We make use of three dimensional hydrodynamical simulations to investigate the effects of granulation on the Cu I lines of Mult. 1 in the near UV, at 324.7 nm and 327.3 nm. These lines remain strong even at very low metallicity and provide the opportunity to study the chemical evolution of Cu in the metal-poor populations. We find very strong granulation effects on these lines. In terms of abundances the neglect of such effects can lead to an overestimate of the A(Cu) by as much as 0.8 dex in dwarf stars. Comparison of our computations with stars in the metal-poor Globular Clusters NGC 6752 and NGC 6397, show that there is a systematic discrepancy between the copper abundances derived from Mult. 2 in TO stars and those derived in giant stars of the same cluster from the lines of Mult. 2 at at 510.5 nm and 587.2 nm. We conclude that the Cu I resonance lines are not reliable indicators of Cu abundance and we believe that an investigations of departures from LTE is mandatory to make use of these lines.