This paper presents an analysis of capillary-tissue exchange system, representing blood by Casson fluid model flowing through a capillary of diameter 5 to 10 times larger than the red blood cells and the suspending medium plasma freely takes part in the exchange processes with surrounding tissue. Governing equations for continuity and momentum have been solved in the two regions (capillary and tissue) for fluid transport in capillary and the tissue subject to boundary and matching conditions at the capillary-tissue interface. The tissue has been identified as porous matrix following Darcy's law. Results for velocity profiles, pressure distributions in the two regions, wall shear stress and fluid flux at the interface have been represented through graphs and discussed. It has been noted that the capillary-tissue fluid exchange plays an important role in oxygen and nutritional transport to the tissue cells. Analysis oxygen transport is being presented in the next paper.