Refer to this document, and particular to Figure 2:

enter image description here

I wonder what is the formula that governs the box culvert partial flow with respect to full flow? In other words, base on what formula the above graph is generated?

The formula for pipe partial flow is as thus:

enter image description here

So essentially I'm trying to find the analogous formula for pipe culvert.


2 Answers 2


Page 2 of the DD_17.pdf document that you linked tells you the answer in the first paragraph.

Manning's equation is what is used. There are different formulas used to solve for the hydraulic radius R for different shape culverts.

This is a fundamental topic for hydraulic engineering can I suggest that you find a text book on this topic? It will offer much more in depth background information than can be offered in this setting.

  • $\begingroup$ It doesn't explain how can the Q/Qf way more than 1, when D:RISE ratio approaches 1. $\endgroup$
    – Graviton
    Apr 4, 2019 at 8:56
  • $\begingroup$ Well that wasn't part of your original question, and I think you might be able to find that answer by digging into some reference texts. $\endgroup$
    – ShadowMan
    Apr 4, 2019 at 15:01

To answer my own question:

The formula for hydraulic radius, velocity and flow remains the same. But what is important to note is the wetted perimeter, $P$.

When the flow is partial flow,


When the flow is full flow,


(because now the top part of the box is also "wetted").

This explains why the "strange" behavior how when $D:RISE$ approaches $1$, $\frac{Q}{Q_f}$ can go way beyond $1$, and when $D:RISE=1$, $\frac{Q}{Q_f}$ goes back to $1$.


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