  We attempt to generalize a theorem of Nash-Williams stating that a graph has
a $k$-arc-connected orientation if and only if it is $2k$-edge-connected. In a
strongly connected digraph we call an arc {\it deletable} if its deletion
leaves a strongly connected digraph. Given a $3$-edge-connected graph $G$, we
define its Frank number $f(G)$ to be the minimum number $k$ such that there
exist $k$ orientations of $G$ with the property that every edge becomes a
deletable arc in at least one of these orientations. We are interested in
finding a good upper bound for the Frank number. We prove that $f(G)\leq 7$ for
every $3$-edge-connected graph. On the other hand, we show that a Frank number
of $3$ is attained by the Petersen graph. Further, we prove better upper bounds
for more restricted classes of graphs and establish a connection to the
Berge-Fulkerson conjecture. We also show that deciding whether all edges of a
given subset can become deletable in one orientation is NP-complete.
