The convective drying of a natural porous material, limestone, is investigated in this study, with both experimental and numerical approaches. The first experimental campaign, which focuses on the influence of samples' slenderness, suggests the presence of a hydraulic connection layer between the porous water and the external environment, in spite of the very fine pore structure of the material. This hydraulic transfer enables the fast water evaporation at the beginning of the drying test, when external conditions drive the kinetics. Furthermore, the results show that this layer does not exceed 30 mm deep from the external surface, given the drying conditions of the test. A second experimental campaign aims to analyse, by mean of an X-ray tomography tool, the internal water content during the drying. It confirms that water transfer takes place within the limestone in two distinct stages. The first stage being faster than the second one with a homogeneous desaturation along the sample, it is consistent with the hypothesis of the hydraulic connection layer. Finally, the finite element modelling makes possible to identify the main mechanisms of water transfer, namely liquid convection and vapour diffusion.