The solution is as follows:

My Java-based solver is viewable here.

The solver is a quasi-brute-forcer with efficient backtracking. The major headache was that I initially assumed both "wells" in piece 9 were simultaneously full or empty. However, no solution exists for either of these cases. Much time was spent trying to figure out what was wrong with the blasted solver that kept (correctly) returning "no solution", "no solution".

After reconsidering my assumption and hard-coding in one of the ugliest hacks in the history of ugly hacks, the solver yielded the above solution in slightly under a second.

Special thanks to Tryth for a flood-fillable graphic, and to Victor, whose lengthy analysis instantly convinced me that converting this problem to SAT (which was my first inclination as well) would not have been a fun way to solve this problem.

The solution is as follows:

My Java-based solver is viewable here.

The solver is a quasi-brute-forcer with efficient backtracking. The major headache was that I initially assumed both "wells" in piece 9 were simultaneously full or empty. However, no solution exists for either of these cases. Much time was spent trying to figure out what was wrong with the blasted solver that kept (correctly) returning "no solution", "no solution".

After reconsidering my assumption and hard-coding in one of the ugliest hacks in the history of ugly hacks, the solver yielded the above solution in slightly under a second.

Special thanks to Tryth for a flood-fillable graphic, and to Victor, whose lengthy analysis instantly convinced me that converting this problem to SAT (which was my first inclination as well) would not have been a fun way to solve this problem.