Underground structures in shales or shaly rocks endure time-dependent swelling effects. Laboratory test results show that the swelling of these shales is dependent on three-dimensional stresses; an external stress on a specimen in one principal direction reduces the swelling not only in that direction but also perpendicularly. The effectiveness of a time-dependent swelling model that considers the three-dimensional stress effect is presented in this paper. A finite element algorithm incorporating the new constitutive model is used for a numerical analysis. The finite element program is used to analyze two tunnels in southern Ontario: the Heart Lake storm sewer tunnel, and the Darlington cold-water intake tunnel. The predicted results agree well with the records of field performance of these tunnels. The comparison between present analyses and the existing closed-form solution shows that the existing solution overestimates the time-dependent swelling effects. The three-dimensional stress effects on swelling are not considered in the closed-form solutions and are the cause of this discrepancy. The pseudo-Poisson's effect is a key parameter for modelling the observed time-dependent swelling. The use of these solutions in design is discussed.Key words: time-dependent swelling, shale, modelling, three-dimensional stress effect, finite element method, tunnel.
Reconstruction of three-dimensional maps based on closed-form solutions of the variational problem of multisensor data registration