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This well is on the road from Medina to Badr. Its name, "ash-Shifa'", means "healing".
No book of hadith or Sira mentions this well or the origin of its name. The stories that circulate online have no source. So nothing special can be attributed to it.
It is a witness to life on this road: for centuries, every water point set the pace of the journey for pilgrims and caravans.