In this study, we examined the rheology of polyurethane foam (PUF) as an analogue system of magma undergoing vesiculation, deformation, and solidification. Solid PUF was formed by mixing two polymeric liquids, followed by chemical reactions of foaming and curing. The transient rheology during the processes was investigated using an instrument based on a stress-controlled rheometer. First, the oscillatory rheology was measured applying small-amplitude oscillations with cyclic frequency sweeps. In the foaming stage, the frequency-dependence of the rheology depends on the dynamic capillary number in a similar way to the existing bubbly flow model. In the curing stage, the viscoelasticity of the liquid becomes dominant, and the rheology of PUF shows Maxwell-type viscoelasticity with relaxation time exponentially increasing with time. When the reaction is rapid because of high temperature, the viscoelasticity during curing is not scaled by the single relaxation time. Instead, PUF passes through a condition called gelation, which is characterized by a power-law relaxation spectrum. Next, we examined the relationship between the dynamic viscosity measured by the small-strain oscillation and the shear viscosity measured with a fixed shear rate at a large strain. We found that PUF obeys the Cox-Merz rule, which states that the two viscosities are equivalent when the shear rate and the angular frequency are equal. Specifically, the shear viscosity tends to be smaller than the dynamic viscosity by a few tens of percent. The difference was shown to be in the range expected for bubbly fluid in the foaming stage, while it went beyond what can be explained by the bubbly fluid model when the viscoelasticity of the liquid becomes significant. X-ray computed tomography analyses revealed that samples in these cases contained large bubbles, indicating bubble coalescence. We infer that the difference is caused by bubble coalescence as well as non-Newtonian rheology during solidification, including gelation. All these processes might occur in actual magma during eruptions. We show that the ranges of the capillary number comparing the deformation time and the bubble shape relaxation time and Deborah number comparing the deformation time and the viscoelastic relaxation time in our experiments are comparable with their ranges expected for magma ascending in an eruptive conduit. The characteristic time of viscosity increase of PUF was also shown to be realizable for magma in natural volcanoes. We conclude that PUF is useful for simulating magma processes in eruptive conduits. (C) 2020 Elsevier B.V. All rights reserved.