A full solution will require an interative program which I will write in the next couple of weeks. But initial results are very discouraging for the young- earth position. Especially if one want to have men living in caves in Europe and Asia only a couple of hundred years after the flood. Christianity needs to seriously reconsider the young-earth position. I will assume laminar flow which greatly simplifies the problem and yet overestimates the speed with which the erosion will occur.
Caves start from small cracks in rocks.
Lets start with a typical but large crack of .01 inch. We will assume that every day it rains and that water drains off the land above within two hours. I will let all the water that wants to, come down the conduit for two hours. Initially the speed with which water can travel the conduit limits the water flow and every day most of the water goes down the creek rather than down the conduit. This is the quadratic equation in Vb which is the velocity the water travels through the conduit.