Large earthquakes alter the stress in the surrounding crust, leading to triggered earthquakes and aftershocks. A number of time-dependent processes, including afterslip, pore-fluid flow and viscous relaxation of the lower crust and upper mantle, further modify the stress and pore pressure near the fault, and hence the tendency for triggered earthquakes. It has proved difficult, however, to distinguish between these processes on the basis of direct field observations, despite considerable effort. Here we present a unique combination of measurements consisting of satellite radar interferograms and water-level changes in geothermal wells following two magnitude-6.5 earthquakes in the south Iceland seismic zone. The deformation recorded in the interferograms cannot be explained by either afterslip or visco-elastic relaxation, but is consistent with rebound of a porous elastic material in the first 1-2 months following the earthquakes. This interpretation is confirmed by direct measurements which show rapid (1-2-month) recovery of the earthquake-induced water-level changes. In contrast, the duration of the aftershock sequence is projected to be approximately 3.5 years, suggesting that pore-fluid flow does not control aftershock duration. But because the surface strains are dominated by pore-pressure changes in the shallow crust, we cannot rule out a longer pore-pressure transient at the depth of the aftershocks. The aftershock duration is consistent with models of seismicity rate variations based on rate- and state-dependent friction laws.