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

Tunnel exploration is a classic problem of decision making under uncertainty. This paper, Part II, dealing with the exploration problem, discusses the sources of uncertainty in tunnel exploration, and provides techniques to formally incorporate uncertainties into the exploration planning. Part I described the state of uncertainty in geology and exploration reliability using assigned probabilities. These probabilities are, however, uncertain. Construction costs are also uncertain. This paper presents tools with which the decision maker is able to assess the effects of these uncertainties on exploration decisions. Decisions are therefore made based on the value of collecting new, uncertain information through “virtual” exploration prior to actually exploring.

Decision Analysis Applied to Tunnel Exploration Planning. II: Consideration of Uncertainty

For hydraulic fracturing operations in the field, the pumping rate is generally controlled but it is not fully understood how this affects the resulting fracture processes. This paper presents the results of an experimental study using a laboratory hydraulic fracturing setup, in which the injection rate was varied across four experiments, and the fracturing process was observed visually as well as with acoustic emission sensors. Prismatic 2 x 4 x 1 in. specimens of Opalinus clay shale, with a pre-cut 0.33 in. width flaw, were subjected to hydraulic pressure under constant biaxial far-field loading conditions. The pressure was measured internally inside the flaw as well as at the pressure volume actuator. Both a high speedand a high resolution camera were used to visually record changes on the face of the specimen (i.e. fracturing). Acoustic emissions were recorded with an array of eight piezoceramic sensors embedded in specialized platens. The results of the experiments were then used in an analysis of the peak pressures and fracture propagation speed related to the injection rate. Higher peak pressures, fracturing speeds and fracturing accelerations were observed with higher injection rates. Additionally, the spectral analysis of the largest AE events showed that the highest injection rates resulted in higher power at lower frequencies. The highest injection rate was also associated with greater AE activity in general.

Effect of Injection Rate on Hydraulic Fracturing of Opalinus Clay Shale

Development of stochastic models for fracture systems

Soil and rock America 2003 : 12th Panamerican Conference on Soil Mechanics and Geotechnical Engineering, 39th US Rock Mechanics Symposium, June 22 - 26, 2003, Cambridge, Massachusetts, USA : proceedings

Herbert H. Einstein

Papers

Decision Analysis Applied to Tunnel Exploration Planning. II: Consideration of Uncertainty

Effect of Injection Rate on Hydraulic Fracturing of Opalinus Clay Shale

Development of stochastic models for fracture systems

Soil and rock America 2003 : 12th Panamerican Conference on Soil Mechanics and Geotechnical Engineering, 39th US Rock Mechanics Symposium, June 22 - 26, 2003, Cambridge, Massachusetts, USA : proceedings

Feasibility analysis for a radioactive waste repository tunnel