John Kemeny

A MODEL FOR NON-LINEAR ROCK DEFORMATION UNDER COMPRESSION DUE TO SUB- CRITICAL CRACK GROWTH

Abstract Time dependency in rock deformation under compression is modelled by considering an elastic body containing cracks that grow under compressive stresses due to sub-critical crack growth. This is considered the prime mechanism for the time-dependent deformation of brittle rocks at low temperatures. The growth of cracks under compressive stresses is formulated using the “sliding crack” model, which considers extensile crack growth due to stress concentrations around pre-existing flaws. Subcritical crack growt is included into the sliding crack model by utilizing the empirical Charles power law relation between crack velocity and the crack tip stress intensity factor. The model is able to predict the dependence of the stress-strain curve on the applied strain rate, and agrees extremely well with experimental data. Also, the model is able to predict the occurrence of both transient and tertiary creep. The transient creep behavior is derived in closed-form, and is found to give creep that depends on the logarithm of time, which is similar to many empirical formulae for creep in brittle rocks. Tertiary creep in the model is due to crack interaction, and is found to occur at a critical value of crack density. This allows time-to-failure predictions to be made, which could be useful for underground structures required to remain open for long periods of time.

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.

Estimating three-dimensional rock discontinuity orientation from digital images of fracture traces

This paper describes a computer approach that has been developed for estimating three-dimensional fracture orientations from two-dimensional fracture trace information gathered from digital images of exposed rock faces. The approach assumes that the fractures occur in sets, and that each set can be described by a mean orientation and a measure of the scatter about the mean. Mathematical relationships are developed that relate the 3D fracture properties with the trace angles that would be measured on one or more rock faces. These algorithms are used in conjunction with a genetic algorithm to invert the trace angles to estimate 3D joint orientation. A number of case studies have been conducted indicating a great potential for the technique.

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.

Method for Automated Discontinuity Analysis of Rock Slopes with Three-Dimensional Laser Scanning

Three-dimensional (3D) laser scanning data can be used to characterize discontinuous rock masses in an unbiased, rapid, and accurate manner. With 3D laser scanning, it is now possible to measure rock faces whose access is restricted or rock slopes along highways or railway lines where working conditions are hazardous. The proposed method is less expensive than traditional manual survey and analysis methods. Laser scanning is a relatively new surveying technique that yields a so-called point cloud set of data; every single point represents a point in 3D space of the scanned rock surface. Because the density of the point cloud can be high (on the order of 5 mm to 1 cm), it allows for an accurate reconstruction of the original rock surface in the form of a 3D interpolated and meshed surface using different interpolation techniques. Through geometric analysis of this 3D mesh and plotting of the facet orientations in a polar plot, it is possible to observe clusters that represent different rock mass discontinu...

An inverse approach to the construction of fracture hydrology models conditioned by geophysical data: An example from the validation exercises at the Stripa Mine

Abstract One approach for the construction of fracture flow models is to collect statistical data about the geometry and hydraulic apertures of the fractures and use this data to construct statistically identical realizations of the fracture network for fluid flow analysis. We have found that this approach has two major problems. One is that an extremely small percentage of visible fractures may be hydrologically active. The other is that on any scale you are interested in characterizing usually a small number of large features dominate the behaviour ([1] Transport Processes in Porous Media. Kluwer Academic, The Netherlands, 1989). To overcome these problems we are proposing an approach in which the model is strongly conditioned by geology and geophysics. Tomography is used to identify the large features. The hydraulic behaviour of these features is then obtained using an inverse technique called “simulated annealing.” The first application of this approach has been at the Stripa mine in Sweden as part of the Stripa Project. Within this effort, we built a model to predict the inflow to the Simulated Drift Experiment (SDE), i.e. inflow to six parallel, closely-spaced holes, the N- and W-holes. We predict a mean total flow of approx. 3.1 (l/min) into the six holes (two-holes) with a coefficient of variation near unity and a prediction error of about 4.6l/min. The actual measured inflow is close to 2l/min.

Rock bench: Establishing a common repository and standards for assessing rockmass characteristics using LiDAR and photogrammetry

Remote sensing methods are now used to assess rockmass characteristics along transportation corridors, in mines and tunnels, and in other areas where rock falls can affect humans and infrastructure. A variety of sensor methods, primarily LiDAR and photogrammetry, have seen recent use with widespread success and state of practice acceptance. Various commercial and custom tools exist to process the resulting data to extract geometry, surface and location based statistics, and to perform kinematic stability assessments. Although there is a widespread need to assess how different sensors and processing workflows actually perform, these are often compared anecdotally solely with the field practices they replace and using site and sensor data unavailable to other researchers. Two principles must be established to move across-the-board comparisons of remote rockmass characterization forward: (i) establishment of accessible, documented test sites, and (ii) test databases that are accessible to all. We propose the establishment of several key sites for equipment tests, including already-studied areas in Europe and North America, as well as an open approach to adding sites and related data to the collection. Site descriptions must include detailed local geology, photographs, LiDAR and/or photogrammetry datasets, and access notes. Second, we describe and provide a prototype data repository for storing this information, and in particular for providing open access to benchmark data into the future. This initiative will allow for meaningful comparisons of sensors and algorithms, and specifically will support better methodologies for benchmarking rock mass data in the geosciences. Data and metadata will be hosted at the www.rockbench.org domain.

Nucleation and growth of dip‐slip faults in a stable craton

We analyze the conditions for nucleation of a dip-slip fault in intact crust and its subsequent growth using linear elastic fracture mechanics and the finite element method. We assume a fault can be modeled as a mode II shear crack in a layered elastic crust, and we investigate fault behavior under conditions of a variable dip and shear fracture energy. Two shear fracture energy gradients, Gc, were evaluated; both have surface values of 1 × 105 J/m2 and increase linearly to values of 1 × 106 or 1 × 107 J/m2 at 15 km depth, the base of the seismogenic zone. We assume that the vertical and horizontal components of the gravitational stress are both equal to the lithostatic load, effectively removing them from the calculation of shear stress on the fault Strain required for failure varied with respect to shear fracture energy gradient and rupture direction but not with dip. Rupture upward from the base of the seismogenic zone required more than twice the initial strain to cause fault growth and would result in very large earthquakes. Rupture downward in models associated with the higher Gc gradient resulted in stable growth of the fault; that is, additional strain was needed to allow for continuous fault growth from the surface to the base of the seismogenic zone. We suggest that fault nucleation and growth are a stable process which would preferentially occur by rupturing downward under stable conditions. Rerupturing of an established fault may initiate at depth or the surface depending on the degree of healing between rupture events and the tectonic stress available. A vast majority of shallow crustal earthquakes nucleate at the base of the seismogenic zone and rupture upwards. This suggests that most events are a reactivation of an old established fault and not growth of a new or developing fault.

Automatic extraction of blocks from 3D point clouds of fractured rock

Abstract This paper presents a new method for extracting blocks and calculating block size automatically from rock surface 3D point clouds. Block size is an important rock mass characteristic and forms the basis for several rock mass classification schemes. The proposed method consists of four steps: 1) the automatic extraction of discontinuities using an improved Ransac Shape Detection method, 2) the calculation of discontinuity intersections based on plane geometry, 3) the extraction of block candidates based on three discontinuities intersecting one another to form corners, and 4) the identification of “true” blocks using an improved Floodfill algorithm. The calculated block sizes were compared with manual measurements in two case studies, one with fabricated cardboard blocks and the other from an actual rock mass outcrop. The results demonstrate that the proposed method is accurate and overcomes the inaccuracies, safety hazards, and biases of traditional techniques.

An Asperity Model to Simulate Rupture along Heterogeneous Fault Surfaces

A model has been developed to simulate the statistical and mechanical nature of rupture on a heterogeneous strike-slip fault. The model is based on the progressive failure of circular asperities of varying sizes and strengths along a fault plane subjected to a constant far-field shear displacement rate. The basis of the model is a deformation and stress intensity factory solution for a single circular asperity under a unidirectional shear stress. The individual asperities are unified through the fault stiffness and the far-field stress and displacement. During fault deformation asperities can fail and reheal, resulting in changes in the local stresses in the asperities, stress drops, and changes in the stiffness of the fault. Depending on how the stress is redistributed following asperity failure and on the strenghts of the neighboring asperities an earthquake event can be the failure of one or more asperities. Following an earthquake event seismic source parameters such as the stress drop, energy change, and moment magnitude are calculated. Results from the model show a very realistic pattern of earthquake rupture, with reasonable source parameters, the proper magnitude-frequency behavior, and the development of characteristic earthquakes. Also the progression ofb-values in the model gives some insight into the phenomenon of ‘self-organized criticality.’