St. Venant Terms in SWMM 5 and how they change for Force Mains

Note:  An explanation of the four St. Venant Terms in SWMM 5 and how they change for Force Mains.  The HGL is the water surface elevation in the upstream and downstream nodes of the link.  The HGL for a full link goes from the pipe crown elevation up to the rim elevation of the node + the surcharge depth of the node.  dq1 is calculated differently based on full or partially full force mains and gravity mains

            dq2 = Time Step * Awtd * (Head Downstream – Head Upstream) /  Link Length  or

            dq2 = Time Step * Awtd * (HGL) /  Link Length

            Qnew = (Qold – dq2 + dq3 + dq4) / (  1 + dq1)

when the force main is full dq3 and dq4 are zero and

Qnew = (Qold – dq2) / (  1 + dq1) 

The dq4 term in dynamic.c uses the area upstream (a1) and area downstream (a2), the midpoint velocity, the sigma factor (a function of the link Froude number), the link  length and the time step or

            dq4 = Time Step * Velocity * Velocity * (a2 – a1) / Link Length * Sigma

the dq3 term in dynamic.c uses the current midpoint area (a function of the midpoint depth), the sigma factor and the midpoint velocity

            dq3 = 2 * Velocity * ( Amid(current iteration) – Amid (last time step) * Sigma

dq1 = Time Step * RoughFactor / Rwtd^1.333 * |Velocity|

The weighted area (Awtd) is used in the dq2 term of the St. Venant equation:

            dq2 = Time Step * Awtd * (Head Downstream – Head Upstream) /  Link Length