Culverts in H2OCalc, background to Culverts in #InfoSWMM and #SWMM5
Introduction: H2OCalc is a hydrology and hydraulics calculator sold by Innovyze that has many of the equations and calculation methods used in InfoSWMM, ICM and SWMM5. This is the H2OCalc information for the FHWA Culvert Equations.
A culvert is a pipe that carries water under or through some feature (usually a road or highway) that would otherwise block the flow of water. The culvert acts as an open channel as long as the flow is partly full. The characteristics of the flow are very complicated because the flow is controlled by many variables, including the inlet geometry, slope, size, roughness, approach and tailwater conditions, etc. Therefore, an adequate determination of the flow through a culvert is generally made by laboratory or field investigation.
Culverts are classified according to which of their ends controls the discharge capacity: inlet control or outlet control. If water can flow through and out of the culvert faster than it can enter, the culvert is under inlet control. If water can flow into the culvert faster than it can flow through and out, the culvert is under outlet control. Culverts under inlet control will always flow partially full. Culverts under outlet control can flow either partially full or full. H2OCalc analyzes culverts using two different approaches: a simplified culvert analysis method and the Federal Highway Administration’s (FHWA) HDS No. 5 method (Normann et al, 1985). Both techniques are described below.
Simplified Method
The simplified method classifies culvert flow into six different types on the basis of the type of control, the steepness of the barrel, the relative tailwater and headwater heights, and in some cases, the relationship between critical depth and culvert size. These parameters are quantified through the use of the ratios in Table 3.4. The six types are illustrated in the following figure.
For culverts flowing full, the friction loss (hf) can be determined using the Darcy formula. For partial flow, the Manning equation can be used. The friction head loss between sections 1 and 2 (see the figure below), for example, can be calculated from Manning’s equation as
where L = the culvert length
K = Conveyance factor and equals
R = hydraulic radius (m, ft); R = A/P
A = crosssectional area of flow section (m2, ft2)
P = wetted perimeter (m, ft)
N = Manning’s coefficient
Table 34: Culvert Flow Classification Parameters
Flow type  (h1–z)/D  h4/hc  h4/D  Culvert slope  Barrel flow  Location of control  Kind of control 
1  < 1.5  < 1.0  ≤ 1.0  steep  partial  Inlet  critical depth 
2  < 1.5  < 1.0  ≤ 1.0  mild  partial  outlet  critical depth 
3  < 1.5  > 1.0  ≤ 1.0  mild  partial  outlet  backwater 
4  > 1.0  > 1.0  any  full  outlet  backwater  
5  ≥ 1.5  ≤ 1.0  any  partial  inlet  entrance geometry  
6  ≥ 1.5  ≤ 1.0  any  Full  outlet  entrance geometry 
Adapted from Lindeburg (2003)
The total hydraulic head available, H, is divided between the velocity head in the culvert, the entrance loss (if considered), and the friction loss as follows
where ke is the local loss for entrance.
Rearranging Equations (46) and (48), velocity through the culvert can be given as
A. Type1 Flow
Water passes through the critical depth near the culvert entrance, and the culvert flows partially full. The slope of the culvert barrel is greater than the critical slope, and the tailwater elevation is less than the elevation of the water surface at the control section.
where Q = discharge from the culvert (m3/s, ft3/s)
Cd = discharge coefficient
v1 = average velocity of the water approaching the culvert entrance
α = velocityhead coefficient (i.e., assumed as 1.0)
dc = the critical depth
Ac = flow area at the critical section, not the culvert area
B. Type2 Flow
As in Type1 flow, flow passes through the critical depth at the culvert outlet, and the barrel flows partially full. The slope of the culvert is less than critical, and the tailwater elevation does not exceed the elevation of the water surface at the control section.
C. Type3 Flow
When backwater is the controlling factor in culvert flow, the critical depth cannot occur. The upstream water surface elevation for a given discharge is a function of the height of the tailwater. For Type3 flow, flow is subcritical for the entire length of the culvert, with the flow being partial. The outlet is not submerged, but the tailwater elevation does exceed the elevation of critical depth at the terminal section.
where A3 is the flow area at section 3 (i.e., the exit).
D. Type4 Flow
As in Type3 flow, the backwater elevation is the controlling factor in this case. Critical depth cannot occur, and the upstream water surface elevation for a given discharge is a function of the tailwater elevation. Discharge is independent of barrel slope. The culvert is submerged at both the headwater and the tailwater.
where Ao is the culvert area. The complicated term in the denominator corrects for friction. For rough estimates and for culverts less than 50 ft long, the friction loss can be ignored.
E. Type5 Flow
Partially full flow under a high head is classified as Type5 flow. The flow pattern is similar to the flow downstream from a sluice gate, with rapid flow near the entrance. Usually, Type5 flow requires a relatively square entrance that causes contraction of the flow area to less than the culvert area. In addition, the barrel length, roughness, and bed slope must be sufficient to keep the velocity high throughout the culvert.
F. Type6 Flow
Type6 flow, like Type5 flow, is considered a highhead flow. The culvert is full under pressure with free outfall.
Note that distance h3is undefined. For conservative first approximations, h3 can be taken as the barrel diameter.
The FHWA Method
The Federal Highway Administration (FHWA) offers equations as well as nomographs that can be used for analysis and design of culverts. Different equations and nomographs are developed for inlet controlled culvert flows and outlet controlled culvert flows. Only equation based analysis and design approaches are described in this section. Readers interested in the FHWA nomographs, for both control types, may refer to Normann et al. (1985).
Culvert design according to FHWA involves analyzing the culvert under both inlet control and outlet control conditions and selecting the control type that yields the worst condition (i.e., larger headwater depth). The design would be acceptable if the governing headwater depth is less than the maximum allowable headwater to avoid flooding of streets and property. Otherwise, the design needs to be revised (e.g., culvert size is increased) to reduce the headwater depth.
Inlet Control
The objective is to determine the headwater depth based on predetermined design discharge and a trial culvert size. The design equation used to determine headwater depth for inlet controlled culvert vary depending on the flow condition at the inlet of the culvert. If the inlet is submerged, the flow type would be orifice flow. Unsubmerged conditions will behave as a weir flow.
If the inlet is submerged (orifice flow), the equation to determine the headwater depth will be
where HWi = headwater depth above the inlet control section invert (ft)
D = diameter of the culvert (ft)
Q = discharge (ft3/s)
A = full crosssectional area of the culvert (ft2)
c, Y = constant from Table 3.5
Z = culver barrel slope term (ft/ft).
For mitered inlets,
and for all other conditions (i.e., inlet types other than mitered inlets),
The unsubmerged flow (weir flow) condition can be evaluated using one of the following two approaches:
1) Based on specific head (Hc) at critical depth
where Hc= specific head at critical depth (ft)
K and M = constants from Table 3.5
dc = critical depth (ft)
Vc =critical velocity (ft/sec)
2) A simpler form that ignores specific head (Hc) at critical depth
Table 35: Constants for Inlet Control Design Equations
Shape and material  Inlet Edge Description  K  M  c  Y 
Circular Concrete  Square edge w/ headwall  0.0098  2.000  0.0398  0.67 
Groove end w/ headwall  0.0078  2.000  0.0292  0.74  
Groove end projecting  0.0045  2.000  0.0317  0.69  
Circular CMP  Headwall  0.0078  2.000  0.0379  0.69 
Mitered to slope  0.0210  1.330  0.0463  0.75  
Projecting  0.0340  1.500  0.0553  0.54  
Circular Ring  Beveled ring, 450 bevels  0.0018  2.500  0.0300  0.74 
Beveled ring 33.70 bevels  0.0018  2.500  0.0243  0.83  
Rectangular Box  300  750 wingwall flares  0.0260  1.000  0.0385  0.81 
900 and 150 wingwall flares  0.0610  0.750  0.0400  0.80  
00 wingwall flares  0.0610  0750  0.0423  0.82  
Rectangular Box  450 wingwall flare  0.5100  0.667  0.0309  0.80 
180  33.70 wingwall flare  0.4860  0.667  0.0249  0.83  
Rectangular Box  900 headwall w/¾ in chamfers  0.5150  0.667  0.0375  0.79 
900 headwall w/ 450 bevels  0.7950  0.667  0.0314  0.82  
900 headwall w/33.70 bevels  0.4860  0.667  0.0252  0.87  
Rectangular Box  ¾ in chamfers, 450 skewed headwall  0.5220  0.667  0.0402  0.73 
¾ in chamfers, 300 skewed headwall  0.5330  0.667  0.0425  0.71  
¾ in chamfers, 150 skewed headwall  0.5450  0.667  0.0451  0.68  
450 bevels, 10450 skewed wall  0.4980  0.667  0.0327  0.75  
Rectangular Box, ¾ in. chamfers  450 non offset wingwall flares  0.4970  0.667  0.0339  0.80 
18.40 non offset wingwall flares  0.4930  0.667  0.0361  0.81  
18.40 non offset wingwall flares, 300 skewed barrel  0.4930  0.667  0.0386  0.71  
Rectangular Box, top bevels  450 wingwall flaresoffset  0.4970  0.667  0.0302  0.84 
33.70 wingwall flares  offset  0.4950  0.667  0.0252  0.88  
18.40 wingwall flares  offset  0.4930  0.667  0.0227  0.89  
Corrugated Metal Boxes  900 headwall  0.0083  2.000  0.0379  0.69 
Thick wall projecting  0.0145  1.750  0.0419  0.64 
Adapted from Normann et al. (1985)
Outlet Control
Headwater for outlet control conditions can be determined using energy equation based on tailwater depth and head loss through the culvert considering entrance loss, exit loss, and friction loss.
where H = total head loss (ft)
ke = entrance loss coefficient (see Table 3.6)
ho = water depth at the outlet of the culvert (ft)
dc = critical depth (ft)
q = unit discharge (discharge per unit width of the culvert) (ft3/s/ft)
Table 36: Entrance Loss CoefficientsOutlet Control, Full or Partly Full
Type of Structures and Design of Entrance  Coefficient ke  
Pipe, Concrete  Mitered to conform to fill slope  0.7 
Endsection conforming to fill slope  0.5  
Projecting from fill, square cut end  0.5  
Headwall or headwall and wingwalls
Square Edge Rounded( radius= 1/12 Culvert Diameter Socket End of Pipe ( grooveend) 

0.5  
0.2  
0.2  
Projecting from fill, socket end (grooveend)  0.2  
Beveled edges, 33.70 or 450 bevels  0.2  
Side or slopetapered inlet  0.2  
Pipe, or PipeArch, Corrugated Metal  Projecting from fill ( no metal)  0.9 
Mitered to conform to fill slope, paved or unpaved slope  0.7  
Headwall or headwall and wingwalls squareedge  0.5  
Endsection conforming to fill slope  0.5  
Beveled edges, 33.70 or 450 bevels  0.2  
Side or slopetapered inlet  0.2  
Box, Reinforced Concrete  Wingwalls parallel ( extension of sides)
Square edge at crown 
0.7 
Wingwalls at 100250 or 300750 to barrel
Squareedged at crown 
0.5  
Headwall parallel to embankment (no wingwalls)
Squareedged on three edges Rounded on three edges to radius of 1/12 barrel dimension, or beveled edges on three sides 
0.5
0.2 

Wingwalls at 300750 to barrel
Crown edge rounded to radius of 1/12 barrel dimension, or beveled to edges 
0.2  
Side or slopetapered inlet  0.2 
Adapted from Normann et al. (1985)