When there are many non-intersecting paths between two vertices on a network, the connectivity is fault-tolerant. Because of no common vertices on these paths, they can be emergently used in avoiding destroyed parts on the usual paths by any disasters or attacks. It gives a tolerance index whether the combination of non-intersecting paths is many or few. However, to enumerate such paths is an intractable combinatorial problem, no practical algorithm has been known. On the other hand, many socio-technological infrastructure networks are embedded on the surface of Earth. Thus, as an approximate solution, we extendedly apply the counting method based on a path matrix with our proposed mapping to directed acyclic graphs from a planar network according to each pair of source and terminal vertices. The tendency of many or few combinations of the paths is clearly investigated through computer simulations for real networks. This approach will be useful for evaluating the existence of substitutive paths to improve the tolerance in risk management.