Rocks as Fractals

Phys. Rev. Focus 3, 22
Figure caption
Eric Weisstein, University of Virginia
Menger sponge. Mathematical fractals like this Menger sponge maintain their complexity (or roughness) over infinitely many length scales. New research shows that rocks are equally rough over at least three length scales.

Complex yet strangely beautiful, fractals are mathematical curves and surfaces that look equally rugged no matter how many times you magnify them. For this reason, they have been used to describe all sorts of irregular shapes in nature, including coastlines, arteries, ferns, and rocks. But scientists have done rather badly at proving any of these items are truly fractal–in other words, that they are self-similar over many different size scales. In the 12 April PRL, a team of six physicists reports that they have found the best experimental proof of fractality yet, in sedimentary rocks.

In most experimental systems, there is a limit to how many times you can “zoom in” before the geometry smooths out. A network of arteries, for example, might look very intricate under tenfold or hundredfold magnification, but if you zoom in again all you see is an individual capillary. Until now, physicists have not found a single object that maintains the same level of complexity over more than three zooming-in steps, or “decades.” An international team of physicists, led by Andrzej Radlinski of the Australian Geological Survey Organization in Canberra, has now surpassed this mark by using a beam of neutrons to “x-ray” (in the colloquial sense) the interior of a rock.

To the low-energy neutrons, the rock is nearly transparent. “It’s like shining a light at a piece of glass,” Radlinski says. But when there are air bubbles in a piece of glass, the light bounces off the boundary interface between the air and the glass, and the glass appears cloudy. Similarly, the pores of air, water and oil in a rock deflect the neutron beam ever so slightly, with different sizes of pores scattering the beam at different angles. To obtain over three size scales, the experimenters had to measure extraordinarily small angles of deflection, using a one-of-a-kind instrument at the Oak Ridge National Laboratories called a double crystal diffractometer. By plotting the intensity against the angle of scatter, Radlinski and his colleagues estimated the fractal dimension, a measure of the roughness of the interface between the pores and the rock. It was very rough indeed, with a dimension of 2.82. (A classical smooth interface would have a dimension of 2, and the “Menger sponge” shown here has dimension 2.73.)

Although calculating the fractal dimension might seem like a purely mathematical hobby, Radlinki says that the method has already proven to be useful in prospecting for oil. “Immature” shales, which have the right chemical composition to produce oil but have not done so yet, 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 oil and rock.

According to Arthur Thompson, a statistical physicist at Exxon Production Research Corporation in Houston, the constancy of the fractal dimension over many scales is “astounding,” considering what a messy, heterogeneous material sedimentary rock appears to be. “This will enhance the credibility of the idea that you can describe rock with simple concepts,” Thompson adds. A previous flurry of interest in fractals in the mid-1980s died out due to the lack of data to support the models, but now, he says, “The emotionalism associated with fractals is over. There will be certain bona fide uses of fractal concepts, and one of them will be in the geological sciences.”

–Dana Mackenzie

Dana Mackenzie is a freelance science writer.

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