Andrew Simon USDA-ARS National Sedimentation Laboratory, Oxford, MS

andrew.simon@ars.usda.gov

Equilibrium, Shear Stress, Stream Power and Trends of Vertical

Adjustment

Non-Cohesive versus Cohesive Materials

• Non-cohesive: sands and gravels etc.Resistance is due solely to particle size, weight, shape and “hiding”.

• Cohesive: silts and clays

Resistance is derived from electro-chemical inter-particle forces under zero normal stress

Shields Diagram

Denotes uncertainty

CohesiveMaterials

Shields Diagram by Particle Diameter

Excludes cohesives

Heterogeneous Beds

ks = 3* D84

Need for a means to determine critical shear stress (c) and the erodibility coefficient (k) in-situ for soils and sediments.

National Sedimentation Laboratory

Erosion of Cohesives by Hydraulic Shear

Erosion Rate is a Function of Erodibility and Excess Shear Stress

= k (o- c) = erosion rate (m/s)

k = erodibility coefficient (m3/N-s)

o = boundary shear stress (Pa)

c = critical shear stress (Pa)

(o-c) = excess shear stress

Critical shear stress is the stress required to initiate erosion.

Obtained from jet-test device

Impinging Jet Applies Shear Stress to Bed

Jet Nozzle

National Sedimentation Laboratory

Impinging Jet Applies Shear Stress to Bed

As scour hole depth increases, shear stress decreases.

Jet Nozzle

National Sedimentation Laboratory

From Relation between Shear Stress and Erosion We Calculate c and

Time

Eros

ion

Dep

th,

cm

c

National Sedimentation Laboratory

(cm3/Pa/sec)k

General Relation for Erodibility and Critical Shear StressErodibility, m3/N-s

k = 0.1 c -0.5

Where; c = critical shear stress (Pa), x, y = empirical constants

CRITICAL SHEAR STRESS, IN Pa

0.01 0.1 1 10 100 1000

EROD

IBILIT

Y COE

FFICI

ENT (

k), IN

cm3 /N-

s

0.0001

0.001

0.01

0.1

1

10

k = 0.09 c -0.48

44

Revised Erodibility Relation

y = 1.3594x-0.8345

R2 = 0.5253

0.0001

0.0010

0.0100

0.1000

1.0000

10.0000

100.0000

1.0E-07 1.0E-06 1.0E-05 1.0E-04 1.0E-03 1.0E-02 1.0E-01 1.0E+00 1.0E+01 1.0E+02 1.0E+03CRITICAL SHEAR STRESS (Pa)

ERO

DIB

ILIT

Y C

OEF

FIC

IEN

T (k

)

Distributions: Critical Shear Stress

0

10

20

30

40

50

60

70

80

90

100

0.1 1.0 10.0 100.0 1000.0CRITICAL SHEAR STRESS (Pa)

PER

CEN

TILE Yalobusha River System

Kalamazoo RiverJames CreekShades CreekMissouri RiverUpper Truckee RiverW. Iowa, E. NebraskaN Fork Broad RiverTualatin River SystemTombigbee RiverS Branch Buffalo RiverAll Data

Distributions: Erodibility Coefficient

0

10

20

30

40

50

60

70

80

90

100

0.001 0.010 0.100 1.000 10.000 100.000ERODIBILITY COEFFICIENT (k)

PER

CEN

TILE

Yalobusha River SystemKalamazoo RiverJames CreekShades CreekMissouri RiverUpper Truckee RiverW. Iowa, E. NebraskaN Fork Broad RiverTualatin River SystemTombigbee RiverS Branch Buffalo RiverAll Data

Mapping Critical Shear Stress: Yalobusha River Basin, Mississippi

National Sedimentation Laboratory

Idealized Adjustment TrendsIdealized Adjustment Trends

For a given discharge (Q)

VS

Se

n

c

d

National Sedimentation Laboratory

Adjustment: Boundary Shear Stress

Adjustment: Increasing Resistance

Adjustment: Increasing Resistance

Adjustment: (Excess Shear Stress)Degrading Reach

Boundary Shear Stress: Range of FlowsSh

ear s

tress

, in

N/m

2

Adjustment: Excess Shear Stress

Degrading ReachEx

cess

shea

r stre

ss

Adjustment: (Excess Shear Stress)Aggrading Reach

Adjustment of Force and Resistance

Results of Adjustment

Decreasing Sediment Loads with Time

Toutle River System

Experimental Results

Total and Unit Stream Power = w y V S = Q S = total stream power per unit length of channel = specific weight of water w = water-surface width y = hydraulic depth v = mean flow velocity Q = water discharge S = energy slope

w = / ( w y) = V S where w = stream power per unit weight of water

Adjustment: Unit Stream Power

Flow Energy• Total Mechanical EnergyTotal Mechanical Energy

H = z + y + (H = z + y + ( v v22 / 2 g)/ 2 g)where H = total mechanical energy (head)where H = total mechanical energy (head)

z = mean channel-bed elevation (datum head)z = mean channel-bed elevation (datum head) = coefficient for non-uniform distribution velocity= coefficient for non-uniform distribution velocityy = hydraulic depth (pressure head)y = hydraulic depth (pressure head)g = acceleration of gravityg = acceleration of gravity

• Head Loss over a reach due to FrictionHead Loss over a reach due to Friction hhff = [z = [z11 + y + y11 + ( + (11 v v1122 / 2g)]- [z/ 2g)]- [z22 + y + y22 + ( + (22 v v2222 / /

2g)]2g)]• Head, Relative to channel bedHead, Relative to channel bed EEss = y + ( = y + ( vv22 / 2g) =/ 2g) = y + [y + [ Q Q22 / (2 g w / (2 g w22 y y22)])]

As a working hypothesis we assume that a fluvial system has been disturbed in a manner such that the energy available to the system (potential and kinetic) has been increased. We further assume that with time, the system will adjust such that the energy at a point (head) and the energy dissipated over a reach (head loss), is decreased.

Now, for a given discharge, consider how different fluvial processes will change (increase or decrease) the different variables in the energy equations.

Adjustment: Total Mechanical Energy

Adjustment: Energy Dissipation

Minimization of energy dissipation

Trends of Vertical Adjustment and Determining Equilibrium

Determining Equilibrium

Recall definitionA stream in equilibrium is one in which over a

period of years, slope is adjusted such that there is no net aggradation or degradation on the channel bed (or widening or narrowing)

ORThere is a balance between energy conditions at

the reach in question with energy and materials being delivered from upstream

Causes of Channel Incision

Trends of Incision: Channelization

Trends of Incision: Below Dams

Bed-level Trends Along a Reach

Bed-level Trends Along a Reach

Empirical Functions to Describe Incision E = a t b

E = elevation of the channel beda = coefficient; approximately, the pre-disturbance elevationt = time (years), since year before start of adjustmentb = dimensionless exponent indicating rate of change on the bed (+) for aggradation, (-) for degradation

E/ Eo = a + b e-kt

E = elevation of the channel bedEo = initial elevation of the channel bed

a = dimensionless coefficient, = the dimensionless elevation a > 1 = aggradation, a < 1 = degradationb = dimensionless coefficient, = total change of elevation b > 0 = degradation, b < 0 = aggradationk = coefficient indicating decreasing rate of change on the bed

Empirical Model of Bed-level Response

Comparison of the Two Bed-level Functions

A Natural Disturbance (Toutle River System)

Bed-Level Response

Bed Response: Toutle River System

Upstream disturbance, addition of potential energy, sub-alpine environment

Comparison with Coastal Plain Adjustment

Downstream disturbance, increase in gradient, coastal plain environment

Model of Long-Term Bed Adjustment