A new model for normal deformation of single fractures under compressive loading

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, δ_1/2, is introduced and is used, in addition to the maximum fracture closure, Δv_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, K_ni, 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 δ_1/2 and Δv_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.