We have investigated a magnet-free, nonreciprocal surface wave absorber based on high impedance surfaces (HISs) using a spatial-temporal modulation approach. By controlling embedded switches with a travelling wave, the HIS metasurface is modulated to break the time and spatial symmetry, which enables surface waves to propagate in one direction but be absorbed when propagating in the reverse direction. The nonreciprocity has been demonstrated by an EM-circuit co-simulation. We envision that this could be possibly applied in future communication systems that preferably transmit unidirectionally but absorb interference from the reverse direction caused by reflections or other devices.