Solve The Mystery

I have recently kept seeing these little pieces of paper weighted down by stones on bridges and culverts. Are they a form of offering to the river/stream as Hua suggests? Or is it some sort of rural messaging system? Suggestions to be written on a piece of paper and placed under a stone on the nearest bridge.

Probably more easy to solve is the identity of these little brown birds which we often see running through the desert bushes. This is when I wish I had a good bird book in the hand.

P.S  Several years later I discovered these were chukar partridges.