APPLICABILITY OF FRACTAL CHARACTERIZATION AND MODELLING TO ROCK JOINT PROFILES

Abstract Synthetic records of fractional Brownian motion ( f Bm) with known dimensions were generated using random midpoint displacement, spectral synthesis and deterministic fractal geometry methods. The validity of the methods used to measure the fractal dimensions of self-affine records was evaluated by examining the calculated dimensions of the synthetic f Bm records with known fractal dimensions. Two methods of fractal analysis (i.e. variogram and periodogram) were employed to determine the fractal dimensions of the profiles. Analysis of periodograms of f Bm records revealed a higher correlation between the expected and calculated dimensions of pre-whitened data. Variograms of those f Bm records indicated a better accuracy while smaller lag (i.e. approx. 10% of the profile length) of the synthetic profile length was used for determination. Laser profilometry was utilized as well to provide the measurement of joint roughness. Several natural rock joints were cored parallel to the joint faces. The cores were then analyzed with a laser profilometer for asperity measurement. The fractal dimensions of the natural joint profiles determined by using variograms and periodograms ranged from 1.3 to 1.4. The periodogram method, however, produced slightly higher dimensions than did the variogram method. Synthesized joint profiles of known dimensions were generated using fractal interpolation functions to provide a qualitative comparison with the natural joints. The computer generated profiles were found to mimic many of the visual characteristics of natural rock joints.

[1]  B. Mandelbrot.  Self-Affine Fractals and Fractal Dimension , 1985 .

[2]  J. Carr,et al.  The fractal dimension as a measure of the roughness of rock discontinuity profiles , 1990 .

[3]  W. R. Dearman,et al.  Characterization of rock joint surfaces by fractal dimension , 1987 .

[4]  K. Aki,et al.  Fractal geometry in the San Andreas Fault System , 1987 .

[5]  Michael F. Barnsley,et al.  Fractals everywhere , 1988 .

[6]  R. Reyment,et al.  Statistics and Data Analysis in Geology. , 1988 .

[7]  S. Hough.  On the use of spectral methods for the determination of fractal dimension , 1989 .

[8]  Donald S. Fussell,et al.  Computer rendering of stochastic models , 1982, Commun. ACM.

[9]  R. Voss.  Random Fractal Forgeries , 1985 .

[10]  J. Elliot,et al.  An investigation of the change in surface roughness through time on the foreland of Austre Okstindbr , 1989 .

[11]  Christopher H. Scholz,et al.  Fractal analysis applied to characteristic segments of the San Andreas Fault , 1987 .

[12]  C. Fox.  Empirically derived relationships between fractal dimension and power law form frequency spectra , 1989 .

[13]  P. G. Earwicker,et al.  Fractal properties of computer-generated and natural geophysical data , 1989 .

[14]  Stephen R. Brown,et al.  Broad bandwidth study of the topography of natural rock surfaces , 1985 .

[15]  Stephen R. Brown.  A note on the description of surface roughness using fractal dimension , 1987 .

[16]  Stephen R. Brown,et al.  Correlation between the surfaces of natural rock joints , 1986 .

[17]  B. Mandelbrot.  How Long Is the Coast of Britain? Statistical Self-Similarity and Fractional Dimension , 1967, Science.

[18]  D. Mark,et al.  Scale-dependent fractal dimensions of topographic surfaces: An empirical investigation, with applications in geomorphology and computer mapping , 1984 .

[19]  P. Burrough.  Fractal dimensions of landscapes and other environmental data , 1981, Nature.

[20]  C. G. Fox.  An inverse Fourier transform algorithm for generating random signals of a specified spectral form , 1987 .

[21]  B. Mandelbrot.  Stochastic models for the Earth's relief, the shape and the fractal dimension of the coastlines, and the number-area rule for islands. , 1975, Proceedings of the National Academy of Sciences of the United States of America.

[22]  James R. Carr,et al.  Relationship between the Fractal Dimension and Joint Roughness Coefficient , 1989 .

[23]  D. Saupe.  Algorithms for random fractals , 1988 .

[24]  Richard F. Voss,et al.  Fractals in nature: from characterization to simulation , 1988 .

[25]  S. Orey.  Gaussian sample functions and the Hausdorff dimension of level crossings , 1970 .

[26]  A. Malinverno.  Testing linear models of sea-floor topography , 1989 .