Changes in rhyolite melt viscosity during magma decompression and degassing exert a first order control on ascent through the crust and volcanic eruption style. These changes have as yet unknown hazard implications for geothermal drilling in pursuit of particularly hot fluids close to magma storage regions. Here, we exploit the situation at Krafla volcano in which rhyolite has both erupted at Earth's surface and been sampled at shallow storage depths via drilling of the 2009 IDDP-1 and 2008 KJ-39 boreholes. We use differential scanning calorimetry to constrain that the IDDP-1 magma quenched to glass at similar to 700 K, at a rate of between 7 and 80 K.min(-1). We measure the equilibrium viscosity of the IDDP-1 rhyolite at temperatures close to the glass transition interval and show that the rhyolite viscosity is consistent with generalized viscosity models assuming a dissolved H2O concentration of 2.12 wt%. We couple these results with micro-penetration and concentric cylinder rheometry over a range of potential magma storage temperatures to constrain the response of surficial Krafla rhyolites to stress. The surficial rhyolites at Krafla match the same viscosity model, assuming a lower dissolved H2O concentration of 0.12 wt%. Our results show that at a storage temperature of 1123-1193 K, the viscosity of the stored magma is similar to 3x10(5) Pa.s. At the same temperature, the viscosity following degassing during ascent to the surface rises to similar to 2x10(9) Pa.s. Finally, we use high-stress compression tests on the Hrafntinnuhryggur surface obsidian to determine the onset of unrelaxed behavior and viscoelastic melt rupture or fragmentation pertinent to understanding the melt response to rapid pressure changes that may be associated with further (near-) magma exploration at Krafla. Taken together, we characterize the relaxation and viscosity of these magmas from source-to-surface.