  In a digraph, a dicut is a cut where all the arcs cross in one direction. A
dijoin is a subset of arcs that intersects each dicut. Woodall conjectured in
1976 that in every digraph, the minimum size of a dicut is equal to the maximum
number of disjoint dijoins. However, prior to our work, it was not even known
whether at least $3$ disjoint dijoins exist in a digraph whose minimum dicut
size is arbitrarily large. By building connections with nowhere-zero (circular)
$k$-flows, we prove that every digraph with minimum dicut size $\tau$ contains
$\frac{\tau}{k}$ disjoint dijoins if the underlying undirected graph admits a
nowhere-zero (circular) $k$-flow. The existence of nowhere-zero $6$-flows in
$2$-edge-connected graphs (Seymour 1981) directly leads to the existence of
$\frac{\tau}{6}$ disjoint dijoins in any digraph with minimum dicut size
$\tau$, which can be found in polynomial time as well. The existence of
nowhere-zero circular $(2+\frac{1}{p})$-flows in $6p$-edge-connected graphs
(Lov\'asz et al 2013) directly leads to the existence of $\frac{\tau p}{2p+1}$
disjoint dijoins in any digraph with minimum dicut size $\tau$ whose underlying
undirected graph is $6p$-edge-connected.
