1B Clastic SedimentsLecture 27

SEDIMENT TRANSPORT

Onset of motionMode of transport

Estimation of bedload transportEstimation of suspended load transport

NH 01-2007

SEDIMENT CONTINUITY

For constant sedimentconcentration C,

(qs)1 < (qs)2: erosion of channel bed.

(qs)1 > (qs)2: deposition on channel bed.

Bed volume change in time dt:

dydx = -1/(1-) dqsdt ,

where dy is thickness eroded/aggraded is porosity of bed material.

Bed volume change may also result from change in sediment concentration:

dydx = -1(1-) (hdC)dx ,

where h is flow depth.

dydt = -1/(1-) (dqs/dx + hdC/dt)

Net change in bed elevation over time is related to down-stream change of sedimenttransport rate and the changein suspended sedimentconcentration in time.

TRESHOLD OF SEDIMENT MOVEMENT

Stream power = 0u

is rate of work done on channel bed.

Below threshold shear stress c:no sediment motion and transport.

Effective shear stress: – c

FORCES ON PARTICLE ON BED

Cylinder on horizontal bedbelow inviscid fluid. No viscosity: no drag force.

Stream lines converge and then diverge over cylinder in symmetrical fashion: pressure distribution resultsin net upward force: lift force.

FORCES ON PARTICLE ON BED

Cylinder on horizontal bedbelow inviscid fluid. No viscosity: no drag force.

Stream lines converge and then diverge over cylinder in symmetrical fashion: pressure distribution resultsin net upward force: lift force.

Cylinder under viscous flow:Flow separation behind cylindercauses drag force in addition to lift.

Resultant fluid pressure force FF has upward and downstream component.

FORCES ON PARTICLE ON BED

Resisting grain motion:Gravitational force FG; neighbouring grains.Entrainment of grain by rotation about pivot.Angle of easiest movement: .Threshold of movement: balance of momentsabout pivot point:

FG(sin )a1 = FF(cos )a2 ,

where a1 and a2 are moment arms.FG = c1D3’ ,

where ’ = (s – fg, and constant c1 accounts for flow variability grain characteristics.

FF = c2D20 ,

where constant c2 accounts for grain shape and packing.Ignore lift force: vanishes after entrainment.

FORCES ON PARTICLE ON BED

Threshold of movement: balance of momentsabout pivot point:

FG(sin )a1 = FF(cos )a2 ,

FG = c1D3’ ,

FD = c2D20 .

Combine and regroup:

[a1c1/a2c2 tan = 0/D’ .

~ grain characteristicslocal flow:boundary Reynolds no.

Definition of threshold condition relies on experimentation.

DIMENSIONAL ANALYSIS OF MOTION THRESHOLD

Variables:Critical bed shear stress, c [ML-1T-2] repeatFluid density, f [ML-3] repeatFluid viscosity, [ML-1T-1]Grain diameter, D [L] repeatSubmerged specific weight of grain, ’ [ML-2T-2]

Sought: balance of inertial and viscous forces (Reynolds number),balance of gravitational and fluid forces.

Combine ’ with repeating variables:

0/’D = (Shields’ stress).

Combine with repeating variables:

(D√f√0)/ = (fu*D)/ = u*D/ = Re* ,

where is the kinematic viscosity.Remember: shear velocity u*

2 = 0/ .Re* is boundary Reynolds number.

and Re* fully characteriseonset of sediment motion.Their relation was constrainedexperimentally by Shields.

SHIELDS’ DIAGRAM

Re* < 10: fine grain sizes: well-packed, cohesive sediment, enclosed within viscous sublayer. Entrainment more difficult than fine sand. Shields’ stress increases with decreasing Re*

Re* > 10: Non-cohesive silt and sand. Entrainment more difficult with increasing grainsize. Expected: Shields’ stress increases with increasing Re*

Experimental results: flat trend.

Shear stress and grain size on both axes.

YALIN’S DIAGRAM

In Shields’ diagram: shear stress and grain size on both axes.Solve by combining boundary Reynolds number with Shields’ to eliminate shear stress:

= Re*2/ .

Yalin’s plot of against √ has same general form as Shields’ curve.

MODES OF SEDIMENT TRANSPORT

BED LOADSliding, rolling, saltation

SUSPENDED LOAD

Mode of transport depends on grain densitygrain sizeflow hydraulics

Conditions vary in space and time:Modes of transport change frequently.Distinction between bed load andsuspended load is not easy.

Transport stage

TRANSPORT STAGE

u*/w ,

where u* is the shear velocity, 0/w is the settling velocity (cf. Stoke’s law).

With increasing shear velocity, proportion of load moving in suspensionincreases.

Therefore dimensionless grain velocityug/U increases with transport stage.

Here, ug is the grain velocity, and U is theflow velocity.

u* = w approximates saltation –suspension threshold.

When u* > w, then grains move with approximately the velocity of the flow.Results shown for quartz sand in flow

48 mm deep.

BEDLOAD TRANSPORT

Bedload transport rate ~ stream power, 0u .

conversion factor to be constrained empirically

Prediction of bedload transport complicated by:bed armouring and consolidation of gravels.resistance of bedforms in sand and gravel rivers.lack of constraints on threshold of sediment motion.unsteadiness in high stage flows.

has dimensions [ML2T-3] over unit area of stream bed [L2]:Proportional to the cube of (excess) flow velocity.

Distribution of flow velocity in open channels.

FLOW RESISTANCE IN BEDLOAD RIVERS

Force driving flow down inclined plane: downslope component of gravity acting on the mass of water.Flow resistance: frictional energy loss during flow on bed and banks.

Parameters of the problem:Flow depth h Combine two length scales in problem inflow velocity u dimensionless variable: h/ks, the relative roughness.fluid density f

fluid viscosity Express fluid flow in a Reynolds number: Re = fuh/ .basal shear stress 0

roughness height ks Make shear stress dimensionless by dividing by u2:80/fu2 = f f is the friction factor.

Using 0 = gSh, fgh sin = 1/8 fu2f .

Solving for u, u = √(8 gh sin )/f .

Usually written as: u = C(h sin )0.5 , C = (8g/f)0.5

Chezy coefficient.

Flow depth h is inadequate length scale.

FLOW RESISTANCE IN BEDLOAD RIVERS

Only bed and banks exert friction.Together they are termed:Wetted perimeter.

Hydraulic radius RH =channel cross section area wetted perimeter

RH is better length scale for calculation of flow resistance.

u = (RH2/3 sin 1/2)/n ,

where n is Manning’s roughness coefficient.

n = 1/C RH4/3 ; C = (8g/f)0.5 ; f = 80/fu2

Use to estimate formative discharges in (paleo) channels:

Measure channel slope from exposed geometry or terrace form.Largest clasts are assumed to represent maximum discharge conditions.Critical bed shear stress is estimated from particle size using c = ’D .

SUSPENDED LOAD

The distribution of suspended sediment in a flow can be treated asdiffusion problem, with high concentration at bed, and low concentrationnear surface. Mass flux Q is linearly proportional to concentration gradient:

Q = -∂C/∂y , where is a diffusivity constant.

Assuming conservation of mass,

∂C/∂t = ∂Q/∂y .

Combined: ∂C/∂t = ∂/∂y (∂C/∂y).

If diffusivity constant in y, then ∂C/∂t = (∂2C/∂y2).If concentration is constant in time, then ∂C/∂t = 0, and ∂2C/∂y2 = 0.

Concentration profile obtained by twice integrating for boundary conditions:C = K1y + K2

It can be shown that K1 = -C0/h, where C0 is the concentration at the bed.

K2 = C0 .

C = C0 (1 – y/h) This profile is linear in depth.

SUSPENDED LOAD

Mississippi River at St. Louis

C = C0 (1 – y/h)This profile is linear in depth.Observed suspended sediment concentrationprofiles are not linear in depth.

We have ignored the settling of grains.Concentration profile reflects balance ofupward diffusion and gravitational settlingof grains.

When C is constant in time, then any loss of sediment due to settling is balanced by upward diffusion of sediment. Settling flux is wC. Upward diffusion flux Q = -dC/dy.

wC = - dC/dy ,or dC/C = -wdy/t , where t = , and is the kinematic eddy viscosity.

≈ 1, and = u*[(h-y)/h]ky . Von Karman’s k = 0.4 .

dC/C = whdy/[ku*(h-y)y] .

SUSPENDED LOAD

dC/C = whdy/[ku*(h-y)y]

At reference height a, C = Ca . Integrate:

C/Ca = [h-y/y × a/h-a]w/ku*.

This gives suspended sediment concentration at any depth in flow in relation to concentrationat reference depth. Only need to know a and Ca.

The grouping w/ku* is the Rouse number. Since ≈ 1, and k = 0.4, w = u* for a Rouse number of 2.5.This is the criterion for suspension. Rouse number > 2.5: w > u* with bedload transport dominant.

High bed roughness → high shear velocity → high suspended sediment conc.High viscosity → low settling velocity → high suspended sediment conc.

SUSPENDED LOAD

dC/C = whdy/[ku*(h-y)y]

At height a close to bed, C = Ca Then, integration gives:

C/Ca = [h-y/y × a/h-a]w/ku*

This gives suspended sediment concentration at any depth in flow in relation to concentrationat reference depth. Only need to know Ca.

Suspended sediment transport rate is product of the mass of suspended sediment ms in a column of water over a unit area of bed and the depth-averaged flow velocity U at the station:

Qs = msUAssumes that all sediment is mobilized from bed.