A model is presented for propagation along a flexible pipe whose thickness is not small in comparison with its diameter across a partial blockage with varying size and material properties. Comparison is made with Flugge's well-known thin-shell theory for a propagation along a pipe where it is found that the additional computational complexity found in the current model becomes necessary when the thickness of the shell exceeds 10% of the pipe radius. Comparison is also made with experiment both for the propagation characteristics of the pipe and for the reflection from a partial blockage. Two reflection models are presented: a crude area change model with compensation for the mass of the blockage and a more accurate model using high-order modes and matching to a flexible blockage using co-location. Reasonable agreement is found for both with the more accurate model giving better agreement but at the expense of computational efficiency. The work is useful both for blockage detection and for detecting stenosis in blood vessels.