The cosmic web contains a large part of the total gas mass in the Universe, but it is difficult to detect at most wavelengths. Synchrotron emission from shock-accelerated electrons may offer the chance of imaging the cosmic web at radio wavelengths. In this work we use 3D cosmological ENZO-magnetohydrodynamic simulations (combined with a post-processing renormalisation of the magnetic field to bracket for missing physical ingredients and resolution effects) to produce models of the radio emission from the cosmic web. In post-processing we study the capabilities of 13 large radio surveys to detect this emission. We find that surveys by LOFAR, SKA1-LOW, and MWA have a chance of detecting the cosmic web, provided that the magnetisation level of the tenuous medium in filaments is of the order of ~1% of the thermal gas energy.