We consider the probability model of edge-fault tolerance of a network in the sense of connectivity with link faults. Using graph-theoretical notation, we define the edge-fault (EF) and Menger-type edge-fault (MEF) tolerances of a graph as the probabilities that the graph is connected and strongly Menger edge-connected when each edge has a certain failure probability, respectively. We derive an upper bound on the EF tolerance for regular graphs, which reveals an asymptotical behavior when graphs and edge failure probability are large enough. We also perform a simulation experiment on a number of randomly generated regular graphs and some typically well-used graphs. The numerical results show that, in addition to their well-structured properties for networks, Hypercubes, Möbius Cubes, Ary-Cubes and Circulant graphs have also higher EF and MEF tolerance in general. In particular, the Möbius Cube has both the highest EF and MEF tolerance among all involved graphs. The numerical results also hint that, in contrast to MEF tolerance, the EF tolerance of regular graphs is not strongly effected by the graph structure.