I wish I could see a picture. So this black steel pipe with the cap on top that sticks out of the ground 3 feet is considered the casing (annulus)? Does it go all way down the shaft, or at some point does it connect to another casing that goes further into the well? Isn't there also a liner in there somewhere. What I'm trying to wrap my head around is how/why surface water can be getting inside the casing when it is 3 feet above ground?
To install maybe 5" casing it is necessary to drill about an 8" hole. The annular space is the area between the 5" casing and the 8" hole. It doesn't really matter if the casing goes all the way down or just part way. The annular space between the 5" casing and 8" hole must be sealed with cement from the surface down far enough to prevent rainwater from getting into the well. If this concrete seal was not done properly or the concrete has cracked over time, rainwater can take any contaminates on the ground down the outside of the casing.
A tent over the well might be a good idea. If this prevents the brown water after a rain, the annular space is not sealed properly. But I have seen gopher holes many feet away from the well that will channel rainwater past the cement seal and cause this problem. In other words if the tent doesn't stop the brown water, it maybe coming in from a distance away from the well. If this is the case the gopher hole must be found and plugged.