The usable gradient of an earth channel is boxed in by two limits: it must be steep enough to keep water moving, and gentle enough that the water does not scour the bed. For a 30 cm wide channel with 20 cm of water, Manning’s equation puts the erosion limit near 0.23% in fine sand, 0.32% in sandy loam, 0.41% in silt loam, 0.65% in firm loam and 1.45% in stiff clay, so the usual 0.1-0.5% range is safe only in the firmer soils.
Part of Aqueducts and Channels
Optimal gradient is the slope at which an open channel carries water fast enough to stay clear of silt and weeds but slow enough not to erode its own bed. It is found by comparing the water speed a slope produces with the speed a soil can resist.
The Slope Sets The Speed, And The Soil Sets The Limit
The aqueducts page states the rule: 0.1% to 0.5% for an earth channel (1-5 m of fall per kilometre), 0.5% to 2% for a lined one. This page works out where that rule comes from.
Speed from slope. Manning’s equation in SI units gives the mean speed of water in an open channel:
V = (1 ÷ n) × R^(2/3) × S^(1/2)
where V is speed (m/s), n is a roughness coefficient, R is the hydraulic radius (flow area ÷ wetted perimeter, in m) and S is the slope as a fraction (0.002 = 0.2%). For a trapezoid with bottom width b, depth y and side slope z (horizontal run per unit of rise): A = (b + z·y)·y and P = b + 2y·√(1 + z²).
Roughness. Published values (Chow, Open-Channel Hydraulics, 1959) for excavated earth channels, straight and uniform: clean, recently completed 0.016-0.020; clean, after weathering 0.018-0.025; with short grass and few weeds 0.022-0.033. This page uses n = 0.025, a cautious figure for a hand-dug channel.
Soil limit. The classic table of maximum permissible velocities for clear water in aged, straight channels is Fortier and Scobey (1926), reproduced in Chow:
| Soil | Limit (ft/s) | Limit (m/s) |
|---|---|---|
| Fine sand, colloidal | 1.50 | 0.46 |
| Sandy loam, noncolloidal | 1.75 | 0.53 |
| Silt loam, noncolloidal | 2.00 | 0.61 |
| Ordinary firm loam | 2.50 | 0.76 |
| Stiff clay, very colloidal | 3.75 | 1.14 |
Rearranging For The Steepest Safe Slope
Set V equal to the soil limit and solve for S:
S = (V × n ÷ R^(2/3))²
These are the two limits. For the aqueducts-page channel, 30 cm wide (b = 0.30 m, z = 1, y = 0.20 m): A = (0.30 + 0.20) × 0.20 = 0.100 m²; P = 0.30 + 2 × 0.20 × √2 = 0.866 m; R = 0.100 ÷ 0.866 = 0.1155 m; R^(2/3) = 0.2371.
Sandy loam: S = (0.533 × 0.025 ÷ 0.2371)² = (0.0562)² = 0.00316, or 0.32%.
| Soil | Steepest safe slope (y = 0.20 m, n = 0.025) |
|---|---|
| Fine sand | 0.23% |
| Sandy loam | 0.32% |
| Silt loam | 0.41% |
| Firm loam | 0.65% |
| Stiff clay | 1.45% |
So the 0.5% upper bound is safe in firm loam and clay, but it exceeds the limit for fine sand, sandy loam and silt loam. In those soils, build to about 0.2-0.3%, or line the channel. The limit also moves with depth: at 0.15 m the same soils tolerate a slightly steeper slope (0.31%, 0.42%, 0.55%, 0.86% and 1.93%), because a shallower stream is slower at the same slope.
The Lower Limit: Why Not Flatter Than 0.1%
At 0.1% the same channel flows at 0.26 m/s with 15 cm of water and 0.30 m/s with 20 cm (n = 0.025). That is slow. When the channel carries only 2-5 L/s at 0.1%, water stands just 4.3-7.4 cm deep and moves at 0.14-0.18 m/s. Slow, shallow water drops silt and grows weeds. The aqueducts page gives no minimum speed, and none is claimed here; use the 0.1% floor as a practical limit and clean the channel (Sediment Management).
Limits Of This Method
The tabulated velocities are for clear water in aged, straight channels of modest depth. Bends, silt-laden water, loose construction and poor compaction all reduce the real limit. Treat the numbers as a screening check, then confirm on a short trial reach, watching for scour after the first flows.
What You Need
- Soil type at the route, identified by feel: sandy, silty, loamy, clayey
- The planned cross-section and depth
- A survey capable of the precision needed (Water Tube Level)
- A trial reach to test before committing the whole length
Method: Check The Slope Against The Soil
Step 1: Identify the soil and read its speed limit.
Step 2: Compute R for the intended depth.
Step 3: Compute S from the formula and compare it with the planned slope.
Step 4: If the plan exceeds it, flatten the grade, line the channel (Lined Channels), or add drops.
Common Mistakes
- Using the 0.1-0.5% range for all soils. It overshoots in sandy ground.
- Ignoring depth. A deeper flow is faster at the same slope.
- Flattening below 0.1% to be safe. The channel then silts.
What’s Next
- Learn next: Channel Cross Section — sizing the section the slope acts on
- Combine with: Earth Channels — building the unlined channel