Hello,
I don't quite understand what you exactly do, especially the culvert through the hole. A culvert is simply two points in the mesh, preferably not on boundaries, otherwise it should be an open liquid boundary (on a solid boundary it would go against the no-flux condition of that boundary). The two points are treated as sources, except that their discharge is deduced from the free surfaces and the culvert characteristics. The direction of the flow goes from the point with highest free surface elevation to the other. The flow depends on the characteristics of the culvert, but cannot go from low level to high level. To prevent drying, there is a relaxation coefficient that limits the flow at the first time steps (this can explain a small flow at the beginning). I hope this can help.
Jean-Michel Hervouet