N.G.W. Cook

Natural joints in rock: Mechanical, hydraulic and seismic behaviour and properties under normal stress

Abstract The mechanical and hydraulic properties of many rock masses are affected significantly by the additional mechanical compliance and fluid conductivity that result from joints, fractures or faults. The effects of these features, generally referred to as joints, can be so pronounced in many problems in geology or geophysics, mining or petroleum engineering, hydrogeology and waste management that it is important to be able to locate and characterize them remotely within a rock mass using geophysical methods. The effect of joints on seismic wave propagation, therefore, becomes important also. Experimental measurements of the deformation of natural joints are analyzed in terms of theories concerning the roughness of the two joint surfaces and their deformation under stress. For many rocks, the deformation of the surfaces is reversible after the first few cycles of loading and unloading and is, therefore, elastic. The highly non-linear stress-deformation curves for joints must be a result of changes in the geometry of the areas of contact between asperities brought about by the elastic deformation of adjacent voids in response to changes in the applied stress. The flow of fluids between the surfaces of a joint must also depend upon the geometry of the void space between these surfaces. Measurements using liquid metal porosimetry, show that when the two surfaces of a natural joint are in contact with one another, this geometry becomes so complex that fluid through the joint cannot be approximated as laminar flow between parallel surfaces leading to a cubic relation between flux and aperture. At low stresses, fluid flow through a joint decreases much faster than the cube of the joint closure. This is shown to be a result of changes in contact area, tortuosity and hydraulic aperture brought about by deformation of the void space between the two surfaces of the joint. At high stresses, fluid flow through a joint asymptotes to an irreducible level, largely independent of joint closure and further changes in stress. The slope of a tangent to the curve relating the average closure between the two surfaces of a joint to the stress across the joint defines a specific stiffness for the joint at that stress. The effect of joints on seismic waves can be analyzed by using this specific stiffness as a boundary condition in the seismic wave equation. Displacements across this boundary are discontinuous while stresses are continuous. Although the specific stiffness of the joint and the properties of the rock on each side of it are assumed to be completely elastic, the displacement discontinuity leads to frequently-dependent reflection and transmission coefficients for compressional and shear waves as well as frequency-dependent group time delay for the transmitted waves. This concept can be extended to include a velocity discontinuity across the joint, where the contained fluid or the properties of the rock provide viscous as well as elastic coupling across a joint. Theoretical predictions based on the displacement discontinuity and velocity discontinuity theories agree very well with the results of laboratory measurements of seismic pulses transmitted across natural joints with different specific stiffnesses.

Analysis of compressive fracture in rock using statistical techniques: Part I. A non-linear rule-based model

Abstract A new non-linear rule-based model for the fracture in compression of heterogeneous brittle materials such as rock is presented and used to study crack nucleation and propagation at the grain scale. We have used the model to simulate uniaxial compression tests of rock samples and results underscore the importance of crack interaction in extensile cracking of rock in compression even at low crack densities. Moreover, the model produces non-linear stress–strain behavior similar to that observed in laboratory tests. We have analyzed the stress–strain behavior and found that in these simulations fracture occurs in the following way. First, initial damage occurs by random cracking. When approximately 15% of the sites are broken, cracks start to interact and coalesce to form larger cracks which then may propagate a significant fraction of the array length. Crack extension may be followed by crack arrest and subsequent formation of a damage zone ahead of a crack tip. Finally, a series of cracks will link and form a fracture that eventually causes failure. The model shows decreasing compressive strength with increasing size following a power-law relationship with an exponent that is similar to that determined from the study of laboratory and field-test results. The model can also incorporate heterogeneity in the strength and geometry of rock fabric; in part II, the model is used to investigate how microscale heterogeneity in these parameters affects extensile crack growth in compression.

Deformation and fracture around cylindrical openings in rock—I. Observations and analysis of deformations

Abstract Elastic and inelastic deformation, fracture and failure around underground openings were investigated through experiments on thick-walled hollow cylinders of Berea sandstone and Indiana limestone, incorporating plane strain loading, the application of different stress paths, transference of the external pressure to infinity, and “freezing” of the fracture geometry under stress through metal saturation. This first of two parts discusses elastic and inelastic deformations on both an experimental and theoretical level. Most elastic (recoverable) deformations exhibit non-linearity and hysteresis. Hole deformations due to internal (in the hole) pressure are controlled by different overall moduli than those due to external pressure. These overall deformations are successfully explained by a radially anisotropic moduli model, which allows different overall moduli for different stress paths. A radial-pressure-dependent modulus model cannot adequately explain these measurements. There is some evidence, however, that the stiffness of these rocks is greater with higher pressures. Evaluation of non-constant modulus models reveals that the rate of modulus increase near the hole is critical in determining whether the maximum tangential stress will occur at the hole wall or away from the wall. The apparent strength of the rock adjacent to the unsupported holes is two to three times the uniaxial compressive strength. Significant raadial dilation occurs in the failing material, as evidenced by hole closure measurements both during loading and after test completion. Proper modelling of this failure requires the incorporation of extreme dilation and extreme strength loss. A support pressure in the hole greatly strengthens and stabilizes the rock, and also reduces the amount of hole closure and dilation upon failure. Hole closure is time-dependent, especially for the unsupported holes.

Effective moduli, non-linear deformation and strength of a cracked elastic solid

Abstract Utilizing the principles of Linear Elastic Fracture Mechanics (LEFM), the effective elastic moduli, the stability, and the strength of a solid containing a random distribution of interacting cracks is calculated. In order to account for the effects of interacting cracks, the “external crack” model is introduced, as a high crack density complement to non-interacting crack models. The behaviour of rock may be seen as progressing from the non-interacting crack models to the external crack model as cracks extend, interact, and coalesce. In rock mechanics, it is more common to encounter boundary conditions other than pure load controlled, and therefore we utilize the Griffith locus, which can determine the onset of fracture and the manner in which fractures extend, under any combination of load-controlled and displacement-controlled boundary conditions. Stress intensity factors are also calculated for random distributions of interacting cracks under displacement-controlled boundary conditions. The external crack model is found to exhibit sub-critical strain softening behaviour, and this gives a mechanism, not found in the non-interacting crack models, for the ultimate failure of brittle rock.

Observations of crack growth in hard rock loaded by an indenter

Abstract A description is given of the processes of rock fragmentation induced by circular flat-bottomed punches from 5 to 20 mm dia loading orthogonal to the flat surface of cylindrical specimens of Sierra granite of 89 mm dia and confined by a steel belt, and 100 mm cube specimens confined in a biaxial frame. It is found that the rock deforms elastically until the applied punch load exceeds 45% of the maximum load that the rock cna sustain. At loads greater than 45% of the maximum, a crack is initiated around the perimeter of the punch and this crack propagates in the well-known conical Herizian manner. When the crack begins to propagate, the location of its front is reasonably well defined. However, as it extends with continued application of the load, an intense zone of microcracks surrounds and conceals the crack tip. Fractures initiating within this microcracked region propagate to the surface and form rock chips. The effects of punch size and confining stress, orthogonal to the punch, on the rock indentation strength are examined. Some decrease in punching strength with increasing punch size is noted; there is an increase in strength with increasing confining stress. At low confining stress, specimens split by cleavage through the axis of the punch. The theory of elasticity is used to show that this could occur with a punch loading against a semi-infinite rock surface and may therefore be an important phenomenon, not just an artifact of finite laboratory specimens.

Micromechanics of Deformation in Rocks

Laboratory testing of rocks subjected to differential compression have revealed many different mechanisms for extensile crack growth, including pore crushing, sliding along pre-existing cracks, elastic mismatch between grains, dislocation movement, and hertzian contact. Micromechanical models based on fracture mechanics have been developed for these different mechanisms by many different researchers. In this paper, the KI solutions for these micromechanical models are reviewed. Because of the similarity in rock behavior under compression in a wide range of rock types, it is not surprising that these micromechanical models have many similarities. This may explain the success of models based on certain micromechanisms in spite of the lack of evidence for these mechanisms in microscopic studies. Based on these similarities, a generic micromechanical model is proposed that in some way takes into account all of the above phenomena. It is demonstrated how the KI solutions from the micromechanical models can be used to derive nonlinear stress-strain curves that exhibit strain-hardening and strain-softening, dilatation, σ2 sensitivity, and rate dependence. By using subcritical crack growth, transient and tertiary creep behavior can also be predicted. Also, it is shown how these micromechanical models can form the basis for continuum damage models using the finite element method.

The fractal dimension of pores in sedimentary rocks and its influence on permeability

Abstract Perimeter-area power-law relationship of pores in five sedimentary rocks are estimated from scanning electron micrographs of thin sections. These relationships for the pores of four sandstones were found to lie between 1.43 and 1.49, while that of an Indiana limestone was found to be 1.67. We show how the perimeter-area power-law relationship of pores, along with a pore-size distribution, can be used to estimate the hydraulic permeability. A discussion is given of how the fractal dimension of the pore perimeter derived by Mandelbrot for islands whose boundaries are fractal: P = ϵ D A D/2 , where ϵ is some constant that depends on the length of the measuring grid size and D is the fractal dimension of the pore perimeter, influences permeability.

The fractal geometry of flow paths in natural fractures in rock and the approach to percolation

The distributions of contact areas in single, natural fractures in quartz monzonite (Stripa granite) are found to have fractal dimensions which decrease from D = 2.00 to values near D = 1.96 as stress normal to the fractures is increased from 3 MPa up to 85 MPa. The effect of stress on fluid flow is studied in the same samples. Fluid transport through a fracture depends on two properties of the fracture void space geometry: the void aperture; and the tortuosity of the flow paths, determined through the distribution of contact area. Each of these quantities change under stress and contribute to changes observed in the flow rate. A general flow law is presented which separates these different effects. The effects of tortuosity on flow are largely governed by the proximity of the flow path distribution to a percolation threshold. A fractal model of correlated continuum percolation is presented which quantitatively reproduces the flow path geometries. The fractal dimension in this model is fit to the measured fractal dimensions of the flow systems to determine how far the flow systems are above the percolation threshold.

Analysis of compressive fracture in rock using statistical techniques : Part II. Effect of microscale heterogeneity on macroscopic deformation

Abstract We have performed a parameter-sensitivity analysis to evaluate the relative importance of different types of grain-scale heterogeneity on fracture processes and compressive strength in simulated compression tests of brittle, heterogeneous materials such as rock. This was done using a non-linear, rule-based model described in a companion paper. Results presented here indicate that heterogeneity in local stress field (due to grain shape and loading) has a first-order effect on macroscopic properties and is much more important than heterogeneity in site strength or location. In particular, increasing local stress heterogeneity lowers the mean ultimate strength following an inverse power law. Increasing heterogeneity in the lattice-site locations (i.e. irregular lattice spacing) decreases crack localization and decreases normalized crack strain energy. This result is consistent with the postulate that systems with increasing disorder require more energy to break. Heterogeneity in site-strength distribution had a relatively minor effect on macroscopic behavior. Peak strength is dependent on the mean site strength, not on the width of the site-strength distributions. This study also revealed that percolation thresholds are much lower than those predicted from stochastic fracture models. Consequently, statistical models for rock fracture must consider alternative percolation algorithms such as directed-bond percolation, because the standard percolation models may not be appropriate for analyzing systems where crack interaction dominates behavior at a low fraction of sites broken.

Considerations of Autocatalytic Criticality of Fissile Materials in Geologic Respositories

Potential routes to autocatalytic criticality in geologic repositories are systematically assessed. If highly enriched uranium (HEU) or {sup 239}Pu are transported and deposited in concentrations similar to natural uranium ore, in principle, criticality can occur. For some hypothesized critical configurations, removal of a small fraction of pore water provides a positive feedback mechanism that can lead to supercriticality. Rock heating and homogenization for these configurations can also significantly increase reactivity. At Yucca Mountain, it is highly unlikely that these configurations can occur; plutonium transport would occur primarily as colloids and deposit over short distances. HEU solute can move large distances in the Yucca Mountain setting; its ability to precipitate into critical configurations is unlikely because of a lack of active reducing agents. Appropriate engineering of the waste form and the repository can reduce any remaining probability of criticality.