There is an interesting question in the Algorithm Design textbook: how to find a min flow in a network with a source s and a sink t, and each edge has a lower bound l but no upper bound(inif). Here is a solution, we can first find a feasible flow f large enough to satisfy every lower bound and then decrease this feasible flow by finding a max flow g from t to s in the residual graph Gf, and the sum of f and g will be the minimum feasible flow in original graph. But, I am confused about that, what if the original graph has both lower bound and capacity on each edge? I tried first to find a feasible flow in that graph, but failed. Because after replacing the lower bounds by demands on nodes, there are both a super source s and some nodes with negative demands, and a super sink t and some nodes with positive demands. That is pretty strange, how to find a feasible flow in that network? If we can find one, how we can minimize the flow to get a minimum feasible flow to satisfy both lower bounds and capacities. 
How minimize the max flow in a networl? refer to Algorithm Design
network flow