A multirelaxation model for conditioning and relaxation of modulus in sandstones introducing fast and slow dynamics contributions

Elastic modulus of sandstones exhibits a non trivial dependence on strain. Wavespeed decreases when a strain is applied, with a complex hysteretic/butterfly dependence on strain amplitude and the softening is not instantaneous but increases progressively with time (conditioning). On a time scale of several minutes, velocity recovers the original value when strain is removed (relaxation). Here, we propose a multirelaxation model, which allows reproducing hysteresis, cumulative conditioning and relaxation. We introduce a non-equilibrium strain, given as the superposition of contributions which are evolving on different time scales (from very fast to very slow) and weighted according to a given distribution of relaxation times. Each component evolves in time due to a conditioning term and a relaxation term, acting simultaneously. The balance between them determines the state of the material at a given time and strain amplitude. The ratio of relaxation times and wave period allows to identify which components contribute to fast and which to slow dynamics. We validate the model by comparison with experimental data and apply it to obtain predictions for cases, which are difficult to realize in experiments.