What This Solves
Computes water surface profiles in open channels where depth changes gradually along the reach, classifying profile types (M1, M2, S1, S2, etc.) using the Direct Step Method.
Best Used When
- You need to calculate backwater effects upstream of a dam, bridge, or culvert
- You want to determine how far upstream a downstream control affects water levels
- You are analyzing drawdown curves approaching a free overfall or steep reach
Do NOT Use When
- Flow is uniform and you only need normal depth — Use Normal Depth Calculator
- You need to route a flood hydrograph through a channel reach over time — Use Muskingum Routing Calculator
Key Assumptions
- Flow is steady (not changing with time) but non-uniform (depth changes with distance)
- Channel slope, roughness, and cross-section change gradually (no abrupt transitions)
- Energy slope is approximated using Manning's equation at each step
- The channel is prismatic or varies gradually between cross-sections
- No lateral inflow or outflow along the reach
Input Quality Notes
The starting depth (boundary condition) must be known — typically normal depth or critical depth at a control section. Step size affects accuracy; use smaller steps where depth changes rapidly.
Compute open-channel water surface profiles — backwater and drawdown curves — using the Direct Step Method. Enter the discharge, channel geometry and a known control depth to get the profile, its classification (M1, M2, S1, …) and the normal and critical depths.
Water Surface Profile Classifications
Mild Slope (yn > yc)
- M1: Backwater (y > yn)
- M2: Drawdown (yc < y < yn)
- M3: Supercritical (y < yc)
Steep Slope (yn < yc)
- S1: Backwater (y > yc)
- S2: Drawdown (yn < y < yc)
- S3: Backwater (y < yn)
Other Classifications
- C1, C3: Critical slope
- H2, H3: Horizontal bed
- A2, A3: Adverse slope
Ready to Calculate
Enter channel parameters and depth range to compute the water surface profile.
For educational purposes only. Not a substitute for professional engineering judgment.
How it works: the Direct Step Method
Gradually varied flow is steady, non-uniform flow where the depth changes slowly enough that the pressure distribution stays hydrostatic. The water surface is found by integrating the energy equation between sections. This calculator uses the Direct Step Method, which solves for the distance Δx between two specified depths:
Δx = (E₂ − E₁) / (S₀ − S̄f)
where the specific energy at each section is
E = y + α·V² / (2g)
and the friction slope is obtained from Manning's equation, with S̄f taken as the average over the step:
Sf = ( n·V / (k·R^(2/3)) )²
Starting from a control section with a known depth, the method marches through small depth increments, summing each Δx to trace the full profile. The calculator also solves the two reference depths that classify the profile:
- Normal depth (yn) from Manning's equation: Q = (k/n)·A·R^(2/3)·S₀^(1/2)
- Critical depth (yc) from the critical-flow condition: Q²/g = A³/T
| Symbol | Variable | Notes |
|---|---|---|
| Δx | Distance between sections | ft or m |
| E | Specific energy = y + αV²/2g | ft or m |
| y | Flow depth | ft or m |
| V | Mean velocity = Q / A | fps or m/s |
| S₀ | Channel bed slope | ft/ft or m/m |
| S̄f | Average friction slope (Manning) | dimensionless |
| R | Hydraulic radius = A / P | ft or m |
| α | Energy (velocity) coefficient | typically 1.0 |
| n | Manning's roughness coefficient | dimensionless |
| k | Manning units constant | 1.486 (US), 1.0 (SI) |
| g | Gravitational acceleration | 32.174 ft/s² (US), 9.81 m/s² (SI) |
Method after Chow (1959), Open-Channel Hydraulics, Ch. 10–11; consistent with USACE EM 1110-2-1601 and FHWA HDS-4.
Water surface profile classifications
The profile type is set by the slope class (letter) and the zone of the actual depth relative to normal depth (yn) and critical depth (yc) (number). The calculator classifies the slope as mild when yn > yc and steep when yn < yc.
| Profile | Slope class | Depth zone | Description |
|---|---|---|---|
| M1 | Mild (yn > yc) | y > yn | Backwater curve (e.g. behind a dam) |
| M2 | Mild | yc < y < yn | Drawdown curve (e.g. free overfall) |
| M3 | Mild | y < yc | Supercritical, depth below critical |
| S1 | Steep (yn < yc) | y > yc | Backwater above critical |
| S2 | Steep | yn < y < yc | Drawdown between critical and normal |
| S3 | Steep | y < yn | Backwater below normal |
| C1, C3 | Critical (yn = yc) | above / below yc | Critical slope profiles |
| H2, H3 | Horizontal (S₀ = 0) | above / below yc | Horizontal-bed profiles (no normal depth) |
| A2, A3 | Adverse (S₀ < 0) | above / below yc | Adverse-slope profiles (uphill bed) |
Classification system after Chow (1959), Fig. 10-2. Zone 1 = depth above both yn and yc; zone 2 = between them; zone 3 = below both.
Assumptions and limitations
Assumptions
- Steady flow conditions
- Prismatic channel (constant cross-section)
- Gradually varied flow (hydrostatic pressure distribution)
- Small channel slope (cos θ ≈ 1)
- No lateral inflow or outflow
Limitations
- Not suitable for rapidly varied flow (e.g. hydraulic jumps)
- Cannot handle channel transitions directly
- Requires a known control depth
- May be numerically unstable near critical depth
Results are for preliminary analysis and design verification. Final designs should be reviewed by a licensed professional engineer.
Frequently asked questions
What is gradually varied flow (GVF)?
Gradually varied flow is steady, non-uniform open-channel flow in which the depth changes slowly along the channel — for example the backwater curve upstream of a dam or weir, or the drawdown approaching a free overfall. Because the change in depth is gradual, the pressure distribution stays effectively hydrostatic, so the flow can be analyzed step-by-step with the energy equation. This contrasts with rapidly varied flow (such as a hydraulic jump), where depth changes abruptly over a short distance.
How does the Direct Step Method work?
The Direct Step Method solves the energy equation for the horizontal distance between two known depths: Δx = (E₂ − E₁) / (S₀ − S̄f), where E is the specific energy at each section, S₀ is the channel bed slope, and S̄f is the average friction slope over the reach (from Manning’s equation). Starting from a control section with a known depth, the calculator marches through a series of small depth increments, summing each Δx to build the full water-surface profile. It is well suited to prismatic (constant cross-section) channels.
What do profile classifications like M1, M2 and S1 mean?
The letter denotes the channel slope relative to critical: M = mild (normal depth yn greater than critical depth yc), S = steep (yn less than yc), C = critical, H = horizontal, and A = adverse. The number denotes where the actual depth sits relative to yn and yc: zone 1 is above both, zone 2 is between them, and zone 3 is below both. So an M1 profile is the classic backwater curve behind a dam on a mild slope, while an M2 is a drawdown curve approaching a free overfall.
What is the difference between normal depth and critical depth?
Normal depth (yn) is the uniform-flow depth at which the channel would carry the given discharge if the water surface were parallel to the bed — found from Manning’s equation Q = (k/n)·A·R^(2/3)·S₀^(1/2). Critical depth (yc) is the depth of minimum specific energy, where the Froude number equals 1 and Q²/g = A³/T. Comparing the two classifies the slope: yn > yc is mild, yn < yc is steep. Their values bound the profile zones the calculator reports.
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Last verified: February 2026