Lattice Boltzmann methods (LBM) for one- and two-phase flows are used to model the capillary pressure-driven infiltration of fluids into porous spheres and rectangular beds. This simulates the soaking of fine particle agglomerates and powder compacts, respectively. The sensitivity of the infiltration behavior to physical parameters (such as porosity, permeability, degree of saturation and dimensions of the porous solid, as well as the dynamic viscosity and surface tension of the fluid) was investigated. Results of the LBM approach show good consistency with those from continuum models describing infiltration behavior. Additionally, the one-phase LBM was applied to investigate the influence of pore network structure on the infiltration behavior.