Lane's relation, modified Lane relation, fluvial hydraulics, fluvial geomorphology, erosion and sedimentation, Victor Miguel Ponce

Fig. 1 Erosion to bedrock downstream of sediment retention dam.

THE LANE RELATION REVISITED
by Victor M. Ponce
Professor of Civil Engineering
San Diego State University
[080414]

INTRODUCTION
This article revisits the Lane relation (1955) of fluvial hydraulics:
Q_s d_s ∝ Q_w S_o
The relation is expressed as an equation, with the particle size (d_s) replaced by the relative roughness d_s/R. The derivation follows.

RESISTANCE AND TRANSPORT FUNCTIONS
The quadratic resistance equation in open-channel flow is:
τ_o = ρ f v²
A general sediment transport equation is:
q_s = ρ k₁ v^m
According to Colby (1964), the exponent m varies in the range 3 ≤ m ≤ 7, with the lower values corresponding to high discharges, and the higher values to low discharges.
Assume m = 3 as a first approximation (high water and sediment discharge)
In this case, the sediment transport function is:
q_s = ρ k₁ v³
where k₁ is a dimensionless parameter.
The unit-width discharge:
q = v d
The sediment concentration:
C_s = q_s/q
C_s = ρ k₁ v³ / (v d) = ρ k₁ v²/d
The Froude number:
F² = v²/(gD)

Thus, the sediment concentration is:
C_s = ρg k₁ F²
C_s = γ k₁ F²

DIMENSIONLESS CHEZY EQUATION
The Chezy equation:
v = C (RS_o)^1/2

S_o = v²/(C²R)
S_o = f (D/R) F²
where
f = g/C²
is a dimensionless Chezy friction factor (1/8 of the Darcy-Weisbach friction factor).
The sediment concentration is:
C_s = γ k₁ (R/D) (S_o/f)
In SI units:
f = gn²/R^1/3
In U.S. Customary units:
f = gn²/[1.486² R^1/3]
f = k₂ n²/R^1/3
In SI units:
k₂ = g = 9.81
In U.S. Customary units:
k₂ = g/1.486² = 32.17 / 2.208 = 14.57

STRICKLER'S RELATION
Strickler's relation is:
n = k₃ d₅₀^1/6
In SI units:
k₃ = 0.0417
with d₅₀ in meters.
In U.S. Customary units:
k₃ = 0.0342
with d₅₀ in feet.
Assume d_s = d₅₀
n = k₃ d_s^1/6
n² = k₃² d_s^1/3
f = k₂ k₃² (d_s/R)^1/3

SEDIMENT CONCENTRATION
C_s = γ k₁ (R/D) S_o/ [k₂ k₃² (d_s/R)^1/3]
C_s = [γ k₁/(k₂ k₃²)] (R/D) [S_o/(d_s/R)^1/3]
C_s = Q_s/Q_w
Q_s/Q_w = [γ k₁/(k₂ k₃²)] (R/D) [S_o/(d_s/R)^1/3]
For a wide channel, R ≅ D. Then:
Q_s/Q_w = [γ k₁/(k₂ k₃²)] [S_o/(d_s/R)^1/3]

MODIFIED LANE'S RELATION
The Lane relation (1955):
Q_s d_s ∝ Q_w S_o
The Modified Lane relation:
Q_s (d_s/R)^1/3 ∝ Q_w S_o
The sediment transport equation is:
Q_s (d_s/R)^1/3 = [k₁/(k₂ k₃²)] γ Q_w S_o
In SI units:

Q_s (d_s/R)^1/3 = [k₁/(9.81 × 0.0471²)] γ Q_w S_o
Q_s (d_s/R)^1/3 = 45.95 k₁ γ Q_w S_o
Q_s (d_s/R)^1/3 = 46 k₁ γ Q_w S_o
In U.S. Customary units:

Q_s (d_s/R)^1/3 = [k₁/(14.57 × 0.0342²)] γ Q_w S_o
Q_s (d_s/R)^1/3 = 58.68 k₁ γ Q_w S_o
Q_s (d_s/R)^1/3 = 59 k₁ γ Q_w S_o
The sediment transport parameter k₁ varies typically in the range 0.005 ≤ k₁ ≤ 0.02, with a central value k₁ = 0.01.

APPLICATIONS
Assume pre- and post-development cases, with subscripts 1 and 2, respectively. Define:
a = Q_s₂/Q_s₁
b = d_s₂/d_s₁
c = R₂/R₁
d = Q_w₂/Q_w₁
e = S_o₂/S_o₁
From the modified Lane relation:
a (b/c)^1/3 = d e
Thus, the slope change is:
e = (a/d)(b/c)^1/3
Example 1
A water diversion with d = 0.9, and a = 0.99, b = 1., and c = 0.95, will result in e = 1.12 (aggradation).
Example 2
A sediment retention basin with a = 0.3, and b = 1., c = 0.95, and d = 0.9, will result in e = 0.34 (degradation). In practice, the latter may be limited by geologic controls (armoring or bedrock).

REFERENCES
Colby, B. R., 1964. Discharge of sands and mean velocity relations in sand-bed streams. U.S. Geological Survey Professional Paper No. 462-A, Washington, D.C.
Lane, E. W., 1955. The importance of fluvial morphology in hydraulic engineering. Proceedings, American Society of Civil Engineers, No. 745, July.

NOTATION
a, b, c, d, e = ratios of post- and pre-development variables;
C = Chezy constant;
C_s = sediment concentration;
D = hydraulic depth;
d_s = particle size;
d₅₀ = mean particle size;
f = dimensionless Chezy friction factor;
F = Froude number;
g = gravitational acceleration;
k₁ = dimensionless sediment transport parameter;
k₂ = friction parameter;

k₃ = Strickler coefficient;
n = Manning's friction coefficient;
q = unit-width [water] discharge;
q_s = unit-width sediment discharge;
Q_w = [water] discharge;
Q_s = sediment discharge;
R = hydraulic radius;
S_o = bottom slope;

v = mean velocity;
γ = unit weight of water;
ρ = mass density of water;
τ_o = bottom shear stress;

Fig. 2 Sediment deposition at upstream entrance to [tail of] reservoir.