In hydrology there’s two terms we’re interested in here. Water Table, and Potentiometric Surface.
The water table is simply the depth that the top of the water sits at (e.g. 10m below surface). Groundwater is under pressure, sorta like if you filled a baggie with water and then put a bunch of books on top. The weight of all the rock and stuff above it pressurizes the water. The Potentiometric Surface is the height that the water would rise *if that pressure was removed in an area*, like if you poked a pinhole in your baggie of water while the books are on top, it would spray upward.
An Artesian spring, which is what you’re describing, is simply a place where the Potentiometric Surface is **above** the ground surface. So when there’s a weak spot like a well, or cracks in the Earth, the water bubbles up out of the ground until gravity and other forces pull it back down.
Another thing that can happen is when water seems to flow out the side of a hill. What’s going on here is that under the water table beneath that hill there’s a layer of impenetrable material that water can’t pass through easily called a ‘confining layer’, or aquitard. The water can’t go down anymore, so it goes off to the side, going downhill. When it hits the side of the hill, it still can’t go down because of that aquitard, so it just dribbles out the side of the hill. You can see this if you look at a hillside and part way up there’s suddenly a bunch of greenery with nothing above it. The plants took root there because there’s lots of water percolating through the soil there.
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