Acquisitions of the Sentinel-1 satellite are processed and comprehensively analyzed to investigate the ground displacement during a three-year period above a double gas storage site (Lussagnet and Izaute) in Southwestern France. Despite quite low vertical displacements (between 4 and 8 mm) compared to the noise level, the cyclic motion reflects the seasonal variations due to charge and discharge during summer and winter periods, respectively. We can simulate the ground deformation at both storage sites by a simple mechanical model. However, ground movements of low-magnitude may be also induced by natural factors, such as the temperature or the soil moisture. Using a wavelet-based analysis, we show there is a soil expansion in the Lussagnet zone that contrasts both in phase and period with the seasonal deformation and that is linked to the surface soil moisture measured by the SMOS satellite. This other displacement is consistent with the water infiltration in the unsaturated zone followed by the swelling of a clay layer. This work reveals the combination of two different processes driving the ground displacement with the same order of magnitude (about 6 mm), namely the pressure variation of a deep gas reservoir and the swelling/shrinking of the shallow subsurface.