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The DF pre embark aquifer indicator error (i.e. showing no aquifer when there is one) affects the lower half of the world tiles, i.e. the line from the NW corner to the SE corner downwards, including the line. This means that embarks above that line aren't affected by that bug (there's another, older, bug that can show the presence of an aquifer when there is none, as well as clay when there is none, due to DF failing to take soil erosion caused by high elevations into account).
If you're using DFHack, the Embark Assistant plugin shows aquifers (and clay) correctly.
Aquifers appear whenever an aquifer supporting material makes up the layer at a depth of 3 or deeper. All soil types except true clay ones (ones that can be used as a source of clay, rather than the "fakes" that have clay in their names but aren't real clays) support aquifers, and a small number of rocks do (sandstone and conglomerate as a minimum).
DF generates 4 layers of soil normally, although it sometimes generates only 3 in deserts, which means there usually is an aquifer at a depth of 3 and/or 4 (blocked only if they're both clay when soil). At high elevations soil erosion can reduce the number of soil layers to 2/1/0, which means the soil is too shallow to allow for an aquifer. However, you can always end up with one or more layers of aquifer bearing rock under the soil.
It can also be noted that 5% of all aquifers are heavy one (the "old" type) while 95% of them are the light one. The wiki page on the double slit method is still the go-to resource for learning how to deal with heavy aquifers (it's quite tedious with all the cancellations).
Dealing with light aquifers is fairly easy, although it's still possible to screw up (I've done it, so I know it's possible):
- Time is of the essence. It's not extremely urgent, but make sure the dorfs involved in the aquifer penetration work aren't distracted by a lot of other tasks. Personally I use the standard two miners to do all of the work (and I use them as the sole workers for heavy aquifers as well), which means they need to have construction work enabled, and you shouldn't have any other construction tasks active during this time.
- Dig a single tile staircase down to a depth of at least 6 (aquifers leak down through the roof of the level below, so you need a solid level of (non aquifer bearing) rock (or clay) in between the lowest aquifer level and the first level you're going to use. Once you're past that barrier level you can cancel the further digging downwards if you want. At the bottom level, dig out an evaporation area of something like 5*5 (i.e. a room, or the beginning of a room, which can sit at the end of a corridor).
- Once you've dug out your evaporation area you can then start to both expand the staircase if you want a wider one, and plug the leaks. I use a 3*3 staircase with a "pillar" in the middle, but you use whatever structure you like. It's important that the expansion and plugging is done only one level at a time, since if you do all the digging first and perform the plugging afterwards you may have too high a water flow for the evaporation area to cope with it (it's possible to dig out an evaporation area large enough to deal even with a heavy aquifer, but that's a rather large area, and when you realize that the area you have is too small, you're really in a hurry to enlarge it, and it may already be too late).
So, dig out the top level. Since that level usually is soil, dig away the tiles perpendicular to any staircase tiles (including the center tile if you have one), but ignore the corners (aquifers don't leak diagonally. Water flows diagonally, but it somehow doesn't leak diagonally), and then build walls there (I use wood, but it would work with stone from below, if you've got enough of it. Obviously you need enough building material before embarking on this step, but I always start cutting down trees immediately on embark, so I've got plenty of it). Once you've built all the walls (don't forget the center tile, if you have one) you start with the next level below that one. If it's soil as well, just repeat the process. If, however, it's aquifer bearing rock, order the rock to be smoothed (don't forget to enable stone detailing on the workers intended for the task). You don't have to smooth the corners, but I do just because it looks nicer. Note that you don't have to do anything to the bugger level just below the aquifer.
- You're done. You can then start to build your fortress below the aquifer (although you may hit the first cavern, of course. I change the advance world gen parameters to give me a 15 level gap between the surface and the first cavern, so I can fit my fortress in there, unless the rock turns out to be aquifer bearing, in which case I have to adapt).