I'm not sure at all what you're dealing with, but my first idea would be to find a way to use it as it is or with minor modifications. Perhaps plant something like willow or bamboo that is water tolerant, build a little pond, or alternatively create ridges that rise above the boggy area but that can take advantage of the water.

Throughout much of the US Midwest a lot of agricultural fields were drained using field tile; back in the day it was clay tiles and today they use a special plastic pipe that has holes in it. There is a special shovel called a tile shovel or tile Spade that is long and narrow just for digging trenches to bury things like this.