Submerged Culvert Inlet (Orifice) Headwater

A culvert has two possible personalities and the design question is always which one it is wearing. Under inlet control the barrel could carry more than the opening will admit, so the entrance is the bottleneck and neither the length, the slope nor the roughness of the pipe matters at all. Under outlet control the barrel and the tailwater govern, and all three of those matter. Once the inlet is submerged, inlet control behaves as an orifice: $Q = C_d A\sqrt{2g(HW - D/2)}$. A one metre pipe with 2 metres of ponding above the invert passes $0.6 \times 0.785 \times \sqrt{2 \times 9.807 \times 1.5} = 2.56$ m³/s.

The head is measured to the centroid of the opening rather than to the invert or the crown, which is where the $D/2$ comes from, and it is the term most often dropped. The discharge coefficient carries the entrance geometry and it is worth real money: a square-edged pipe projecting from a fill runs about 0.5, a headwall with a bevelled or grooved edge about 0.7, and that difference is roughly 40 percent more capacity through the same pipe for the price of a concrete end section. Improving an inlet is one of the few upgrades in drainage that costs almost nothing.

Use this as the screening calculation it is. FHWA's HDS-5 gives the full inlet-control equations with fitted coefficients per inlet type, and those are what a design submission needs; this orifice form is the physics underneath them and is close enough to size a pipe before you open the chart. Note also that below submergence, at roughly $HW/D < 1.2$, the inlet acts as a weir instead and this equation does not apply. The practical rule is to compute both inlet and outlet control headwater for the design flood and take the larger, because the culvert will do whichever is worse, and to check the answer against the allowable headwater set by the road shoulder and by whatever sits upstream of the crossing.