Investigation concerning the bioinspired pumping flow of viscous fluids in the porous region using Darcy's law is demonstrated in the present article. The rhythmic membrane contraction propels fluids in the porous microchannel. The periodic contraction of the membrane is utilized in the present analysis to introduce the unique pumping mechanism. For small pattern, width to channel height ratio (i.e., the channel is substantially longer than its width) and at low Reynolds numbers, the governing equations are solved by an analytical approach. In light of porous effects, we noticed the implications of rheological limitations on pumping and trapping processes. The porosity has a dynamic role in the augmentation of membrane-based pumping. These outcomes may be productive in various bioengineering (drug delivery schemes) applications.