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

Characterizing rock joint geometry with joint system models

Cracking and coalescence behavior has been studied experimentally with prismatic laboratory-molded gypsum and Carrara marble specimens containing two parallel pre-existing open flaws. This was done at both the macroscopic and the microscopic scales, and the results are presented in two separate papers. This paper (the first of two) summarizes the macroscopic experimental results and investigates the influence of the different flaw geometries and material, on the cracking processes. In the companion paper (also in this issue), most of the macroscopic deformation and cracking processes shown in this present paper will be related to the underlying microscopic changes. In the present study, a high speed video system was used, which allowed us to precisely observe the cracking mechanisms. Nine crack coalescence categories with different crack types and trajectories were identified. The flaw inclination angle (β), the ligament length (L), that is, intact rock length between the flaws, and the bridging angle (α), that is, the inclination of a line linking up the inner flaw tips, between two flaws, had different effects on the coalescence patterns. One of the pronounced differences observed between marble and gypsum during the compression loading test was the development of macroscopic white patches prior to the initiation of macroscopic cracks in marble, but not in gypsum. Comparing the cracking and coalescence behaviors in the two tested materials, tensile cracking generally occurred more often in marble than in gypsum for the same flaw pair geometries.

Crack Coalescence in Molded Gypsum and Carrara Marble: Part 1. Macroscopic Observations and Interpretation

Systematic evaluation of cracking behavior in specimens containing single flaws under uniaxial compression

A series of uniaxial compression tests were performed on gypsum specimens with preexisting fractures to study the failure mechanism of fractures and rock bridges in fractured rock masses. The coalescence mechanism of two parallel and offset fractures was investigated by monitoring the process of fracture initiation and propagation with a video camera. The tests showed that two inclined parallel fractures can coalesce by shear failure and/or tensile failure under a uniaxial load. The coalescence path and mechanism mainly depend on the relative position of the two fractures. For instance, when the two fractures are coplanar or slightly offset, coalescence is generated by shear failure; when they are overlapping in the loading direction, coalescence is generated by mixed shear and tensile failure. Two types of preexisting fractures, one without initial surface contact and hence frictionless and another with surface contact and friction, were used to study the influence of fracture contact conditions on the coalescence path and load. It was found that coalescence of fractures with surface contact and friction requires loads as much as 35% higher than that for coalescence of fractures without contact and friction. A stress analysis was conducted in this study to explain the different coalescence mechanisms. The analytical work indicated that different fracture geometries produce significantly different stress fields in the rock bridge area and hence result in different failure modes.

Herbert H. Einstein

Papers

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

Characterizing rock joint geometry with joint system models

Crack Coalescence in Molded Gypsum and Carrara Marble: Part 1. Macroscopic Observations and Interpretation

Systematic evaluation of cracking behavior in specimens containing single flaws under uniaxial compression

Coalescence of fractures under shear stresses in experiments