There are many examples of granular materials with grain sizes spanning several orders of magnitude. At geophysical scales – of particular interest to the author of this paper – rock fragments in geological faults, icebergs from calving glaciers, and ice floes forming the polar sea ice pack are composed of fragments with sizes from sub-meter up to hundreds or even thousands of meters. In all these materials, fragmentation processes often result in tapered power law (TPL) grain-size distributions (GSD), p(d)~d^(-α) exp⁡(-d/β), where d is a measure of grain size, and α and β are related to the number of spatial dimensions in the system, its size, and other factors specific for a given material. Materials with scale-invariant GSDs have unique properties, different from those of their more-widely studied weakly polydisperse counterparts. Therefore, rheological models developed for materials with narrow GSDs perform poorly for TPL-type GSDs. In this paper, DEM modelling is used to analyse shear deformation of 2D granular materials with three different GSDs: TPL (with size span s>0.99), bidisperse (BID; s≈0.7), and uniform (UNI; s≈0.5). It is shown that materials with the same mass-weighted mean grain size d_m, but different GSDs, follow different μ(I),ϕ(I) curves (where μ, ϕ and I denote the bulk friction coefficient, packing fraction and inertial number, respectively), indicating that d_m, although widely used, is not a good measure of grain size in the definition of I. Similarly, materials with TPL and narrow GSDs exhibit very different behaviours in the limit I→0. Based on DEM results and theoretical arguments,, the reasons behind those differences are analyzed, and alternatives to d_m are proposed, based on the mean grain-grain contact length and other properties of the contact networks. In order to make this “improved inertial number” and rheology based on it useful in practice (i.e., without DEM simulations), the possibilities to predict those properties from the GSD alone are explored.