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
Victor M. Ponce
Professor of Civil Engineering
San Diego State University
[080619]

ABSTRACT
The Lane relation of fluvial hydraulics is derived from basic principles of sediment transport, and expressed as follows:
Q_s (d_s/R)^1/3 ∝ Q_w S_o
Unlike the original Lane relation, this new relation is completely dimensionless.

INTRODUCTION
This article revisits the Lane relation of fluvial hydraulics (Lane, 1955):
Q_s d_s ∝ Q_w S_o
The relation is derived from theory, and expressed as a dimensionless equation, with the particle size (d_s) replaced by the relative roughness function (d_s/R)^1/3. The derivation follows.

FRICTION FUNCTION
The quadratic friction law is (Ponce and Simons, 1977):
τ_o = ρ f v²
in which f is a friction factor equal to 1/8 of the Darcy-Weisbach friction factor.
The bottom shear stress in terms of hydraulic variables is (Chow, 1959):
τ_o = γ R S_o
from which
S_o = f v²/(gR)

The Froude number is:
F = v/(gD)^1/2
Then:
S_o = f (D/R) F²

For a hydraulically wide channel, D ≅ R.
Then:

S_o = f F²

SEDIMENT TRANSPORT FUNCTION
A general sediment transport equation is (Ponce, 1988):
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 is:
q = v d
The sediment concentration is:
C_s = q_s/q
C_s = ρ k₁ v²/d
For a hydraulically wide channel, d ≅ D.
Thus, the sediment concentration is:
C_s = ρg k₁ F²
C_s = γ k₁ (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 Manning 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
Then:
f = k₂ k₃² (d_s/R)^1/3

SEDIMENT CONCENTRATION
C_s = γ k₁ (S_o/f)
C_s = γ k₁ S_o/[k₂ k₃² (d_s/R)^1/3]
C_s = [γ k₁/(k₂ k₃²)] [S_o/(d_s/R)^1/3]
Q_s/Q_w = [γ k₁/(k₂ k₃²)] [S_o/(d_s/R)^1/3]
Q_s (d_s/R)^1/3 = [k₁/(k₂ k₃²)] γ Q_w S_o

MODIFIED LANE'S RELATION
The Lane relation (Lane, 1955) is:
Q_s d_s ∝ Q_w S_o
The Modified Lane relation is:
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 parameter k₁ varies typically in the range 0.005 ≤ k₁ ≤ 0.02.

APPLICATIONS
Assume pre- and post-development cases, with subscripts 1 and 2, respectively. Further 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 river reach downstream of a channel diversion intake, with d = 0.9, and a = 0.99, b = 1., and c = 0.95, will result in e = 1.12 (aggradation).
Example 2
A river reach downstream of 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
Chow, V. T. 1959. Open-channel hydraulics. Mc-Graw-Hill, New York.
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.
Ponce, V. M., and D. B. Simons, 1977. Shallow wave propagation in open channel flow. American Society of Civil Engineers Journal of the Hydraulics Division, Vol. 103, No. HY12, December.
Ponce, V. M. 1988. Ultimate sediment concentration. Proceedings, National Conference on Hydraulic Engineering, Colorado Springs, Colorado, August 8-12, 1988, 311-315.

NOTATION
a, b, c, d, e = ratios of post- and pre-development hydraulic variables;
C = Chezy coefficient;
C_s = sediment concentration;
d = flow depth;
D = hydraulic depth;
d_s = particle size;
d₅₀ = mean particle size;
f = friction factor equal to 1/8 of Darcy-Weisbach friction factor;
F = Froude number;
g = gravitational acceleration;
k₁ = dimensionless sediment transport parameter;
k₂ = friction parameter;

k₃ = coefficient in Strickler's Manning relation;
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 = density of water; and
τ_o = bottom shear stress.

Fig. 2 Sediment deposition at tail of reservoir.