A new semi-empirical model that predicts fracture deformation under normal compressive loading is presented. The development of a simple exponential model is given first after which a modified and more general exponential model, with an additional degree of freedom in the model parameters, is presented. The simple and the modified exponential models are then compared to available fracture closure models, namely the empirical Barton-Bandis hyperbolic model, and a power-law model based on Hertzian contact theory, to determine how good they fit the results of fracture closure experiments conducted under monotonically increasing normal compressive loading. A new parameter called the half-closure stress, s½, is introduced and is used, in addition to the maximum fracture closure, ¿¿m, in the model fitting procedures for the Barton-Bandis and the simple and generalized exponential models. The half-closure stress is shown to be related to the initial normal stiffness, Kni, used in the original Barton-Bandis model. An additional parameter, n, is used in fitting the modified exponential model to the experimental data. Of the models presented herein, the modified exponential model was found to provide the best fit to the experimental data, for the same values of s½ and ¿¿m, over the entire range of compressive stresses. The power-law model based on Hertzian contact theory was found to be unsuitable for accurate prediction of fracture normal deformation behavior.
Fracture deformability resulting from normal compressive stress is of fundamental importance to the study of the hydraulic and mechanical behavior of rock discontinuities. Fracture deformation directly affects the factors that govern the hydraulic conductivity of single fractures such as aperture distribution, contact area distribution, and spatial connectivity of the apertures [1,2]. Since fracture geometry networks and fluid flow through single fractures govern the hydraulic behavior of fractured rock masses, it follows directly that deformability of single fractures due to the action of compressive stress would affect the hydraulic properties of a rock mass. It is also generally understood that the mechanical behavior of rock masses is controlled significantly by the deformation of discontinuities .
The most fundamental properties of the bounding surfaces of a fracture affecting fracture deformation include the rock type, weathered state and matedness of the surfaces, and the spatial and size distributions of asperities on the surfaces. The roughness of each bounding fracture surface is directly related to the size and spatial distributions of the surface asperities, whereas the aperture size and spatial distributions, and the contact area distribution are functions of the cross-correlation of the surface asperity spatial and size distributions. The mechanical strength and deformability of the asperities are functions of the rock type and weathered state of the bounding surfaces. It is also a well known experimental observation that the shear displacement of the bounding surfaces of a fracture, that governs fracture mating, drastically affects the aperture and contact area distributions, even for zero to moderate normal loads, for which the asperity spatial and size distributions of the individual surfaces are practically unchanged [4,5].