Dr. Pete is Lost

Dr. Pete, the renowned adventurer and expert in all things obscure, has gone and gotten himself stuck down a well in the middle of Nowheresville, USA. His trusty map has been reduced to a soggy mess, and his flashlight battery is dead.

He's trying to get out, but the walls are slippery, and he's not exactly the most athletic of individuals. Help him out by solving the puzzle below!

Well-Solving Puzzle