Question
What will happen if you take a chain saw to one of the running roots off a water oak?  Is there a tap root and this will not affect the tree?  Or will the tree die?

Answer
If you cut only one of the roots it will not effect the overall health of the oak. The oak does not have a tap root as such the root system is spread out from the trunk about 1 1/2 the distance of the foliage spread. If general rule is if you cut more than 30% of the roots the tree will suffer die back of the limbs on the side of the root cutting.
In this case I would think there will be no effect to the tree by cutting one root. It will not die from this.