  We reveal the generic characteristics of wave packet delocalization in
two-dimensional nonlinear disordered lattices by performing extensive numerical
simulations in two basic disordered models: the Klein-Gordon system and the
discrete nonlinear Schr\"{o}dinger equation. We find that in both models (a)
the wave packet's second moment asymptotically evolves as $t^{a_m}$ with $a_m
\approx 1/5$ ($1/3$) for the weak (strong) chaos dynamical regime, in agreement
with previous theoretical predictions [S.~Flach, Chem.~Phys.~{\bf 375}, 548
(2010)], (b) chaos persists, but its strength decreases in time $t$ since the
finite time maximum Lyapunov exponent $\Lambda$ decays as $\Lambda \propto
t^{\alpha_{\Lambda}}$, with $\alpha_{\Lambda} \approx -0.37$ ($-0.46$) for the
weak (strong) chaos case, and (c) the deviation vector distributions show the
wandering of localized chaotic seeds in the lattice's excited part, which
induces the wave packet's thermalization. We also propose a
dimension-independent scaling between the wave packet's spreading and
chaoticity, which allows the prediction of the obtained $\alpha_{\Lambda}$
values.
