When not flowing full, water surface profiles within a culvert are generally calculated using equations that describe Gradually Varied Flow (GVF) conditions.  The GVF equations account for gravitational and frictional forces acting on the water, and are used to calculate water depths throughout the culvert.  A GVF profile is also known as a water depth profile and applies to steady-state, or constant flow, conditions.

Limitations of Gradually Varied Flow equation:

1.       Steady State Flow

2.       One Dimensional (can only calculate average cross sectional water velocity)

General form of the Gradually Varied Flow equation is:

Where:

S_o = Bottom slope, positive in the downward direction

S_f  = Friction slope, positive in the downward direction

y  = Water depth, measured from culvert bottom to water surface

x  = Longitudinal distance, measured along the culvert bottom

Fr = Froude number

The Friction slope is approximated from Mannings Equation:

Where:

S_f = Friction slope, positive in the downward direction

n    = Mannings roughness coefficient

V  = Average cross section velocity

j  = Constant equal to 1.49 for English units and 1.00 for SI units.

R  = Hydraulic radius, (Wetted Area / Wetted Perimeter)

Numerical Integration of the GVF Equation:

The GVF differential equation does not have an analytical solution.  Therefore, FishXing uses numerical integration to generate a water surface profile.   Numerical integration is a technique of dividing the channel, or culvert, into numerous short reaches and then performing the computations from one end of the reach to the other. 

FishXing primarily uses the Standard Step Method of numerical integration.  The following form of the equation is used:

Where:

DE  = Change in specific energy from one end of the reach to the other

S_fave = Average friction slope across the reach

Dx = Longitudinal distance from one end of the reach to the other

y   = Depth of water

Q = Flow rate

g = Gravitational acceleration

A = Wetted cross sectional area

Since the friction slope and wetted area are functions of depth, solving for depth at a given distance (Dx) requires an iterative solution. FishXing uses a bisection method to find the solution.

Backwater and Frontwater Calculations

The water surface profile can be calculated from downstream going upstream (backwater calculations) or from upstream going downstream (frontwater calculations).  The direction depends on the classification of the water surface profile (hydraulic slope and type of curve). For Mild, Critical, Adverse, and Horizontal slopes FishXing performs a backwater calculation beginning at the downstream boundary. Frontwater calculations are performed for Steep slopes, beginning at the upstream boundary.  If a Steep slope culvert is backwatered (S1 curve), FishXing also performs a backwater calculation and identifies the location of the hydraulic jump (if one occurs).

Other Considerations

If only a portion of the culvert becomes pressurized, FishXing will switch between the GVF and full flow equations.

See Boundary Conditions, Rapidly Varied Flow, Headwater Calculation