A semi-empirical approach to model pressure dependence of elastic moduli in granular media accounting for variations of coordination-number and Poisson-ratio

The effective medium theory based on the Hertz–Mindlin contact law is the most popular theory to relate dynamic elastic moduli (or elastic velocities) and confining pressure in dry granular media. However, many experimental results proved that the effective medium theory predicts pressure trends lower than experimental ones and over-predicts the shear modulus. To mitigate these mispredictions, several evolutions of the effective medium theory have been presented in the literature. Among these, the model named modified grain contact theory is an empirical approach in which three parametric curves are included in the effective medium theory model. Fitting the parameters of these curves permits to adjust the pressure trends of the Poisson ratio and the bulk modulus. In this paper, we present two variations of the modified grain contact theory model. First, we propose a minor modification in the fitting function for the porosity dependence of the calibration parameters that accounts for non-linearity in the vicinity of the critical porosity. Second, we propose a major modification that reduces the three-step modified grain contact theory model to a two-step model, by skipping the calibration parameter–porosity fit in the model and directly modelling the calibration parameter–pressure relation. In addition to an increased simplicity (the fitting parameters are reduced from 10 to 6), avoiding the porosity fit permits us to apply the model to laboratory data that are not provided with accurate porosity measurements. For this second model, we also estimate the uncertainty of the fitting parameters and the elastic velocities. We tested this model on dry core measurements from literature and we verified that it returns elastic velocity trends as good as the original modified grain contact theory model with a reduced number of fitting parameters. Possible developments of the new model to add predictive power are also discussed.