 

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):
in which f is a friction factor equal to 1/8 of the DarcyWeisbach friction factor.
The bottom shear stress in terms of hydraulic variables is (Chow, 1959):
Combining Eqs 2 and 3:
S_{o} = f v^{2} / (gR)
 (4) 
The Froude number is (Chow, 1959):
Combining Eqs. 4 and 5:
S_{o} = f (D/R) F^{2}
 (6) 
For a hydraulically wide channel: D ≅ R.
Therefore:
3. THE SEDIMENT TRANSPORT FUNCTION
A general sediment transport function is (Ponce, 1988):
q_{s} = ρ k_{1} 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 k_{1} v^{3}
 (9) 
where k_{1} is a dimensionless constant.
Using Eq. 2:
q_{s} = (k_{1}/f) τ_{o} v
 (10) 
from which:
i.e., the sediment transport rate, per unit of channel width, is proportional
to the stream power τ_{o}v, as documented by Simons and Richardson (1966) in connection with the prediction of forms of bed roughness
in alluvial channels.
The unitwidth discharge is:
The sediment concentration is:
Combining Eqs. 9 and 12:
C_{s} = k_{1} v^{2}/(gd)
 (14) 
For a hydraulically wide channel, d ≅ D. Combining Eqs. 5 and 14, the sediment concentration is:
Combining Eqs. 7 and 15:
C_{s} = k_{1} (S_{o}/f)
 (16) 
The relationship between f and Manning's n is, in SI units (Chow, 1959):
f = gn^{2} / R^{1/3}
 (17) 
In U.S. Customary units:
f = gn^{2} / (1.486^{2} R^{1/3})
 (18) 
Thus, in general:
f = k_{2} n^{2} / R^{1/3}
 (19) 
In SI units:
In U.S. Customary units:
k_{2} = g/1.486^{2} = 32.17 / 2.208 = 14.568
 (21) 
4. THE STRICKLER RELATION
The Strickler relation between Manning n and mean particle size d_{50} is (Chow, 1959):
n = k_{3} d_{50}^{1/6}
 (22) 
In SI units:
with d_{50} in meters.
In U.S. Customary units:
with d_{50} in feet.
Assume d_{s} = d_{50}:
n = k_{3} d_{s}^{1/6}
 (25) 
n^{2} = k_{3}^{2} d_{s}^{1/3}
 (26) 
Combining Eqs. 19 and 26:
f = k_{2} k_{3}^{2} (d_{s}/R)^{1/3}
 (27) 
5. THE SEDIMENT CONCENTRATION
The sediment concentration is:
C_{s} = k_{1} (S_{o}/f)
 (16) 
Substituting Eq. 27 on Eq. 16:
C_{s} = k_{1} S_{o}/[k_{2} k_{3}^{2} (d_{s}/R)^{1/3}]
 (28) 
Thus:
C_{s} = [k_{1}/(k_{2} k_{3}^{2})] [S_{o}/(d_{s}/R)^{1/3}]
 (29) 
Therefore:
Q_{s}/(γQ_{w}) = [k_{1}/(k_{2} k_{3}^{2})] [S_{o}/(d_{s}/R)^{1/3}]
 (30) 
and:
Q_{s} (d_{s}/R)^{1/3} = [k_{1}/(k_{2} k_{3}^{2})] 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} = [k_{1}/(k_{2} k_{3}^{2})] γ Q_{w} S_{o} (R/d_{s})^{1/3}
 (33) 
In SI units:
Q_{s} = [k_{1}/(9.81 × 0.04169^{2})] γ Q_{w} S_{o} (R/d_{s})^{1/3}
 (34) 
Q_{s} = 58.7 k_{1} γ Q_{w} S_{o} (R/d_{s})^{1/3}
 (35) 
In U.S. Customary units:
Q_{s} = [k_{1}/(14.568 × 0.0342^{2})] γ Q_{w} S_{o} (R/d_{s})^{1/3}
 (36) 
Q_{s} = 58.7 k_{1} 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 k_{1}
is the only one to be determined by calibration. Experience shows that this parameter
varies typically in the range 0.001 ≤ k_{1} ≤ 0.01.
7. APPLICATIONS
Assume pre and postdevelopment cases, with subscripts 1 and 2, respectively. Further define:
From the modified Lane relation (Eq. 32):
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).
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).
Fig. 2 Erosion to bedrock downstream of sediment retention dam.

8. ONLINE CALCULATOR
Given: Water discharge Q_{w} = 100 m^{3}/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 k_{1} 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 k_{1} 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 k_{1} 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. Openchannel hydraulics. McGrawHill, New York.
Colby, B. R., 1964. Discharge of sands and mean velocity relations in sandbed streams. U.S. Geological Survey Professional Paper No. 462A, 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 812, 1988, 311315.
Simons, D. B., and E. V. Richardson, 1966.
Resistance to flow
in alluvial channels. U.S. Geological Survey Professional Paper 422J, Washington, D.C.
NOTATION
a, b, c, d, e = ratios of post and predevelopment hydraulic variables;
C = Chezy coefficient;
C_{s} = sediment concentration;
d = flow depth;
D = hydraulic depth;
d_{s} = particle size;
d_{50} = mean particle size;
f = friction factor equal to 1/8 of DarcyWeisbach friction factor;
F = Froude number;
g = gravitational acceleration;
k_{1} = dimensionless sediment transport parameter;
k_{2} = friction parameter;
k_{3} = coefficient in the Strickler relation;
n = Manning's friction coefficient;
q = unitwidth water discharge;
q_{s} = unitwidth 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.
