Water issuing from a spring is what is commonly called living water. Water collected from rain in pools and cisterns is not called living water. It may have originally flowed from a spring; yet if it collects in some place and is left to stand without any connection to its source, separated, as it were, from the channel of the spring, it is not called “living water.” Water is designated as “living” when it is taken as it flows. This is the kind of water that was in that fountain.

Saint Augustine, Tractates on the Gospel of John, 15.12, in Joel C. Elowsky (ed).  John 1-10 (Ancient Christian Commentary on Scripture)149-150.

One must investigate what is meant by “will thirst” in the statement “Everyone who drinks of this water will thirst again.” … What is meant in the first place would be something like this: he who partakes of supposedly profound thoughts, even if he is satisfied for a little while and accepts the ideas that are drawn out and that he thinks he has discovered to be most profound, will, however, when he has reconsidered them, raise new questions…. But [the Word] says, I have the teaching that becomes a fountain of living water in the one who has received what I have declared. And he who has received of my water will receive so great a benefit that a fountain capable of discovering everything that is investigated will gush forth within him. The waters will leap upward. His understanding also will spring up and fly as swiftly as possible in accordance with this briskly flowing water, the springing and leaping itself carrying him to that higher life that is eternal.

Origen, Commentary on the Gospel of John, 13.13,15-16, in Joel C. Elowsky (ed). John 1-10 (Ancient Christian Commentary on Scripture) 152.

Today, of course, is the Sunday of the Samaritan woman.