The pipe failure during gravity flow through a partially (or completely) full pipe occurs due to the internal friction of the fluid (viscosity) and with the wall. The algorithm is applicable for water or liquids with kinematic viscosity equal to it.

Gauckler–Manning–Strickler Ecuation(*):

where
v: average flow velocity in the section, [m/s] k: conversion factor, its value is 1 when the equation has SI units, [m1/3] n: Manning coefficient – ​​dimensionless
RH: hydraulic radius, [m] SH: hydraulic slope – dimensionless
The Manning coefficient is determined empirically and depends on the shape of the flow section, the nature of the wall material, the sinuosity of the route, etc. Usual values(**) for circular flow sections and straight routes, for different materials from which the pipe is made:

The hydraulic radius is calculated with the formula:

where:
A: area of ​​the flow section, [m2] P: wetted perimeter of the flow section, [m] The area of ​​the flow section and the wetted perimeter of the flow section are a function of the degree of filling and are calculated with the calculation formulas corresponding to the circle segment.
The hydraulic slope is:

where:
SH: hydraulic slope, dimensionless
hi: pipe height at the beginning of flow, [m] hf: pipe height at the end of flow, [m] L: flow length, [m]

If we take into account that the flow rate is a function of the flow velocity:

we can calculate the flow rate with the formula:

where:
Q: flow rate, [m3/s]

(*)
http://en.wikipedia.org/wiki/Manning_equation)
Handbook of PE pipe (http://plasticpipe.org/publications/pe_handbook.html), cap. 6, pag. 186
(**)
http://www.engineeringtoolbox.com/mannings-roughness-d_799.html