Backwater Curves: Why the Water Rises Before a Structure

Put a weir, a gate, or a culvert in a canal and the water does not jump to its new depth at the structure—it eases into it over a distance, sometimes for hundreds of metres or more. That rising water surface upstream of an obstruction is a backwater curve, one family of gradually varied flow (GVF). This walkthrough shows how to trace it with FlowStudio’s gradually varied flow worksheet (SI units).

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What you are solving

In uniform flow the depth sits at normal depth y_n and never changes along the reach. A control structure breaks that: it fixes the depth at one point to some other value, and the channel then blends from that control depth back toward normal depth. The blend is governed by the gradually varied flow equation:

dy/dx = (S₀ − S_f) / (1 − Fr²)

Here S₀ is the bed slope, S_f is the friction slope from Manning’s equation, and Fr is the Froude number. FlowStudio integrates this along the channel from an initial depth you supply, using Manning friction on a prismatic (rectangular or trapezoidal) section.

Two reference depths set the shape

Every GVF profile is read against two lines: normal depth y_n (where Manning balances) and critical depth y_c (where Fr = 1). On a mild slope (y_n > y_c), which covers most irrigation canals, the three profiles are:

• M1 — the classic backwater curve. Depth is above y_n, caused by a downstream weir, gate, or reservoir holding the water up. The surface rises smoothly upstream and approaches y_n asymptotically.
• M2 — a drawdown curve. Depth is between y_n and y_c, as the flow accelerates toward a free overfall or a steep reach.
• M3 — supercritical. Depth is below y_c, for example the fast jet just downstream of a sluice gate, rising until it meets a hydraulic jump.

Knowing which one you expect tells you where to start the integration and which way to march. If you have not yet found y_n and y_c for your channel, get them first from the rectangular or trapezoidal normal-depth worksheet.

Step 1 — Create the worksheet

Sign in if prompted, then create or open a project. Add a new worksheet and choose gradually varied flow. Open the sheet so you see the intro note (it states the GVF equation and available integration methods), the Section type selector, and the input fieldsets for geometry, initial conditions, and integration.

Figure 1. The freshly opened GVF worksheet — the “About this worksheet” note, the Rectangular/Trapezoidal selector, and the empty input grid.

Step 2 — Enter channel geometry and flow

Pick Rectangular or Trapezoidal (for a trapezoid, add side slope z, H:V). Then enter SI values: bottom width b, Manning n, bed slope S₀ (m/m), and discharge Q (m³/s). For a reproducible mild-slope demo, try a rectangular canal: b = 5 m, n = 0.025, S₀ = 0.0004, Q = 12 m³/s. For these values normal depth is roughly y_n ≈ 2.6 m and critical depth y_c ≈ 0.84 m—so y_n > y_c, a mild slope, and any pond-up behind a structure is an M1 curve.

Figure 2. Channel geometry and flow entered: b = 5 m, n = 0.025, S₀ = 0.0004, Q = 12 m³/s, with control depth y₀ = 3.0 m at x₀ = 0.

Step 3 — Set the control depth as the initial condition

This is the step that makes a backwater curve. Under Initial conditions, put the structure at x₀ = 0 and set y₀ to the depth the structure imposes—for example a weir that holds the water at y₀ = 3.0 m, above normal depth. Because a subcritical backwater is controlled from downstream, the profile must be integrated upstream: set Direction = Upstream (−x).

Rule of thumb for direction: subcritical flow (M1, M2) is controlled from downstream, so march upstream from the control; supercritical flow (M3, e.g. below a gate) is controlled from upstream, so march downstream.

Figure 3. Initial conditions and Integration: method = RK4, reach L = 2000 m, 48 steps, Direction = Upstream (−x), and the Compute profile button.

Step 4 — Choose a method and reach, then compute

Under Integration, keep Runge–Kutta 4 (fixed Δx) for a smooth, accurate curve (Midpoint, Euler, and a fixed-Δy Standard-step method are also available). Set a reach length L long enough to see the surface relax back toward y_n—on a mild slope a backwater can stretch a long way, so try L = 2000 m with 40–80 steps—then click Compute profile.

Step 5 — Read the profile and the table

The long-section chart plots depth versus distance. Starting from the control depth at the structure, the M1 surface should sag gently toward normal depth as you move upstream, flattening out and running nearly parallel to the bed far from the structure. The results table lists each station’s x, depth y, bed/stage z, Froude number Fr, friction slope S_f, and the local slope dy/dx—handy for seeing the curve go asymptotic (dy/dx → 0 as y → y_n).

Figure 4. The computed long-section profile: control depth at x = 0 relaxing toward normal depth upstream — the M1 backwater shape (y_n ≈ 2.56 m, y_c ≈ 0.84 m).

Figure 5. The results table (x, y, z, Fr, S_f, dy/dx); reference values read y_n = 2.564 m and y_c = 0.837 m.

Common pitfalls

• Wrong direction. If a subcritical profile blows up or runs away instead of settling to y_n, you are almost certainly marching the wrong way—subcritical is controlled from downstream, so integrate upstream.
• Starting on the wrong side of y_n/y_c. The initial depth decides which profile you get. Above y_n → M1; between y_n and y_c → M2; below y_c → M3. Check your control depth against both reference lines first.
• Near-critical stalling. The GVF equation is singular at Fr = 1 (the denominator 1 − Fr² → 0), so integration slows or stops near critical depth. A true transition through critical is a rapidly varied phenomenon (a jump or a control), not GVF—model it with the sluice-gate worksheet or a dedicated control instead of pushing GVF through y_c.
• Reach too short. If the profile has not yet flattened to y_n at the end of L, lengthen the reach. Backwater lengths on flat canals are often surprisingly large.

Wrap-up

You’ve traced a backwater curve from a control depth back to normal depth using the same GVF closure FlowStudio integrates. Swap in your own b, z, n, S₀, and Q, set the control depth your structure imposes, and read off how far upstream the influence reaches—the number that tells you whose fields get wetter when you raise a gate.

Run the sheet live at https://flow.syncster.dev.