Seismic gradient measurements from distributed acoustic sensors and rotational sensors are becoming increasingly available for field surveys. These measurements provide a wealth of information and are currently being considered for many applications such as earthquake detection and subsurface characterizations. 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 2-D numerical test. In particular, we study the impact of the wavefield gradient measurement sensitivity to heterogeneities that are much smaller than the minimum wavelength. Indeed, as shown through the homogenization theory, small-scale heterogeneities induce an unexpected coupling of the strain components to the wavefield gradient measurement. We further 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 structures. 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.