  We investigate the growth of the entanglement entropy $S_{\textrm{ent}}$
following global quenches in two-dimensional free fermion models with potential
and bond disorder. For the potential disorder case we show that an intermediate
weak localization regime exists in which $S_{\textrm{ent}}(t)$ grows
logarithmically in time $t$ before Anderson localization sets in. For the case
of binary bond disorder near the percolation transition we find additive
logarithmic corrections to area and volume laws as well as a scaling at long
times which is consistent with an infinite randomness fixed point.
