We present an experimental study of dissolution-driven convection in a three-dimensional porous medium formed from a dense random packing of glass beads. Measurements are conducted using the model fluid system MEG/water in the regime of Rayleigh numbers, Ra=2000-5000. X-ray computed tomography is applied to image the spatial and temporal evolution of the solute plume non-invasively. The tomograms are used to compute macroscopic quantities including the rate of dissolution and horizontally averaged concentration profiles, and enable the visualisation of the flow patterns that arise upon mixing at a spatial resolution of about (2x2x2). The latter highlights that under this Ra regime convection becomes truly three-dimensional with the emergence of characteristic patterns that closely resemble the dynamical flow structures produced by high-resolution numerical simulations reported in the literature. We observe that the mixing process evolves systematically through three stages, starting from pure diffusion, followed by convection-dominated and shutdown. A modified diffusion equation is applied to model the convective process with an onset time of convection that compares favourably with the literature data and an effective diffusion coefficient that is almost two orders of magnitude larger than the molecular diffusivity of the solute. The comparison of the experimental observations of convective mixing against their numerical counterparts of the purely diffusive scenario enables the estimation of a non-dimensional convective mass flux in terms of the Sherwood number, Sh=0.025Ra. We observe that the latter scales linearly with Ra, in agreement with both experimental and numerical studies on thermal convection over the same Ra regime.