This paper describes some numerical experiments that were performed on wave propagation in a randomly generated anisotropic heterogeneous elastic media. By comparison with more classical, homogeneous me- dia, several numerical issues arise that are presented in detail. In particular, an appropriate parameterization has to be chosen for the field of mechanical properties, and the Perfectly Matched Layers have to be mod- ified for such media, lest the stability be lost. Some open questions will also be raised. Finally, we will present ongoing research on the characterization of the physical phenomena taking place over long distances depending on the type of random elasticity model used.