OPEN-CHANNEL HYDRAULICS: LECTURE 114 - THE LANE RELATION

1. LANE RELATION
1.01
In 1955, Emory W. Lane presented a relation between sediment load and hydraulic variables:

Q_s d_s ∝ Q_w S_o

Eq. 1

1.02
This relation states the proportionality between the product of sediment discharge and particle size on the left-hand side and the product of water discharge and bottom slope on the right-hand side.
1.03
At equilibrium, this relation holds.
1.04
Under nonequilibrium conditions, when any of these variables changes, the relation indicates the direction of the changes which are necessary in one or more of the other variables to restore equilibrium.
1.05
For instance, if the sediment load is decreased, equilibrium can be restored by decreasing the water discharge or the bottom slope, or by increasing the sediment size.
1.06
The sediment discharge is the bed material load. It excludes the fine-material load, or wash load, which is transported through the system largely without affecting the river hydraulics.
1.07
This photo shows the effect of releasing sediment-free water from a dam. A reduction in sediment load caused a reduction in bottom slope downstream of the dam, triggering channel degradation exceeding 3 m.
Fig. 01
Degradation below Fort Sumner dam, Pecos river, New Mexico (Lane).

1.08
This photo shows the effect of a sediment retention dam in a small creek. A reduction in sediment load downstream of the dam caused the stream to erode down to bedrock, about 3 m, within a short timespan.
Fig. 02

Degradation to bedrock below sediment retention dam, Aguaje de la Tuna, Tijuana, Baja California, Mexico.

Fig. 03

Degradation to bedrock below sediment retention dam, Aguaje de la Tuna, Tijuana, Baja California, Mexico.

1.09
Thus, as the Lane relation predicts, decreases in sediment load downstream of a dam result in "hungry water."
1.10
"Hungry water" leads to channel degradation until a new equilibrium is reached.

2. DIMENSIONLESS LANE RELATION
2.01
The Lane relation can be expressed in dimensionless form by using suitable hydraulic and sediment transport relations.
1. The dimensionless Chezy friction formula
2. The Colby sediment transport formula, and
3. The Strickler relation for Manning's n.
2.02
The following derivations are strictly applicable to hydraulically wide channels.
2.03
The dimensionless Chezy formula is:

S_o = f F²

Eq. 2

2.04
The Colby sediment transport formula states the proportionality between unit-width sediment discharge and mean velocity.

q_s = k₁ ρ v^m

Eq. 2

2.05
where ρ is the mass density of the water, and k₁ and m are coefficient and exponent, respectively, of the power function.
2.06
In general, the exponent m varies from 3 to 7.
2.07
Values in the lower range are associated with high flow rates, while values in the upper range correspond to the lower flow rates.
2.08
For m = 3, that is, under high flow rates, the coefficient k₁ is dimensionless, and the sediment transport equation is theoretically based.

q_s = k₁ ρ v³

Eq. 3

2.09
The water discharge is:

q_w = vd

Eq. 4

2.10
The sediment concentration is:

       q_s
C_s =

       γq_w

Eq. 5

2.11
Therefore:

       k₁v²
C_s =

        gd

Eq. 6

C_s = k₁ F²

Eq. 7

          S_o
C_s = k₁

          f

Eq. 8

2.12
The relation between f and Manning's n is:

       g n²
f =

      k² R^1/3

Eq. 9

2.13
k is equal to 1 in SI units, and 1.486 in U.S. Customary units. Therefore:

       k₂ n²
f =

        R^1/3

Eq. 10

2.14
in which k₂ is equal to g/k²:

       g
k₂ =

       k²

Eq. 11

2.15
k₂ is equal to 9.81 in SI units, and 14.568 in U.S. Customary units.

k₂ = 9.81 (SI units)
k₂ = 14.568 (U.S. Customary units)

Eq. 12

2.16
The Strickler relation for Manning's n is:

n = k₃ d₅₀^1/6

Eq. 13

2.17
k₃ is equal to 0.04169 in SI units, with d₅₀ in meters, and 0.0342 in U.S. Customary units, with d₅₀ in feet.

k₃ = 0.04169 (SI units)
k₃ = 0.0342 (U.S. Customary units)

Eq. 14

2.18
Assume that the mean particle diameter d₅₀ is the representative particle size d_s. Then:

n = k₃ d_s^1/6

Eq. 15

n² = k₃² d_s^1/3

Eq. 16

2.19
Replacing n² into the equation for f:

             d_s
f = k₂ k₃² (
)^1/3
             R

Eq. 17

2.20
Replacing f into the equation for C_s:

        k₁
C_s =
S_o (R/d_s)^1/3
       k₂k₃²

Eq. 18

2.21
Therefore, the sediment concentration is:

       Q_s       k₁
C_s =
=
S_o (R/d_s)^1/3
       γQ_w     k₂k₃²

Eq. 19

2.22
Therefore, the dimensionless Lane relation is:

                 k₁
Q_s (d_s/R)^1/3 =
γ Q_w S_o
                k₂k₃²

Eq. 20

2.23
Unlike the original Lane relation, which was a proportionality, this new Lane relation is a dimensionless equation; thus, it is properly a sediment transport function.

3. SEDIMENT TRANSPORT FUNCTION
3.01
The sediment transport function is:

        k₁
Q_s =
γ Q_w S_o (R/d_s)^1/3
       k₂k₃²

Eq. 21

3.02
It reduces to:

Q_s = k₁ k₄ γ Q_w S_o (R/d_s)^1/3

Eq. 22

3.03
in which k₄, a constant regardless of the system of units, is a function of k₂ and k₃.

        1
k₄ =

       k₂k₃²

Eq. 23

              1
k₄ =
= 58.7
       9.81 (0.04169)²

Eq. 24

3.04
Therefore, the dimensionless sediment transport function is:

Q_s = 58.7 k₁ γ Q_w S_o (R/d_s)^1/3

Eq. 25

3.05
This sediment transport function is applicable in either SI or U.S. Customary units.
3.06
Having assumed m = 3, the only other parameter to be determined experimentally is k₁.
3.07
Lower values of k₁ are associated with higher values of the exponent m.
3.08
Assume the following example:

Discharge Q_w = 10 m³/s
Bottom slope S_o = 0.001
Hydraulic radius R = 1 m
Particle size d_s = 0.5 mm

Sediment transport parameter m = 3
Sediment transport parameter k₁ = 0.01
Unit weight of water γ = 9810 N/m³

Eq. 26

3.09
The application of the sediment transport function leads to:

Q_s = 58.7 (0.01) (9810) (10) (0.001) (1/0.0005)^1/3
Q_s = 726 Newtons/second

Eq. 27

Fig. 03

Degradation to bedrock below sediment retention dam, Aguaje de la Tuna, Tijuana, Baja California, Mexico.

Fig. 01

Degradation below Fort Sumner dam, Pecos river, New Mexico (Lane).

Fig. 02

Degradation to bedrock below sediment retention dam, Aguaje de la Tuna, Tijuana, Baja California, Mexico.

Fig. 03

Degradation to bedrock below sediment retention dam, Aguaje de la Tuna, Tijuana, Baja California, Mexico.

Narrator: Victor M. Ponce
Music: Fernando Oñate
Editor: Flor Pérez

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