The smallest microscopic blood vessels in the human lung are organized into sheet-like networks. These sheets form the walls of the 300 million alveoli in which air flows due to breathing. Each sheet may be idealized into a channel bounded by two thin layers of porous media. This paper is concerned with the blood flow in the channel and water movement in the porous wall. The Reynolds number in the lung alveolar sheets is very low, so creeping flow is assumed. Analytical and numerical results on the velocity and pressure distribution in the porous layers are presented.