Vertical structure of estuarine fine sediment suspensions


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Vertical structure of estuarine fine sediment suspensions
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xix, 188 leaves : ill. ; 28 cm.
Ross, Mark Allen, 1959-
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Subjects / Keywords:
Sedimentation and deposition   ( lcsh )
Suspended sediments   ( lcsh )
Sediment transport   ( lcsh )
Estuaries   ( lcsh )
bibliography   ( marcgt )
theses   ( marcgt )
non-fiction   ( marcgt )


Thesis (Ph. D.)--University of Florida, 1988.
Includes bibliographical references.
Statement of Responsibility:
by Mark Allen Ross.
General Note:
General Note:

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University of Florida
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oclc - 20117446
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Full Text









My deepest and most heartfelt appreciation is extended to my

chairman, Dr. Ashish J. Mehta, Professor of Coastal Engineering. In his

capacity as advisor, educator and friend, he has shown me many lofty

values by example. My cochairman, Dr. Robert G. Dean, Graduate Research

Professor, an individual of unmatched character and inspiration, receives

a lion's share of my gratitude.

Special thanks are extended to my committee members and teachers,

Drs. Dave Bloomquist, Wayne Huber, Jim Kirby, and Dan Spangler, who

served so patiently and were responsible for many fruitful ideas.

My Ecuadorian research partner and friend, Eduardo Cervantes, proved

to be a source of much assistance, insight and camaraderie.

Honorable mention must be made of the tireless crew at the Coastal

Engineering Laboratory especially Vernon Sparkman and Chuck Broward for

their technical assistance. Helen Twedell of the Coastal Engineering

Archives also was very helpful.

Perhaps, most importantly for me is the great sense of honor to

which I have been imbued by my family. The result of love, patience,

encouragement and support shown by my beautiful wife and parents. Their

belief in me never faltered.

Finally, financial support for this work was derived from a research

grant extended by the U.S. Army Engineers, Waterways Experiment Station,

Contract No. DACW 39-87-P-1064. Particularly, technical and

administrative assistance and input provided by Allen Teeter is

gratefully acknowledged.







ABSTRACT. .. . . . xviii


1 INTRODUCTION. .. . . 1

1.1 Problem Significance . 1
1.2 Objective and Scope. . 4
1.3 Outline of Presentation. . 5


2.1 Introduction . . 8
2.2 Typical Concentration Profile. . 9
2.3 Problems Related to Defining The Bed 12
2.3.1 Bed Formation Concepts. . 13
2.3.2 Effective Stress. . ... 16
2.4 Fluid Mud. . . .. 20
2.4.1 Stationary Fluid Mud. . 22
2.4.2 Mobile Fluid Mud. . ... 27
2.5 Lutoclines . . ... 30


3.1 Introduction . .... 34
3.2 Mass Conservation Equation . .... 35
3.3 Diffusive Transport . .. 39
3.3.1 Turbulent Diffusion . .... 39
3.3.2 Gravitational Stabilization . 43
3.4 Settling . . ... 49
3.4.1 Free Settling . .... 50
3.4.2 Flocculation Settling . 52
3.4.3 Hindered Settling . 54

3.5 Vertical Bed Fluxes. .
3.5.1 Bed Erosion .
3.5.2 Deposition. .
3.6 Fluid Mud Entrainment. .
3.7 Horizontal Fluid Mud Transport .


4.1 Introduction . .
4.2 Flume Study. . .
4.2.1 Objectives. . .
4.2.2 Mud Characterization. .
4.2.3 Equipment, Facilities and Techniques.
4.2.4 Summary of Test Conditions. .
4.2.5 Results . .
4.2.6 Discussion. . .
4.3 Settling Column Tests. . .
4.3.1 Historical Approaches .
4.3.2 Concentration Profile Approach. .


5.1 Introduction . .
5.2 Settling . . .
5.2.1 Quiescent Settling. . .
5.2.2 Turbulence-Enhanced Settling. .
5.3 Wave Resuspension. . .
5.4 Lutocline Evolution in Severn Estuary. .
5.5 Fluid Mud Transport. . .
5.5.1 Wave Tank Fluid Mud Transport .
5.5.2 Avon River Fluid Mud Transport. .


6.1 Conclusions. . .
6.2 Recommendations. . .







. .147

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. .153

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. .159


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. .188

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. .103

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. .138
. .142


Table Page

2-1 Fluid Mud Definition by Density/Concentration 21

3-1 Summary of Coefficient Values for Turbulent Vertical
Diffusion of Momentum in Continuously Stratified Flow 46

A-I Wave Data (Period, Length, Height and MWS Elevation),
Run 1 . . . 159

A-2 Visual Bed Elevations (cm), Run 1 ... .159

A-3 Wave-Averaged Bed Pressures (kPa), Run I. ... .160

A-4 Dynamic Pressure Amplitudes (0.1 kPa), Run 1. .160

A-5 Sediment Bed Concentrations (g/1), Run 1. ... .160

A-6 Sediment Concentrations Station A (g/1), Run 1. .161

A-7 Sediment Concentrations Station B (g/1), Run 1. .161

A-8 Sediment Concentrations Station C (g/1), Run 1. .161

A-9 Sediment Concentrations Station D (g/1), Run 1. .161

A-10 Sediment Concentrations Station E (g/1), Run 1. .162

A-11 Wave Data (Period, Length, Height and MWS Elevation),
Run 2 . . . 162

A-12 Visual Bed Elevations (cm), Run 2 ... .163

A-13 Wave-Averaged Bed Pressures (kPa), Run 2. ... .164

A-14 Dynamic Pressure Amplitudes (0.1 kPa), Run 2. .165

A-15 Sediment Bed Concentrations (g/1), Run 2. ... .165

A-16 Sediment Concentrations Station A (g/1), Run 2. .166

A-17 Sediment Concentrations Station B (g/1), Run 2. .166


A-18 Sediment Concentrations Station C (g/1), Run 2. .. .167

A-19 Sediment Concentrations Station D (g/1), Run 2. .167

A-20 Sediment Concentrations Station E (g/1), Run 2. .. .168


Figure Page

2-1 Typical Instantaneous Concentration and Velocity
Profiles in High Concentration Estuarine Environments 10

2-2 Schematic Representation of Bed Formation Process 14

2-3 Bed Formation Process According to Imai (1981). 15

2-4 Definition Sketch of Bed Stress Terminology 17

2-5 Effective Stress Profiles in a Settling/Consolidation
Test (reprinted with permission from Been and Sills,
1986) . . . 19

2-6 Mud Dynamic Viscosity Variation with Concentration. 24

2-7 Bingham Yield Strength Variation with Concentration 27

2-8 Settling Velocity Variation with Concentration
Severn Estuary Mud (adapted from Mehta, 1986) 28

2-9 Vertical Settling Flux Variation with Concentration
(reprinted with permission from Ross et al., 1987). 29

2-10 Typical Suspended Concentration Profile Showing
Multiple Lutocline Stability Over 10 min. Period
(Kirby, 1986) . . ... 32

3-1 Diffusion Flux vs. Concentration Gradient 49

3-2 Ratio C/Co of Instantaneous to Initial Suspended Sediment
Concentration Versus Time for Kaolinite in Distilled
Water (after Mehta, 1973) . .... 61

3-3 Simplified Description of Density Stratified
Entrainment (after Narimousa and Fernando, 1987). 64

4-1 Grain Size Distribution of Hillsborough Bay Mud 78

4-2 Flume Configuration . .. 80


4-3 Example of Pressure Gage Calibration. . 83

4-4 Example of Wave Gage Calibration. . 84

4-5 Suspended Sediment Siphon Sampler . 85

4-6 Wave-Average Bed Pressures at Various Times for Run 1 87

4-7 Wave-Average Bed Pressures at Various Times for Run 2 88

4-8 Temporal Response of Effective Stress for Run 1 90

4-9 Temporal Response of Effective Stress for Run 2 90

4-10 Structural and Visual Bed Elevations for Run 1. 91

4-11 Structural and Visual Bed Elevations for Run 2. 91

4-12 Concentration Versus 1 Pa Effective Stress Elevation. 92

4-13 Bed Concentration Variation With Time . 94

4-14 Visual Bed Elevation Variation With Time for Run 1. 95

4-15 Visual Bed Elevation Variation With Time for Run 2. 95

4-16 Bed Dynamic Pressure Amplitudes With Time for Run 1 96

4-17 Bed Dynamic Pressure Amplitudes With Time for Run 2 96

4-18 Concentration Profiles at Station C for Run 1 98

4-19 Concentration Profiles at Station C for Run 2 98

4-20 Local Mean Settling Velocity as a Function of Time for
Bentonite Clay and Alum in Water (adapted from
Mclaughlin, 1958) . ... .105

4-21 Scale Drawing of Settling Column. . .107

4-22 Grid Index used in the Settling Velocity Calculation
Program . . .109

4-23 Settling Velocity Variation with Concentration of
Tampa Bay Mud . .. ..110

5-1 Settling Velocity and Flux Versus Concentration
for Tampa Bay Mud . . .115

5-2 Model Simulated vs. Measured Settling Column
Concentrations Initial Concentration, Co= 1 g/l .119

5-3 Model Simulated vs. Measured Settling Column
Concentrations Initial Concentration, Co= 2 g/l .119

5-4 Model Simulated vs. Measured Settling Column
Concentrations Initial Concentration, Co= 4 g/l .120

5-5 Model Simulated vs. Measured Settling Column
Concentrations Initial Concentration, Co= 5.5 g/l .120

5-6 Model Simulated vs. Measured Settling Column
Concentrations Initial Concentration, Co= 7 g/1 .121

5-7 Model Simulated vs. Measured Settling Column
Concentrations Initial Concentration, Co= 8 g/l .121

5-8 Model Simulated vs. Measured Settling Column
Concentrations Initial Concentration, Co= 12 g/l. .122

5-9 Model Simulated vs. Measured Settling Column
Concentrations Initial Concentration, Co= 17 g/l. .122

5-10 Conceptual Model of Concentration "Thinning" in Low
Concentration Flocculation Settling ... .124

5-11 Conceptual Model for Constant Settling in Moderate
Concentration Range of Flocculation Settling. .124

5-12 Simulated Field Settling of Parrett Estuary
Suspensions. . . ..127

5-13 Model Simulated Versus Measured Concentrations --
Run 1 . . . 129

5-14 Model Simulated Versus Measured Concentrations --
Run 2 . . . 129

5-15 Model Simulated and Measured Lutoclines -- Severn
Estuary . . . 133

5-16 Model Simulated and Measured (Kirby, 1986)
Concentration Profiles 0900 hrs . .134

5-17 Model Simulated and Measured (Kirby, 1986)
Concentration Profiles 1100 hrs . .134

5-18 Model Simulated and Measured (Kirby, 1986)
Concentration Profiles 1300 hrs . .135

5-19 Model Simulated and Measured (Kirby, 1986)
Concentration Profiles 1530 hrs . .135

5-20 Model Simulated and Measured (Kirby, 1986)
Concentration Profiles 1700 hrs . .136

5-21 Normalized Velocity Profiles -- Severn Estuary
(data from Kirby, 1986) . .. 137

5-22 Total Fluid Mud Transport in Five Minutes -- Run I. .139

5-23 Non-Dimensional Bed Shear Stress (tb) versus Wave
Steepness (H/Lo) (reprinted with permission from Dean,
1987) . . .. 140

5-24 Calculated Fluid Mud Velocity Profile -- Run .. .142

5-25 Measured Fluid Mud Concentration, Velocity and
Horizontal Flux -- Avon River (data from Kendrick and
Derbyshire, 1985) . . 144

5-26 Calculated and Measured Horizontal Fluid Mud
Velocities. . . .. 146



b Buoyancy jump across density interface

b Body force per unit mass tensor

C Sediment suspension concentration (mass/unit volume)

C, Concentration at upper fluid mud interface

Cb Concentration at mobile/stationary fluid mud interface

Cc Concentration at bed surface

Cd Drag coefficient for sphere fall velocity

Ceq Equilibrium concentration during deposition

Ch Interference settling velocity concentration

ChT Hindered settling (flux) concentration

Ci Concentration of class i for deposition

Cm Characteristic maximum concentration

Css Steady state concentration after deposition

CT Total concentration (sum of components)

Co Initial concentration for settling; deposition

C1 Cohesive (class) sediment concentration

C2 Non-cohesive (class) sediment concentration

C Time mean concentration

C' Instantaneous concentration component about mean

C' Non-dimensional concentration C/Cm


d Equivalent spherical diameter of sediment grain

D Molecular diffusivity

d50 Sediment grain size diameter of 50% greater than fraction

Eij Turbulent diffusivity components in i,j direction

Em Turbulent momentum diffusion rate (eddy viscosity)

E* Entrainment coefficient, ue/u*

f Darcy-Weisbach friction factor

Fb Vertical sediment bed flux (Fe+Fp)

Fd Vertical sediment flux from diffusion

Fe Vertical sediment flux from erosion

Fp Vertical sediment flux at the bed from deposition

Fpi Class i vertical sediment flux from deposition

Fs Vertical sediment flux from settling

g Acceleration of gravity

h Water depth

H Wave height

Hb Breaking wave height

Ho Deep water wave height

k Wave number (2rn/L)

K Turbulent mixing tensor

Kn Local neutral mixing rate

Ks Local mixing rate in presence of stratification

Kx,y,z Turbulent mixing components (cartesian)

Kx,y,z' Non-dimensional turbulent mixing components (cartesian)

k1 Flocculation settling velocity constant

k2 Hindered settling velocity constant


1 Mixing length scale of turbulence

L Lutocline layer; wave length

Lo Deep water wave length

m Mass flux of sediment across bed boundary

n1 Flocculation settling velocity constant

n2 Hindered settling velocity constant

P Pressure variable used in the horizontal momentum equation

P Relative Probability for deposition rate expression

Pi Relative Probability for deposition (class i)

Ph Hydrostatic pressure

Ppy Pore water pressure

P' Non-dimensional pressure P/yH

q Mass flux vector

R Reynolds' number of sediment grain (wsd/v)

Ri Gradient Richardson number

Ri* Bulk Richardson number (bh/ui)

Riu Richardson number based on average velocity

Rw Wave Reynolds number

RI Shear stress ratio, ty/to

Sc Turbulent Schmidt number

t Time variable

to Characteristic time scale

t' Non-dimensional time t/to

T Wave period

T Stress tensor

u Velocity component in x-direction


Uo Characteristic velocity scale

U Imposed velocity on the sheared turbid layer

Ue Entrainment rate

ub Maximum near-bed orbital velocity

Ue Entrainment rate (dh/dt)

Au velocity jump across stratified layer

u* Friction velocity (ot7/p)

u Time mean velocity

u' Non-dimensional velocity u/uo

u' Instantaneous velocity component about mean

v Velocity component in y-direction

w Velocity component in z-direction

Wg Sediment settling velocity

ws' Non-dimensional settling velocity

wsm Characteristic maximum settling velocity

wso Richardson-Zaki reference settling velocity

Wsol Stokes' settling velocity

wso2 Reference settling velocity for average floc size

x Longitudinal (horizontal) cartesian coordinate direction

x' Non-dimensional horizontal direction x/L

y Lateral cartesian coordinate direction

z Elevation variable (positive upwards)

z' Non-dimensional vertical direction

Zg Upper fluid mud interface elevation

Zb Mobile/stationary fluid mud interface

Zc Bingham plastic yield elevation

Zc Bed elevation

a Wave diffusivity constant

as '2/11

ay Yield strength calculation constant

a P Viscosity/concentration constant

a' Munk and Anderson constant

al,02 Erosion rate constants

a5 Viscosity ratio, p24/1

B Settling velocity constant

Oi Settling velocity constant for sediment class i

Pg Exponential diffusivity constant (mass diffusivity)

OH Holtzman constant

BMA Munk and Anderson constant

BOR Odd and Rodger constant

aRM Rossby and Montgomery constant

By Yield strength calculation constant

BP Viscosity/concentration constant

0' Munk and Anderson constant mass diffusivity

Bg Coefficient used in fluid mud calculations

6 Intermediate entrainment layer

6c Similarity variable (zc(t)/2/vit)

6 6c /ac s

6fm Mobile fluid mud thickness

8i Upper entrainment layer thickness

Ss Shear layer thickness; Similarity variable (z/2/vt)

8 6s/vas
Fluid shear rate (8u/az)

K von Karman constant (0.4)


p Density of water

Pb Bulk density of suspension

Po Fluid reference density for stratification

Ps Granular density of dry sediment

Pw Density of suspension fluid (water)

p' Non-dimensional density p/pm

PW', Dynamic viscosity of suspension fluid (water)

Pm Dynamic viscosity of mud suspension

Vm Kinematic viscosity of mud suspension (pm/p)

v' Non-dimensional kinematic viscosity

Y Odd and Rodger peak gradient Richardson number

c Munk and Anderson constant mass diffusivity

Eo Erosion rate constant

o' Bed effective stress

a Total stress; wave frequency (2nr/T)

Tb Applied (time-mean) bed shear stress

tbm Critical bed shear stress for partial deposition

tcd Critical bed shear stress for total deposition

to Bed shear stress

Ts Bed shear strength for erosion

xZ, Shear stress component acting in x-direction on z-face

"ty Yield strength of bed deposits

X Log average of sediment concentration

V Vector operator


Abstract of Dissertation Presented to the Graduate School
of the University of Florida in Partial Fulfillment of the
Requirements for the Degree of Doctor of Philosophy



Mark Allen Ross

August 1988

Chairman: Ashish J. Mehta
Major Department: Civil Engineering

Fine sediment suspension concentrations in estuaries vary with depth

depending on sediment settling and mixing processes, which are in turn

dependent on the turbulent flow field and the type of sediment. Two

important phenomena, fluid mud and lutoclines, are characteristic of high

concentration suspensions. Understanding the physical significance of

these phenomena is of paramount importance to quantifying the mixing

process and the rate of material transport advected with the prevailing

currents. This research investigated the physical characteristics

(vertical structure) of estuarine fine sediment suspension profiles

within a comprehensive descriptive framework. Suspension related

mechanisms of erosion, entrainment, diffusion (in the presence of

buoyancy stabilization), advection, settling and deposition were examined

in this context. A vertical mass transport model developed from

functional relationships between the above processes was used to explain

some of the important physical characteristics.

Lutoclines, sharp steps (gradients) in the concentration profile,

are regions where the local mixing rate is minimal. The mechanisms for

their formation have been shown to be the non-linear relationships

between 1) vertical diffusion and concentration gradient and 2) vertical


settling and concentration. The effect of sediment settling is to

further stabilize the lutocline layer thereby making it much more

persistent in high energy environments than other pychnoclines (e.g.,

haloclines). Application of the vertical transport model to data from

settling column tests, wave flume resuspension tests, and estuarine field

investigations provided reasonable predictive agreement for lutocline


Fluid mud, a near-bed, high concentration layer with negligible

structural integrity, results from high bed erosion or fluidization rates

relative to upward entrainment fluxes and from rapid deposition.

Sensitive pore pressure and total pressure measurements made in a

laboratory flume have been used to demonstrate that waves, for example,

provide one mechanism for fluid mud formation by rapid destruction of

effective stress in the sediment bed. The upper interface of the fluid

mud layer, by definition a lutocline, represents a local maximum in net

downward settling flux (i.e., maximum settling minus diffusive flux).

The fluid mud layer thus forms (and grows) from rapid deposition whenever

the depositional flux at the bed exceeds the rate at which the sediment

can develop effective stress (usually very low).

Fluid mud has been shown to be either horizontally mobile or

stationary depending on the depth of horizontal momentum diffusion

vertically downward into the high concentration layer. Fluid mud tends

to occur over a density range between 1.01 1.1 g/cm3 but due to the

dependence on hydrodynamic action near the bed a precise definition

cannot be made on the basis of density alone.


1.1 Problem Significance

Fine-grained, cohesive sediment is transported in suspension from

fluvial and marine sources to depositional environments including

navigation channels and harbors. This sediment affects water quality by

transport of sorbed nutrients (or pollutants) and light penetration

(Hayter, 1983). Shoaling is often one other critical issue. In the

continental United States alone, the cost of maintenance dredging of

coastal waterways, including estuarine ports and harbors, is

approximately one-half billion dollars per year (Krone, 1987). Estimates

of contaminant removal or dredging requirements are dependent upon a

knowledge of the rates of horizontal transport of the suspended material

over periods ranging from days to years. The accuracy of predictions,

typically via numerical solutions of the sediment mass transport

equation, is therefore strongly contingent upon an understanding of the

structure of the vertical profile of sediment concentration and

interaction with the turbulent flow field.

Present day modeling of cohesive sediment transport is limited by

knowledge of the fundamental transport processes of erosion, entrainment,

settling, deposition and consolidation of these sediments. In

particular, the dynamics of estuaries with relatively high concentration

suspensions typical of macro-tidal (tidal range > 4 m) environments are


poorly understood (Parker, 1987). In this context engineers and other

scientists are beginning to deal with the important question of fluid

mud, loosely defined as a high concentration slurry transported in the

form of a relatively thin suspension layer near the bed by the prevailing

currents. At present, there are difficulties associated with measuring

the slurry concentration and transport velocity. The result is that

large errors often occur in calculating the associated horizontal flux of

sediment transport over the water column.

Fluid muds also occur in meso- (2-4 m) and micro-tidal (< 2 m)

estuaries and along the open coasts where waves play a more important

role than in macro-tidal environments. Wells and Kemp (1986) observed

that waves traveling over nearshore mud shoals principally acted as an

agent for softening and fluidizing the muddy bed. Maa and Mehta (1987)

made similar observations in laboratory flume tests. In nearshore areas

waves can thus significantly assist currents in transporting fluidized

material to sites prone to sedimentation. Consequently, in micro-tidal

waters the generation and transport of fluid mud is far more episodic

than under macro-tidal conditions.

Understanding the dynamics of fluid mud is central to the issue of

understanding the response of the vertical concentration profile to

hydrodynamic forcing by currents and waves. Unlike the boundary of beds

composed of cohesionless material (e.g., sandy beds), the cohesive bed

boundary is often poorly defined as it is not evident, e.g., from echo

sounder data, at what depth the near-bed suspension ends and the bed

begins. Parker (1986) noted ambiguities when lead lines, echo sounders

or nuclear transmission or backscatter gauges are used to identify the


bed. In fact, Ross et- al. (1987) noted that due to the dynamic nature of

the cohesive bed boundary which responds significantly to hydrodynamic

forcing, the density of the suspension by itself cannot be used either to

identify the cohesive bed boundary or the fluid mud layer which occurs

immediately above this boundary. An understanding of the interaction

between the concentration (or density) profile with the flow field is

critically important.

Kirby (1986) recently published a summary of extensive field

observations made in the Severn estuary, a macro-tidal estuary (maximum

tidal range 14.8 m) on the west coast of England. Large mass transport

rates via fluid mud generation regularly occur in this estuary. The

dynamic interaction between the concentration field and flow field are

further complicated by the extremely high concentrations. Surface

concentrations reach 1000 mg/l easily, which may be compared with 20

mg/l in Florida's coastal waters (Mehta et al., 1984). A significant

observation in the Severn was the generation of rather sharp gradients in

concentration termed lutoclines, which rise and fall through the water

column depending upon the flow condition. According to Kirby (1986),

lutoclines, which are analogous to other types of pycnoclines, e.g.,

haloclines, seem to occur where the suspension concentration exceeds *

500 mg/1. However, they differ from other pycnoclines by the added

process of sediment settling. Sediment settling further supports

stabilization and resulting high density gradients. For this reason

lutoclines are much more persistent than for example haloclines in high

energy environments. An example of a lutocline is the upper level of

fluid mud within which concentrations can typically exceed 10,000 mg/l.


Above this level, lutoclines often show up as multiple "steps," which

represent local complex imbalances between diffusive and settling fluxes.

Kirby (1986) observed that lutoclines often are not simulated properly by

numerical models with resulting errors in the estimates for the rates of

mass transport.

The aforementioned issues illustrate the strong need to examine the

entire question of the vertical structure of concentration and its

interaction with the flow field within a comprehensive framework. An

attempt is made in this thesis to approach the problem via analysis of

laboratory and field measurements within a descriptive framework for the

vertical concentration structure. New definitions are proposed and the

dynamics of the concentration profile are adduced through relatively

simple mathematical models which are verified by laboratory and field

data. The objectives and scope are accordingly as follows.

1.2 Objective and Scope

The objectives of this study were to

1. Define the physical characteristics of fine sediment
suspension profiles in estuaries including lutoclines,
fluid mud, and the cohesive bed within a comprehensive
descriptive framework.

2. Determine the important physical mechanisms and processes
which influence these characteristics.

3. Develop simple but useful qualitative and quantitative
descriptions for these processes which could be used in a
predictive capacity to model suspensions in the prototype

To meet these objectives the scope of this research was as follows:

1. Laboratory tests were conducted, using natural estuarine


sediment, to measure the parameters important to cohesive
bed and suspension profile definitions.

2. For the simple vertical structure model development, only
vertical transport fluxes were considered. Analysis of
turbulent diffusion was based on classical mixing length
approximations and gradient Richardson number bouyancy
stabilization relationships. Sediment settling velocity
expressions were concentration dependent.

3 Horizontal transport in the fluid mud layer was calculated
from consideration of momentum diffusion resulting from
applied interfacial shear stress.

4. Verification of model applicability was limited to
comparisons with selected field and laboratory data (e.g.,
time series concentration profiles).

1.3 Outline of Presentation

The study is presented in the following order. Chapter 2 can be

regarded as a description and definition chapter. Specific justification

is presented for delineating processes influencing vertical suspended

sediment structure. Physically based, qualitative definitions are given

for lutoclines, stationary and mobile fluid mud layers, and bed

elevation. This chapter also describes many of the complexities

associated with defining the cohesive bed from theoretical and applied


Chapter 3 presents the theoretical development of the vertical

transport and momentum diffusion models. For the transport model, the

advection-diffusion equation is given and the individual terms are

discussed. Entrainment, diffusion, settling and bed fluxes are

addressed. For the fluid mud momentum diffusion model, theoretical

formulations are presented with assumptions concerning theological and

temporal responses.


Chapter 4 presents the objectives, procedures and results of

laboratory experiments with three specific themes: cohesive bed dynamics

associated with wave-induced bed fluidization and delineation of the

cohesive bed boundary; wave resuspension with emphasis on the evolution

of the suspension profile with time; and settling velocity determination.

The natural estuarine sediment used in the tests is characterized in

Section 4.2.2. Historical approaches toward settling velocity

determinations are discussed in the context of strengths and weaknesses.

Section 4.3.2 presents an improved procedure for determining cohesive

sediment settling velocity concentration relationships using settling


The application of the vertical transport model is presented in

Chapter 5. An early attempt was made to progressively verify the

individual routines in the model, before concurrent simulation. Thus,

for example, the settling routine was first tested by reproducing

quiescent (column) settling results (a nearly pure settling condition,

see, for example, Yong and Elmonayeri, 1984, for diffusion in quiescent

settling). Next, the diffusion, erosion and deposition routines were

added and tested. Predictions of lutocline formations in field

conditions together with fluid mud layer development in a wave-tank

illustrating the model's ability to handle sediment fronts (i.e., sharp

concentration gradients) are also shown in Chapter 5.

The fluid mud horizontal transport model results also are included

in Chapter 5. Steady and unsteady simulations of wave-tank data

(presented in Chapter 4) and field data (published by Kendrick and

Derbyshire, 1985) are shown.


Conclusions, recommendations for future research and miscellaneous

closing comments are given in Chapter 6.

Appendix A presents a dimensional analysis to determine the

important terms in the transport equation. Appendix B contains the

tabulated laboratory data taken during wave resuspension tests. Appendix

C is a printout of the l-D vertical transport model developed for this study.


2.1 Introduction

Suspended sediment concentration in estuaries varies greatly with

depth, the highest concentrations being usually found nearest the bed.

Simply stated, this variance is because gravitational flux (associated

with settling) counteracts mixing and prevents the sediment from becoming

uniformly mixed as is the case with neutrally buoyant or dissolved

constituents. In an equilibrium profile (profile not changing with time)

the vertical flux associated with settling is everywhere equal to the

vertical flux associated with upward diffusion (typically turbulent

mixing). For sediment with a constant settling velocity and mixing based

on a Prandtl/von Karman mixing length approximation, analytical solutions

for the concentration profile follow the classical works of O'Brien

(1933) and Rouse (1937). In a fully developed turbulent flow in the

absence of significant density gradients the mixing rate, which is

directly proportional to the eddy scale of turbulence, is smallest near

the bed and increases upward reaching a maximum approximately at mid-

depth (Schlichting, 1979). However, sediment in suspension can greatly

increase the bulk density of the water. If high concentration (density)

gradients develop, turbulent mixing will be greatly damped locally

(Fischer et al., 1979). This has been well documented for stratified

flows associated with dissolved salt- and temperature-induced density


gradients (Turner, 1973), but has been generally overlooked by classical

solutions of vertical sediment transport (Rouse, 1937; Raudkivi, 1967).

This chapter presents the physically based definitions for the

vertical structure (or vertical characteristics) of fine-grained

suspended sediment profiles. Much of the terminology and descriptions

used have been liberally applied in broad contexts in previous related

and unrelated studies. Most were disparate in their objective. The

following pages will help clarify the usage.

2.2 Typical Concentration Profile

Figure 2.1 shows a typical instantaneous concentration profile as

might be observed in a high sediment load environment such as the Thames

River (UK), San Francisco Bay or the Severn Estuary. While the values

are assumed, they are representative of those commonly reported in the

literature (Parker and Kirby, 1979).

It is noted that there is a 4-5 order of magnitude range in

concentration from water surface to bed surface. While most sediment

transport models focus primarily on calibrations of the upper water

column concentrations, the significance of neglecting the near bed layers

should be obvious, but will be shown in detail.

The largest layer is the mobile suspension layer which extends down

to reference level, Za. This is the layer that is most often turbulent.

It is also generally, dominated by pressure gradient driven flow

associated with water surface elevation gradients resulting from tides

and freshwater discharge. Concentrations in the upper mobile suspension


10 103 104 105


0.50 0.75

VELOCITY, u(m/s)




Figure 2-1. Typical Instantaneous Concentration and Velocity Profiles
in High Concentration Estuarine Environments

o r








layer are usually 1-1000 mg/l but in rare cases exceed 10,000 mg/l (in

the lowest portions) during extreme tides or storm conditions (Parker and

Kirby, 1979).

At various levels in the mobile suspension layer there can exist

sharp increases in concentration which result from and further support

local minima in mixing and upward vertical diffusion. These are termed

lutocline layers which are one form of pycnoclines (regions of sharp

density gradients). There can be multiple lutocline layers but more than

2-3 is rare. Such multiple layering in salinity or thermal structure is

called finestructure (Posmentier, 1977).

Below Za there is a sharp increase in concentration above 10,000

mg/1 to 100-300 g/l. This is the so-called "fluid mud" layer defined in

Section 2.4. Thus, Za represents a lutocline between upper column mobile

suspensions and near-bed fluid mud. This is often mistaken as the bed on

echo sounder records (Kirby, 1986). Depending on the theological

properties of the mud, the magnitude and duration of the applied

interfacial (lutocline) shear stress, and/or the internal pressure

gradients, a portion of this fluid mud layer is mobilized to flow in a

direction with the applied forcess. The interface between the mobile

and stationary mud suspensions is labeled reference level Zb. The

symbol, Zb, will not necessarily be identifiable from concentration

profiles but instead must be identified from accurate measurements of the

velocity profile. Below the fluid mud layer at reference level Zc there

exists a definable sediment interface below which the sediment exhibits

bed properties based on classical soil mechanical definitions. This is

the cohesive bed elevation, above which only suspension occurs (discussed


in Section 2.3). Strictly speaking, the stationary fluid mud layer (Zb-

Zc) may not necessarily behave as a fluid (i.e., not supporting shear

stresses), but since it fits the general definition for fluid mud (i.e.,

near-bed, high-concentration layer) the terminology is nevertheless


A typical velocity profile is also shown in Figure 2-1 for

reference. It is shown to be of almost logarithmic form in the mobile

suspension layer--indicative of turbulent flow. Near the fluid mud layer

turbulence is dampened out and there is a transition layer which,

proceeding down with depth, gives way to a shear flow viscous layer.

This is analogous to stratified flows of salts (Yih, 1980; Narimousa and

Fernando, 1987) and is described further in Chapter 3.

2.3 Problems Related to Defining the Bed

When trying to determine the bed elevation, Zc, to do so on the

basis of concentration only is imprecise. As pointed out by Sills and

Elder (1986) and in this report, bed properties (i.e., development of

effective stess as defined in Section 2.3.2) can exist in concentrations

as low as 70-80 g/1l, depending primarily on the fluid suspension and bed

dynamics (stress, strain and strain rate) at any particular time. Thus,

under field conditions, a precise identification of the cohesive bed

interface would not be possible without dynamical data (e.g.,

measurements to determine effective stress). Bed definition is also

dependent on previous formation conditions, wave and current actions and

consolidation properties of the particular sediment. These phenomena

provide justification for a brief discussion of bed formation and


consolidation concepts followed by a subsection (2.3.2) on the concept of

bed definition related to effective stresses.

2.3.1 Bed Formation Concepts

To characterize the process of bed formation in a laboratory or

field setting it is important to distinguish the mode of deposition.

According to Parchure and Mehta (1985), in the laboratory, bed formation

can be in the form of a "placed" or "deposited" bed. Placed beds are

those developing from high concentration slurries. Deposited beds result

from lower concentration, particle by particle deposition. Placed beds,

therefore, are more uniform vertically, whereas deposited beds are non-

uniform and dewater relatively rapidly. The specific character of each

bed type is most pronounced earliest after formation, decaying with time

until the properties are nearly indistinguishable. Because of the time

scales involved, placed beds are probably more typical of laboratory

conditions; however, rapid fluid mud deposition in an estuary would have

similar characteristics.

In the field, an alternative to considering the bed based on

depositional mode is to examine in detail the physics of mud deposition

and bed formation. A schematic representation of the bed formation

process is shown below (Figure 2-2).

There are basically two mechanisms responsible for bed formation:

sedimentation (deposition) and consolidation. Sedimentation can be

defined as the process by which particles or masses of particles leave

suspension and settle onto the bed under gravity. Consolidation in a

fully saturated environment results from particle framework (mineral


Increasing Rate of
Soft-Rapidly Concentration Vertical
Consolidating Movement


Figure 2-2. Schematic Representation of Bed Formation Process

skeleton) deformation under applied stress. The applied mechanical

forces can be either due to net negative buoyancy (self-weight) or

imposed overburden (surcharge) loading.

Imai (1981) gave a description and graphical model of the bed

formation processes. Figure 2-3 shows this description.

The flocculation stage in Figure 2-3 actually includes the complex

process of particle destabilization by doublelayer suppression in the

presence of available cations and subsequent aggregation by interparticle

collision and cohesion. The floc formation process takes place under

settling conditions as pointed out by Krone (1962). The settling zone

shown in Figure 2-3 would be more appropriately labeled hindered (or

zone) settling. No further discussion of settling will be given here as

the settling process is discussed further in Chapter 3. Between times tl

SF I Settling: ..t. Soil Formation Line
> Flocculation.' Zone .: ,

: /Consolidation Zone

o t t2

Figure 2-3. Bed Formation Process According to Imai (1981)

and t2, sediment flocs settle to form a soft bed. The bed is continually

built up by continuous deposition of these flocs but simultaneously

undergoes dewatering and consolidation. During this time, bed properties

begin to change with depth due to particle rearrangement and larger floc

breakdown (Krone, 1962). After the settling stage, consolidation

continues and the bed slowly begins to "harden" as depth-variation in bed

properties (i.e., density, effective stresses, etc.) become more

pronounced. This self-weight consolidation approaches a steady state

condition exponentially.

Since it is not within the scope of this study to discuss the

details of consolidation, it will suffice to conclude this section by

stating that a vast amount of geotechnical literature on


sedimentation/consolidation theories is available. The pioneering work

by Terzaghi (1923) using one-dimensional finite strain theory now has

evolved into complex multi-dimensional, non-linear finite strain

theories. The reader is directed to the paper by Schiffman et al.

(1986), which presents a noteworthy historical, theoretical, and applied

account of one-dimensional sedimentation and consolidation.

2.3.2 Effective Stress

Given that the porous medium (the cohesive bed) is a two-phase

system consisting of a deformable mineral skeleton filled with an

incompressible liquid (water), the effective stress, o', is defined as

the difference between the total stress, a, and the pore water pressure,

Pp at any point:

o' = o Ppw (2.1)

Empirically, it is found to be the controlling parameter in determining

soil strain, deformation and strength (Schiffman et al., 1986).

Classically, one type of soil failure is defined as a "quick" condition

in which the effective stress tends toward zero (Sowers, 1976).

Another important parameter is excess pore pressure, Au. This is

the difference between actual pore water pressure, Ppw (e.g., as measured

by a manometer), and hydrostatic pressure, Ph. Under dynamic conditions,

if the sum of excess pore pressure, Au, and hydrostatic pressure, Ph,

approaches the total stress, a, liquefaction occurs (Perloff and Baron,



Figure 2-4 is a definition sketch for these terms.

Water Surf;

Mobile Suspension
0 _\Fluid Mud Surf-
Fluid Mud
> _Bed Surft


Figure 2-4.

Definition Sketch of Bed Stress Terminology

In a non-fluidized sandy (or any large porosity) sediment bed the

effective stress everywhere is non-zero. The total pressure is the

integral of the density profile over depth (acted on by gravity) and the

pore pressure is everywhere hydrostatic. For finer sediments which are

much less permeable, pore pressures easily increase to above hydrostatic.

In the upper bed where the pore water pressure is equal to the total

pressure, the sediment is in suspension and the water bears the weight of

the sediment (increased bouyancy through higher bulk densities). When


Au + Ph 4 a


the pore pressure drops below the total vertical stress, there is

particle interaction. Thus, a weak structure forms that is able to

support some of the weight of the sediment. Therefore, the development

of effective stress provides a fundamental distinction between suspension

and structural bed, i.e.,

a' = 0 ; Pp = a Suspension

a' > 0 ; Ppw < a Bed (2.3).

The elevation of the (structured) bed, Zc in Figure 2-1, therefore,

should be based on the development of effective stress below this

elevation. Within the bed, the interaction between sediment flocs

provides a resistance to erosion due to frictional and electrochemical

bonding. A reduction in effective stress, therefore, leads also to the

reduction in the yield strength and the critical shear stress for

erosion. No effective stress means no inter-aggregate contact or

friction. This important distinction was pointed out by Sills and Elder


Been (1980) and Been and Sills (1981) made extensive laboratory

measurements of the development of effective stress in quiescent

settling/consolidation of fine sediments. A representative plot of

measured effective stress profile is shown in Figure 2-5.

In their experiments on Combwich mud using different initial

concentrations, no unique concentration was found at which effective

stress developed. The concentration range over which structural

development occurred was between 80 and 220 g/l depending on the initial


1200- 1200-

Pressure plotted above

E 800- 800 \ o Total stress
E E Pore pressure

400- 400-

0 ------I 0-
1-0 11 12 0-2 04 0-6 0-8 1-0 1-2
Density Mg/mr Pressure kN/m2
Initial density 1-09 Mg/mn A hour profile

Figure 2-5. Effective Stress Profiles in a Settling/Consolidation Test
(reprinted with permission from Been and Sills, 1986)

concentration of the slurry. One significant observation was that

effective stress existed always in concentrations greater than 220 g/l.

This observation seems to imply that structural phase development is

dependent on sedimentation rate especially in low concentration quiescent

conditions. It is noted in the following sections that structural phase

development is also dependent on hydrodynamic agitation. This dependence

is demonstrated in laboratory tests of wave erosion as described in

Chapters 4 and 5.

2.4 Fluid Mud

As stated in Chapter 1, fluid mud is defined as a near-bed, high

density, cohesive sediment suspension layer (Ross et al., 1987). In


areas with large bed slopes, fluid mud loosened by currents or waves can

flow down the slope by gravitational forces similar to mudslides and

debris flow on hillsides (Odd and Rodger, 1986). For this reason,

navigation channels and basins are especially vulnerable to this type of


Many investigators have identified fluid mud in terms of a range of

bulk density (or concentration) of the sediment-fluid mixture, as noted

in Table 2-1. It should be pointed out that these investigations were,

in general, disparate in terms of their aims, dealing with field

observations or laboratory tests. Nevertheless, there seems to be some

agreement amongst the proposed densities initially suggesting perhaps an

approximate range of 1.03 to 1.20 g/cm3 (concentration range of 10 to 320


To provide a quantitative definition for fluid mud based on a

discrete concentration range is not possible because, as pointed out in

the previous section, the effect is not simply dependent on concentration

but on the flow conditions and sediment settling properties. Therefore,

the values given in Table 2-1, without qualifying the particular flow

conditions and sediment settling behavior under which the ranges apply,

are not amenable to developing a general definition applicable in all

cases. All that can be deduced from the tabulated data is that fluid mud

seems to occur within a rather wide concentration range of between 3 and

500 g/l (two orders of magnitude).

Fluid mud can form during rapid erosion or deposition. During

erosion, if initially the erosion rate greatly exceeds the turbulent

entrainment rate, i.e., the rate at which sediment is mixed by turbulence


into the upper column mobile suspension layer, the near-bed high

concentration further dampens turbulent mixing and the near-bed

Table 2-1. Fluid Mud Definition by Density/Concentration

aConversion between density and concentration
density of 2.65 g/cm3.

based on assumed sediment

suspension can be stabilized as a stratified flow. This effect is often

the case in wave erosion (Maa and Mehta, 1987). This is discussed in

Chapter 3 and later shown in laboratory wave resuspension tests

documented in Chapter 4. During deposition, if the sediment deposition

flux exceeds the rate of pore fluid transport upward (dewatering rate of

the suspension), dense near-bed suspensions are formed that grow upward

and only slowly consolidate (see Section 2.3.1).

As shown in Figure 2.1, fluid mud can occur as a mobile or

stationary suspension. This distinction was first made by Parker and

Kirby (1977). Both conditions are discussed in the following sections.

Density/Conc. Range
Bulk Degsity Concentration
(g/cm ) x 10 (mg/1)

Inglis and Allen (1957) 1.03 1.30 10 480

Krone (1962) 1.01 1.11a 10 170

Wells (1983) 1.03 1.30 50 480a

Nichols (1985) 1.003 1.20 3 320

Kendrick and Derbyshire (1985) 1.12 1.25a 200 400

2.4.1 Stationary Fluid Mud

Within the fluid mud layer there are typically two distinct regions

separated by a level below which no horizontal motion takes place. In

the definition sketch (Figure 2-1) this was elevation Zb. For instance,

this elevation might be considered to be the applicable elevation of the

bottom boundary condition for a horizontal transport model. However,

this level is quite sensitive to the theological response of the mud from

imposed stress (e.g., the lower extent of the vertical momentum diffusion

resulting from an applied horizontal shear stress at the upper fluid mud

interface). For the purpose of describing why this layer exists and how

it is differentiated from the mobile layer above, several simple

arguments are presented here and are more formally posed in Chapter 3

(Section 3.7).

It is possible to estimate the stationary layer elevation, Zb, by

making several simplifying assumptions. As a first approach, analogy can

be made between flow in the fluid mud layer and unsteady couette flow

development beneath an infinite plate moving with a constant velocity

after being started from rest. A shear stress results on the upper fluid

mud interface, elevation Za in the definition sketch (Figure 2-1),

because of an imposed velocity in the upper column (mobile suspension).

Momentum diffusion then occurs over a finite thickness, Sfm, in the fluid

mud layer. For a constant kinematic viscosity, vm of the mud, the

temporal response (for relative time t after imposing the shear stress)

of the mobile fluid mud layer thickness, 6fm, is

6fm = 86 (.t


where $g is a constant (Eskinazi, 1968). However, this leads to

calculations of layer thicknesses of many meters after periods of

minutes--unreasonably large values. Also, the viscosity of these high

concentration layers is not constant but concentration (and thus depth)

dependent (Krone,1962).

Another approach is to consider the slurry to have a concentration

dependent dynamic viscosity, pm(C). The density of the bulk suspension,

Pb, is given by a simple linear relation to the (mass/unit volume)

concentration, C, as

Pb Pw + C (1 -) (2.5)

where pw is the density of the suspending fluid (water) and ps is the

granular density of sediment (typically 2.65 g/cm3).

A summary of empirical relationships for dynamic viscosity variation

with concentration is shown in Figure 2-6. The trend, in the fluid mud

range (10 C 200 g/1), seems to be of an exponential or polynomial

form: i.e.,

Pm = Pw e (2.6a)


PM = w(l1 + Op C ) (2.6b)

where pm is the dynamic viscosity of mud suspension, p, is the dynamic

80 Krone(1963)
60 -

=40- j
>/ e
S20 Engelund-Zhaohul(1984)
C 0/ / y ^ (Bentonlle)

0) 10- Delft Hyd. Lab (1985)
S 8
U. 6 -

4- Engelund-Zhaohul(1984)
rr 2^

0 100 200 300 400 500


Figure 2-6. Mud Dynamic Viscosity Variation with Concentration

viscosity of clear (free from suspension) water, C is the concentration

of suspended sediment, a, a1 and P0 are empirical coefficients.

Engelund and Zhaohui (1984) proposed a relationship of the form of Eqn.

(2.6b) for kaolinite suspensions. They found 5P/a =- 0.206 and a= P 1.68

for kaolinite concentration (in percent) and fresh water (p, .001 N-

s/m2). Equation 2.6b represents a truncated approximation of a power

series expansion of pm(C). For a more general form for pm(C), the reader

is directed to the discussion by Krone (1963).

However, it must be pointed out that data published by Krone (1963)

showed that multiple values for Rp, ap and aP are possible for a

particular sediment, depending on the shear rate and degree of


aggregation. Therefore, caution must be advised concerning the validity

of Eqn. 2.6 for more detailed application.

By considering the mobile fluid mud as depth varying viscous

Rayleigh flow (Stokes' first problem with variable viscosity; see

Schlichting, 1979), a numerical solution of the flow and boundary layer

thickness can be obtained. This is a non-steady state approach to

determining the horizontal transport layer detailed in Section 3.7.

An alternative approach to determining the mobile/stationary

interface is considering the non-Newtonian theological properties of high

concentration suspensions. Past research has indicated that

concentrations in the fluid mud range behave as Bingham plastics or

pseudoplastics (Krone, 1963; Kirby and Parker, 1977; Faas, 1981; 1987;

Nichols, 1985). Over short (tidal) time periods the designation of

effective yield strengths may be appropriate. In this case, the data

seem to suggest a concentration power law relationship. Figure 2-7 shows

a very approximate linear (on log-log paper) relationship between yield

strength, ty, and concentration through the data sets shown. The

expression to relate this functional dependence is of the form,

ty = Oy C (2.7)

where Py and ay are empirical constants. The data in Figure 2-7 suggest

that By = 8.7 x 10-7 Pa (1 Pa = 1 N/m2) and ay = 2.55. With this

assumption it is possible to estimate the lower penetration distance

(lower extent of horizontal motion), the mobile/stationary fluid mud

layer interface, Zb, based solely on equating the applied bed shear to

Q San Francisco Bay Mud
Wilmington Mud
Brunswick Mud
2 Grundlte
1 0 Kaollnlte Suspension


E o

z a y

U ) yR =p y C ay

10' 2
1 /

1d0-2 -

10 "

/ /

10 1024


Figure 2-7. Bingham Yield Strength Variation with Concentration


the level of equal yield strength. However, field observations of fluid

mud flows (Kendrick and Derbyshire, 1985) do not seem to support this

approach (see Section 5.5). This is because flow occurs when the applied

shear stress is less than the reported shear strength (Figure 2-7). This

suggests that the behavior is more pseudoplastic than Bingham. Further

discussion of possible means of determining the elevation, Zb, is

described in Section 3.7.

2.4.2 Mobile Fluid Mud

The mobile fluid mud layer as described in Section 2.2 is that part

of the fluid mud layer which is advected along with the mobile suspension

layer current. It may also be gravitational slump flow along a sloping

bed (Kendrick and Derbyshire, 1985). The elevation, Za, (Figure 2-1)

which defines the upper bound of the layer represents a local maximum in

net downward vertical flux.

The settling velocity of cohesive sediments varies with

concentration in suspension, ws(C). Initially constant, the velocity

rises with increasing concentration (due to flocculation) to a level

where it becomes constant again then rapidly drops. An example of the

settling characteristic of a natural estuarine sediment is shown in

Figure 2-8. The point beyond which no further increase in settling

velocity occurs has been termed "hindered settling" (Owen, 1970; Imai,

1980; Teeter, 1986a). For purposes that will become clear with the

following arguments, it is important to distinguish hindered settling

velocity from hindered settling flux. The details of free, flocculation

and hindered settling velocity are discussed in Chapter 3 (Section 3.4).

100.0 1
kj = 0.513
E n1 = 1.29
10.0- W n Wso= 2.6 mm/s
k2 =0008
So Ws= kiC k g = 0.008
n n 2 = 4.65

1.0- *r
o *

> 0.1- 0y

W= W3o (1-k2 C)n 2
0.01 -
u, 4

0.001 -
0.01 0.10 1.0 10.0 100.0


Figure 2-8. Settling Velocity Variation with Concentration
Severn Estuary Mud (adapted from Mehta, 1986)

The vertical flux of sediment (mass per unit area per unit time)

from settling, F5, is the product of the local settling velocity and

concentration as

Fs(C) = ws(C) C (2.8)

For the data of Figure 2-8 (source: Thorn, 1981) the vertical flux, Fs,

is plotted against concentration, C, in Figure 2-9 below.

From Figures 2-8 and 2-9 it is observed that the peak flux occurs at

a much higher concentration (i.e., 20 g/1) than that at which the peak

settling velocity occurs (i.e., 3 g/1). This is due to settling

velocity being either constant or only slightly decreasing from the

60- 2
F0 m= 40 g/m-s

CA= 2x104 mg/l


D) I


20- -
L. / I

103 10 4 CA 105


Figure 2-9. Vertical Settling Flux Variation with Concentration
(reprinted with permission from Ross et al., 1987)

maximum (' 2-3 g/1) over a wide concentration band (2-10 g/1). The peak

settling flux (i.e., 20 g/1) represents a more reasonable definition for

hindered settling than that based on the peak settling velocity. Beyond

this point, the actual vertical mass flux from settling diminishes

rapidly with increasing concentration.

The upper elevation, Za, of the fluid mud layer under settling

conditions therefore occurs at the "hindered" (defined on the basis of

flux) concentration. A discrete interface forms because the sediment

accumulates at this level because the flux is increasing above and

decreasing below this interface.

After all the sediment in the upper suspension layer has settled

onto the fluid mud layer, the interface settles according to the


interfacial settling region (shown in Figure 2-3). When the flow in the

upper suspension layer is turbulent, diffusion and entrainment at the

interface reduce the overall downward vertical flux and the interfacial

concentration, Ca, drops from that given by pure settling conditions.

Thus, C, has a maximum value given by the hindered (flux) concentration.

In the presence of mixing, the mobile fluid mud layer, 6fm = Zb Za,

does not necessarily become thicker (by becoming more diffuse). Due to

the sharp density gradients resulting from the high suspension

concentrations, turbulent mass and momentum diffusion across the fluid

mud layer is greatly damped. This results in a stable stratification,

often termed buoyancy or gravitational stabilization (Fischer et al.,

1979). In this case, upward entrainment, which is dependent on the

degree of stratification and relative turbulent intensity (Yih, 1980)

becomes the dominant mixing mechanism. Stratification development is

discussed in greater detail in the next section. Mixing in the presence

of gravitational stabilization is discussed in Section 3.3.2.

2.5 Lutoclines

Lutoclines are defined as pronounced "steps" in the vertical

concentration profile resulting from complex mixing-settling processes.

The upper fluid mud interface, by this definition, is also a lutocline

(shown as L1 in Figure 2-1). However, lutoclines can as well occur in

the mobile suspension layer (shown as L2 and L3). Lutoclines have a

vertical scale (distance between steps) dependent on the local vertical

scale of turbulence (Posmentier, 1977). Therefore, only a limited number

can exist and over limited periods. The origin of this term stems from


the Latin word lutum which means mud (Kirby, 1986). Lutoclines are

analogous to other types of density stratification (pycnoclines) from

sharp salinity gradients (haloclines) and temperature gradients

(thermoclines) with the exception that suspended sediment exhibits

settling independent of the fluid. They are easily recorded by high

frequency echo sounders and are characteristically observed in high

sediment (> 500 mg/1) environments.

Figure 2-10 shows a typical suspended sediment profile showing the

relative temporal stability of two lutoclines.

The velocity data also shown in the figure together with the

concentration profiles suggest turbulent, well mixed flow between


The physics of lutocline genesis, growth, and decay is governed by

the dynamic interaction between the counteracting processes of turbulent

mixing and gravitational settling. Simply stated, lutoclines occur

because sediment is heavier than water and it tries to settle out under

quiescent conditions. Due to flocculation and hindered settling, fine

sediment suspended at large concentration settles as a sharp interface,

as opposed to concentration "thinning" (Bosworth, 1956). Turbulent

eddies impinging on the interface exchange "parcels" of sediment-laden

fluid. However, due to the potential energy difference of each "parcel"

with its surroundings, they are returned to near origin levels with only

modest mixing. This is in sharp contrast to the rapid mixing which takes

place in the low density gradient regions (qualitatively defined below).

Thus, the moderate mixing at the interface is counter-balanced by the

sediment settling, and the interface remains stable.

Water surface --



i *

****** 15:50
--- 15:55

' I I + I I I I
0 2 4 6
i i ii i I I I
0 .5 1.0


Time of profile

5 minute mean velocity

Figure 2-10. Typical Suspended Concentration Profile Showing
Multiple Lutocline Stability Over 10 min. Period
(Kirby, 1986)

One means of relating the relative magnitudes of gradients in

kinetic energy, 8(pu2/2)/az, to potential energy, a(pgz)/8z, is through

the local gradient Richardson number defined as

Ri = & 2 (au)-2
p az 8z


where g is the acceleration of gravity, p is the fluid density, u is the

horizontal velocity and z is the vertical coordinate direction (positive




Thus, the implications of equation 2.9 are

Ri > 0 High rate of kinetic energy dissipation relative
to low potential energy gradient Rapid Mixing

Ri >> 1 Low rates of kinetic energy dissipation relative
to high potential energy gradient Minimal

Ri < 0 Density gradient, ap/az, > 0 the system is
unstably stratified Overturning

As an example, letting the local mixing rate in a neutrally

stratified condition (no density stratification) be defined as Kn(z)

implying that it is variable with depth, the simplest relationship for

mixing in the presence of stratification, based on Richardson number, is

Ks(z) = Kn(z) (1 + Ri)-1 (2.10)

where Ks is the local mixing rate dampened by stratification. The limits

on mixing meet the above requirements as it can be seen that Ks(z) -*

Kn(z) for Ri 0 and Ks(z) 0 if Ri 4 -. This is then one means of

quantifying the stratification dampened mixing.

A more general form for the above equation, a review of literature,

and a discussion of applicability of buoyancy stabilization are given in

Section 3.3.2.


3.1 Introduction

The physics related to the vertical structure of fine sediment

suspension can be addressed by considering the important components of

the advection-diffusion equation. This equation, simply an Eulerian

conservation of sediment mass expression, relates the temporal changes in

sediment concentration to the spatial gradients in fluxes. Simple

arguments show that for the present purpose the important coordinate in

the equation is the vertical, z (positive upwards from the bed),

direction. Furthermore, gravitational forces which influence the

diffusion and settling flux terms are responsible for the complex

structure of lutoclines and fluid mud as defined in Chapter 2.

Theoretical and rationally based relationships for settling

velocity, neutral turbulent diffusivity, and buoyancy stabilization are

presented in this chapter for explanation and predictive purposes. A

simple one-dimensional numerical model, developed from these

relationships and the advection-diffusion equation, is used to explain

laboratory and field data presented later in this report. Finally, to

distinguish the lower layer of mobile fluid mud, a simple numerical model

based on momentum diffusion is developed to evaluate the dynamic and

steady state characteristics of this layer and to estimate horizontal

sediment transport rates.


3.2 Mass Conservation Equation

In Cartesian coordinates (x, longitudinal; y, lateral; and z,

directed vertical upwards positive from the water surface), the

instantaneous Eulerian conservation of mass equation for (scalar)

sediment suspension concentrations C(x,y,z,t) (mass of sediment/volume of

suspension) can be written as

C -Voq (3.1)

where q is the resultant mass flux vector (from diffusion and settling)

and V is the vector operator.

For Fickian molecular diffusion, the mean mass flux vector is

qm = DVC (3.2)

where it is assumed that the molecular diffusivity, D, is isotropic

(Fischer et al., 1979).

Since it is implausible to track particles in suspension on an

instantaneous, infinitesimal scale, and because natural flows are

typically turbulent, it is usual to express equation (3.1) in terms of

time averaged values (e.g., time average velocity, u, and concentration,

C) where the averaging time is sufficiently long to negate turbulent

fluctuations but short enough to track longer period temporal behavior

(Vanoni, 1975; McDowell and O'Connor, 1977). However, time averaging

greatly increases the diffusive mass flux vector. Fortunately, as an


approximation, turbulent diffusion can be expressed analogous to Fickian

diffusion in the form

qt = K.VC (3.2a)

where K is the turbulent mixing vector with Cartesian (x,y,z) coordinate

components (KX,Ky,KZ). Since turbulent mixing is much greater (2-8

orders of magnitude) than molecular diffusion, the latter is often

neglected (McCutcheon, 1983). Simple perturbation analyses, i.e.,

letting the velocity (vector) and concentration (scalar) components be

represented by mean (e.g., u,C ) and fluctuating values (e.g., u',C'),

have been used to support this result mathematically. The reader is

directed to Hayter (1983) or French (1985) for this derivation.

The mass flux vector from settling is, simply,

qs = Fs = -wsCj (3.3)

where ws is the mean sediment settling velocity and j is the unit vector

directed along the z axis.

The resultant mass flux vector for suspended sediment is then

approximated by

q = qt + qs 0 < z < Zb (3.4)

away from the boundaries (water surface z=0 and bed surface, z=Zb).


For the purposes of considering vertical structure, only the

vertical transport terms need to be evaluated. Horizontal gradients in

concentration are (typically 3-4) orders-of-magnitude smaller than

vertical gradients.

8x 8z ay az

Non-dimensional scaling arguments have been used to determine the

relative importance of the individual terms in Eqn. 3.1 This analysis is

included for reference in Appendix A. For typical estuarine conditions

(see Appendix A) horizontal and vertical advective fluxes and horizontal

diffusive fluxes can be neglected for first order analysis. The

governing equation for considering the vertical structure of fine

suspended sediments is now reduced to

a z = C + K < z < Z (3.5)
-t az az S + zaz b

where qz is the resultant vertical flux from settling and vertical

diffusion away from the boundaries (z=0 and z=Zb) shown by the bracketed.

terms in Eqn. 3.5.

The boundary conditions which must be imposed on Eqn. 3.5 are

Bed Flux Boundary Condition. Application of Eqn. 3.5 at z = Zb

requires that a bed flux term, Fb (mass of sediment per unit bed area per

unit time), containing both erosion, Fe, and deposition, Fp, fluxes as


Fb = Fe Fp (3.6)

be defined. In addition, the diffusion and settling flux terms at the

bed are zero. Thus qz(Zb,t) = Fb and w. = Kz = 0 at z = Zb is the

appropriate bed boundary condition. Fb is dependent on sediment and

hydrodynamic conditions. Section 3.5 presents a detailed discussion of

bed fluxes (erosion and deposition) used in the vertical transport model.

Surface Boundary Condition. The boundary condition at the water

surface, z=0, is a no net flux boundary. This means that there is no net

transport across the free surface and diffusion flux is always

counterbalancing settling flux i.e.:

qz(0,t) = (wC) + {K } = 0 (3.7)

The diffusion flux term Fd = f{KC} must include entrainment and

gravitationally stabilized mixing. In the absence of well defined

hydrodynamics (i.e., perhaps the results from a full turbulence model

simulation), functional forms for the vertical turbulent diffusivity, KZ,

based on first-order closure modeling using mixing length approximations

can be used (McCutcheon, 1983). This assumes that the mass diffusivity

can be related to the momentum diffusivity. Furthermore, due to

differences in time scales, spatial variability, and kinetic energy

dissipation, the functional forms for highly oscillatory currents (e.g.,

waves) are quite different from those for unidirectional flows.

Stratification, in general, dampens turbulent mixing by the

mechanisms described in Chapter 2. Through local gradient Richardson


number relationships of the Munk and Anderson (1948) form, buoyancy

(gravitational) stabilization can be modeled. Stabilized diffusivity is

treated separately in Section 3.3.2.

On the subject of mixing and stratification it must be pointed out

that surface waves can create interfacial waves which can build to

breaking, thereby greatly enhancing interfacial mixing (Yih, 1970; Dean

and Dalrymple, 1984). Due to the limited scope of this research and

because this phenomenon was not observed in laboratory or field data for

this study, no further discussion is provided. The reader can find

additional information on this topic in Lamb (1945), Yih (1976), and Yih


The settling flux (wsC) as written in Eqn. (3.5) allows for spatial

variability in both unknowns, settling velocity and concentration. In

general, for both cohesive and non-cohesive sediments, settling velocity

is a function of concentration, ws(C). The settling behavior of cohesive

and non-cohesive sediment is covered in Section 3.4.

3.3 Diffusive Transport

3.3.1 Turbulent Diffusion

In turbulent flows mixing occurs mainly because the time-averaged

products of the velocity and concentration fluctuations i.e., u'C', are

non-zero. Through adequately measuring the simultaneous fluctuations in

velocities and concentrations, turbulent mixing can be precisely

quantified. Then, for predictive purposes, correlations to flow

parameters such as bottom friction, mean velocity and pressure gradient

are required. Reasonable success is beginning to be achieved in the area


of turbulence modeling (Zeman and Lumley, 1977; Sheng, 1983). However,

in light of the difficulties in precise measurement of these

fluctuations, verification poses difficulties.

For fine sediment suspensions the turbulent diffusion of sediment

mass, Ks, is approximately equal to that of the diffusion of momentum,

Em. The turbulent Schmidt number, Sc (Daily and Harleman, 1966), which

is the ratio of mass to momentum diffusivity is equal to one (Teeter,

1986b) as

Sc = 1 (3.8)

In turbulent flows momentum diffusion is by Reynolds stress,

tij = -p ulul, gradients where the time mean product of the velocity

fluctuations is nonzero. For mass diffusion, the time mean product of

the concentration and velocity fluctuation is nonzero analogously.

This observation (Reynolds' analogy) allows the use of a wide body

of literature on first-order closure modeling based on the coefficient of

eddy viscosity, relating the Reynolds stress to mean velocity gradient as

Pij = -pEij a (3.9)

where Eij is the i,j component of the momentum diffusivity (eddy

viscosity) tensor. It can be seen from Eqn. (2.9) that the eddy

viscosity, in general, must be a function of mean shear rate and shear

stress. It is also common to assume that turbulent diffusion is


isotropic (i.e., Eij = Eji = E, Fischer et al., 1979) in the absence of


The most commonly applied expression of vertical variation in eddy

diffusivity is the formulation given by Rouse (Vanoni, 1975). By

following von Karman's assumptions of a linear shear stress distribution

with depth leading to a logarithmic velocity profile, the following

expression is found:

E(z) = KU*Z (1- -) (3.10)

where K is von Karman's constant, u* is the friction velocity (vto/lp) and

h is the flow depth. While this expression may be sufficient for

describing turbulent-logarithmic uni-directional flows, it does not

describe highly oscillatory flows such as under waves. Maa (1986),

Kennedy and Locher (1972), and Hwang and Wang (1982) have reviewed

currently popular expressions for diffusion coefficients under waves.

There seems to be little consistency in the forms. One of the most

promising expressions based on energy dissipation is that developed by

Hwang and Wang (1982). Their model, applicable outside the wave boundary

layer, is of the form

E(z) = aH2a sinh 32
2sinh2kh (3.11)

where a is a constant, H is the wave height (twice the amplitude), a is

the wave frequency (2n1/T, T = wave period), and k is the wave number

(2n/L, L = wave length).

Thimakorn (1984) found success using a coefficient similar to that

given by Hwang and Wang (1982) to predict vertical profiles of natural

clay concentration during resuspension in a wave flume. It should be

pointed out that the concentrations reported were small (<1000 mg/1) and

any buoyancy stability effects therefore were likely to be negligible.

Next to the bed boundary layer effects greatly increase vertical

mixing under waves due to the relatively large velocity gradients and

shear (Neilson, 1979). Orbital particle trajectories are significantly

altered from those predicted for example by linear wave theory (inviscid

potential flow) because viscous (or turbulent) effects dominate.

However, this layer is small (o/(2v) < 1 cm) and is often neglected (Maa,

1986). Further upward, the velocity amplitude gradients increase with

distance above the bottom to a maximum at the surface. This is the basis

for the Hwang and Wang (1982) form shown above. Maa (1986) conducted dye

diffusion tests under waves which showed larger lateral spreading rates

near the surface and immediately near the bottom. This is indicative of

higher energy dissipation in those regions which would support the

proposition of higher vertical mixing rates.

In the presence of density stratification the form of the neutral

diffusivity is not as important as the form of the stability coefficient

(French, 1985), which provides the basis for a discussion of mixing in

the presence of density stratification.


3.3.2 Gravitational Stabilization

In the previous section, theoretical and empirical based expressions

for the vertical turbulent diffusivity under current and waves were

mentioned. In a continuously, stably stratified flow the vertical

diffusion of both momentum and mass is inhibited by stratification, and

significant modification of the turbulent diffusivity occurs.

Furthermore, the diffusivity of momentum and mass are not affected in the

same manner. In the presence of density stratification, the eddy

viscosity (i.e., the turbulent momentum diffusity) is larger than the

eddy diffusivity of heat and mass (French, 1985). Progress has been made

towards estimating values and obtaining expressions for mass and momentum

diffusion in a continuously stratified flow. However, it must be

emphasized that at the present time an expression does not exist for

either eddy viscosity or diffusivity which is considered universally

valid. French (1985) provides a summary of several popular forms

developed for uni-directional flow only. A brief review of those plus

others is given here for the purpose of explaining vertical structure.

Rossby and Montgomery (1935) first proposed an equation relating

vertical eddy viscosity for stratified flow, Es, to the corresponding

value for homogeneous or neutral conditions, En, of the form

E- (1 + BRMRi) (3.12)

where 8RM is an empirical coefficient and Ri is the local gradient

Richardson number, Eqn. (2.9). They assumed that the change in kinetic

energy per unit mass in going from a neutral or unstratified condition to


a stably stratified condition is equal to the potential energy change due

to displacement over the mixing length from the equilibrium position with

a different density.

Holzman (1943) suggested a somewhat different relationship

E- = (1 OHRi) (3.13)

where OH is a coefficient. Note the change in sign of the coefficients.

Munk and Anderson (1948) proposed a generalized form of the Rossby

and Montgomery (1935) and Holzman (1943) equations as

E a
= (1 + OMARi) (3.14)

where BMA and aMA are free coefficients.

Kent and Pritchard (1957) also used a conservation of energy

argument to develop an equation of the Munk and Anderson (1948) form;

however, they argued that a=-2 on a theoretical basis.

Delft Hydraulics Laboratory, (DHL) (1974), reported that the ratio

of E, to En should decrease exponentially with increasing values of Ri or

E -0 Ri
E e (3.15)

where 0e is an empirical coefficient.

Finally, Odd and Rodger (1978) used the original hypothesis of

Rossby and Montgomery (1935) to define equations applicable for two

specific cases:


Case 1. Stratified flow with a significant peak in the vertical profile

of Ri at a distance z = z0 from the bottom boundary where T is the peak

gradient Richardson number, then

s -= (1 + BORT)


for y 1 I

for Y > 1



where BOR = a coefficient.

Case 2. No significant peak exists in the vertical profile of Ri: then

E- = (1 + BORRi)

E- = (I + O0R)

for Ri 1I

for Ri > I

Equations (3.16) through (3.19) are applied throughout the vertical

dimension, but near the boundaries, if Es > En then En is used. Note

that Es/En = constant (not a function of depth) for all cases except

conditions when Eqn. (3.18) applies. This is significantly different

from the previously proposed forms (Eqns. 3.12 3.15) which are

everywhere depth variable.




The problem with all the above methodologies is that, in general,

they cannot be shown to be universally valid. Suggested values for some

of the coefficients used in the above equations are summarized in Table

3-1 below.

Table 3-1. Summary of Coefficient Values for Turbulent
Vertical Diffusion of Momentum in Continuously
Stratified Flow

Equation a Source

3.12 2.5 -- Nelson (1972)
5.0 -- DHL (1974)
30.3 -- French and McCutcheon (1983)
3.14 10 -0.5 Munk and Anderson (1948)
30 -0.5 DHL (1974)
3.16-3.19 0.31 0.747 French (1979)
0.062 0.379 French and McCutcheon (1983)
140-180 Odd and Rodger (1978)

With regard to the data summarized in Table 3-1, the following should be


1. Nelson (1972) used published oceanographic, atmospheric, and

pipe flow data for his analysis, and the same was true of the

analysis by the Delft Hydraulics Laboratory (1974). Thus,

these investigators had no control over the quality of their


2. The data used by French (1979) were taken under laboratory

conditions, but the flume used for these experiments had a

small width-to-depth ratio, and the results may have been

unduly affected by this fact.

3. Odd and Rodger (1978) used field data from a reach of tidal

channel. Their data set is perhaps the best data presently


available regarding the turbulent vertical diffusion of

momentum under stratified conditions.

4. French and McCutcheon (1983) used the Odd and Rodger (1978)

data set for their analysis. The coefficients for Eqns. (3.16

3.19) used in their work differ from that of Odd and Rodger

(1978) due to differences in the definition of reasonable fit.

5. In the past, Eqn. (3.12) has been the most commonly used method

of estimating E. (Nelson, 1972). It is more theoretically

justifiable than the methods of Odd and Rodger (1978), French

and McCutcheon (1983) or French (1979).

6. Delft Hydraulics Laboratory (1974) concluded that when Ri <

0.7, the scatter of the data available is so great that no

best-fit equation can be selected.

A number of models for the eddy (mass) diffusivity in stratified

flow have also been proposed. Most have been based on the results from

momentum diffusion; however, under stratified conditions, questions arise

as to the applicability of this assumption (e.g., see Oduyemi, 1986).

One of the most frequently used expression is of the form

s -a'

where Kn and Ks are the vertical mass diffusivities for homogeneous and

stratified flows, respectively, and c, 0', and a' are coefficients. Munk

and Anderson (1948) estimated that c = 1, a' = 1.5, and 0' = 3.33.

It is interesting to note that stratification apparently also acts

to reduce the value of the turbulent transverse diffusion coefficient by


turbulence damping; however, the results presently available in this area

(see, for example, Sumer, 1976) are inconsistent and are not relevant for

vertical structure considerations.

When gravitational stability is considered (e.g., by Eqn. 3.20),

nonlinearity between diffusive flux, Fd, and vertical concentration

gradient, C,, (note ,z denotes differentiation with respect to z),

develops. Without regard to stabilization (Kz Kn), by Fickian

diffusion, the diffusive flux is linearly proportional to concentration

gradient, Fd = K=.C,z. However, from theoretical results presented for

gravitational stabilization, the turbulent mass diffusion coefficient (Kz

= Ks) was shown to be inversely proportional to the gradient Richardson

number, Ri (given by Eqn. 2.9), to a power, a' > 1. The gradient

Richardson number, of course, is directly proportional to the density

gradient, p,z, which is a function of the concentration gradient because

bulk density is a function of sediment concentration (Eqn. 2.5). The

resulting dependence of diffusion flux to concentration gradient is

therefore highly nonlinear.

Figure 3-1 shows an example of the nonlinearity resulting between Fd

and C,z using Eqn. 3.20 with Munk and Anderson values for stability

coefficients (i.e., E = 1, a' = 1.5, and 0' = 3.33). For this case the

flux initially increases with C,z reaches a maximum and then slowly

decreases. For a given flux below the maximum two values of C,z

(corresponding to the two roots) satisfy the equation. In the absence of

settling, discontinuities in concentration profile (two distinct C,z's)

are theoretically possible (because Fd,z = 0) and relatively stable as

long as Fd is constant with time (e.g., near the bed during steady

X 3-

X 2If~ / (1+4.17Ri2
d (1+3.33Ri")1.5

0.0 0.2 0.4 0.6 0.8

Concentration Gradient, LC (kg/m2

Figure 3-1. Diffusion Flux vs. Concentration Gradient

erosion). For salinity concentrations in estuarine environments, this

has been pointed out to be a likely cause of salinity finestructure

(Postmentier, 1977). For suspended fine sediment, the nonlinearity in

diffusion flux, Fd, (with C,z) has the effect of promoting lutocline

growth and stability--in addition to the nonlinearity between settling

flux, Fs, and concentration, C (pointed out in Section 2.4.2). The

settling properties of estuarine fine sediment is presented in the

following section (3.4).

3.4 Settling

The predominant distinction between fine sediment suspensions and

other density altering constituents (e.g., salt, temperature, etc.) is


that suspended sediment is negatively buoyant and settles independent of

the suspending fluid which surrounds it. This counteracts mixing to the

extent that under quiescent conditions partial or total clarification is

possible only to be later well mixed again under high flow conditions.

While the settling characteristics of non-cohesive sediments (e.g.,

sand) are reasonably well behaved, i.e., not so strongly dependent on

concentration, salinity, etc., cohesive sediments are very sensitive to

these variables.

It is convenient to start by discussing the settling characteristics

of individual particles and work into high concentration (>20,000 mg/1)

settling suspensions.

3.4.1 Free Settling

Free settling was defined in Chapter 2 as the concentration range

over which individual settling sediment particles (both dispersed primary

particles and aggregates) do not physically interfere with one another.

For cohesive sediments, the upper concentration limit is in the range of

300-500 mg/l (Krone, 1962) but for non-cohesive sediments it is one to

two orders-of-magnitude higher (McNown and Lin, 1952).

Individual sediment particles settle at a terminal fall velocity

which results in a force balance between form and skin friction (viscous)

drag and net negative buoyancy. For a spherical particle of diameter, d,

settling in a viscous fluid with kinematic viscosity, v, the settling

velocity, ws, is

w Ed s w 2 (3.21)
3 CD Pw


where g is the acceleration of gravity, CD is the drag coefficient and ps

and p, are the sediment and fluid densities, respectively. The

coefficient of drag, CD, is a function of the Reynolds' number of the

sphere (R wsd/v), but cannot be determined analytically for R > 1 (see,

for example, Vanoni, 1975).

In the viscous or Stokes' settling range (R < 0.1) the drag

coefficient is given by CD = 24/R and the settling velocity is

gd2 (P (3.22)
s 18v p,

Fine estuarial sediment in dispersed or quiescent conditions typically

falls well within this range. Therefore, no further discussion of the

deviations from Stokes settling will be presented here with one minor

exception: fine estuarine sediment is not generally spherical. In

dispersed form, cohesive size sediment is plate-like with a large surface

area to volume ratio (Van Olphen, 1963). This results in a higher drag

coefficient and slower settling velocity than spherical sediment of the

same volume. Very fine (d < 1 pm) dispersed sediment may not settle at

all due to the increased relative importance of Brownian motion.

Aggregates, although irregularly shaped, are generally more spherical

(and substantially larger than dispersed primary particles). For both

particles, it is typical to define an "effective" particle diameter based

on measured settling velocity and specific density (ps). A more thorough

discussion of the effects of particle shape on settling velocity as well

as other deviations from Stokes settling can be found in Vanoni (1975).

3.4.2 Flocculation Settling

In the presence of small amounts of dissolved salts (< 1 ppt NaC1)

cohesive sediment in suspension can flocculate greatly, thus changing the

settling properties. Flocculation of cohesive sediment particles is the

consequence of inter-particle collision and cohesion. Cohesion and

collision are discussed in detail by Einstein and Krone (1962), Krone

(1962), Partheniades (1964), O'Malia (1972), and Hunt (1980) and reviewed

by Hayter (1983).

Cohesion depends primarily on the mineral composition and the

availability and charge of cations in the suspended fluid. Colloidal

particles have both attractive and repulsive forces (Van Olphen, 1963).

The attractive forces predominate when the coulombic repulsive forces are

suppressed by sorbed cations near the particle surfaces. A measure of

the relative cohesiveness of a particular colloidal sediment is the

cation exchange capacity, CEC. A high CEC indicates a highly cohesive

sediment. Montmorillinitic sediments have a higher CEC and thus are more

cohesive than illitic or kaolinitic sediments with lower CECs.

Collision intensity and frequency are dependent on three mechanisms:

Brownian motion, fluid shearing, and differential settling. Brownian

motion is the natural thermal agitation of the sediment particles in the

suspending medium. Particle movement from Brownian motion is erratic,

the collisions are weak and the resulting flocs are "fluffy" (of

relatively low density and weakly bound). This motion becomes much less

apparent as the floc size grows. Brownian motion in estuaries is the

least significant collision mechanism of the three (Krone, 1962).

Particle collision from fluid shearing, however, becomes much more


significant as the size of the flocs grows. The result is a greater

intensity of collision and stronger flocs. Differential settling becomes

increasingly more important as the distribution of the size widens.

Under quiescent conditions e.g., at the time of slack water, with a

natural non-uniform sediment this becomes the primary collision

mechanism. The frequency of all three means of collision increases with

increasing concentration.

Two characteristics of flocculated sediment which differ from the

dispersed form and which affect the settling velocity are particle

density and shape. First, because of interstitial trapped water, the

relative particle densities are reduced. This effect alone would lead to

reduced settling velocity in the flocculated state. However, because of

the larger size and more spherical shape, a decrease in viscous drag

results. Since the reduction in drag is much more significant than the

reduction in density, the settling velocity of the flocs are up to 4

orders of magnitude larger than dispersed particles (Bellessort, 1973).

This can result in rapid sedimentation and shoaling in upper estuaries

where flocculation (by introduction of dissolved salts) is first


Krone (1962) reasoned that the average (median, by weight) settling

velocity of flocculating Mare Island Strait (San Francisco Bay) sediment

for equal flocculation time was proportional to the sediment

concentration raised to the 4/3 power,


ws a C4/3


His reasoning was based on consideration of collision probability and

average floc size. He further supported this argument with data taken in

settling column and flume studies.

Burt (1986) used a general relationship for flocculation enhanced

variation with concentration as

s -= k1 C (3.24)

where k1 depends on sediment composition and n, can vary from about 1 to


3.4.3 Hindered Settling

As the concentration of sediment in suspension increases beyond the

flocculation settling range, the mean sediment settling velocity begins

to drop. Aggregates are so closely spaced as to form a continuous

network, and the interstitial fluid is forced to escape through smaller

and smaller pore spaces. This is commonly termed "hindered settling" in

the literature (Mehta, 1986, Lavelle and Thacker, 1978). However, the

inadequacies of this definition were pointed out in Section 2.4.2. The

pioneering work of Richardson and Zaki (1954) on the settling of uniform

glass spheres resulted in a widely accepted relationship for the settling

velocity as a function of concentration of the form,


ws wso (1 k2C)



where wso is the initial or reference settling velocity, k2 is a

coefficient which depends on the sediment composition and n2 5. The

coefficient k2 can be considered to be the reciprocal of the hypothetical

concentration where hindered settling gives way to primary, first-stage

consolidation. This is typically in the neighborhood of 120-160 g/1

(Mehta 1986, Einstein and Krone, 1962). For fine sand-coarse silt the

reference velocity, wso, is given by Stokes' Law. For cohesive

flocculated sediment the reference velocity, wso, is the maximum velocity

of the flocculation range. Teeter (1986b) found that most natural fine

bay sediments fit this relationship well.

Lavelle and Thacker (1978) used an expression of this type in

steady-state analysis of the high concentration data of Einstein and

Chien (1955) for coarse-grained sediment. Including a term of (I-C)0 in

the Rouse (1938) equation allowing for finite and reasonable

concentrations at the bed (z=0), they found success in predicting the

near-bed high concentration data of Einstein and Chein (1955).

3.5 Vertical Bed Fluxes

The bed flux boundary condition for solution of Eqn. (3.5) plays a

critical role in the evolution of the vertical suspension profile as the

overall source and sink component of sediment mass in suspension. Bed

fluxes can be either erosional or depositional. Both are discussed in

the following paragraphs. It is important to point out that defining the

elevation at which the erosion or deposition process takes place is, in

itself, a formidable task. From a practical viewpoint, simultaneous

continuous profiling of concentration, velocity and bed stresses


(pressures and shear) are required in the upper-bed to near bottom layers

to define the interface elevation with time and hydrodynamic action. As

was pointed out in Chapter 2, it is very important to distinguish the

stationary bed material from the fluid mud layer. Additionally, erosion

relationships developed for bed/mobile suspension interfaces may not be

adequate for erosion and fluidization of the bed beneath a fluid mud

layer. This is a possible limitation of the proposed erosion/deposition

functions used in the vertical structure model and presented in the

following subsections.

3.5.1. Bed Erosion

Bed erosion occurs when the resultant hydrodynamic lift and drag

forces on the sediment at or below the bed interface (Zc in Figure 2-1)

exceed the resultant frictional, gravitational and physico-chemical

bonding forces of the sediment grain or particle. Continuous inter-

particle contact ceases and individual or groups of aggregates become


There are two modes of erosion (Mehta, 1986), surface or particle by

particle erosion and mass or bulk erosion. In surface erosion,

individual particles break free of the bed surface as the hydrodynamic

erosive force (i.e., instantaneous turbulent shear stress acting on the

particle surface) applied to them exceeds the resultant gravitational,

frictional and cohesive bed bonding force. Under mass erosion, failure

occurs well below the bed surface resulting in large chunks of sediment

being broken from the bed structure and, subsequently, resuspended. Bed

fluidization is mass erosion where large structural breakdown occurs with


an initially minimum change in density. Surface erosion is more typical

of low concentration, low energy environments while mass erosion occurs

under higher flow and higher concentration conditions (Mehta, 1986).

Surface waves and other highly oscillatory currents have a

particularly pronounced influence on erosion in comparison with

unidirectional currents. Because of the increased inertial forces (e.g.,

"added mass" drag) associated with a local change in linear momentum, the

net entrainment force is much greater than with turbulent unidirectional

flows. Much more significant is the effect bed "shaking" and "pumping"

can have under highly oscillatory flows. "Shaking" or bed vibrations

occur because of the oscillatory bed shear stress which is transmitted

elastically (while at the same time damped) down through the bed.

"Pumping" occurs from oscillatory fluid hydrostatic pressure at the bed

which, given the low permeability of cohesive sediments, can lead to

internal pore pressure build up and liquefaction, similar to earthquake

failure of saturated terrigenous soils (Seed, 1976). This effect can

cause destruction of effective stress in larger layers depending on the

bed characteristics leading to mass erosion and fluid mud formation

(Alishahi and Krone, 1964; Wells et al., 1978; and Maa and Mehta, 1987).

The destruction of effective stress under waves is documented, perhaps

for the first time, in laboratory measurements presented in Chapter 4.

Erosion (particles leaving the bed surface) precedes scour

(resulting decrease in bed elevation) which will continue under constant

loading until the bed shear stress and the bed shear strength are equal.

The bed shear strength is a function of the deposition and consolidation


history plus the physico-chemical characteristics of the sediment. The

shear strength, in general, increases with depth into the bed.

The rate of erosion (= flux of sediment from the bed), Fe, from

surface erosion is linearly related to the "excess shear" stress, tb-ts,

for spatially and temporally uniform bed properties (Kandiah, 1974) as

F a, s) (3.26)
e ts

where al, is an empirical rate constant, tb is the applied (time-mean)

bed shear stress and ts is the bed shear strength for erosion. For a

given al, which is related to the type of flow and sediment

characteristics, the erosion rate, Fe, is constant. For non-uniform beds

(e.g., soft, partially consolidated) the rate of erosion can be found by

(Parchure 1984, Parchure and Mehta, 1983).

Fe = co exp{a2 [tb ts(z)]11/2 (3.27)

where cE and a2 are constants (determined empirically). Since ts

increases with depth below bed, the erosion rate, Fe, decreases as scour


No currently unique expression exists for mass erosion since it must

involve dynamic bed data (i.e., bed stresses and pressures) as well as

imposed shear.

For mass erosion under waves the practice is to increase the

coefficients to account for the larger magnitude erosion. Maa (1986)

showed success with this procedure and demonstrated that the coefficients


were as much as an order-of-magnitude larger for wave erosion than for

what has been found for the uni-directional case. Under pure wave flow

condition it is difficult to distinguish bed erosion from fluid mud

entrainment. Even though wave erosion has a greater ability to break the

bonding forces, without high momentum diffusion or turbulent entrainment

rates the fluid mud may not become mobile.

3.5.2 Deposition

Sediment particles or aggregates in suspension will redeposit on the

bed if the bed shear stress drops below some critical threshold value,

tcd-. cd is the shear stress below which all initially suspended

sediment deposits eventually. In general, it takes lower turbulent bed

shear stress to keep cohesive sediment in suspension than it does to

erode it (i.e., tbm < TO). tbm is the shear stress above which no

deposition occurs and it is generally larger than the limit for total

deposition, tbm > tcd, (Mehta, 1986). This is because after deposition

interparticle bonding and orientation are time-dependent, as well as

dependent on consolidation mechanics (e.g., overburden, etc. as discussed

briefly in Chapter 2) and the critical shear stress for erosion increases

with time. For a uniform sediment tcd = tbm*

For uniform sediment, in a depositional environment (i.e., tb <

tcd), the rate of sediment deposited (= flux of deposited sediment), Fp,

on the bed is related to the average aggregate settling velocity, ws, the

near-bed concentration in suspension, C, and the relative probability, P,

that the sediment will stay on the bed as

Fp = s C P tb < rcd (3.28)


The probability, P, that the sediment will stick to the bed is related to

the relative shear stress (Krone, 1962) as

P = (1 -) (3.29)

As observed, this relationship indicates no deposition when tb tcd and

rapid settling when the bed shear goes to zero (tb = 0). Krone (1962)

and Mehta (1973) conducted deposition experiments under steady flows

using natural estuarine sediments and commercial kaolinite. tcd was

found to depend on sediment composition, varying from 0.04 to 0.15 N/m2.

Mehta (1986) made the distinction for critical shear stress for

deposition of non-uniform sediment. He pointed out that while deposition

proceeds when tb < tbm, not all of the sediment in suspension deposits

when tb > tcd* This is illustrated by the data in Figure 3-2. Mehta

(1986), in reanalyzing earlier data (Mehta, 1973), pointed out that even

after long periods the ratio Ceq/C, of ultimate equilibrium

concentration, Ceg, to initial concentration, Co, was only a function of

tb (i.e., Ceq/Co = f(tb)), not of Co.

This, then represents a fundamental distinction between cohesive and

cohesionless sediment since for cohesionless sediment the equilibrium

concentration, Ceq, is dependent on tb and independent of initial

concentration, Co, (i.e., Ceq = f(tb)). For cohesionless sediment, the

equilibrium concentration represents a balance between the rates of

erosion and deposition, whereas for cohesive sediment simultaneous

erosion and deposition did not occur under test conditions relative to

C 0.6 o


0 k-..* I I I I I t I I t
0' 2 4 6 8 10 12 14 16 18 21

TIME (hrs)

Figure 3-2. Ratio C/Co of Instantaneous to Initial Suspended Sediment
Concentration Versus Time for Kaolinite in Distilled Water (after Mehta,

Figure 3-2. Thus Ceq, for cohesive soils was the steady state

concentration, Css. For cohesive soils, winnowing (coarser material

settling out first) is a likely cause of the variable steady state

ratios, Css/Co (Mehta and Lott, 1987). Thus, the steady state

concentration, Css, results in a suspension with a mean particle grain

size finer than the original suspension. For modeling purposes,

discretizing non-uniform suspended sediment into a finite number of

classes, Ci, and treating erosion and deposition for each class

separately would be one means of handling the winnowing (and resulting

bed layering) phenomena. The vertical structure model considers

independent settling and deposition of multiple classes of suspended


particles after it was found to be significant in settling column tests

of natural bay sediment (see Section 4.3). For discretizing the non-

uniformity of the deposition Eqn. 3.28 (originally developed for uniform

sediment) is assumed to be valid as

Fpi wsi Ci P tb < tcdi (3.30)

where the i subscripted variables must be defined for each class.

3.6 Fluid Mud Entrainment

Once a fluid mud layer is formed, either from high erosion or

deposition rates, entrainment of this high concentration sediment

suspension can occur at the upper, mobile fluid mud interface (see Figure

2-1). Entrainment is markedly distinguished from bed erosion in that the

sediment is already in suspension. Fluid mud entrainment results from

interfacial instabilities and dissipation of kinetic energy with, as yet,

limited theoretical analysis. However, because it is believed to behave

analogously to two-layer density stratified flows associated with salt or

temperature gradients, a relatively larger literary and theoretical base

exists for these cases (Yih, 1980).

Velocity shear at the interface accounts for the primary mixing

mechanism. Unlike mixing in homogeneous or weakly stratified shear

layers, strong stratification characterized by a high Richardson Number

is composed of events such as interfacial wave generation and breaking,

interchange of energy between waves and the mean flow, and local shear-

instabilities (Narimousa and Fernando, 1987).


Kato and Phillips (1969) in laboratory experiments of entrainment of

linearly stratified fluids found that the entrainment coefficient,

E* = ue/u*, where ue is the entrainment rate (dh/dt) and u* is the

friction velocity, decreased with increasing stratification. They found

an inverse relation between entrainment rates and bulk Richardson Number,

Ri* = Abh/u*2 in which Ab = g(p-po)/po is the buoyancy jump, p is the

fluid density, po is a fluid reference density, g is the gravitational

acceleration, and h is the average depth of mixed layer. They suggested

E* Ri*-1/2. Other research indicates that it should be related to mean

velocity, u, in the mixed (i.e., upper) layer (Price, 1979; Thompson,

1979) as

E= -e = f(Riu)-4 (3.31)

where Riu = Abh/u2.

Still other researchers (Phillips, 1977; Price, 1978; Narimousa and

Fernando, 1987) showed supporting evidence for using the velocity jump,

Au, across the interface defined as the difference between the mean flow

velocities in each layer. Later researchers reasoned that the major

portion of the energy for turbulent mixing at the density interface

results from shear production at the entrainment zone itself and,

therefore, Au is the significant velocity scale to obtain a measure of

the energy dissipation rate.

Narimousa and Fernando (1987) presented a graphical depiction of the

entrainment process which is qualitatively descriptive enough to warrant

reproduction here.

Mixed layer


Non-turbulent layer

Figure 3-3. Simplified Description of Density Stratified Entrainment
(after Narimousa and Fernando, 1987)

Figure 3.3 shows the entrainment process based on experimental

observations (Narimousa and Fernando, 1987). The upper turbulent layer

of thickness, h, is well mixed and the lower layer is initially

stationary. An intermediate entrainment layer, 8, separates the two

layers and is the region characterized by high energy dissipation and

buoyancy gradients. In the upper entrainment layer, the mean shearing

rate, du(z)/dz, increases downward reaching a maximum at 61, and then

decreases as viscous dominant momentum diffusion penetrates deeper and

deeper into the stationary layer. The shear layer thickness is shown as

Ss. The highest density gradients occur in the entrainment layer of

thickness 6, which is inside the shear layer, 6s, where turbulence

dampening is sufficient to eliminate turbulent penetration into the


lowest layer. The momentum diffusion (viscous) layer, of thickness 68,

can be dynamic (growing with time) or relatively constant with respect to

the interface. Also shown are the flattening of large eddies (with

turbulent velocity components ul and wl) at the density interface and

local scouring and internal waves of height 6w in the intermediate

entrainment layer by the mixed layer eddies of mixing length scale, 1

(proportional to the mixed layer depth, h).

As can be deduced from the number of characteristics in the above

description, entrainment of density stratified flows of single phase

fluids is, in itself, an interesting and challenging field. Add to this,

particle settling associated with the two-phase sediment/fluid mixture

and one can see that fluid mud entrainment deserves fundamental research.

No effort has been made to distinguish fluid mud entrainment from general

lutocline mixing in this research. Nevertheless, despite this

limitation, reasonable success has been achieved in explaining the

observed physical behavior of prototype and field vertical profiles, as

shown in Chapter 5. Further research in fluid mud interfacial

entrainment is required before a more refined understanding and usable

results are obtained.

3.7 Horizontal Fluid Mud Transport

Several approaches to solving for the horizontal transport of mobile

fluid mud and the relative thickness of the mobile layer are available.

These approaches are based on different simplifying assumptions

concerning the theological and temporal behavior of the fluid/sediment

system. The solution approximations (for velocity profile in the fluid


mud layer) together with limitations are presented in order of increasing

complexity, beginning with the analytical solution of viscous boundary

layer development under an imposed shear stress. The following titles

are given for solution approaches:

A. Constant Viscosity Rayleigh Flow

B. Constant Viscosity Unsteady Bingham Flow

C. Variable Viscosity Steady Bingham Flow

D. Variable Viscosity Rayleigh Flow

E. Variable Viscosity Unsteady Bingham Flow

Applicable solution techniques were applied to field and laboratory data,

the results of which are presented in Chapter 5.

The constitutive equations which govern fluid mud transport are the

conservation of momentum (Cauchy's Equation) and mass (continuity)

equations. The Cauchy Equation of motion written in tensor notation is

(Malvern, 1969)

p = pb + V*T (3.31)

where p is the local fluid density u is the velocity vector, b is the

body force per unit mass vector, T is the stress tensor, and V is the

vector operator. The first term in Eqn. (3.31) is the time rate of

change of momentum per unit volume. The other terms are the body force

per unit volume and stress tensor gradient, respectively.

For an incompressible viscous fluid, the conservation of horizontal

momentum equation in Cartesian coordinates is


du aP 8 au a au
P + (3.32a)
dt ax 8z 8z ay ay

where P is the pressure. The dynamic viscosity, p, is assumed here to be

isotropic but, in general, a function of concentration, p=p(C). Together

with the continuity equation,

8u 8v 8w
S+ + I (3.32b)
ax 8y az

sufficient boundary and initial conditions (outlined below), the problem

is said to be closed and formally defined.

For a tractable solution to the horizontal flow problem, somewhat

far reaching assumptions must be made. First, one-dimensional horizontal

flow in the x direction is assumed (no v and w components in the velocity

vector). Next, the assumption of lateral uniformity is made. Then, by

continuity, the horizontal velocity component must only vary in the z

direction, u = u(z,t). The third assumption is by far the most

stringent. It is assumed, analogous to the laminar sub-layer next to a

boundary (Schlichting, 1979) and the shear layer in a stratified fluid

(Narimousa and Fernando, 1987), that the horizontal pressure gradient is

much smaller than the vertical shear stress gradient,

aP at
S<< --XZ (3.33a)
Tx az

To more formally show the conditions under which this assumption is

valid, scaling arguments are used to evaluate the relative magnitude of


the terms of Eqn. 3.32a for dynamic momentum diffusion into the fluid mud

layer. First, defining non-dimensional (primed) variables as

t' -' -x x' -? z -
to uo L 6

S P (3.33b)
yH po o

where to and uo are the characteristic maximum time and velocity, L is

the length scale of the estuary, & is the length scale of the fluid mud

layer depth, H is the differential height of the water surface over L, y

is the specific weight of mud, po and p. are the characteristic mud

density and dynamic viscosity. Substituting the above variables in Eqn.

3.32a and considering only vertical shear gives

Uo du' H 1 aP' rouo 1 8 au'
[ I =- H a[P- I + [Pu L ,,a' (3.32c)
to dt' poL p'ax' po 62 paZI aZ'

where all terms not in brackets, [ ], are order 1. Multiplying
Eqn. 3.33c through by [-] gives,

du' gHto 1 aP' too1 Su
[1] -[ ] + [p ] {P' } (3.32d)

Substituting typical numerical values for fluid mud layers in estuaries

of g = 101 m/s2, H = 100 m, to = 103 s, L = 105 m, uo = 100 m/s,

vo = P0/pO = 10- m2/s, and 6 = 10-1 m, the order of magnitude of the

pressure gradient term is

(101 m/s )(100 m)(103 s) ] = [10-1] (3.33d)
(105 m)(10 m/s)

The viscous shear term is

(104 m2/s)(IO3 s) 1] (3.33e)
(10 m)

which, for the particular set of conditions, is two orders of magnitude

greater than the pressure gradient magnitude. Hence, neglecting

horizontal pressure gradients in Eqn 3.32a for qualitative understanding

of the dynamic momentum diffusion depth is justified, albeit weakly. It

must be emphasized that under fully developed steady flow, the order of

magnitude of the viscous shear stress and horizontal pressure gradient

terms are the same (since they are the only two non-zero terms in the


Under the above constraints the momentum equation becomes

au a au (334)
at p az az

The equation is now in a form in which analytical and simple numerical

solutions are possible with careful specification of initial and boundary

conditions and theological behavior.


A. Constant Viscosity Rayleigh Flow. For the case of constant mud

viscosity, pm, and unsteady shear flow, an analytical solution proposed

by Stokes (see Schlichting, 1979) is appropriate. With the boundary

conditions, 1) imposed velocity, U, at the upper interface u(z)=U @ z=Za

(in Figure 2-1), and 2) U(z)=0 @ z4-. The solution for the horizontal

flow velocity is (Eskinazi, 1968)

u = U (1 erfc6s) (3.35)

where 6Ss is the similarity variable z/2/vEt and erfc is the complementary

error function defined as

erfc6s = 1 2- s dn (3.36)

The penetration depth of the mobile fluid mud layer (Zb defined in Figure

2-1) can be found by considering the boundary layer thickness,

defined by u/U = 0.01 which is

6 = 3.64 /vtf (3.37)

The inadequacy of this solution is that even for viscosities ten

times higher than water (i.e., 10-5 m2/s), the predicted boundary layer

thickness over several hours is too large; e.g.,

6 = 0[3.64(10-5.104)1/21 = 0(3.64 m)



Additionally, the approach does not adequately represent the rapid depth

variation in concentration (i.e., increasing concentration with depth) of

the fluid mud.

B. Constant Viscosity Unsteady Bingham Flow. For the steady state

flow of a Bingham plastic with constant viscosity (and constant yield

strength), an analytic similarity solution has been presented by Phan-

Thien (1983). He assumed a two layer system with properties as

t ty (3.39)

where y is the time rate of shearing, z-, and ty is the Bingham yield

strength. Denoting the velocities in layer i (i=1,2) as

ui(z,t) = 2 2/vi Ui(6S) (3.40)

where to is the imposed shear stress, vi, pi are the kinematic and

dynamic viscosities and 6s is the similarity variable.

6s = (3.41)

Ui(5s) is the similarity solution of Eqn. 3.34 given as

6 -z2 -62
S 1 s
[J e0 dz + e ]
U1(6 ) = 6 (i-R ){ _________ (3.42)

c dz


6' -z2 62
r s 1 S
[6 e dz + a ]
U 2(6) = R {6 c } (3.43)

{e dz
where Sc = zc(t)/2/vti7 aS P2/P1 (ag -' for an ideal Bingham fluid),

S; = 56/-/oS, 6' = Sc /aS, and R, = ty/to. Additionally, since the

velocity field is continuous at z = zc(t) and is represented as

U2(6c) = a Ul(6c) (3.44)

a value for Sc is determinable and consequently for zc(t). But,

unfortunately, the inability to treat concentration variation with depth

as was the case in approach A is still undesirable.

C. Variable Viscosity Steady Bingham Flow. Neglecting the obvious

error involved with omission of the pressure gradient term, for the

steady flow from applied shear stress of an ideal Bingham plastic with

concentration dependent yield strength and viscosity, the constitutive

equation is

to = P(z) yto y
y 0, u(z)=0 to < ty

In the region where the yield strength is exceeded, the flow velocity is

analogous to Couette flow with depth varying viscosity. For the region


where the shear strength exceeds the applied shear stress, no motion

occurs (the mud behaves as a solid).

For a solution to Eqn. 3.34 for this case, a concentration

relationship for the viscosity and yield strength must be specified. For

example, one approximation for the viscosity/concentration relationship,

based on data presented in Chapter 2 (see Figure 2-6), is of the form

P(C) = W(1' + SPC) (3.46)

where p, is the viscosity of the suspending fluid (water), and Bp and aP

are empirical constants.

A power law expression for yield strength, ty, (also presented in

Chapter 2, Figure 2-7) is

ry = yC (3.47)

where 0y and ay are empirical constants.

The boundary conditions are 1) an imposed shear stress at the upper

interface, t = to @ z = Za and 2) no-slip at the lower yield elevation,

u = 0 @ z = Zb.

Additionally, for steady flow the depth varying shear stress, ty(z),

is everywhere equal to the imposed interfacial shear stress, to, down to

the stationary interface, zb, where ty = to.

This approach has been presented for comparison only since the

aforementioned error would be appreciable under quasi-steady flows in

estuaries. The absence of time-dependence is also a drawback to this


approach. Depending on the depth of the fluid mud layer and the imposed

shear stress, the velocity profile can take minutes to hours to reach

steady state form. For imposed shear which is continuously changing, as

is the case in tidal flows, steady flow is never reached.

The last two approaches offer the most promise in providing for

realistic spatial and temporal variability of horizontal momentum

diffusion into a fluid mud layer.

D. Variable Viscosity Rayleigh Flow. This approach describes

unsteady flow of a fluid mud layer with depth varying viscosity initially

subjected to horizontal motion at the upper interface, Za. The governing

equation is still Eqn. (3.34) but no specification is made regarding the

overall lower extent of the boundary layer, 6fm* In general, a

numerical solution is warranted. An explicit, finite difference

approximation (with j time and i direction index), for example,

j+l = u + t 1 p(ui u (u- u ) (3.48)
Ai+l i i-l i 1 2

where pi = (pi+l+ Mi)/2, gives an easily obtainable solution path. The

boundary conditions are those given in A. Additionally, proper concerns

for numerical stability and convergence must be addressed (i.e., At

Az2 P/2max)'

The last approach offers the most realistic simulation of

theological and temporal variability of the approaches presented thus


E. Variable Viscosity Unsteady Bingham Flow. A numerical solution

of a form similar to Eqn. 3.48 above is employed for the region where the


mud is sheared. Additionally, the lower interface is tracked by

considering the temporal response of the shear stress and yield strength

at each layer. This is written

u(z) Eqn. (3.48) for t(z,t) t y(C,t)
0 for t(z,t) < ty(C,t)

The boundary conditions are the same as those of C. Comments concerning

the numerical technique, stability and convergence mentioned in D, also

apply for the shear layer here.

With regard to Bingham plastic vs. Newtonian fluid (with viscosity

which varies with concentration) behavior, the data in Figure 2-7 suggest

yield strengths which are sufficiently large to preclude flow under mild

bed shear stress (e.g., C = 100 g/l corresponds to ty = 0.1 N/m2).

However, field and laboratory data used to verify the above approaches in

Chapter 5 (Section 5.5) show evidence of relatively high flows under very

mild imposed shear stress. For this reason, care must be taken in

application of the above approaches using a functional relationship for

yield strength such as Eqn. 3.47, that the empirical coefficients ay and

By fit a particular sediment behavior. It is suggested that non-

Newtonian pseudo-plastic behavior (where viscosity is a function of shear

rate) may be a more reasonable model than Bingham plastic for fluid mud

flows. However, no further supporting arguments or discussion are made

in this report since application of the Newtonian models showed

reasonable results.


4.1 Introduction

Laboratory experiments were conducted at the University of Florida's

Coastal Engineering Laboratory. These experiments consisted of two flume

tests and settling column tests. The flume tests were designed to

evaluate the dynamical effects of wave action on a partially consolidated

natural estuarine sediment bed. Bed erosion (by fluidization) and upper

column suspension concentrations were measured. Settling tests were

performed to obtain the concentration dependent settling properties of

natural flocculating fine sediment. New settling column tests were

devised to provide development and verification data needed for the

vertical profile model.

4.2 Flume Study

4.2.1 Objectives

The objectives of the wave flume study were as follows:

1. To use advanced pressure sensor instrumentation to measure and

document the effective stress breakdown (fluidization) in a

partially consolidated cohesive bed subjected to wave loading.

2. To observe, record and determine factors characterizing fluid

mud formation and stability (during wave erosion) presented in

Chapters 2 and 3.


3. To measure wave resuspension concentrations related to

hydrodynamical data (i.e., wave height, water depth, fluid mud

and bed thickness) for the purpose of verification of the

descriptive vertical transport model.

4. Investigate the role of wave resuspension in the overall

sediment transport process in the prototype setting.

4.2.2 Mud Characterization

The estuarine sediment selected for use in the flume and settling

column studies was mud from Tampa Bay, Florida. Collection was from a

site adjacent to a Hillsborough Bay navigation channel. It was

predetermined by a bay mapping study (City of Tampa, 1986) to be an area

of predominately fine sediment (clay and fine silt) and relatively high

sedimentation rates (0.3-1 m/year). Grain size distribution of dispersed

free particles, obtained by standard ASTM hydrometer method, is shown in

Figure 4-1. It can be seen that d50 = 2.6 pm, which indicates that 50%

of the sediment sample was finer than the upper limit of clay size

particles (2 pm). Furthermore, less than 10% by weight of the sediment

sample was coarse silt to fine sand.

The flocculated sediment was pumped into 55 gallon drums in the

field then to washing and storage tanks in the laboratory. The sediment

was then mixed and decanted several times to equilibrate with tap water

until a slight background salinity (1 ppt) remained. Details of this

procedure can be found in Cervantes (1987). The slight salinity was

sufficient to maintain the flocculated state of the cohesive (< 20pm)

particles. Characterization tests were conducted at the University of

0.001 0.010 0.100
Equivalent Grain Size (mm)

Figure 4-1. Grain Size Distribution of Hillsborough Bay Mud

Florida Soils Science Laboratory. X-Ray diffraction revealed that the

clay size fraction was primarily made up of montmorillinite (91%) and

very small amounts of kaolinite (4%) and quartz (5%). A cation exchange

capacity (CEC) test reported 197.2 meq/100g (an unrealistic and suspected

erroneously high value). Percent organic carbon content, determined by

standard combustion technique (e.g., ASTM 500*C incineration) indicated

that 5% by weight of the sediment sample was of detrital (organic)

origin. Chemical composition of the fluid (tap water) can be found in

Dixit (1982).


4.2.3 Equipment, Facilities and Techniques

Facilities. The flume was a 20 m long plexiglass tank fitted with a

plunging type wavemaker at one end and a sloping beach at the other. The

width and depth dimensions were 48.5 cm and 45 cm, respectively. The mud

bed section was 8 m long with sloping sides to contain the mud during

consolidation. The wave maker had a variable stroke from 5-20 cm and a

variable period down to 0.8 seconds. Cervantes (1987) has presented a

detailed description of the laboratory equipment and procedures used for

the specific purpose of wave resuspension measurements. Figures 4-2a and

4-2b show the flume configuration for the two tests. The sediment trough

slopes were reduced for the second test to help minimize boundary effects

associated with a steep drop in depth (wave reflection and vortex

generation). Vertical concentration profiles were taken at five

locations labeled A-E. Supplemental data were collected to quantify

hydrodynamic parameters, bed response to wave forces, and data pertinent

to the temporal behavior and/or ultimate equilibrium form of the vertical

suspension profile. These are described in detail in the following


Data Acquisition. Intensive simultaneous data collection of spatial

and temporal variability of suspended sediment concentration, bed profile

and density changes, flow and bed kinematics, bed dynamics including pore

and total pressure profiling, spatial variability in wave height and

length, and water temperatures down to the time scale of the wave period

required sensitive data acquisition hardware and software. A sixteen

channel Metrabyte DASH-16 analog to digital interface card installed in

o Suspension Sampling Station
* Wave Gauge Location

z= 14.3 a = 12.3

z = 9.2


a= 6.5

z= 4.0


Z = 14.4 I X = Q.1
Bed Sampling Station 4--
Dimensions in meters

a. Run 1

0 Suspension Sampling Station

* Wave Gauge Location

r = 14.2 x = 12.3 x = 9.2

V T*- 9

a = 6.5 z = 4.6

@0 ,

a = 15.4 Bed Sampler Station z = 4.3 z

Dimensions in meters

b. Run 2

Figure 4-2. Flume Configuration


an IBM personal computer (512k RAM) with interstitial R/C low pass

filters met the requirement in an economical fashion. Maa (1986) has

given details of the data acquisition system. For wave periods of one

second, software was developed to read analog signals from the 16

channels every fifteen minutes for 30 second durations at a 20Hz

frequency. For a ten hour laboratory test, this required more than

393,600 data points to be analyzed per run. Numerical filtering

described by Kassab (1984) was used to minimize noise passed through the

R/C hardware filter and to determine wave mean and rms amplitude values

for each channel monitored. Pertinent data from the two flume studies

are included in a reduced form in Appendix B.

Velocities. Horizontal and vertical velocities at three elevations

(2, 6, 10 cm) above the bed were obtained with a Marsh McBirney (model

523) electromagnetic current meter. The meter was previously calibrated

for time constant and system coefficient for oscillatory flow by Maa

(1986). Care was taken to keep the transducer sufficiently far (- 4 cm)

from the mud bed, water surface and other instruments to prevent density

interface and electromagnetic feedbacks from distorting the data.

Bed Pressures. Bed total and pore pressures were measured at

various elevations below the mud surface so that effective stresses and

bed elevation could be quantified. Pore pressures were measured with

Druck model PDCR81 miniature pore pressure gauges with saturated ceramic

stones with maximum pore diameters of 1 pm. The pore pressure gauges had

a 15 my per psi output voltage operation range. This was too low for the

set range of the data acquisition system (2.5 v). So, the gauges were

fitted with specially designed 100x signal amplifiers. Total pressures