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.

/pdf/scaling-of-fracture-systems-in-geological-media-1uq6op1ap8.pdf

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

Crack Coalescence Tests on Granite

Laboratory Experiments On Geometric And Hydromechanical Characteristics of Three Different Fractures In Granodiorite

The Decision Aids for Tunneling (DAT) is a computer based tool with which one can obtain distributions of the tunnel constructin cost and time considering uncertainties of the geology and construction process for a given tunnel. As construction progresses, geologic information is checked based on the excavated part of the tunnel and, therefore, the uncertainty about this part of the tunnel will disappears. This new information can be used to update the geology of the tunnel in order to obtain more precise prediction during construction. In this paper, the new study on the application of the DAT to update the geology of the given tunnel is introduced. DAT simulation is updated based on face-mapping data and two approaches are used for the geologic updating of the given tunnel. The results of simulations for the initial prediction and the updated prediction will be compared in order to see how the construction time and cost distributions of the given tunnel are changed after the geologic updating.

Application of the Decision Aids for Tunneling (DAT) to update excavation cost/time information

Stochastic Analysis of Surface Roughness, Aperture And Flow In a Single Fracture

Pore pressures below hydrostatic (underpressures) have been measured at the Wellenberg site in Switzerland, that Nagra (Swiss National Cooperative for the Disposal of Radioactive Waste) has selected as a potential repository for short-lived radioactive waste. The rock consists of overconsolidated Cretaceous marls and Tertiary shales, characterized by low to very low hydraulic conductivity. The maximum measured underpressures are between 4.8 and 7.1 MPa below hydrostatic and occur at depths of ∼ 800 m below ground. Results from this investigation show that mechanical unloading is capable of producing the observed underpressures. Unloading can be produced in conjunction with denudation (uniform erosion), differential erosion, and glaciation. The effect of these processes, of relevant material parameters, and of boundary conditions are investigated in two phases. In the first phase, simple one-dimensional and two-dimensional “column” models are used under uniform erosion (denudation) conditions. These investigations show that within the range of the investigated parameters and physical processes, the hydraulic conductivity, the stiffness, and the drainage boundary conditions are more important than erosion rate, horizontal stresses, and nonlinear material behavior. However, although underpressures are produced with parameters and conditions corresponding reasonably well to observations, uniform rates of denudation alone cannot account for the actually encountered magnitude of underpressure. The second phase in which differential erosion and, particularly, the effect of glaciation are considered produces very realistic results. This is established by examining a number of combinations of denudation, differential erosion, and glacial loading and unloading.

https://agupubs.onlinelibrary.wiley.com/doi/pdf/10.1029/2001JB000327

Herbert H. Einstein

Papers

Crack Coalescence Tests on Granite

Laboratory Experiments On Geometric And Hydromechanical Characteristics of Three Different Fractures In Granodiorite

Application of the Decision Aids for Tunneling (DAT) to update excavation cost/time information

Stochastic Analysis of Surface Roughness, Aperture And Flow In a Single Fracture

Generation and evolution of hydraulic underpressures at Wellenberg, Switzerland