Regime, Silting and Lining of an Unlined Distributary in Gwalior District
An unlined distributary taking off from a canal system in Gwalior district must carry 12 m³/s at full supply. The bed material is non-cohesive sandy silt with a permissible velocity of 0.75 m/s; silt factor f = 1.0; side slopes are fixed by land width at 0.5H:1V. The section on the drawing has a bed width of 12 m and a full supply depth of 2.0 m, and the existing bed slope is 1 in 5,000. The Executive Engineer wants to know whether the drawn section is in regime, whether the reach will silt or scour in the first year, and whether steepening the slope to 1 in 2,500 will let the section be reduced. Separately, a lining proposal at ₹400 per m² has been received, against an estimated seepage loss of 1.25 m³/s per million m² of wetted area and water valued at ₹2.50 per m³, with the canal running 120 days a year.
1. Check whether the proposed section (b = 12 m, d = 2.0 m) is in regime. Will it silt or scour?
Read the worked answer
Lacey regime values for Q = 12 m³/s and f = 1.0:
- Regime velocity v = (Q f²/140)^(1/6) = (12/140)^(1/6) = 0.664 m/s.
- Regime area A = Q/v = 12/0.664 = 18.1 m².
- Regime perimeter P = 4.75√Q = 4.75 × 3.464 = 16.45 m.
- Regime hydraulic radius R = A/P = 1.10 m.
Proposed section:
- A = bd + 0.5d² = 24 + 2.0 = 26.0 m².
- P = b + 2d√(1 + 0.5²) = 12 + 4 × 1.118 = 16.47 m.
- R = 26.0/16.47 = 1.58 m; V = 12/26.0 = 0.46 m/s.
The section is about 44 per cent larger in area than regime, R is 44 per cent above regime and the mean velocity is 30 per cent below regime. A channel carrying less than regime velocity cannot transport the regime silt charge, so the reach will silt, not scour. Verdict: the section is oversized and not in regime; it will progressively shallow up, requiring repeated dredging.
2. Design the regime section for the same Q and f, and comment on the land-width constraint.
Read the worked answer
Solve the two Lacey equations simultaneously for 0.5H:1V:
- A = bd + 0.5d² = 18.1
- P = b + 2.236d = 16.45, so b = 16.45 - 2.236d
Substituting: 16.45d - 2.236d² + 0.5d² = 18.1, giving 1.736d² - 16.45d + 18.1 = 0. d = [16.45 ± √(270.6 - 125.7)]/(2 × 1.736) = [16.45 ± 12.05]/3.472, so d = 1.27 m (rejecting 8.2 m). b = 16.45 - 2.84 = 13.6 m.
Check: A = 13.6 × 1.27 + 0.5 × 1.27² = 17.3 + 0.8 = 18.1 m², V = 0.66 m/s, R = 1.10 m. The velocity of 0.66 m/s is below the 0.75 m/s permissible value, so a non-cohesive sandy silt bed will be stable and will not scour.
If land acquisition forces b = 12 m, solve 12d + 0.5d² = 18.1, giving d ≈ 1.42 m and R ≈ 1.19 m. The area and velocity match regime closely, the mismatch in R (about 8 per cent) being within normal design tolerance. The point is that the section is made smaller by reducing area to regime, not by steepening the slope.
3. Comment on the 1 in 2,500 slope proposal and assess whether lining is economically justified.
Read the worked answer
Slope proposal. Lacey regime slope S = f^(5/3)/(3340 Q^(1/6)) = 1/(3340 × 1.513) = 1/5,054, essentially the existing 1 in 5,000. The channel is already at the regime slope; doubling it to 1 in 2,500 doubles the tractive force and velocity, so the bed will scour and the section will widen until it re-attains regime, with siltation downstream. The proposal should be rejected; only lining, which raises the permissible velocity, allows a smaller section.
Lining economics. Wetted area per km = 16.45 × 1000 = 16,450 m². Seepage = 1.25 × 0.01645 = 0.0206 m³/s, i.e. 0.0206 × 120 × 86,400 = 2.13 × 10⁵ m³ per season, worth 2.13 × 10⁵ × 2.50 ≈ ₹5.3 lakh per year. Lining cost = 16,450 × 400 = ₹65.8 lakh, giving a simple payback of about 12 years (about 18 years on a discounted basis) against a lining life of 25 years.
Judgement: seepage recovery alone is marginal, but add the savings on annual silt clearance (typically ₹1 lakh or more per km), weed control, land saving from a smaller lined section, and more reliable supply to tail-end outlets, and lining the first 2 to 3 km is defensible. If funds are limited, the correct immediate action is to reshape the unlined channel to the 13.6 m × 1.27 m regime section rather than to steepen it.