IV. Fragmentation of Rock and Fractal Particle Size Distribution

     

Matthew Otwinowski

Non-linear waste rock modelling

IV. FRAGMENTATION OF ROCK AND 
FRACTAL PARTICLE SIZE DISTRIBUTION

A basic concept useful for understanding the fractal nature of fragmented rock is illustrated in Fig. 4.1. A zero-order cubic cell with dimensions h is divided into eight cubic elements each with dimension h/2. Two diagonally opposite cubes are retained at each scale. We have N₁=2 of particles with R₁=h/2, N₂=12 particles with R₂=h/4; fractal dimention d_f=2.58. This value of the fractal dimension (d_f=2.58) is very close to the experimental values for fragmented rock¹, presented in Fig. 4.2.

Fig. 4.1. An idealized model for fractal fragmentation: R₁=h/2, N(>R₁)=2; R₂=h/4, N(>R₂)=14; R₃=h/8, N(>R₃)=86; …

The number N(>R) of fragments with a linear dimension greater than R is described by a relation

The experimental data for the number N(R) of fragments with a size greater than R are presented in Fig. 4.2 for: broken coal², broken granite³ from a 61 kt underground nuclear detonation, and impact ejecta due to 2.6 km/s projectile impacting on basalt⁴.

Fig. 4.2. Experimental data on the fragmentation of rock for different types of rock.

The distribution function N(R) is obtained from N(>R) as

According to the fractal size distribution formula, in waste rock piles the number of particles grows dramatically with a decrease of particle size. We have used our results on the particle size dependence of leaching rates and the particle size distribution to derive geostatistical formulae for waste rock reactivity.

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¹ If at a certain later stage the geometric division rule was changed, we would obtain a set of particles which cannot be described by a single fractal dimension value. Such a set is usually called a multifractal. The basic concepts of fractal geometry are presented in Appendix A.

² J.G. Bennet, "J. Inst. Fuel" ("Journal of the Institute of Fuel") 10, 22 (1936).

³ J.E. Schoutens, Empirical analysis of nuclear and high-explosive cratering and ejecta, in Nuclear Geoplosics Sourcebook, 55, Def. Nuclear Agency, 1979.

⁴ A.G. Fujiwara A., G. Kamimoto and A. Tsukamoto, "Icarus" 31, 277 (1977).