The Lane relation revisited, with online calculation, fluvial geomorphology, sediment transport function, Victor M. Ponce, San Diego State University

The Lane relation revisited,
with online calculation

Victor M. Ponce

26 February 2015

ABSTRACT

A new Lane relation of fluvial hydraulics is derived from basic principles of sediment transport. It is expressed as follows:

Q_s (d_s/R)^1/3 ∝ γ Q_w S_o

Unlike the original Lane relation, this new relation is dimensionless. An online calculator is developed to solve the sediment transport equation arising from the new Lane relation.

1. INTRODUCTION

This article revisits the Lane relation of fluvial hydraulics (Lane, 1955):

Q_s d_s ∝ Q_w S_o (1)

The new 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.

2. THE FRICTION FUNCTION

The quadratic friction law is (Ponce and Simons, 1977):

τ_o = ρ f v² (2)

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 (3)

Combining Eqs 2 and 3:

S_o = f v² / (gR) (4)

The Froude number is (Chow, 1959):

F = v / (gD)^1/2 (5)

Combining Eqs. 4 and 5:

S_o = f (D/R) F² (6)

For a hydraulically wide channel: D ≅ R. Therefore:

S_o = f F² (7)

3. THE SEDIMENT TRANSPORT FUNCTION

A general sediment transport function is (Ponce, 1988):

q_s = ρ k₁ v^m (8)

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 (9)

where k₁ is a dimensionless constant.

Using Eq. 2:

q_s = (k₁/f) τ_o v (10)

from which:

q_s ∝ τ_o v (11)

i.e., the sediment transport rate, per unit of channel width, is proportional to the stream power τ_ov, as documented by Simons and Richardson (1966) in connection with the prediction of forms of bed roughness in alluvial channels.

The unit-width discharge is:

q = v d (12)

The sediment concentration is:

C_s = q_s/(γq) (13)

Combining Eqs. 9 and 12:

C_s = k₁ v²/(gd) (14)

For a hydraulically wide channel, d ≅ D. Combining Eqs. 5 and 14, the sediment concentration is:

C_s = k₁ F² (15)

Combining Eqs. 7 and 15:

C_s = k₁ (S_o/f) (16)

The relationship between f and Manning's n is, in SI units (Chow, 1959):

f = gn² / R^1/3 (17)

In U.S. Customary units:

f = gn² / (1.486² R^1/3) (18)

Thus, in general:

f = k₂ n² / R^1/3 (19)

In SI units:

k₂ = g = 9.81 (20)

In U.S. Customary units:

k₂ = g/1.486² = 32.17 / 2.208 = 14.568 (21)

4. THE STRICKLER RELATION

The Strickler relation between Manning n and mean particle size d₅₀ is (Chow, 1959):

n = k₃ d₅₀^1/6 (22)

In SI units:

k₃ = 0.04169 (23)

with d₅₀ in meters.

In U.S. Customary units:

k₃ = 0.0342 (24)

with d₅₀ in feet.

Assume d_s = d₅₀:

n = k₃ d_s^1/6 (25)

n² = k₃² d_s^1/3 (26)

Combining Eqs. 19 and 26:

f = k₂ k₃² (d_s/R)^1/3 (27)

5. THE SEDIMENT CONCENTRATION

The sediment concentration is:

C_s = k₁ (S_o/f) (16)

Substituting Eq. 27 on Eq. 16:

C_s = k₁ S_o/[k₂ k₃² (d_s/R)^1/3] (28)

Thus:

C_s = [k₁/(k₂ k₃²)] [S_o/(d_s/R)^1/3] (29)

Therefore:

Q_s/(γQ_w) = [k₁/(k₂ k₃²)] [S_o/(d_s/R)^1/3] (30)

and:

Q_s (d_s/R)^1/3 = [k₁/(k₂ k₃²)] γ Q_w S_o (31)

6. THE MODIFIED LANE RELATION

The Lane relation (Lane, 1955) is:

Q_s d_s ∝ Q_w S_o (1)

Following Eq. 31, the Modified Lane relation is:

Q_s (d_s/R)^1/3 ∝ γ Q_w S_o (32)

The sediment transport relation is:

Q_s = [k₁/(k₂ k₃²)] γ Q_w S_o (R/d_s)^1/3 (33)

In SI units:

Q_s = [k₁/(9.81 × 0.04169²)] γ Q_w S_o (R/d_s)^1/3 (34)

Q_s = 58.7 k₁ γ Q_w S_o (R/d_s)^1/3 (35)

In U.S. Customary units:

Q_s = [k₁/(14.568 × 0.0342²)] γ Q_w S_o (R/d_s)^1/3 (36)

Q_s = 58.7 k₁ γ Q_w S_o (R/d_s)^1/3 (37)

The sediment transport function is dimensionless; therefore, independent of the system of units.

The sediment transport parameter k₁ is the only one to be determined by calibration. Experience shows that this parameter varies typically in the range 0.001 ≤ k₁ ≤ 0.01.

7. APPLICATIONS

Assume pre- and post-development cases, with subscripts 1 and 2, respectively. Further define:

a = Q_s₂/Q_s₁ (38)

b = d_s₂/d_s₁ (39)

c = R₂/R₁ (40)

d = Q_w₂/Q_w₁ (41)

e = S_o₂/S_o₁ (42)

From the modified Lane relation (Eq. 32):

a (b/c)^1/3 = d e (43)

Thus, the channel slope change is:

e = (a/d) (b/c)^1/3 (44)

Example 1

A river reach entering a reservoir, with a = 0.95, b = 0.95, c = 5., and d = 1., will result in e = 0.55 (aggradation in the reservoir) (Fig. 1).

Sediment deposition at tail of reservoir

Fig. 1 Sediment deposition at tail of reservoir.

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) (Fig. 1).

Sediment deposition at tail of reservoir

Fig. 2 Erosion to bedrock downstream of sediment retention dam.

8. ONLINE CALCULATOR

Given: Water discharge Q_w = 100 m³/s; bottom slope S_o = 0.001; hydraulics radius R = 2 m; particle size s_s= 1 mm. Use default k = 0.001. Use onlinemodifiedlane.php to calculate sediment discharge.

Result of the online calculator: Sediment discharge Q_s = 6,389.9 M.Tons/day.

9. EXTENSION TO m > 3

In the general case for the exponent of the sediment transport function, i.e., for m > 3, Eq. 8 remains:

q_s = ρ k₁ v^m (8)

The general Modified Lane relation is:

Q_s (d_s/R)^1/3 ∝ γ Q_w S_o v^{m - 3} (45)

The general sediment transport relation is:

Q_s = 58.7 k₁ γ Q_w S_o (R/d_s)^1/3 v^{m - 3} (46)

10. SUMMARY

A new Lane relation of fluvial hydraulics is derived from basic principles of sediment transport. It is expressed as follows:

Q_s (d_s/R)^1/3 ∝ γ Q_w S_o (32)

Unlike the original Lane relation:

Q_s d_s ∝ Q_w S_o (1)

the new relation (Eq. 32) is dimensionless. A sediment transport equation is derived from the modified Lane relation, particularly for the case of sediment rating exponent (Eq. 8) m = 3:

Q_s = 58.7 k₁ γ Q_w S_o (R/d_s)^1/3 (35)

An online calculator is developed to solve the sediment transport equation.

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.

Simons, D. B., and E. V. Richardson, 1966. Resistance to flow in alluvial channels. U.S. Geological Survey Professional Paper 422-J, Washington, D.C.

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 the Strickler 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.

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