Pumps for the most part.
Some quarries are high enough and have natural drainage to allow water to drain naturally but many others require full time pumping to keep them dry. Many will have a sump hole blasted deeper than the current working floor elevation to keep the water level below the floor.
Some smaller quarries I work in shut down over the winter and they allow them to fill up and when they start back up in the spring they fire large 6 inch pumps back up to drain them out again.
Some quarries have water seeping in or out, but most have pumps.
As the quarries get deeper, they have to pump part way up and then use another pump to get all the way up.
Some big open pit mines/quarries actually drill wells around the outside of the pit, and use pumps to lower the water table in the ground before it seeps into the pit. They do it this way because the water that comes out is clean groundwater, so it can be pumped to a nearby river or lake. If they pump the water out of the pit, it is considered dirty and has to go to a settling pond before it can go into the environment in
I feel like most people here are talking about rainwater filling a quarry. Often, it really is about the existing groundwater table “filling” the quarry, not precipitation. Think of it more like digging a hole in sand near the water at the beach. If you dig far enough down, you encounter water. The water didn’t run in from the top, or fall in from rain, it was already there. Now, you can use a bucket to quickly scoop out the water faster than it seeps back in. This is basically what is happening at a quarry. The pumps remove the water that is seeping in from the surrounding rock faster than it seeps in. The difference between the sand at the beach and the rock at the quarry is that water typically travels through loose sand much faster. However, if you stop pumping or managing the seeping water all together, the water level will slowly rise until it’s equal to the surrounding water table.
Culverts, pumps, geo-engineering, gravity. The real answer is maintenance. There are a bunch of systems that can keep a quarry dry, but if nobody is getting paid to maintain them they’re gonna get filled with muck and beaver dams and eventually you maybe get a superfund site, or a fence, or most likely nobody gives a shit and the thing sits for years, providing a place for cliche high school drama for years to come.
Occasionally people stick boats on the thing and then you 10x some property values, that’s legit but not common. Mostly you 10x the property value for mosquitos until somebody tosses a few gold fish in, then end up with a carp pond.
Active quarries have pumps to help them keep dry. In addition to the pumps a lot of the water is soaked up by the masses being brought out of the quarry. But even abandoned quarries does not always fill with water. Some places the bedrock is porous enough due to cracks that the water can drain through the bedrock and out towards the ocean faster then the quarry can fill up.
If drainage is faster than accumulation of water from rainfall or streams, then a lake will not form.
At least some of that drainage will likely take the form of streams above ground itself, but it can also drain through the Earth directly If there are places where the rock layer is not complete or the rock is very porous.
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