SUCCESS!!!!!!
1. Hmm, never actually had this happen. Usually the pressure from falling keeps everything at the lowest level. Was your drainage shaft the same dimentions as your staircase? Were you draining into the caves or off the side of the map? How many levels down did you dig your drain pipe? Not really sure what happened if it started to fill the shaft with water. Usually only the level of the drain fills up to 7/7 and the rest of the shaft is almost empty.
The drainpipe was five levels downward, and drained to the north of the map, which was slightly closer than the other sides. It was four wide, as the staircase is four wide on an east-west and three thick on a north-south. I have been following your instructions, you see, or it would have been square instead of rectangular, I'm weird like that. I dug down under the hatch four levels, then the dwarf got tired and went to bed (No idea why staircases are so tiring, perhaps he'd had a big day) and it promptly filled the lowest section of the stair with 4/7 water. I got another dwarf to dig sideways from the third level down below the aquifer and then dig down to where I had set up the drain. He then, when he'd finished digging the drain with his narcoleptic friend, went back and fixed up the little sump problem I had with the sideways staircase, so it looked exactly the same as your pic, then I dug each level of staircase below the drain in a singular sheet, 5, 4, 3, 2, WATER. Other than a series of really annoying cancellations when I was smoothing the walls, it all went well, but I didn't know that it would at the time.
The old drain is now being layed out as my jewelers and decorating quarters until I can get a few things worked out down below.
2. Taking down the waterwheel. There are plenty of tutorials explaining how to build a waterwheel. There are none explaining how to take it DOWN, more than a brief warning that it can fall. So I tried to stop it spinning first. That... that didn't work out too well, but no one was actually injured, as my hospital is still empty. So yeah. I think next time I'll try deconstructing it whilst it is spinning. Oops. 
You're on the right track there, deconstructing it while spinning is indeed the best way to do it so that they don't collapse into the drain.
Whups. Well, I thought it was the opposite. Damn.
3. I didn't realise that I couldn't dig out the layer below the aquifer. I almost did, due to sheer exuberance at getting there. Um. The result is flooding. Got it all fixed up now. Looks like that's why you said to make the drain a few levels lower than the aquifer.
Yup, aquifers drain in all directions except up. I usually smooth the walls around the stairs of first layer of stone under the aquifer just as a reminder that I don't want to dig there.
...I'm going to do that now too.
For various reasons I ended up with a whole pile of clay bricks when I was still messing around trying to work out how to get all the way through the aquifer (I also made a more robust military) so I've paved the walls of the shaft with earthenware. It looks really good.
THANKYOU!!!
You are welcome 
. As I said before, I totally love aquifers. Most of the time, I won't even consider an embark site unless it has one. Once you have an aquifer tamed, it's so useful to have that it's worth the inconvenience of having to pierce it.
I agree it will be so useful! I did really want an aquifer tamed, as I wanted a constant fountain-and-waterfall-waterpark system in my "dwarf entertainment area", and I keep choosing places to live which are frozen for part of the year. So far I'd only been able to BOOM the hell out of the aquifers, which was a clumsy process at best, and I've pounded rather a lot of dwarfs deep into the earth with it so far.
Now to find out if my well cistern idea will work! Also, I need to work with some overpressure systems for bubbly fountains... Which is my backup idea for the well cistern... Heh.
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Topic: Aquifer Pierce Pump-Stack Method Tutorial (Read 6211 times)