The present paper addresses the issue of finding key parameters that may lead to the occurrence of non-classical decay regimes for fractal/multiscale generated grid turbulence. To this aim, a database of numerical simulations has been generated by the use of the eddy-damped quasi-normal Markovian (EDQNM) model. The turbulence production in the wake of the fractal/multiscale grid is modelled via a turbulence production term based on the forcing term developed for direct numerical simulations (DNS) purposes and the dynamics of self-similar wakes. The sensitivity of the numerical results to the simulation parameters has been investigated successively. The analysis is based on the observation of both the time evolution of the turbulent energy spectrum $\def \xmlpi #1{}\def \mathsfbi #1{\boldsymbol {\mathsf {#1}}}\let \le =\leqslant \let \leq =\leqslant \let \ge =\geqslant \let \geq =\geqslant \def \Pr {\mathit {Pr}}\def \Fr {\mathit {Fr}}\def \Rey {\mathit {Re}}E(k,t)$ and the decay of the flow statistical quantities, such as the turbulent kinetic energy $\mathcal{K}(t)$ and the energy dissipation rate $\varepsilon (t)$. A satisfactory agreement with existing experimental data published by different research teams is observed. In particular, it is observed that the key parameter that governs the nature of turbulence decay is $\alpha ={d/U_{\infty }}\, {(\varepsilon (0)/\mathcal{K}(0))}={d/L(0)} \, {(\sqrt{\mathcal{K}(0)}/U_{\infty })}$ (with $d$ the bar diameter and $U_{\infty }$ the upstream uniform velocity), which measures the ratio of the time scale largest grid bar $d/U_{\infty }$ to the turbulent time scale $\mathcal{K}(0)/\varepsilon (0)$. Two asymptotic behaviours for $\alpha \rightarrow + \infty $ and $\alpha \rightarrow 0$ are identified: (i) a fast algebraic decay law regime for rapidly decaying production terms, due to strongly modified initial kinetic energy spectrum and (ii) a real exponential decay regime associated with strong, very slowly decaying production terms. The present observations are in full agreement with conclusions drawn from recent fractal grid experiments, and it provides a physical scenario for occurrence of anomalous decay regime which encompasses previous hypotheses.