What is gradually varied flow?

Gradually varied flow is steady open-channel flow in which the depth changes slowly along the channel, so pressure stays hydrostatic and friction can be taken from Manning's equation at each section. It covers backwater behind a structure and drawdown towards a drop. Hydraulic jumps and the flow over a weir crest change too quickly and are treated as rapidly varied flow.

What is the difference between the direct step and standard step methods?

The direct step method fixes the depth at each step and solves for the distance, with no iteration, which suits a prismatic channel. The standard step method fixes the distance and iterates for the depth, which suits stations set by surveyed cross-sections or reaches whose geometry changes. Both apply the same energy balance with the average friction slope over the step.

How far upstream does backwater extend?

In theory forever, because an M1 profile approaches normal depth asymptotically, so backwater length is quoted to a tolerance. In the worked example on this page, a weir that raises a 20 m³/s channel by 0.81 m at a slope of 0.0008 leaves the water 10% above normal depth 1.34 km upstream and 1% above it about 2.8 km upstream. Flatter slopes give longer profiles.

How do I know whether I have an M1, M2 or S1 profile?

Compare normal depth with critical depth to classify the slope: mild if normal depth is greater, steep if it is smaller. Then compare the control depth with both. A depth above normal depth on a mild slope gives M1, between normal and critical gives M2, and below critical gives M3. On a steep slope a depth above critical depth gives S1.

Can this replace HEC-RAS?

For a prismatic channel or a canal made of uniform reaches, it solves the same steady-flow energy equation that HEC-RAS steps through, with far less set-up. For a natural river with irregular surveyed sections, overbanks, bridges, culverts or junctions, use HEC-RAS, the US Army Corps of Engineers' free river analysis program, which is built for those cases.

Gradually Varied Flow Calculator: Water Surface Profiles

Profile summary
Slope class	Mild, yn > yc
Critical slope, ScBed slope at which this flow runs at critical depth	0.01023
Froude number at the control	0.16
Profile lengthEnds 0.5% from normal depth	3 200m
Depth at the far end	2.004m
Within 10% of normal depth	1 339m from the control
Within 1% of normal depth	2 809m from the control
Computed stations	501

Depth, velocity and Froude number along the profile
Distance (m)	Depth (m)	Velocity (m/s)	Froude
0	2.800	0.67	0.16
250	2.653	0.73	0.18
500	2.519	0.79	0.20
750	2.400	0.85	0.21
1 000	2.299	0.91	0.23
1 250	2.217	0.96	0.25
1 500	2.152	1.00	0.26
1 750	2.103	1.03	0.27
2 000	2.068	1.06	0.28
2 250	2.044	1.08	0.29
2 500	2.027	1.09	0.29
2 750	2.016	1.10	0.30
3 000	2.008	1.10	0.30
3 200	2.004	1.11	0.30

Using the calculator above

The calculator runs the GVF Profiles engine on one prismatic reach. It opens on the worked example further down this page: a trapezoidal channel with a 5.0 m bed and 2H:1V sides, n = 0.030, a bed slope of 0.0008 and 20 m³/s, held at 2.80 m by a weir at the downstream end, computed by the direct step method over 4000 m in 500 steps.

Section, n, slope and flow. These set normal and critical depth, which classify the slope. A slope of 0 is a horizontal bed; a negative slope is adverse.
Control. The end of the reach where the depth is known, and that depth: downstream for subcritical flow (a weir, culvert or reservoir), upstream for supercritical flow (a sluice gate). If the depth is on the wrong side of critical depth for that end, the calculator says so.
Reach length and number of steps. How far from the control to compute, and the depth increments (direct step) or distance increments (standard step).

The results give the profile class, normal and critical depth, a long section, and the depth, velocity and Froude number along the reach.

The gradually varied flow equation

Combining the energy and continuity equations for a prismatic channel gives the rate at which depth changes along it:

\frac{dy}{dx} = \frac{S_0 - S_f}{1 - Fr^2} \qquad\qquad S_f = \frac{n^2 V^2}{R^{4/3}} \ \ \text{(SI)}, \qquad S_f = \left(\frac{nV}{1.49\,R^{2/3}}\right)^2 \ \ \text{(US customary)}

Symbol	Meaning	Units
$y$	Flow depth	m (ft)
$x$	Distance along the channel	m (ft)
$S_0$	Bed slope	m/m
$S_f$	Friction slope, the slope of the energy line	m/m
$Fr$	Froude number, $V / \sqrt{g A/T}$	–
$n$	Manning’s roughness coefficient	–
$V$ , $R$	Mean velocity and hydraulic radius at depth $y$	m/s, m

Two depths organise everything that follows. At normal depth $y_n$ the friction slope equals the bed slope and the numerator is zero: the flow is uniform. At critical depth $y_c$ the Froude number is 1 and the denominator is zero: the equation breaks down, which is where jumps and free overfalls occur. Both come from the same engine as the Manning’s equation calculator.

Profile classes

The slope is classified by comparing $y_n$ with $y_c$ , and the profile by where the depth sits relative to them. Zone 1 is above both lines, zone 2 between them and zone 3 below both.

Slope	Test	Zone 1	Zone 2	Zone 3
Mild (M)	$y_n > y_c$	M1: backwater behind a weir, dam or culvert	M2: drawdown towards a free overfall or a steeper reach	M3: below a sluice gate, ends in a jump
Steep (S)	$y_n < y_c$	S1: behind an obstruction, after a jump	S2: drawdown below a change from mild to steep	S3: below a sluice gate, rises to normal depth
Critical (C)	$y_n = y_c$	C1	–	C3
Horizontal (H)	$S_0 = 0$	–	H2: drawdown towards an overfall	H3: below a gate, ends in a jump
Adverse (A)	$S_0 < 0$	–	A2	A3

Horizontal and adverse beds have no normal depth, so only zones 2 and 3 exist. The class also fixes the direction of computation. Subcritical profiles (M1, M2, S1, C1, H2, A2) are controlled from downstream, so the tool computes them upstream from the control. Supercritical profiles (M3, S2, S3, C3, H3, A3) are controlled from upstream and computed downstream.

Direct step and standard step

Both methods apply the energy equation between two sections a distance $\Delta x$ apart, with the friction slope averaged over the step:

\Delta x = \frac{E_2 - E_1}{S_0 - \bar{S}_f} \qquad\qquad z_1 + y_1 + \frac{V_1^2}{2g} = z_2 + y_2 + \frac{V_2^2}{2g} + \bar{S}_f\,\Delta x

where $E = y + V^2/2g$ is specific energy and section 1 is upstream of section 2. The direct step method (left) picks the depths and solves for the distance in one line per step, with no iteration. It is the default for a single prismatic reach. The standard step method (right) picks the stations and iterates for the depth at each one. It is the method for reaches whose geometry, slope or roughness changes, and it is what the tool’s multi-reach mode marches through a canal.

Inputs and where they come from

Section, n and slope. From the design drawings or a survey. The Manning’s n table on the Manning’s equation calculator page lists typical values.
Flow. The design flow, for example from the rational method calculator or a unit hydrograph peak.
Control depth. A depth you know independently of the profile: the crest height plus the head at a weir (from the weir flow calculator), the headwater at a culvert inlet (from the culvert calculator), critical depth at a free overfall, or a reservoir or confluence level.
Distance limit and number of steps. The tool stops when the profile reaches its asymptote, 0.5% above normal depth for an M1, or at the distance limit. Studio’s default limit is 1000 m with 500 steps, so lengthen the limit for long backwater curves.

Worked example: backwater behind a weir

A trapezoidal earth channel with a 5.0 m bed and 2H:1V side slopes (n = 0.030) runs at a slope of 0.0008 and carries 20 m³/s. A weir holds the depth immediately upstream of it at 2.80 m.

Normal depth is 1.994 m and critical depth 1.020 m, so the slope is mild.
The control depth of 2.80 m is above normal depth: the profile is M1, computed upstream from the weir.
At the weir the flow area is 29.68 m², velocity 0.674 m/s, Froude number 0.16, specific energy 2.823 m and friction slope 2.02 × 10⁻⁴.

One direct step by hand: at $y_2$ = 2.792 m, $E_2$ = 2.8153 m and $S_f$ = 2.05 × 10⁻⁴. The average friction slope is 2.04 × 10⁻⁴, so $\Delta x$ = (2.8153 − 2.8231) / (0.0008 − 0.000204) = −13.1 m. The minus sign means the section lies upstream of the weir.

Run with a 4 km limit and the default 500 steps, the profile is:

Distance upstream of the weir	Depth	Above normal depth
0 m	2.800 m	0.81 m
250 m	2.653 m	0.66 m
500 m	2.519 m	0.53 m
1000 m	2.299 m	0.31 m
2000 m	2.068 m	0.07 m
3000 m	2.008 m	0.01 m

The water surface is within 10% of normal depth 1340 m upstream, within 5% at 1815 m and within 1% at about 2810 m, and the profile ends at 3200 m. Running the same channel through the multi-reach standard step mode at 10 m spacing returns the same depths to the millimetre. The weir’s 0.81 m rise is still 0.31 m a kilometre upstream, so a culvert, bridge or property within the first 2 km needs its levels checked against this profile, not against normal depth.

Limitations

A single-reach run is one prismatic channel with one n and one slope. Natural rivers with irregular sections, overbanks, bridges and junctions need HEC-RAS.
Flow is steady. Flood waves and storage routing are outside the tool.
The step method stops as the profile approaches critical depth, where the equation breaks down. A single-reach run does not place a hydraulic jump; the multi-reach mode does.
The profile approaches normal depth asymptotically, so any backwater length depends on the tolerance. State the tolerance you used.
The result is only as good as the control depth. A wrong weir head or tailwater level moves the whole profile.

What a free Studio account adds

More sections: circular, triangular and surveyed custom sections, a Manning’s n lookup table, and the section, n and slope imported from a saved Channel Analysis run.
A multi-reach canal mode: chain reaches with their own section, roughness, invert levels and lining, or import a longitudinal profile and cross-sections from CSV. The tool marches the standard step through mild and steep reaches and locates hydraulic jumps with their sequent depth, energy loss and type, from undular to strong.
Lining checks: velocity and boundary shear at every station checked against the permissible values for each reach’s lining, with the reaches that exceed them flagged. For the lining design itself, use the HEC-15 channel lining calculator.
Saving and reports: the profile saved to a project with every computed station, and exported as a Word, Excel or PDF report with the water surface, velocity and Froude number charts and the station table.

Gradually varied flow calculator