Multifractal Phenomena in Porous Rocks
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MULTIFRACTAL PHENOMENA IN POROUS ROCKS
J. *
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MULLER , J.P. HANSEN , A.T. SKJELTORP AND J.MCCAULEY Institutt for energiteknikk, 2007 Kjeller, Norway
Physics Department, University of Houston, Houston, TX
77004
ABSTRACT The multifractal spectra of porespace of different types of porous sedimentary rocks are obtained using optical microscopy and a boxcounting technique. The computed f(a) functions were based on the canonical formalism, and can be used as a tool for rock characterization. 1. Introduction Porous sedimentary rocks (see Figure 1, where black regions are pores, and white regions are grains) like many other geometrically complex objects in nature are now known to exhibit statistical selfsimilarity. For these systems, the fractal dimensions are nonintegral, but the fragmentation is too irregular to be well-3 self-similar)'geometrically (i.e. characterized by uniform fractal. The formalism of multifractals 4 - 6 provides a general framework for analysing nonuniform Cantor-like sets where the usual fractal dimension appears as the peak of the f(a) curve, thus suggesting that the the multifractal spectrum may be general enough to analyze much of the statistical irregularity that is found in nature. Therefore, it is important to provide further examples which can help to clarify the extent of applicability and usefulness of the new methods. For example, in turbulence, the inertial range cascade was first modelled as lognormal 7 ,and only later was shown to be fractal 8 , then multifractal' 9. There is an analogous development with the porespace statistics of rocks. There, the distribution of porevolume sizes is broad and was earlier thought to be lognormall°'ll. Later, it was discovered that the porevolume can be characterized by a fractal dimension'2-14. But the fractal dimension is only one number, and many different fractals with entirely different physical properties can have the same fractal dimension. On the other hand the f(a) spectrum provides a complete thermodynamic description of the physical object.
Fig. 1. A typical image of thin section of a porous sedimentary rock. Mat. Res. Soc. Symp. Proc. Vol. 176. ©1990 Materials Research Society
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In this work, we show that porevolume statistics of sedimentary rocks are multifractal, and that the resulting f(a) spectrum can be used as a tool for rock characterization"'16. The extraction of rock's petrophysical properties (such as permeability, conductivity) from the multifractal functions' 7 , and a porous rock modal based on a multifractal Sierpinski carpet' 8 will be published elsewhere. 2. Method Figure 1 shows a typical digitized image of a sedimentary rock. The rock sample is a polished thin section (thickness 30 Vm) bound to a glass substrate. The images of rock samples were digitized using a video frame grabber with 512 x 512 pixel resolution. The canonical f(a) curves were determined for both samples by using a box-counting program based on the multifractal theory. The range where the multifractal scaling is valid was found to be in the
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