Shehiyah restrictions by Water:
The Shehiyah regulations apply likewise to water. This applies whether one wishes to heat it for drinking purposes, or for the purpose of cleaning vessels with it on Shabbos, nonetheless, all the Shehiyah restrictions apply. Thus, it is forbidden to leave water over an uncovered flame into Shabbos unless it reaches the state of Ben Drusaiy [i.e. half cooked] before Shabbos. This means that the water reaches half of its desired heat before Shabbos. Unlike the law by raw meat [see Halacha C], there is no allowance to place the water on the flame right before Shabbos. If one transgressed and placed water there, whether inadvertently or whether advertently, and it was not heated to 1/3 of its [desired] heat from before Shabbos, then it is forbidden to be used until enough time has passed after Shabbos to be able to heat it, just as is the law by all other cases.
Electric water urns: These systems do not contain a method of being covered, and thus must have their water heated up to 110 degrees from before Shabbos [i.e., sunset]. [To note that boilers with a Shabbos setting are considered to have adjustable temperatures and hence the water must be cooked to the point of Ben Drusaiy before Shabbos begins.]
What is the definition of half cooked water with regards to being allowed to leave it on an uncovered flame over Shabbos?
In a time of need may one leave water on an uncovered flame if it is only 1/3 cooked?
 Admur 254:14
 Why water has the Shehiyah restrictions if it can be drunk without cooking? Now, although water can be drunk even cold [and thus it should be permitted as explained regarding raw fruits], nevertheless [since] it is not as good [to drink it cold] as is to eat fruits raw as explained above [therefore it is not allowed]. [Admur ibid]
 Bedieved, or in a time of need it suffices if it has/had enough time to cook 1:3 of its desired heat before Shabbos.
 Other opinions: [This is] according to our custom which is lenient like the opinions which allow Shehiyah when the food is cooked to the point of Ben Drusaiy. However according to those which prohibit to leave food on an oven unless it is fully cooked to the point that further condensing will damage it, then here too the water must be placed on the oven with enough time for it to fully heat up to the point that it condenses in a damaging way. [Admur ibid and 253:9] Meaning that according to the stringent opinion, [as opposed to our custom] the water must be heated to the point that further cooking damages it, such as that it causes it to evaporate.
 Meaning half of its boiling point [100 Celsius], which is thus 50 Celsius. See Q&A1
 The reason: As water heats up quickly, [and thus it can be ready for the night meal, and there is thus worry that one may come to stoke the coals to speedy its heating]. [Admur ibid]
 Tzaruch Iyun why the Alter Rebbe here uses the term 1:3 cooked as opposed to Ben Drusaiy.
 Shabbos Kehalacha Vol. 1 pages 317-319
 There are two options: Either half of Yad Soledes [45 Celsius, or 110 Fahrenheit], which is 55 Fahrenheit or half of its boiling point [100 Celsius, or 212 Fahrenheit] which is 50 Celsius [122 Fahrenheit]. The Igros Moshe strongly proves like the latter, as if the water is only a third [or half] of Yad Soledes then it is barely even warm, and thus it’s not logical to say it refers to Yad Soledes. Rather he explains that it refers to a third [half according to us] of the amount that a person usually desires his water to heat up to, which is half of boiling point.
Rav Farkash supports this ruling and infers it from the wording here of the Alter Rebbe “half of its heat” as opposed to half of its “cooking”.
The Tehila Lidavid learns similarly in the Alter Rebbe that it refers to half of one’s desired heat for the water.
 The Ketzos Hashulchan questions whether if it suffices if water heated for tea is not boiled before Shabbos. This is because a person does not really want to use the water until it is boiling, and it is thus not considered fit to eat prior boiling point. He leaves this query without a conclusion. Rav Farkash however suggests in the supplements [page 338] that even in such a case it is allowed.
 Shabbos Kehalacha 1 p. 315
 This is based on the rule in Poskim that a pressing situation is equal to a Bedieved situation. So, rules also Mishneh Berurah [253:38], Elya Rabah, and Shemiras Shabbos Kihilchasa 1:63.