Abstract
The distributions of contact area and void space in single fractures in granite rock have been determined experimentally by making metal casts of the void spaces between the fracture surfaces under normal loads. The resulting metal casts on 52 cm diameter core samples show a complex geometry for the flow paths through the fracture. This geometry is analyzed using finite-size scaling. The spanning probabilities and percolation probabilities of the metal casts are calculted as functions of observation scale. Under the highest stresses of 33 MPa and 85 MPa there is a significant size-dependence of the geometric flow properties for observation scales smaller than 2 mm. Based on this data, the macroscopic percolation properties of the extended fracture can be well represented by relatively small core samples, even under normal stresses larger than 33 MPa. The metal casts also have rich multifractal structure that changes with changing stress.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Armitrano, C., Coniglio, A., andDi Liberto, F. (1986),Growth Probability Distribution in Kinetic Aggregation Processes, Phys. Rev. Lett.57, 1016.
Benzi, R., Paladin, G., Parisi, G., andVulpiani, A. (1984),On the Multifractal Nature of Fully Developed Turbulence and Chaotic Systems, J. Phys. A17, 3521–3531.
Bhatti, F. M., andEssam, J. W. (1986),Series Expansion Study of the Distribution of Currents in the Elements of a Random Diode-insulator Network, J. Phys. A19, L519-L525.
Blumenfield, R., Meir, Y., Harris, A. B., andAharony, A. (1986),Infinite Set of Exponents Describing Physics on Fractal Networks, J. Phys. A19, L791.
Brown, S. R., andScholz, C. H. (1985),Closure of Random Elastic Surfaces in Contact, J. Geophys. Res.90, 5531–5545.
Brown, S. R., Kranz, R. L., andBonner, B. P. (1986),Correlation between the Surfaces of Natural Rock Joints, Geophys. Res. Lett.13, 1430–1434.
de Arcangelis, L., Redner, S., andConiglio, A. (1985),Anomalous Voltage Distribution of Random Resistor Networks and a New Model for the Backbone at the Percolation Threshold, Phys. Rev. B31, 4725.
de Arcangelis, L., Redner, S., andConiglio, A. (1986),Multiscaling Approach in Random Resistor and Random Superconducting Networks, Phys. Rev. B34, 4656.
Dullien, F. A. L. (1981),Wood's Metal Porosimetry and its Relation to Mercury Porosimetry, Powder Technology29, 109–116.
Feder, J.,Fractals (Plenum Press, New York 1988).
Frisch, U., andParisi, G. (1983),Turbulence and Predictability of Geophysical Flows and Climate Dynamics, Varenna Summer School LXXXVIII.
Grassberger, P., andProcaccia, I. (1983),Characterization of Strange Attractors, Phys. Rev. Lett.50, 346–349.
Grassberger, P., andProcaccia, I. (1984),Dimensions and Entropies of Strange Attractors from a Fluctuating Dynamics Approach, Physica D13, 34–54.
Halsey, T. C., Jensen, M. H., Kadanoff, L. P., Procaccia, I., andShraiman, B. I. (1986),Fractal Measures and their Singularities: The Characterization of Strange Sets, Phys. Rev. A33, 1141.
Hao, L., andYang, Z. R. (1987),Lacunarity and Universality, J. Phys. A., Math Gen.20, 1627–1631.
Hopkins, D. L., andCook, N. G. W.,Fracture stiffness and aperture as a function of applied stress and contact geometry. InRock Mechanics (eds. Farmer, I., Daemen, J., Desai, C., Glass, C., and Neuman S.) (A. A. Balkema, Rotterdam 1987) pp. 673–680.
Hopkins, D. L., Cook, N. G. W., andMyer, L. R.,Normal joint stiffness as a function of spatial geometry and surface roughness. InRock Joints (eds. Barton, N., and Stephansson, O. (A. A. Balkema, Rotterdam) pp. 203–210.
Hoshen, J., andKopelman, R. (1976),Percolation and Cluster Distribution. I. Cluster Multiple Labeling Technique and Critical Concentration Algorithm, Phys. Rev. B14, 3438.
Lin, B., andYang, Z. R. (1986),A Suggested Lacunarity Expression for Sierpinski Carpets, J. Phys. A., Math Gen.19, L49-L52.
Mandelbrot, B. B.,The Fractal Geometry of Nature (W. H. Freeman and Company, New York 1983) 468 pp.
Mandelbrot, B. B. (1989),The Multifractal Measures, Especially for the Geophysicist, Pure and Appl. Geophys.131 (1/2), 5–42.
Meakin, P., Coniglio, A., Stanley, H. E., andWitten, T. A. (1986),Scaling Properties for the Surfaces of Fractal and Nonfractal Objects: An Infinite Hierarchy of Critical Exponents, Phys. Rev. A34, 3325.
Neuzil, C. E., andTracey, J. V. (1981),Flow Through Fractures, Water Resources Res.17 (1), 191–199.
Nolte, D. D. (1989),Invariant Fixed Point in Stratified Continuum Percolation, Phys. Rev. A40 (8), 4817–4819.
Nolte, D. D., Pyrak-Nolte, L. J., andCook, N. G. W. (1989),The Fractal Geometry of Flow Paths in Natural Fractures in Rock and the Approach to Percolation, Pure and Appl. Geophys.131 (1/2), 271.
Nolte, D. D., andPyrak-Nolte, L. J. (1991),Stratified Continuum Percolation: Scaling Geometry of Hierarchical Cascades, Phys. Rev. A44, 6320–6333.
Paladin, G., andVulpiani, A. (1984),Characterization of Strange Attractors as Inhomogeneous Fractals, Lett. Nuovo Cimento41, 82–89.
Pfeuty, P., andToulouse, G.,Introduction to the Renormalization Group and to Critical Phenomena (John Wiley and Sons, London 1975).
Pyrak-Nolte, L. J., Meyer, L. R., Cook, N. G. W., andWitherspoon, P. A.,Hydraulic and mechanical properties of natural fractures in low permeability rock. InProceedings, International Society for Rock Mechanics, 6th International Congress on Rock Mechanics, Montreal, Canada, August 1987 (A. A. Balkema, Rotterdam 1987) pp. 225–231.
Pyrak-Nolte, L. J. (1991),The Feasibility of Using Wood's Metal Porosimetry to Measure the Fracture Void Geometry of Cleats in Coal, Topical Report, Gas Research Institute GRI-91-0373, 22 pp.
Rammal, R., Tannous, C., Breton, P., andTremblay, A. M. S. (1985),Flicker (1/f) Noise in Percolation Networks: A New Hierarchy of Exponents, Phys. Rev. Lett.54, 1718.
Rammal, R., Tannous, C., andTremblay, A. M. S. (1986),1/f Noise in Random Resistor Networks: Fractals and Percolating Systems, Phys. Rev. A31, 2662.
Raven, K. G., andGale, J. E. (1985),Water Flow in a Natural Rock Fracture as a Function of Stress and Sample Size, Internat. J. Rock Mechanics and Mining Sc. and Geomech. Abstracts22 (4), 251–261.
Redner, S. (1990),Random Multiplicative Processes: An Elementary Tutorial, Am. J. Phys.58, 267–273.
Stanley, H. E.,Introduction to Phase Transitions and Critical Phenomena (Oxford University Press, New York 1971).
Stauffer, D.,Introduction to Percolation Theory (Taylor and Francis, London 1985).
Swan, G. (1983),Determination of Stiffness and Other Joint Properties from Roughness Measurements, Rock Mechanics and Engineering6, 19–38.
Swanson, B. F. (1979),Visualizing Pores and Non-wetting Phase in Porous Rock, J. Petroleum Technology31, 10–18.
Taguchi, Y. (1987),Lacunarity and Universality, J. Phys. A., Math Gen.20, 6611–6616.
Tsang, Y. W., andWitherspoon, P. A., (1983),The Dependence of Fracture Mechanical and Fluid Flow Properties on Surface Roughness and Sample Size, J. Geophys. Res.88 (B3), 2359–2366.
Wilson, K. G., andKogut, J. (1974),The Renormalization Group and the Epsilon Expansion, Physics Reports12C, 75–200.
Wilson, K. G. (1975),The Renormalization Group: Critical Phenomena and the Kondo Problem, Adv. Phys.47 (4), 773–840.
Wilson, K. G. (1979),Problems in Physics with Many Scales of Length, Scientific America241 (2), 158–179.
Witherspoon, P. A., Amick, C. H., Gale, J. E., andIwai, K. (1979),Observations of a Potential Size Effect in Experimental Determination of the Hydraulic Properties of Fractures, Water Resources Res.15 (5), 1142.
Yadav, G. D., Dullien, F. A. L., Chatzis, I., andMacDonald, I. F. (1984),Microscopic Distribution of Wetting and Non-wetting Phases in Sandstones during Immiscible Displacements, SPE 13212, Society of Petroleum Engineers of AIME, Dallas, Texas.
Zheng, Z. (1989),Compressive Stress-induced Microcracks in Rocks and Applications to Seismic Anisotropy and Borehole Stability, Ph.D. Thesis, University of California, Berkeley.
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Pyrak-Nolte, L.J., Myer, L.R. & Nolte, D.D. Fractures: Finite-size scaling and multifractals. PAGEOPH 138, 679–706 (1992). https://doi.org/10.1007/BF00876344
Accepted:
Issue Date:
DOI: https://doi.org/10.1007/BF00876344