In the present study, generation of two vortex rings and their cut-and-connect process were numerically simulated by solving three-dimensional and time-dependent Navier-Stokes equations, under conditions similar to the laboratory experiment. In order to explain the mechanism of cut-and-connect, velocity, vorticity, pressure, helicity density and energy dissipation were examined for the flow field of the cut-and-connect of vortex rings. The present study revealed that energy dissipation is an essential process for circulation cancellation during vortex tubes cutting. Based on this energy dissipation mechanism, it is concluded that the cut-and-connect of vortex tubes may occur in the limit of inviscid flows. This conclusion is particularly important to the three-dimensional discrete vortex method for computing high Reynolds number flows. Features of helicity in the three-dimensional flow field of cut-and-connect process of vortex tubes were also investigated. The relation between the helicity density and the energy dissipation function in the three-dimensional flow field was examined.