The volume of pipes used for detention storage is simply the area times the length of the pipe. To support pipes on a slope, we break up the pipe into various components and use average end areas to come up with a rating curve. The following sketch shows the various sections.
Where
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VolA = Volume of pipe fully occupied by water
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VolB = Volume of pipe partially occupied by water
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H = Water depth measured from the I.E. of the inlet end of the pipe
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HA = Wetted water depth at the downstream end.
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HB = Wetted water depth at the upstream end.
Circular
The volume of a circular pipe is simply the area of the circle multiplied by the length.
Arch
The volume of an arch section is simply the area of the arch multiplied by the length.
Ellipse
The volume of an ellipse section is simply the area of the ellipse multiplied by the length.