This work presents a numerical modelling approach of particle packing consolidation, at the particle scale, based on specific numerical methods implemented in a high-performance computing framework. Typically, the sintering process triggers several mass transport paths, thermally activated, that are driven by geometrical as well as physical gradients and laplacians. Computing precisely such major characteristics is of paramount importance but represents a real scientific challenge, which have not been fully solved yet but which must however be tackled to gain precious insights into sintering mechanisms which are seldom accessible at this scale. An Eulerian-based formulation is then proposed here to deal with the strong topological changes related to particle sintering. Also, a specific attention is paid to the precise and robust computation of high-order derivatives which are known to control the physics of surface solid diffusion, namely the surface laplacian of the curvature. Besides, the hydrostatic pressure gradient is known to control the volume diffusion path, it results from the coupled fluid-solid mechanical equilibrium, including surface tension, which must be solved precisely. Furthermore, a mesh adaptation technique allows the particles surface description to be improved, while the number of mesh elements is kept reasonable. Once verified on test-cases, this numerical approach is applied to several 3D granular packings undergoing micro-structural changes under combined surface and volume diffusion.