>After an initial phase where the dam fills up shouldn’t there be same amount of water running out of it as before?
That is not true.
The first point is that you can change when during the year water flow. So when water flows it is not the same. If you use the dam for hydroelectric power then you can lower the water level by using more water then is coming in to it when there is a high power demand. You can then release less when demand is lower and you need to refill it.
The next part is the surface area of the water above the dam has increased. This can be a lot. It looks like Lake Powel in the colorado river loses 6% of the total water flow by evaporation. Add to that the evaporation from Lake Meed at hoover dam. [https://www.glencanyon.org/lake-powell-reservoir-a-failed-solution/](https://www.glencanyon.org/lake-powell-reservoir-a-failed-solution/) The climate where the dam is and its surface area vs volume will have an effect on how much water evaporates
This is all assuming no water is removed as drinking water or for irrigation.
So when water flow during the year will change and evaporation will reduce the total amount of flow in a year.
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