Could someone perhaps make a simplified step by instruction list with a rough accompanying diagram?
The diagrams below are a side view. A is the aquifer layer, R is non-aquifer. First you dig out (from above) the up-down stairs of the first layer:
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RRRRRRRRRRThen build pumps, and channel out a reservoir to the right and the closest stairs, so that you can pump water from those squares back into the aquifer.
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AXXX AA A
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RRRRRRRRRRThat keeps those nearest squares dry enough to dig down into the next layer, discovering in this case that it is another aquifer layer.
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AXXX AA A
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RRRRRRRRRRThis row of stairs provides drainage to dig the next row, and so on across the layer.
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AXXX AA A
AXXXXAAAAA
RRRRRRRRRRNow there's near-instant drainage over the entire area you've dug out, allowing you to smooth the walls (or dig out and rebuild the walls, if it's a soil layer).
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SXXX SA A
AXXXXAAAAA
RRRRRRRRRRPrepare for the next layer of pumps, constructing floors to support them, and channeling spaces for (A) pumping water up from the layer below, and (B) transmitting power down to the next lower layer of pumps, if necessary.
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S . SA A
A X XAAAAA
RRRRRRRRRRBuild the pumps, transmitting power from the pumps above through a horizontal axle and gear assembly.
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S %% SA A
A X XAAAAA
RRRRRRRRRRThat will keep those spaces on the left side dry enough to dig down into the next layer, here finding that it is a non-aquifer layer. In order to drain it, you need to dig your drain to the edge. Build a hatch to ensure that water doesn't drop and flood your tunnel as you dig it, go down a couple of layers, and dig your drain.
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S %% SA A
A¢X XAAAAA
RXRRRRRRRR
RXRRRRRRRROnce the drain is complete, remove the hatch. The drain lets you dig the stairs down row by row, as you had done above. (Take a little jog to the side and go down again, so that when the process is over you only have to wait for the water to drain out of the central area; the long drainage pipe can take as much time as necessary to finish draining.)
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S %% SA A
A X XAAAAA
RXXXXRRRRR
RXRRRRRRRRAgain having complete drainage of the area, you can smooth the walls over. After that, no more leaks. Let the drain drain away.
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S %% SA A
S X XSAAAA
RXXXXRRRRR
RXRRRRRRRRPretty neat. I had to play back some parts a couple of times to really get the details.