Spontaneous imbibition is known to control early fluid redistribution in heterogeneous porous media, whereas the relationship between rapid capillary percolation and effective pore-volume sweep remains to be fully elucidated. This study uses a multi-relaxation-time lattice Boltzmann method coupled with a color-gradient multiphase model to simulate cocurrent spontaneous imbibition in Ketton carbonate and sandstone digital rocks. The main contribution of this work is to distinguish breakthrough-controlled sweep from late-stage stable saturation and to identify the conditions under which pore structure enables capillary driving to activate additional connected pathways rather than merely accelerating outlet-spanning flow. The effects of wettability and interfacial tension were evaluated at breakthrough, whereas the influence of viscosity ratio was further examined until the wetting-phase distribution approached a stable state. The results show that Ketton carbonate tends to develop preferential invasion channels and pronounced bypassing because of its stronger pore-structure heterogeneity, while sandstone supports broader and more uniform front propagation. Increasing the interfacial tension promotes more cooperative pore filling in Ketton carbonate, leading to a transition from channelized percolation to multi-path invasion. This threshold-like response is less evident in sandstone, because the more uniform pore-throat network already favors distributed imbibition. Viscosity-ratio simulations indicate that lower absolute viscosity accelerates early imbibition, whereas higher absolute viscosity delays interface advancement but may enhance late-stage pore filling in heterogeneous media. The findings of this study provide a pore-scale framework for assessing early-time imbibition efficiency in heterogeneous rocks.