Herbert H. Einstein

Coalescence of fractures under shear stresses in experiments

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.

The effect of discontinuity persistence on rock slope stability

Discontinuity persistence has a major effect on rock mass resistance (strength) but, as direct mapping of discontinuities internal to a rock mass is not possible, persistence is a difficult parameter to measure. As a result, the conservative approach of assuming full persistence is often taken. In this paper a method is developed for relating rock mass stability and hence persistence to the geometry and spatial variability of discontinuities. The method is applied to slope stability calculations in which the probability of failure is related to discontinuity data, as obtained in joint surveys. The complete method makes use of dynamic programming and simulation, but a closed form expression satisfactory for most purposes is also presented.

Probabilistic and statistical methods in engineering geology

SummaryAfter having pointed out weaknesses in traditional approaches to exploration in a preceding paper, it will now be shown what can be gained by using statistical methods in practical application to exploration. In particular, statistical data collection for joint surveys will be presented including methods for error reduction. Extensive treatment will be given to subjective assessment of uncertainty, a methodology that is well suited to engineering geology; its use in describing tunnel geology is given as a practical example. Finally, it will be shown how these methods of rational uncertainty description are employed to plan exploration in an optimum manner, with an example in exploration planning for underground gas storage caverns.

Numerical modeling of fracture coalescence in a model rock material

The crack pattern, as well as crack initiation, -propagation and -coalescence observed in experiments on gypsum specimens with pre-existing fractures in uniaxial, biaxial, and tensile loading are satisfactorily predicted with the numerical model presented in this paper. This was achieved with a new stress-based crack initiation criterion which is incorporated in FROCK, a Hybridized Indirect Boundary Element method first developed by Chan et al. (1990). The basic formulation of FROCK is described, and the code verified for both open and closed pre-existing fractures either with only friction or with friction and cohesion. The new initiation criterion requires only three material properties: σcrit, the critical strength of the material in tension; τcrit, the critical strength of the material in shear; r0, the size of the plastic zone. The three parameters can be determined with the results from only one test. Predictions using this model are compared with experiments on gypsum specimens with pre-existing fractures loaded in uniaxial and biaxial compression performed by the authors. Specifically, wing crack and shear crack initiation, crack propagation, coalescence stress and -type as well as the crack pattern up to coalescence can be modeled. The model can also duplicate experimental results in compression and tension obtained by other researchers. These results show that stress-based criteria can be effectively used in modeling crack initiation and crack coalescence.

Estimating the intensity of rock discontinuities

This paper presents an approach for estimating the intensity of discontinuities and formulating intensity and orientation as a fracture tensor. Specifically the size distribution and the number of discontinuities are estimated first, from which the fracture tensor is then derived. Discontinuity size distribution is inferred from the trace data sampled in circular windows by using a general stereological relationship between the true trace length distribution and the discontinuity diameter distribution assuming circular shaped discontinuities. Because the measured trace lengths are biased, a method is proposed to estimate the true trace length distribution for circular window sampling. Circular window sampling has the advantage of automatically eliminating the orientation bias when estimating the true trace lengths. A method is then presented with which the total number of discontinuities in an objective volume can be estimated from the number of discontinuities observed in normal-size boreholes and using the inferred discontinuity diameter from the circular window sampling on the rock surface. With the derived size distribution and number of discontinuities, the intensity of discontinuities, which is the total surface area of discontinuities per unit volume, can then be calculated and included in a new definition of a fracture tensor. An application of the approach to analyze simulated discontinuities produces satisfactory results.

RISK AND RISK ANALYSIS IN ROCK ENGINEERING

Abstract Risk analysis and decision-making procedures are being applied to large engineering projects with increasing frequency. Risk analysis is a versatile tool in decision-making, whether the decisions are in the form of an engineering design, selecting particular construction procedures, or more general decisions made by private or public entities. An overview of the basic aspects of risk analysis and decision-making are followed by a detailed discussion of three typical rock engineering applications: (1) slope design, (2) flow through fractured media, and (3) tunnelling. The paper includes examples from current tunnel projects (e.g., the Gotthard base tunnel, the Adler tunnel and the Lotschberg base tunnel).

Tunnelling in difficult ground — Swelling behaviour and identification of swelling rocks

SummaryThis brief review first describes typical swelling phenomena in underground openings and then discusses in detail the underlying swelling mechanisms. Mechanical, osmotic and intracrystalline swelling in argillaceous ground, as well as hydration of anhydrite are described. The fact that many of these mechanisms interact and, particularly, that swelling and squeezing may occur simultaneously is emphasized. The paper concludes with a few comments on implications on analysis, design and construction.

A new method for automatic discontinuity traces sampling on rock mass 3D model

A new automatic method for discontinuity traces mapping and sampling on a rock mass digital model is described in this work. The implemented procedure allows one to automatically identify discontinuity traces on a Digital Surface Model: traces are detected directly as surface breaklines, by means of maximum and minimum principal curvature values of the vertices that constitute the model surface. Color influence and user errors, that usually characterize the trace mapping on images, are eliminated. Also trace sampling procedures based on circular windows and circular scanlines have been implemented: they are used to infer trace data and to calculate values of mean trace length, expected discontinuity diameter and intensity of rock discontinuities. The method is tested on a case study: results obtained applying the automatic procedure on the DSM of a rock face are compared to those obtained performing a manual sampling on the orthophotograph of the same rock face.

Cracking processes in Barre granite: fracture process zones and crack coalescence

This paper presents a comprehensive study of the cracking and coalescence behavior of granite specimens with pre-existing flaw pairs. Uniaxial compressions tests were conducted on Barre granite with pre-existing flaw pairs of varying inclination angles