In this study, a small two-medium duct with blast-wave propagation is numerically investigated by a high-resolution Euler/Navier-Stokes solver. The solver has a feature of treatment of the Tait equations of state for two media. One of the two media is water which is envisaged as a blood. The second medium is another liquid, used to simulate body's clot or tissue. The duct wall has a mass diffusion effect in addition to viscous effects. Since two different media are considered, the reflection and transmission of an underwater blast wave passing through the interfaces of the two media with different sound impedances are inevitable. The different properties of liquids may cause numerical oscillation at interfaces for very weak blast waves for a high-resolution scheme such as a 5th-order WENO scheme. In order to overcome this difficulty of numerical fictitious oscillation, a third-order WENO scheme was used. It was found that computed pressures of the transmitted blast wave for four kinds of simulated tissues are in good agreements with those obtained by the acoustic principle. Moreover, for the case of a simulated clot, the pressure force and impulse acted on the clot surface are investigated for different intensities of blast waves.