materials science 01

The fractal geometry of rocks

A. P. Radlinski (Australian Geological Survey Organisation, Canberra), E. Z. Radlinska (The Australian National University, Canberra), M. Agamalian, G. D. Wignall (Oak Ridge National Laboratory), P. Lindner, O. G. Randl (ILL).

Neutron scattering has been used to study the fractal microstructure of sedimentary rocks. The results have proven to be useful in prospecting for oil. “Immature” shales, which have the right geochemical characteristics to produce oil but have not yet generated, typically have a high fractal dimension, like the rocks in this study. By contrast, in “mature” oil-bearing rock, the fractal dimension for large pores drops to nearly 2, reflecting a smooth interface between the grain-surface-covering mobile bitumen and the pore space

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. The fractal structure of these systems is of great interest to oil industry because it could indicate a rock's oil-producing potential.

The small-angle scattering technique (SANS), using cold neutrons, has been applied to a hydrocarbon source rock U116, originating from the Urapunga 4 well in the MacArthur Basin, Northern Territory, Australia [1]. A series of SANS experiments have been performed using the instrument D11 at ILL at the wavelength of 14 Å, as well as with the ORNL facility at 4.75 Å. Recent progress in neutron scattering instrumentation at ORNL allowed the microstructure of rocks to be studied well beyond the conventional limit and into the range of ultra small angle scattering (USANS). In particular, the ORNL ultra-small angle neutron scattering (USANS) facility [2] can probe Q-ranges down to Q ~ 2.10^—5Å^—1(momentum transfer Q = 4p^—1 sinq, scattering angle = 2 ).

Figure 1 shows a combination of SANS data taken on the ILL and ORNL SANS instruments, along with USANS data [3]. It may be seen that all three data sets overlap smoothly with no adjustable parameters and the differential scattering cross section in the log-log representation can be approximated by a straight line above Q = 2.10^—4 Å^—1. Detailed analysis of the data in Fig. 1 shows that the pore-rock fabric interface is a surface fractal with dimension DS = 2.82 over three orders of magnitude of the length scale and ten orders of magnitude in cross section. To our knowledge, these data represent the largest range to date over which fractal behaviour has been observed in a natural system. Such an extent of fractal microstructure in a rock is remarkable, when compared with the limited size range over which the fractal properties are usually observed (typically one order of magnitude) [4].

Figure 1: SANS and USANS data from sedimentary rock showing that the pore-rock interface is a surface fractal (D_s = 2.82) over three orders of magnitude in length scale and ten orders of magnitude in cross section (intensity).

This study extends the widest fractal length range previously observed in sedimentary rocks, which covered 2 decades in length scale and 7.5 decades in intensity [5], and shows that sedimentary rocks are in fact one of the most extensive fractal systems found in nature.
The fractal character of the scattering cross-section eventually breaks down at scales larger than several micrometres. This is reflected in a flattening out of the scattering curve in Fig. 1 for smallest Q-values. In real space, the fractal dimension can be quantified by counting the frequency of structural features of given size on scanning electron microscope (SEM) images of the rock. Figure 2 illustrates statistical data for SEM images of rock U116, with the breakdown of fractal properties at the scale of 4 micrometres clearly observed as a sharp change of slope.

In conclusion, a “mature” (oil bearing) sample would have its fractal dimension decreased compared to the immature sample. Also, due to the decrease of contrast (bitumen-filled pores as opposed to water filled pores), the scattering intensity at a given Q-value is smaller for mature samples that have generated hydrocarbons than for immature samples. Therefore, the technique has the potential to distinguish the source rocks that have produced hydrocarbons from those that have not.

As observed in Physical Review Focus [6], “The constancy of the fractal dimension over so many scales is astounding, considering what a messy, heterogeneous material sedimentary rock appears to be... This study will enhance the idea that you can describe rock with simple concepts ... There will be certain bona fide uses of fractal concepts, and one of them will be in the geological sciences”.

Figure 2: Variation of the average number of SEM features per unit length (N/L) with feature size obtained from SEM images of the cleaved surface of sedimentary rock U116. Note the breakdown of fractality (D_s = 2.8 to 2.9) for length scales larger than 4 micrometres. Peter Lindner (centre) and his collaborators aligning a sample on D11.

[1] A. P. Radlinski, C. J. Boreham, G. D. Wignall and J. S. Lin, Phys. Rev. B53 (1996) 14152. [2] M. M. Agamalian, G. D. Wignall and R. Triolo, J. Appl. Cryst. 30 (1997) 345. [3] A. P. Radlinski, E. Z. Radlinska, M. Agamalian, G. D. Wignall, P. Lindner, O. G. Randl, Phys. Rev. Lett. 82 (1999) 3078. [4] D. Avnir, O. Biham, D. Lidar, O. Malcai, Science 279 (1997) 39. [5] H. D. Bale, P. W. Schmidt, Phys. Rev. Letters 53 (1984) 596. [6] Physical Review Focus, 20/04/99.


Figure 2: Variation of the average number of SEM features per unit length (N/L) with feature size obtained from SEM images of the cleaved surface of sedimentary rock U116. Note the breakdown of fractality (D_s = 2.8 to 2.9) for length scales larger than 4 micrometres.		Peter Lindner (centre) and his collaborators aligning a sample on D11.