GEOPHYSICAL RESEARCH LETTERS, VOL. 17, NO. 6, PAGES 701-704, MAY 1990 FRACTALSTUDY AND SIMULATION OF FRA•RE ROUGHNESS Sunil Kumarand Gudmundur S. Bodvarsson Lawrence Berkeley Laboratory, University ofCalifornia, Berkeley Abstract. This study examines theroughness profiles of the surfaces of fractures and faultsby using concepts from fractal geometry. Relationships between fractal characteris- tics of profiles and isotropicsurfaces are analytically developed anda deterministic representation of therough- ness is examined. Introduction Surfaces of fractures andfaults exhibit two distinct pro- perties that make them amenable for analysis by theappli- cation of the theoryof fractal geometry [Mandelbrot, 1982; Feder, 1988]. The firstis thatroughness profiles of thesur- face seemto be nowhere differentiable thoughthey are continuous. The other is that the profileis either self-similar or self-similar with different scaling along different axis (i.e., se!f-affine) over a large range of length scales [Man- delbrot, 1982; Brown and Scholz, 1985b]. Previous studies in the literature have established the fractal dimension of various geophysical surfaces but have relied on numerical simulations [Fournier et al., 1982] based on random number generators to simulate [Brown, 1989] and study these surfaces [Wang et al., 1988]. The present analysis focuses on a deterministic fractal representation of rough- ness profiles androughsurfaces. The fractal representation has the advantage that it preserves the statistics and the self-similarity exhibited by thesurface roughness. Coupled with mathematical analysis from the literature [Nayak,1973;Wanget al., 1988], it also enables the characteristics of surfaces and apertures to be predicted analytically from the profilecharacteristics. The analytical / deterministic expression of fractal formulation additionally offers a tremendous information compression since the roughness can be uniquely re-created with the selection of a few constants. Other advantages of thedeter- ministic fractal formulation are that the various values of interest (such as contactarea) can be evaluated semi- analytically, andprofiles and surfaces exhibiting a "corner frequency" andweakly anisotropic surfaces canbe easily simulated. FractalCharacteristics of Roughness It has been noted by Hough [1989] that a self-a/fine or self-similar fractal distribution is characterized by three essential characteristics: i) it must have appropriate correla- tion over many length scales, ii) it must becontinuous but notdifferentiable, and iii) it musthavea random phase. The first two criteria can be met by examining the power Copyright 1990 by the American Geophysical Union. Paper number 90GL00679 0094-8276 / 90/ 90GL-00679 $ 03 ß 00 spectrum S((o) which must have a power-law slope corresponding to c0to the power-e wheree mustbe less thanthree (e greater thanthree wouldimply a differentiable profile). Experimentally observed profilesof fracture and other rock surfaces indicate that above is indeed satisfied on the experimental level [Brown and Scholz, 1985a,b]. The third criterion is not discussed further since random phases are introduced in the deterministic representations discussed in the present work. The power spectrum of a profile satisfying the above is [Wanget al., 1988; Hough, !989] 1 S(•)o,: , I<D <2 , (1) C05-2D where D is the fractal dimension of the profile. Usingthe above power spectrum and by assuming thatthe surface is isotropic and that its auto-correlation function is the same as thatof a profile, the powerspectrum of the correspond- ing isotropic surfacecan obtainedby considering the analysis of Nayak [1973] and Wang et al. [1988]. This yields thefollowing predicted surface spectral density Ss(c0) = 5-2D F(3-D )F(0.5) S(0)) . (2) 2re F(3.5-D ) (o The difference in the spectral densities indicates that the statistical parameters, such as mean square height, of a profile are notexpected to be identical to that of a surface. This is physically intuitive since the profile is expected to miss most of the highest peaks anddeepest troughs on the surface. Deterministic Representation of Roughness Profiles The deterministic formulation selected for the represen- tation of surfaceroughness profile is the Weierstrass- Mandelbrot cosine fractal function [Berryand Lewis, 1980; Feder, 1988; Majumdar, 1989]which is expressed as o•cos(2•:13nx) 1 <D <2 (3) z (x )= A Z ' ' ß where I• is a non-integral constant greater than unity, and A is a scaling constant. As the fracta! dimension D increases, heights of nearby points become more independent andthe surface becomes increasingly jagged. The abovefunction is continuous at all points but has no derivative at anypoint [Berry and Lewis, 1980; Feder, 1988]. The difference between the Weierstrass-Mandelbrot function and conven- tional Fourier series is in the frequencymodes. The discrete frequencies in a Fourier series increase in arith- metic progression as multiples of a basic frequency and the phases of several modes coincide at certain points which impart a periodic behavior. On theother hand, the frequen- 701