ABSTRACT: An analytical model that assumes variable height asperities and full interaction between them has been developed and implemented numerically to study the behavior of a single fracture under stress. The model has been used to calculate specific stiffness and aperture profiles for specified fracture geometries. The formulation takes account of the deformation of the half-planes defining the fracture which is shown to lead to changes in aperture geometry that are not predicted by other asperity models. A parameter sensitivity study shows that specific stiffness varies significantly depending on the size of the asperities, their height distribution, and their spatial orientation. Curves of stress vs. specific stiffness generated with the model are found to agree in shape and magnitude with those obtained in the laboratory for rock specimens.
1 INTRODUCTION
Understanding the properties of fractures under stress is important in characterizing most geological sites and predicting the behavior of underground engineering structures. The purpose of the present study is to identify the parameters that play an important role in determining the mechanical response of a single fracture to applied loads. An important application of the work is the description of fracture closure as a function of stress. The ability to describe changes in aperture geometry with changes in stress is crucial to predicting fluid flow through fractured rock (see, e.g., Tsang, 1984 and Brown, 1987). Specific stiffness is property that defines the relationship between applied stress and fracture deformation. More formally, specific stiffness is defined as the average applied stress divided by the average displacement across the fracture interface in excess of the displacement that would occur if the fracture were not present. Laboratory experiments on single fractures in rock indicate that specific stiffness is initially a sharply rising function with stress that levels off and approaches a constant value (e.g., Goodman, 1976; Bandis et al., 1983; and Pyrak-Nolte et al., 1987). A common approach to modeling the mechanical deformation of fractures has been to represent the fracture surfaces as parallel planes separated by asperities of varying height. Greenwood and Williamson (1966) modeled the contact between a plane and a nominally fiat surface covered by a large number of asperities with heights described by a specified statistical distribution. The asperity tips were taken to be spherical and their deformation calculated from the Hertzian solution for an elastic sphere in contact with a plane. Their model was extended by Greenwood and Tripp (1971) to the case of two rough surfaces in contact. Gangi (1978) used what he termed a bed of nails model to describe the permeability of a fractured porous rock as a function of confining pressure. The asperities were modeled as rods with equal spring constants and heights following a power law distribution. Brown and Scholz (1985, 1986), like Greenwood and Williamson, assumed the asperity tips to be spherical and modeled their deformation using the Hertzjan solution. However, they also included a term for tangential stresses arising from the oblique contact of spheres so that the stresses at the contacts are not restricted to be normal.
