This guide explains how to unlock the Sandspout wells in Nier Replicant (Remake/PS4). It covers the steps and how to use them.
How to Unlock the Sandspout Wells - Steps
Clear the Desert Temple

As a prerequisite for using the desert wells, you must clear the Desert Temple. The wells exist before clearing the temple, but you cannot use them.
Get the Royal Compass

After clearing the Desert Temple, talk to the guard in front of the King's Mansion to receive the "Royal Compass". Without the Royal Compass, you cannot enter the sandstorm in front of the Desert Temple, so be sure to get it first.
Talk to the Advisor

By visiting the city of Facade again, the advisor will appear in the King's Mansion. Talking to the advisor will grant you permission to use the desert wells. After this, you will be able to use the desert wells.
Unlock the Wells

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After talking to the advisor, go and actually unlock the wells. They become usable by moving the stone on top of the well. There are a total of four wells: at the desert entrance, in front of the city of Facade, at the Barren Temple, and in front of the Desert Temple. They are useful for traveling during quests, so be sure to unlock them all.
Unlocking Two or More Allows Travel
By unlocking two or more wells, you can warp between them. You can still use the warp in Part Two, so unlocking them early will make things easier later on.