Which flow algorithm on a graph is used when the capacity is a rational number?
Since all steps of the algorithm merely involve addition, you can compute the least common multiple $m$ of the denominators of all weights involved, and creating $G'$ from $G$ by multiplying all weights by $m$.

Then all weights are integers and you can run Ford-Fulkerson. You can divide the results from it by $m$ again to get back your rationals.