Scaling in fracture systems has become an active field of research in the last 25 years motivated by practical applications in hazardous waste disposal, hy- drocarbon reservoir management, and earthquake haz- ard assessment. Relevant publications are therefore spread widely through the literature. Although it is rec- ognized that some fracture systems are best described by scale-limited laws (lognormal, exponential), it is now recognized that power laws and fractal geometry provide widely applicable descriptive tools for fracture system characterization. A key argument for power law and fractal scaling is the absence of characteristic length scales in the fracture growth process. All power law and fractal characteristics in nature must have upper and lower bounds. This topic has been largely neglected, but recent studies emphasize the importance of layering on all scales in limiting the scaling characteristics of natural fracture systems. The determination of power law expo- nents and fractal dimensions from observations, al- though outwardly simple, is problematic, and uncritical use of analysis techniques has resulted in inaccurate and even meaningless exponents. We review these tech- niques and suggest guidelines for the accurate and ob- jective estimation of exponents and fractal dimensions. Syntheses of length, displacement, aperture power law exponents, and fractal dimensions are found, after crit- ical appraisal of published studies, to show a wide vari- ation, frequently spanning the theoretically possible range. Extrapolations from one dimension to two and from two dimensions to three are found to be nontrivial, and simple laws must be used with caution. Directions for future research include improved techniques for gathering data sets over great scale ranges and more rigorous application of existing analysis methods. More data are needed on joints and veins to illuminate the differences between different fracture modes. The phys- ical causes of power law scaling and variation in expo- nents and fractal dimensions are still poorly understood.

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Scaling of fracture systems in geological media

A review of techniques, advances and outstanding issues in numerical modelling for rock mechanics and rock engineering

Empirical estimation of rock mass modulus

The theory of flow through fractured rock and homogeneous anisotropic porous media is used to determine when a fractured rock behaves as a continuum. A fractured rock can be said to behave like an equivalent porous medium when (1) there is an insignificant change in the value of the equivalent permeability with a small addition or subtraction to the test volume and (2) an equivalent permeability tensor exists which predicts the correct flux when the direction of a constant gradient is changed. Field studies of fracture geometry are reviewed and a realistic, two-dimensional fracture system model is developed. The shape, size, orientation, and location of fractures in an impermeable matrix are random variables in the model. These variables are randomly distributed according to field data currently available in the literature. The fracture system models are subjected to simulated flow tests. The results of the flow tests are plotted as permeability ‘ellipses.’ The size and shape of these permeability ellipses show that fractured rock does not always behave as a homogeneous, anisotropic porous medium with a symmetric permeability tensor. Fracture systems behave more like porous media when (1) fracture density is increased, (2) apertures are constant rather than distributed, (3) orientations are distributed rather than constant, and (4) larger sample sizes are tested. Preliminary results indicate the use of this new tool, when perfected, will greatly enhance our ability to analyze field data on fractured rock systems. The tool can be used to distinguish between fractured systems which can be treated as porous media and fractured systems which must be treated as a collection of discrete fracture flow paths.

/pdf/porous-media-equivalents-for-networks-of-discontinuous-3bc2n5am14.pdf

Porous media equivalents for networks of discontinuous fractures

Fracture coalescence in rock-type materials under uniaxial and biaxial compression

Statistical description of rock properties and sampling.

The effect of discontinuity persistence on rock slope stability

Suggested methods for determining the strength of rock materials in triaxial compression: revised version

An improved closed-form solution is developed for calculating tunnel support thrusts, moments, and displacements. The solution directly models the interaction of an elastic, circular support with an elastic, isotropic, infinite ground mass. The major improvements in this solution over other solutions in the literature are: (1)A revised formulation of the compressibility ratio; and (2)the incorporation of the correct excavation unloading conditions for mined tunnels. The sensitivity of the tunnel support loads to variations in the relative support stiffness, support cross section, Poisson's ratio, and initial lateral stress ratio is demonstrated using the revised solution; these results are also compared to those from two other solutions in the literature. Discrepancies of up to 100% are found in the support forces calculated from the different solutions.

Herbert H. Einstein

Papers

Statistical description of rock properties and sampling.

The effect of discontinuity persistence on rock slope stability

Suggested methods for determining the strength of rock materials in triaxial compression: revised version

Simplified analysis for tunnel supports

Artificial neural networks for predicting the maximum surface settlement caused by EPB shield tunneling