Carbonate rocks usually present a wide variation in pore size within a sample and may contain macroscopic pores ranging from a few millimeters to microscopic pores smaller than one micrometer. Therefore, studying the fluid flow inside carbonates presents a challenging problem. This study proposes a methodology to create dual-porosity micromodels for studying single and two-phase fluid flow in multiscale, carbonate-like, rocks. For this purpose, a design technique for Rock-on-a-Chip (ROC) devices based on the Voronoi tessellation was extended to take into account bimodal pore size distributions, allowing the creation of a macroporous system made up of larger channels and vugs that can be filled with distinct microporosity types. The porous media thus generated were then employed to fabricate polymer micromodels by applying the soft lithography technique. Experimental and numerical results show that the microporosity can increase or reduce the permeability, depending on whether it is added to the grains and/or to the large channels. Even when the microporous matrix completely filled the large channels, the addition of vugs did not increase the permeability. Regarding two-phase fluid flow, the location of the steady-state fluids after drainage clearly depends on the proportion and spatial distribution of microporosity, as well as its type. For the micromodel with microporous grains, no significant amount of wetting fluid was displaced from the micropores. In contrast, when microporosities fill the large channels, the injected fluid forces the displacement of the wetting liquid from the micropores, although far from effectively. The novel approach presented in this work represents a step forward in the artificial generation of more representative micromodels for studying fluid flow at the pore scale.