Water Channels on the Temple Mount
Warren map

Finding this map on-line many years after I began this web site
was a wonderful confirmation!
And I believe it is the last puzzle piece needed in putting this picture together.
 

As you can see there are NO water channels leading to the Dome of the Rock area. This of course is because water doesn't flow uphill.
 
 

 The aqueduct coming from Bethlehem (Solomon's Pools) that entered the temple mount over Warren's gate was called Ein Etam by the Jews.

 Abaye, a Jewish sage of the 4th century was quoted as saying;" What comes out is that the Ein Etam (the spring of Etam) is 23 cubits higher than the Azara."

Etem was in reference to the Lower Aqueduct which entered the Mount via the bridge (at Wilson's Arch).  An ancient cubit is debatable but somewhere between 17 to 22 inches; 23 cubits is approximately 32 to 42 feet.
The pool at the end of the western branch of the water system (green arrow on map) would have been just below the level of the Azara (Priest court) which is approxamately 32 feet below Wilson's arch.

Tosefta Psachim, Ch. 3, Par. 12, "How is the Azara cleaned? Seal the area and let the water from the aqueduct enter till it becomes clean like milk."

This of course would also require a duct to drain the water which flooded over the Azara each day.  That drain may have been the same as the duct that the blood of the animals was poured down, sometimes called  "the Blood Channel". It was located near the southwest corner of the ramp leading up to the altar where they burnt the offerings. This duct/channel emptied into, the Kidron Valley.

Brazen Laver
There were two reason for " living water" on the Azarah/priests court; one was to cleanse the court daily and the other was to fill the brazen laver, which was a very large pot or tub on a pedestal which held a great amount of fresh "living" water. It had faucets around the sides of it which could be opened for the cleaning of hands and feet for the purpose of purity.  It had to be emptied each night and refilled in the morning with fresh water.  During the second temple period a new way of filling the laver was devised called the muchni.

The Muchni
Ben Katin, one of the High Priests who served during the era of the Second Temple, devised a system for retrieving the water each morning using the mechanism of the muchni, meaning "machinery,"  The laver was submerged into a specially-made pool under the court. The laver was then hoisted up by the first priest in the morning.

Note: Muchni is also a name for a type of wagon wheel.  Wheels installed under the laver may have been needed to roll it from it's place; between  the altar and the House of God, over to where the underground pool was located under the Azara for refilling it. The "hoisting up" of this large pot full of many, many gallons of water by just one priest would have required an elaborate pulley system.
 

I believe this is the pool that I have marked with the green arrow on the map below.
The water channels  leading down to the temple itself in the southern area are now 30 to 40 feet below the ground level of the mount today. There appears to be four watergates/junction boxes with a pool at the end (green arrow) on the duct leading to where the Temple was located.  This pool would have been used for filling the Brazen Laver.
 
 

I'd like to give my opinion on a the channel running north to south, red arrow, that ends abruptly. Since we know water doesn't flow uphill then this section of the channels must have begun at the aqueduct and flowed downhill from there.   This channel that would have connected to the water aqueduct  where the Aksa fountain stands today.  Perhaps that is where there was a junction box and the Arab builders tapped into when building the fountain.
The purple arrow I believe would have started at an upper cistern and flowed downhill toward the lower cisterns for the purpose of catching rain water on the mount. In other words the water channels involved in the purple circle on the map were part of the suffice water system flowing down from the upper level of the temple mount and not connected to the aqueduct. It seems possible that these were put in place by the Arabs at an later time in history and not by the Hebrews.

Picture of a water channel.