Seismic gradient measurements from distributed acoustic sensors, rotational sensors and other instruments are becoming increasingly available for field surveys. Such measurements provide a wealth of information and are currently being considered for many applications. In this work, using a simple 2-D numerical approach, we tackle the implications of such wavefield gradient measurements on full waveform inversion (FWI) techniques using a simple two-dimensional numerical test. In particular, we study the impact of the wavefield gradient measurement sen- sitivity to heterogeneities that are much smaller than the minimum wavelength. Indeed, as shown previously through the homogenization method, small scale heterogeneities induce an unexpected coupling of the strain components to the wavefield gradient measurement. We fur- ther show that this coupling introduces a potential limitation to the FWI results if it is not taken into account. We demonstrate that a gradient measurement-based FWI can only reach the accuracy of a classical displacement field-based FWI if the coupling coefficients are also inverted. Furthermore, there appears to be no specific gain in using gradient measurements instead of conventional displacement (or velocity, acceleration) measurements to image struc- tures. Nevertheless, the inverted correctors contain fine-scale heterogeneities information that could be exploited to reach an unprecedented resolution, particularly if an array of receivers is used.