How much water is carried for jet engine water injection?
This paper here gives a good overview of water injection. 

It suggests for injecting water into the compressor, a 300 gallon tank is necessary, for a 2.2% water to core air flow ratio (the historical norm), for injection from take-off to top of climb.

For injecting into the combustor, it suggests a 135 gallon tank, for a 0.5:1 water to fuel ratio, again for injection from take-off to top of climb.

For both of these scenarios, the tank water flow rate was designed to achieve a 50% NOx reduction. The paper gives other water quantities required to acheive different goals (eg different NOx reductions and different turbine temperature reductions).

I don't think the paper indicates how long it takes to get to top-of-climb. I assume these are US gallons, as its a US paper.