We numerically study the process of quasistatic invasion of a nonwetting fluid in 2D and 3D porous layers from multiple inlet injection sources and show that a porous layer acts as a two-phase filter as a result of the repeated convergence of flow paths: The probability for a pore at the outlet to be a breakthrough point is significantly lower than the fraction of active injection points at the inlet owing to the merging within the porous layer of liquid paths originating from different inlet injection points. The study of the breakthrough point statistics indicates that the number of breakthrough points diminishes with the system thickness and that the behavior of thin layers, defined here as systems of typical thicknesses of less than 15 lattice spacing units (≈15 pore or grain mean sizes), is distinct from thicker layers. For thicker systems, it is found that the probability of an outlet pore to be a breakthrough pore scales as l(1-d) where l is the system thickness and d is the space dimensionality, whereas, a power law behavior is not obtained with a thin system. Other properties, such as the invading phase occupancy profiles are studied. We also described a kinetic algorithm that allowed us to compute the occurrence times of breakthrough points. The distribution of these times is markedly different in 2D and in 3D.