The standard approach to solving hydraulic equations to calculate depth and velocity in pipe, (given flow, pipe size, roughness, and slope) involves iteratively solving pipe geometry and head loss equations. For large models, this can be slow.

Most pipes in storm and sanitary sewer systems are circular. It is possible to generate solutions to the flow equations for circular pipes and fit the solutions to polynomial equations. These polynomial equations can be solved explicitly (i.e. with no iterations), thereby significantly reducing the time to solve large models with the GVF solvers.

The equation for normal depth in circular pipes can be given by

This equation is accurate to within 1% over most of its range and no worse than 3%. For Yn/D > 0.94, there are actually two solutions but the lower one in Figure 1, which compares the equations to an exact theoretical solution, is used as it is more common.

The equations for critical depth can be given by

Once the critical depth has been determined, the critical slope can be given by

The use of the term "explicit" in these equations refers to the fact that the equations are solved explicitly, not iteratively and differs from discussion of the "Explicit" solver which refers to the numerical scheme for solving dynamic wave equations. These equations are only used in the GVF-convex and GVF-rational solvers because in these, the flow is calculated before the depth.

A description of the derivation of these equations can be found in Jin, M. and Walski, T., 2011, "Efficient Equations for Circular Partly-full Pipe Hydraulics", EWRI Conference, palm Springs, Cal.