As far as i know, big Gallstones (over 2cm) can be broken inside liver ducts during water fast, and that helps with eliminating them in subsequent liver flushing.
My way is to do both in one go, that is why i start my fast on the first day of First Quarter of moon month and finish the fast just after the night of the fullest moon. This way Liver Flushing takes place on the night of the fullest moon, midnight, when gallbladder and liver acupuncture channels are at their strongest. At the same time relaxation, gained after the fifth day of the fast, helps with gallstones elimination.
Regarding moon calender,there are four "weeks" in it, the first week is called New Moon, then goes First quarter, then the week aftr full moon, and the last is Last Quarter week. These weeks slightly differ from usual calender weeks, but many calenders print the main four days of a moon month.
The first and third weeks of moon month are weeks of "contraction", the second and the fourth weeks are weeks of "expansion". This is closely related to all liver ducts, which are at their widest during the night of the fullest moon. That helps with any flushing, as it is easier to flush an object down a wider passage.
Somehow, it applies to Water Fasts too. They are more successfull during the second and, in a lesser extent, during the fourth moon week. That is if a fast is 7-10 days long. If someone fasts for 30 days, it is best to try to finish it on the last day of a moon month, and start eating again on the day of New moon. For fasts of 30 days it is not the moon week that matters, it is a month of the year that rules which organ/system is most affected with the fast.