and shales is approximately fractal over length scales ranging from 10Å to 100 µm, with corresponding fractal dimensions varying between 2.57 and 2.87. Other experimental studies 26,27 have proven that fractured-rock wall surfaces also display fractal and multifractal properties over a range of scales between 0.1 µm to 1 cm.…”

Study on microheterogeneity and seepage law of reservoir after long-term water flooding

“…The pore becomes less disperse, and the separation gets better as the data points move closer to the center of mean value. The rule has not been reported before (Angulo et al 1992;Shen et al 1995Shen et al , 1998Becker et al 1997;Perez and Chopra 1997;He and Hua 1998;Radlinski et al 1999;Perez and Miguel 2001;Libny et al 2001;Bijan 2003;Li 2004;Ma et al 2005;Han et al 2006;Li and Horne 2006;Sondergeld et al 2010;Sigal 2013).…”

“…After long-term water flooding in heavy oil reservoir, fractal dimensions of both large and small pores, variable coefficient, skewness, and characteristic function of pore structure decrease as the water cut increases. It indicates that the microheterogeneity of core decreases after long-term water flooding (Angulo et al 1992;Shen et al 1995Shen et al , 1998Becker et al 1997;Perez and Chopra 1997;He and Hua 1998;Radlinski et al 1999 Quantitative study on variation of microheterogeneity with 3D pore network model 3D pore network model is an important tool to study micropore structure and macroscopic seepage characteristics of reservoir. The network model is composed of throats and connected pore bodies.…”

“…Katz and Thompson (1985) firstly applied the fractal geometry theory in analysis of the structure of porous media, which showed that both pore space and interface in porous media have fractal structure. Angulo et al (1992), Becker et al (1997), Perez and Chopra (1997), Shen et al (1998), He and Hua (1998), Radlinski et al (1999), Miguel 2001, Li (2004), Li and Horne (2006), and Ma et al (2005), put forward various mathematical models of computing the fractal dimension of pore structure based on mercury-injection data with fractal theory and achieved a significant fruit.…”