Lacey's Regime Theory: Formulas, Silt Factor and Canal Design Example
Lacey's regime theory says an alluvial canal is "in regime" when it neither silts nor scours, and gives its section directly from the discharge Q and the silt factor f. The key equations (SI units) are: silt factor f = 1.76√d (d in mm), velocity V = (Q f² / 140)^(1/6), wetted perimeter P = 4.75√Q, hydraulic mean radius R = 5V² / 2f and bed slope S = f^(5/3) / (3340 Q^(1/6)). With these five you can design a canal section without trial and error.
What "regime" means
Gerald Lacey studied stable canals in the Indian subcontinent and argued that a channel flowing in loose alluvium adjusts its own width, depth and slope until the silt it carries is just kept in suspension. He described three states:
- True regime – constant discharge, constant silt charge, uniform incoherent alluvium of the same grade the water carries. An ideal case.
- Initial regime – only the bed slope has adjusted (by silting or scouring); the sides have not. Kennedy's theory applies to this state.
- Final regime – width, depth and slope have all adjusted. Lacey's equations describe this state.
Lacey's equations at a glance
| Quantity | Formula (SI) | Notes |
|---|---|---|
| Silt factor | f = 1.76 √d | d = mean particle size in mm |
| Velocity | V = (Q f² / 140)^(1/6) | V in m/s, Q in m³/s |
| Hydraulic mean radius | R = 5V² / 2f | Also R = 0.47 (Q/f)^(1/3) |
| Area | A = Q / V | m² |
| Wetted perimeter | P = 4.75 √Q | Independent of silt |
| Bed slope | S = f^(5/3) / (3340 Q^(1/6)) | Usually 1 in 3000 to 1 in 6000 |
| Regime flow equation | V = 10.8 R^(2/3) S^(1/3) | Lacey's general flow law |
Silt factor for common sand sizes
| Mean size d (mm) | Silt factor f = 1.76√d |
|---|---|
| 0.10 | 0.56 |
| 0.20 | 0.79 |
| 0.323 | 1.00 (standard silt) |
| 0.50 | 1.24 |
| 1.00 | 1.76 |
Exam tip: P = 4.75√Q depends only on discharge, not on silt. If a question gives only Q and asks for the wetted perimeter, you need nothing else. Also remember the value f = 1 corresponds to d ≈ 0.323 mm, called "standard silt".
Kennedy vs Lacey
| Point | Kennedy (1895) | Lacey (1929 onwards) |
|---|---|---|
| Basic idea | Silt is kept in suspension by eddies rising from the bed only | Eddies from the whole wetted perimeter keep silt in suspension |
| Key velocity | Critical velocity V₀ = 0.55 m D^0.64 | Regime velocity from Q and f |
| Design method | Trial and error (assume D, check V) | Direct – no trial needed |
| Bed slope | Not given; needs Kutter's or Manning's formula | Given by the theory |
| Regime state | Initial regime | Final regime |
Worked example: canal for Q = 50 m³/s
Design a regime channel to carry 50 m³/s in alluvium with mean particle size 0.5 mm. Assume side slopes of ½ H : 1 V (typical for a canal that has silted its sides).
- Silt factor: f = 1.76 × √0.5 = 1.245.
- Velocity: V = (50 × 1.245² / 140)^(1/6) = 0.906 m/s.
- Area: A = Q / V = 50 / 0.906 = 55.19 m².
- Hydraulic mean radius: R = 5 × 0.906² / (2 × 1.245) = 1.65 m. (The alternative form 0.47 (Q/f)^(1/3) gives 1.61 m; the small difference comes from rounded constants.)
- Wetted perimeter: P = 4.75 × √50 = 33.59 m. Check: A / P = 55.19 / 33.59 = 1.64 m ≈ R. ✔
- Section: for ½ H : 1 V sides, P = b + √5·D and A = b·D + 0.5·D². Substituting b = P − √5·D gives 1.736 D² − 33.59 D + 55.19 = 0, so D ≈ 1.81 m and b ≈ 29.5 m.
- Bed slope: S = 1.245^(5/3) / (3340 × 50^(1/6)) = 0.000225, i.e. about 1 in 4450.
- Check with the regime flow law: 10.8 × 1.65^(2/3) × 0.000225^(1/3) = 0.92 m/s, close to the 0.906 m/s found above. ✔
Lacey's scour depth
The same theory is used to estimate the normal scour depth at bridges and weirs. For a wide alluvial river the regime depth is R = 0.47 (Q/f)^(1/3), or, using discharge per metre width q, R = 1.35 (q²/f)^(1/3). Designers then multiply R by a factor that grows with the severity of the bend and is largest (about 2) at the noses of piers, to get the maximum scour depth for foundations. You can try this with the railway bridge scour calculator on our free tools page.
Limitations
- True regime rarely exists; discharge and silt charge vary through the season.
- The silt factor is a single number standing in for a whole grading curve.
- The equations are empirical and were fitted to canals in the Indo-Gangetic plains, so they suit similar alluvium best.
Frequently asked questions
What is the silt factor in Lacey's theory?
It is f = 1.76√d, where d is the mean particle size in mm. f = 1 corresponds to standard silt of about 0.323 mm.
What is Lacey's formula for wetted perimeter?
P = 4.75√Q in SI units, with P in metres and Q in m³/s. It does not depend on the silt factor.
What is the difference between Kennedy's and Lacey's theory?
Kennedy relates critical velocity to depth and needs trial and error; Lacey gives velocity, perimeter, radius and slope directly from Q and f and describes the final regime.
What is Lacey's regime flow equation?
V = 10.8 R^(2/3) S^(1/3), where V is in m/s, R is the hydraulic mean radius in metres and S is the bed slope.
Revising irrigation for AE/JE exams? Our Irrigation Engineering notes cover Kennedy, Lacey, canal falls, weirs and cross-drainage works with solved examples — free preview before you buy.
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