DISCUSSION: Fractal analysis of temperature-induced cracking in clays and rocks. L. E. VALLEJO (2009). Géotechnique 59, No. 3, 283–286 Free

Vallejo (2009) has presented an interesting and useful contribution to the troublesome problem of estimating permeability from images showing the structure of soil or rock. The method is: divide the image into boxes, that is small squares; count the number of boxes which intercept voids; repeat with boxes of a different size; and hence obtain a fractal dimension which characterises the structure. As far as permeability and other geotechnical properties are concerned, this is a two-stage process: first assess the condition of the material; then estimate the value of the property of interest.

For simplicity, the following restrictions are made in the discussion below. The image has been pre-processed to: (a) divide it into solids and voids; and (b) fill in any small voids which should be treated as noise. There is always at least one void remaining after pre-processing. The image is composed of pixels whose centres lie on a square grid. Dimensions are quoted in multiples of the pitch of this grid. The image is square of side L*. The boxes, which are square, are of side L. Integers are represented by L and L*; and the boxes exactly fill the image.

Then, the number of boxes, N, is given by

Taking common logarithms, and putting N* = L*²

The above is represented by the straight line TS in Fig. 4, where T corresponds to L = 1 and N = N*, and S corresponds to L = L* and N = 1. (To avoid too great a slope, the scales of the axes in the figure differ; and the ticks merely illustrate this difference.)

Point P corresponds to L = 1 and N = PN*, where P is the ratio of void pixels to total pixels; that is, P is the apparent porosity. (P will be smaller than the true porosity when some voids are too small to be seen.) Thus, P represents the number of boxes which intercept voids when L = 1. Further, S represents the number of boxes which intercept voids when L = L*. Thus, the number of boxes which intercept voids, V, must lie on a curve connecting P and S. In the simplest case, the curve PS is a straight line, and

where V* = P N*, and x is the version of the fractal dimension which is to be found.

When the curve PS is a straight line

This equation is a necessary condition for the existence of a unique fractal dimension and might provide a useful test when only the first few points along the curve PS have been determined.

An alternative representation is given by considering f = V/N, that is the fraction of boxes which intercept voids, see Fig. 5 (Note that f is non-dimensional, unlike both V and N, which denote numbers per a defined area.) For the ideal case of a unique fractal dimension, the curve PS now becomes the straight line

For any case, ideal or non-ideal, the curve PS in Fig. 5 must rise monotonically from P to S, because the larger the boxes become, the more of them intercept voids. As a rough guide, a box of side L will intercept the nearest void when its centre is 0·6L from the void; this distance varies between L/2 and L/√2 with the inclination of the void. On this basis, the assessment relies on estimates of the proportion of the material within or close to voids. Replacing the square boxes by circular filters would remove the variation with inclination. This could be implemented by using a distance transform to label each solid pixel with its distance from the nearest void pixel; and constructing a curve similar to the non-dimensional version of PS from the histogram of the transformed image after masking its borders. Any good commercial image analysis system would provide the tools; and the method could be easily extended to images of inconvenient size and shape.