[Ka Razy]

Is your C7 a Z51? I hear the "water runing sound" on cold startup also.
Heres what I read. Hope this helps...

Checking Engine Oil/Dry Sump Oil Tank Noise on Startup

For an accurate reading of the dry sump lubrication system, check the oil level only after the engine has been thoroughly warmed up, after at least five minutes but no more than 15 minutes after engine shutoff.

Some customers may comment on a noise similar to running water from the dry sump oil tank when the engine is cold and started. This is a normal sound of oil returning to the tank.