Consider a graph $G=(V,E)$. An integer $k$-flow is a map $\phi: E \to \mathbb{Z}$ such that $|\phi(e)| < k$ and on each site, the sum of $\phi$ on incoming edges equals that on outgoing edges (this requires an orientation, but the orientation is actually irrelevant).
We say that a flow is nowhere zero, if for all edges $e$, $\phi(e) \neq 0$. There is quite a bit of literature on nowhere zero flows. As far as I understand (see for example Kochol 2002 and Beck, Zaslavsky 2003) the number of such flows on a graph G is known to be a polynomial in $k$, but not much more is known about how many there are.
What if we allow all integer $k$-flows, without restriction to those that are nowhere zero? What is the number $N$ of such flows? A trivial bound is $N \leq (2k-1)^{|E|}$, but can one do better? I can't seem to find anything online, so likely this is either hard or trivial.
We can slightly improve on the trivial bound.
To get an upper bound on $N$, we will exactly count a larger set of functions $\phi$. Instead of requiring flow conservation to hold exactly at each vertex (or "site"), we will only require flow conservation to hold modulo $2k-1$.
An important consequence of this approach is that now, if we have a vertex where all edges except one have been assigned a value, then there's a unique value in $\{-(k-1), -(k-2), \dots, k-2, k-1\}$ to give to the remaining edge out of that vertex. Just add/subtract all the assigned edges modulo $2k-1$, and then choose the remaining edge to cancel out this sum: we have $2k-1$ choices for the value of that edge, and they all have different remainders modulo $2k-1$.
With this relaxed condition, a connected graph with $|V|$ vertices and $|E|$ edges has $(2k-1)^{|E|-|V|+1}$ solutions. To see, reserve the $|V|-1$ edges of any spanning tree, and assign the values of $\phi$ on all other edges arbitrarily. Now, repeat the following procedure. Pick a leaf of the tree: a vertex at which all but one of its edges have been assigned a value. Then, give its remaining unassigned edge the unique value of $\phi$ that will achieve flow conservation modulo $2k-1$ at that leaf. This procedure fills in all remaining edges uniquely; at the end, all vertices except one satisfy flow conservation modulo $2k-1$, and by adding up all those conditions, we get that flow is conserved modulo $2k-1$ at the last vertex as well.
More generally, a graph with $|V|$ vertices, $|E|$ edges, and $|C|$ components has $(2k-1)^{|E|-|V|+|C|}$ solutions, by applying this argument on each component separately.