This work focuses on the study of a set of Low Gain Avalanche Detectors (LGADs) produced by CNM, Barcelona (run 7859). Several samples were irradiated with 24-GeV/c protons up to different fluences, ranging between 10$^{12}$ and 10$^{15}$ n$_\textrm{eq}$/cm$^2$. This study concentrates mainly on the LGADs irradiated up to 10$^{14}$ n$_\textrm{eq}$/cm$^2$. The measurements performed to characterise the devices include TCT, edge-TCT, TPA-TCT, and CV/IV measurements. The main goals of these studies were to analyse the voltage required to fully deplete the multiplication layer of LGADs; to measure gain degradation; and to investigate the distribution of the electric field inside the devices after irradiation, as well as the characteristics of the space charge. In order to do so, the measurements were performed under different temperature, read-out and biasing conditions. The obtained data confirm that in highly proton-irradiated LGADs the depletion of the bulk starts from the back electrode, thus shifting the onset of charge multiplication towards higher voltages. This is caused by a space charge sign inversion that is in turn accompanied by the appearance of a triple junction. Furthermore, it was found that annealing causes a recovery of gain along with a reduction of the multiplication-onset voltage towards unirradiated-like values.