The absorption (permeation) of alcohols into porous blocks of calcium carbonate has been studied experimentally and with a computer model, The experimental measurement was of change in apparent weight of a block with time after contact with liquid. The modeling used the previously developed 'Pore-Cor' model, based on unit cells of 1000 cubic pores connected by cylindrical throats. To gain some insight into absorption into voids of complex geometry, and to provide a representation of heterogeneities in surface interaction energy, the cylindrical throats were converted to double cones. Relative to cylinders, such geometries caused hold-ups of the percolation of nonwetting fluids with respect to increasing applied pressure, and a change in the rate of absorption of wetting fluids. Both the measured absorption of the alcohols and the simulated absorption of the alcohols and of water showed significant deviations from that predicted by an effective hydraulic radius approximation. The simulation demonstrated the development of a highly heterogeneous wetting front, and of preferred wetting pathways that were perturbed by inertial retardation. The findings are useful in the design of high-performance, low-waste pigments for paper coatings, and environmentally friendly printing inks, as well as in wider industrial, environmental, and geological contexts.