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First question - It depends on the orientation and diameter of the pipe vs. the cohesive tension of the fluid. If the pipe has sufficient cross-section, some water will leak out initially regardless of the slope if the pipe end is held horizontally.

Second question - The pressure will never reach the far end, because there is no pipe that has zero rigidity, and sidewall flex will take up any pressure you could possibly apply. The frozen water in the pipe at sufficient elevations would preclude it as well.

Third question - Same issue with sidewall flex, and you need to define the variable "midway" - i.e. if the leak is 13,000 miles away, or two feet away.
 

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I worked on installing sprinklers for the new house. 6 zones about 40 sprinkler heads.
Even thou I rented a trencher it was very hard work.
Here is a few questions that I have in mind.

Suppose you have a pipe full of water going around the world, over mountains and valleys.. It ends where it starts and you keep both ends at the same elevation. will the pipe leak water out?
Depends where the ends are. Think of a mercury barometer. Atmospheric pressure will push a column of mercury some 30" up into a tube that is otherwise vacuum. Same applies to water, but the column will be much higher. If the ends of the pipe are at sea level or any altitude lower than the top of the column of water in a water barometer, then Yes. Assuming the full pipe is open to the atmosphere, at any points in higher elevations where the weight of the water in the exceeds the atmospheric pressure at the open ends the water will flow downward creating pockets of vacuum. This also assumes there's a 90 degree elbow in both ends pointing upward, otherwise water will simply flow out both ends much like laying a full straw on a table.

Second question :
The pipe is enclosed at both ends and you apply pressure at one end, how long will it take for the pressure to reach the other end?
Assuming a closed system and vacuum pockets have not been allowed to form, there will already be significant static pressure on both ends depending on altitude of the ends. This will depend on the rigidity of the pipe, If "Absolutely" rigid (impossible) theoretically any changes in pressure will propagate throughout immediately, otherwise it will depend on the "give" of the pipe. Think old OEM brake lines on a GL1000 vs brand-new Stainless lines, in either case you will be able to raise the pressure enough to stop the bike.

Third question:
If you apply a steady pressure at one end, lets say 100Lb/ square inch and the pipe has a leak midway, is the pressure at the end the same or smaller than 100lb?
If You are applying 100 PSI ABOVE any static pressure at the end of the pipe, then if your pressure source can supply more GPM than the leak and you can maintain the 100PSI pressure increase, the 100PSI Pressure INCREASE will be seen throughout the pipe regardless of the leak, absolute pressure at any given point will depend on altitude. The highest point in the pipe will have 100PSI absolute pressure.
I bet this gets some debate going! :ROFL:
 
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