Paper-based microfluidic devices are rapidly becoming popular as a platform for developing point-of-care medical diagnostic tests. However, the design of these devices largely relies on trial and error, owing to a lack of proper understanding of fluid flow through porous membranes. Any porous material having pores of multiple sizes contains partially saturated regions, i.e., regions where less than 100% of the pores are filled with fluid. The capillary pressure and permeability of the material change as a function of the extent of saturation. Although methods to measure these relationships have been developed in other fields of study, these methods have not yet been adapted for paper for use by the larger community of analytical chemists. In the current work, we present a set of experimental methods that can be used to measure the relationships between capillary pressure, permeability, and saturation for any commercially available paper membrane. These experiments can be performed using commonly available lab instruments. We further demonstrate the use of the Richards equation in modeling imbibition into two-dimensional paper networks, thus adding new capability to the field. Predictions of spatiotemporal saturation from the model were in strong agreement with experimental measurements. To make these methods readily accessible to a wide community of chemists, biologists, and clinicians, we present the first report of a simple protocol to measure the flow rates considering the effect of partial saturation. Use of this protocol could drastically reduce the trial and error involved in designing paper-based microfluidic devices.