A little more information on how you are shutting down your amine system would be helpful.
All the amine units I've dealt with shut down something like this: 1) Short circulate until the rich amine has an acid gas loading like lean amine. 2) Drain the amine to storage. 3) Refill with water and establish circulation to flush. 4) Drain the wash water. 5) Steam out.
I'm assuming you are looking to minimize the volume of waste water generated in steps 3 and 4 above. I think the true origin of the water wash steps was to flush out residual amine to prevent cracking non-stress relieved carbon steel equipment when steaming out the unit. If your unit is constructed of all post weld heat treated equipment or higher alloys you could consider foregoing the water wash step entirely.
However, the wash water usually has pretty high nitrogen and COD levels even if you drain every last bit of amine you can get out of the unit before flushing it. I really wouldn't recommend skipping the water wash even if you could steam out your equipment with residual amine present. The water wash just does a really good job of cleaning up the unit. I think the condensate you generate while steaming out could be pretty potent if its not following a water wash; it could prove to be a little much for your water plant to handle depending on how capable it is.
If its important to you to minimize the wash water volume be very careful to add no more volume than is absolutely necessary to establish circulation. You certainly don't want to fill every vessel up to normal operating level. Just barely put in enough water to find the level in the level bridles, or just enough to provide the necessary NPSH to your circulating pumps. Be especially careful not overfill the rich flash drum as it tends to be pretty large. Try to circulate at a fairly low rate since you're just flushing the unit and the residual solvent should disperse in water very readily. The lower flow you use, the less NPSH your pumps will require, and the less volume you can get away with using to establish circulation. Just be careful to keep your pumps above a safe minimum flow while optimizing your levels and circulation rate.
Good luck