Porous substrates have the ability to transport liquids not only laterally on their open surfaces but also transversally through their thickness. Directionality of the fluid transport can be achieved through spatial wettability patterning of these substrates. Different designs of wettability patterns are implemented herein to attain different schemes (modes) of three-dimensional transport in a high-density paper towel, which acts as a thin porous matrix directing the fluid. All schemes facilitate precise transport of metered liquid microvolumes (dispensed as droplets) on the surface and through the substrate. One selected mode features lateral fluid transport along the bottom surface of the substrate, with the top surface remaining dry, except at the initial droplet dispension point. This configuration is investigated in further detail, and an analytical model is developed to predict the temporal variation of the penetrating drop shape. The analysis and respective measurements agree within the experimental error limits, thus confirming the model's ability to account for the main transport mechanisms.