Bottom mud transport due to water waves

UFL/COEL-TR/090

BOTTOM MUD TRANSPORT DUE TO WATER WAVES

by

Feng Jiang

Dissertation

1993

BOTTOM MUD TRANSPORT DUE TO WATER WAVES

By

FENG JIANG

A DISSERTATION PRESENTED TO THE GRADUATE SCHOOL
OF THE UNIVERSITY OF FLORIDA IN
PARTIAL FULFILLMENT OF THE REQUIREMENTS
FOR THE DEGREE OF DOCTOR OF PHILOSOPHY

UNIVERSITY OF FLORIDA

1993

ACKNOWLEDGEMENTS

First of all, I would like to express my deepest gratitude to my advisor and the

chairman of my advisory committee, Professor Ashish J. Mehta, for his construc-

tive direction, advice and support throughout this four year study which has been a

challenging, joyful and unforgettable experience in my life.

I wish to thank Professor Robert G. Dean for his continuously valuable advice,

suggestions and discussions. Thanks also go to the other committee members, in-

cluding Professor Michel K. Ochi, Professor Daniel M. Hanes and Professor Ulrich

H. Kurzweg for their advice, comments and patience in reviewing this dissertation.

Appreciation is extended to all other teaching faculty members in the department for

their creative teaching efforts and inspiration. Technical assistance provided by Mr.

Allen Teeter during the experimental study at WES is specially acknowledged here.

Gratitude is due to Mr. Sidney Schofield, Mr. Jim Joiner, Mr. Chuck Broward

and other staff members in the Coastal Engineering Laboratory for their cooperation

and help during the experimental phase of this research. I am also grateful to Mr.

Subarna Malakar for his technical assistance in dealing with computer facilities. Spe-

cial thanks go to Helen Twedell, Becky Hudson, Cynthia Vey, and Sandra Bivins for

their humor and kindness, which helped directly or indirectly in the completion of

this study.

Support came from many friends and fellow research assistants. Deep appreciation

goes to Dr. Gu Zhihao, with whose help I got through the hard times during the initial

period of my stay in Gainesville. Appreciation is also extended to Paul Work, Jingzhi

Feng and Noshir Tarapore for their assistance and useful discussions.

I am greatly indebted to my wife, Jun Feng, for her love, support, encouragement

and patience; and to my daughter, LiLi Jiang, who grew up together with this study,

for her lovely smiles that made my family life so beautiful. Finally, this work is

dedicated to my parents in China for their great love and support throughout my life.

This study was partially funded by WES Contract DACW39-90-K-0010.

TABLE OF CONTENTS

ACKNOWLEDGEMENTS ....................... .... ii

LIST OF FIGURES .................... ............ vii

LIST OF TABLES .... ............. ............ ...... xiii

LIST OF SYMBOLS .................... ........... xiv

ABSTRACT .................................... xviii

CHAPTERS

1 INTRODUCTION .................... ........... 1

1.1 Problem Statement .................... ........ 1
1.2 Tasks and Scope .................... ......... 4
1.3 Outline of Presentation .......................... 6

2 BACKGROUND .................... ............ 7

2.1 Introduction ................ .. .... 7
2.2 Laboratory and Field Observations ................... 7
2.3 Simulation Models .................... ........ 12
2.3.1 Elastic Model ................... ....... 13
2.3.2 Poro-elastic Model ................... ..... 14
2.3.3 Viscous M odel.................... ...... 16
2.3.4 Viscoplastic Model .... .................. .. 17
2.3.5 Viscoelastic Model ....................... 18
2.4 M ass Transport ................... .......... 21
2.5 Discussion ............................. ...... 23

3 MUD RHEOLOGY ................... .......... 27

3.1 Introduction .. ... .... .. .. .. .. ..... 27
3.2 Viscoelastic Models ......... ................. 28
3.2.1 Voigt Model and Maxwell Model . . . ... 30
3.2.2 Jeffreys Model and Burgers Model . . . .... 30
3.3 Static and Dynamic Tests ........................ 33
3.3.1 Static Test .. .. ... ... .. .. .. .. .. ... 33
3.3.2 Dynamic Test ................... ........ 36

3.5 Discussion .. ... ... ... . . ... .. .. .. ... 48

4 A CONSTITUTIVE MODEL FOR MUD ................. 50

4.1 Mechanical Simulation ......................... 50
4.2 Laboratory Experiments . . . .. . . 53
4.2.1 Test Equipment ..... .. .... ...... ...... 53
4.2.2 Test Samples ........................... 55
4.2.3 Test Procedures and Conditions . . . ... 58
4.3 Results and Analysis ........................... 59
4.4 Summary of Observations ........................ 70

5 WAVE-MUD INTERACTION MODELING . . . .... 79

5.1 Introduction . . . . . . . ..... 79
5.2 A Shallow Water Model ......................... 80
5.2.1 Governing Equations ....................... 82
5.2.2 Solution ................... ......... 85
5.3 A Second Order Model .......................... 90
5.3.1 Boundary Value Problem ..................... 91
5.3.2 Solution Approach ........................ 93
5.3.3 Solution . . . . . . . 97

6 LABORATORY EXPERIMENTS ...................... 101

6.1 Introduction . . . . . . . .. 101
6.2 Experimental Equipment ......................... 101
6.2.1 W ave Flume .................... .... 102
6.2.2 W ave Gage ............................ .. 104
6.2.3 Accelerometer .................... ... 104
6.3 Sediment and Fluid ................... ......... 109
6.4 Test Conditions and Procedures . . . . ... 110
6.4.1 Test Conditions .......................... 110
6.4.2 Procedures ...... ..... .. ....... ..... 111

7 MODEL SIMULATIONS AND ANALYSIS OF LABORATORY RESULTS 127

7.1 Introduction ................... ............ 127
7.2 Model Simulations ................... ....... 127
7.2.1 Model Conditions ........... ....... .. 127
7.2.2 Water Surface Profile ....... .. ....... . 128
7.2.3 Wave Dispersion Relationship . . . ... 128
7.2.4 Wave Attenuation Relationship . . . ... 131
7.2.5 Velocity and Pressure Distributions . . .... 136
7.2.6 Mud Mass Transport ................... .... 138
7.2.7 Interfacial and Surface Set-ups . . . .... 141
7.3 Experimental Results and Analysis . . . ... 146
7.3.1 Wave Attenuation ...................... .. .. 146
7.3.2 Mud Acceleration ...................... .. .. 149

3.4 Previous Studies . .

. . . 4 2

7.3.3 Mass Transport in Mud Layer ..................
7.3.4 Water-Mud Interface Profile . . . . .

8 MODEL APPLICATIONS TO FIELD DATA . . . .

8.1 Introduction .... . .... ... ....
8.2 Lake Okeechobee ....................
8.2.1 Field Measurements ............
8.2.2 Application of the Shallow Water Model .
8.2.3 Application of the Second Order Model .
8.3 Southwest Coast of India . . . .
8.3.1 Field Conditions . . . .
8.3.2 Application of the Second Order Model .

9 SUMMARY AND CONCLUSIONS ............

9.1 Summary and Conclusions . . . .
9.2 Recommendations for Future Studies . . .

. . . 197

197
200

APPENDICES

A MEAN RATE OF ENERGY DISSIPATION . . .

B SOLUTIONS OF THE SECOND ORDER WAVE-MUD MODEL

B.1 First Order Solution ......
B.2 Second Order Solution .....

BIBLIOGRAPHY ............

BIOGRAPHICAL SKETCH ......

202

205

. . 205
. . 208

213

222

LIST OF FIGURES

2.1 Sediment mass transport under wave action (after Lhermitte, 1958). 8

3.1 Typical relationships between shear stress and strain rate in com-
mon constitutive theological models for muds . . .... 29

3.2 Voigt model (left) and Maxwell model (right) . . .... 31

3.3 Jeffreys model (A and B) and Burgers model (C and D). ...... 32

3.4 Responses of the Voigt model and the Maxwell model in creep test. 34

3.5 Responses of the Voigt model and the Maxwell model in relaxation
test . . . . . . . . 35

3.6 Gap between two parallel plates in a pulse shearometer. . 40

3.7 Changing creep-compliance with increasing applied stress (after
James et al., 1987; Williams and Williams, 1989) . .... 44

3.8 Storage modulus, G', and loss modulus, G" of Laponite disper-
sions at different strain amplitudes. Solid symbols denote G", open
symbols G' (after Ramsay, 1985) ................... .. 47

4.1 Typical relationship between 7 (1/min2) and j (1/min) for mud. 51

4.2 A Mechanical Rheological Model. . . . ... 54

4.3 Carri-Med CSL Rheometer: 1) rigid cast metal stand/base; 2) air
bearing support pillar; 3) motor drive spindle; 4) air bearing hous-
ing; 5) drive motor stator; 6) air bearing; 7) optical displacement
encoder; 8) draw rod; 9) shorting strap; 10) inlet/outlet for Peltier
water supply; 11) bottom plate of cone and plate assembly; 12)
automatic sample presentation system; 13) gap-setting microme-
ter wheel; 14) height adjusting micrometer scale; 15) liquid crystal
display; 16) adjustable levelling feet . . . .... 56

4.4 Strain, y, as a function of time, t, and strain rate, y, as a function
of strain, 7, for Lake Okeechobee mud . . . .... 60

4.5 Strain, y, as a function of time, t, and strain rate, -, as a function
of strain, 7, for AK mud .................... 61

4.6 Strain rate, 7, and rate of strain rate, 7, in stage I as functions of
strain for Lake Okeechobee mud. . . . ... 63

4.7 Strain rate, 7, and rate of strain rate, j, in stage II as functions
of strain for Lake Okeechobee mud. . . . ... 66

4.8 Strain rate, 7, in stage I as functions of strain for AK mud. 67

4.9 Slope, S, as a function of density and stress for Lake Okeechobee
m ud. . . . . . . . .. 69

4.10 Instantaneous compliance, Jo, as a function of stress for Lake
Okeechobee mud. ......................... .. 72

4.11 Instantaneous compliance, Jo, as a function of stress for AK mud. 73

4.12 Storage modulus, G', and loss modulus, G", as functions of fre-
quency for different densities and stress amplitudes for Okeee-
chobee mud. ........... ................. 74

4.13 G1, G2 and t as functions of frequency for Lake Okeechobee mud. 75

4.14 G1, G2 and y as functions of frequency for AK mud. ...... 76

4.15 G1, G2 and p as functions of applied stress for Lake Okeechobee
mud. .............. ...... .............. 77

4.16 G1, G2 and p as functions of applied stress for AK mud. . 78

5.1 Schematic of mud bottom response to wave in terms of vertical
sediment density and velocity profiles (after Mehta, 1989). . 81

5.2 Sketch of water-fluid mud system subjected to progressive wave
action. . .. . . . . . .... 81

6.1 Elevation view of the wave flume used in the tests. . ... 103

6.2 Two views of the accelerometer (Vibro-Meter, Model CE508). 105

6.3 Views of two 'boats' used for the calibration of the accelerometer. 106

6.4 Examples of spectra for surface elevation time-series and corre-
sponding acceleration signal time-series; f = 1 Hz. . ... 108

6.5 A view of the neutrally buoyant 'boat' in the mud layer in the flume.109

6.6 Calibration coefficient, Cf, against frequency, f, for the accelerom-
eter. . . . . . . . . 110

6.7 An example of a mud density profile for AK mud. . ... 117

6.8 A typical measured wave envelope (R106; f=l Hz). . ... 118

6.9 Example of variation of wave height with wave propagation dis-
tance in the flume for eight tests (R117-R124; f=1.05 HZ). 119

6.10 Displacement of dyed mud 1 minute after initiation of wave action
in the flume (R305); the height shown was 15 cm. . ... 122

7.1 Surface wave profile; G1 = 150 Pa, G2 = 30 Pa, p = 10Pa.s,
hi = 16cm, h2 = 12cm, f = 0.57 Hz; dashed line: first order
result; solid line: second order result. . . . ... 129

7.2 Dimensionless wave number, khl, against angular frequency, a (rad/s);
dashed line: Airy linear wave theory; solid line: G1 = 150 Pa,
G2 = 30 Pa, p = 10 Pa.s, hi = 16 cm, h2 = 12 cm. . ... 131

7.3 Phase difference between the pressure and the vertical velocity at
the interface as a function of angular frequency, a (rad/s); G1 =
150 Pa, G2 = 30 Pa, p = 10 Pa.s, hi = 16 cm, h2 = 12 cm. . 132

7.4 Wave attenuation coefficient as a function of angular frequency,
a (rad/s), for a viscous mud and a viscoelastic mud; hi = 16 cm,
h2 = 12cm; dashed line: G1 -+ oo, G2 = 0.; solid line: G1 =
200 Pa, G2 = 30 Pa, p = 10 Pa.s. . . . ... 134

7.5 Wave attenuation coefficient as a function of angular frequency,
a (rad/s), for different G1; G2 = 30 Pa, p = 10 Pa.s, hi = 16 cm,
h2 = 18cm; Curve A: G1 > 3000Pa; Curve B: G1 = 300Pa;
Curve C: G1 = 150 Pa. ....................... 135

7.6 Horizontal velocity profiles as a function of wave frequency; a =
3.0cm, G1 = 150 Pa, G2 = 30 Pa, p = 10Pa.s, hi = 15cm,
h2 = 15cm; Curve A: o = 3.5rad/s; Curve B: a = 6.0rad/s;
Curve C: o- = 10.radls. ....................... 137

7.7 Vertical distribution of dynamic (normalized) pressure, P/pga;
f = 0.57Hz, a = 2.5cm, G1 = 150Pa, G2 = 30Pa, p =
1.26 Pa.s, hi = 15 cm, h2 = 10cm. . . . ... 138

7.8 Model simulation of the vertical distribution of mud transport
velocity for AK mud: a = 3.5 cm, f = 1.0 Hz, hi = 14 cm, h2 =
17 cm; dashed line represents first order result; solid line is second
order . . . . . . . . 140

7.9 Sketch showing forces in a two-layered, wave-forced system. 142

7.10 Measured attenuation coefficient, ki, against angular frequency,
o (rad/s), for the AK mud; asterisk sign: hi = 16 cm, h2 = 18 cm;
plus sign: hi = 16cm, h2 = 12cm. ................. 147

7.11 Measured attenuation coefficient, ki, against angular frequency,
a (rad/s, for hi = 16 cm, h2 = 12 cm; asterisk sign: ABK mud;
plus sign: AK mud. ......................... 148

Measured attenuation coefficient, ki, against wave height for AK
mud; f = 1.05 Hz, hi = 16 cm, h2 = 18 cm . . .

Attenuation coefficient, ki, against angular frequency, ao (rad/s),
for AK mud: theoretical results represented by solid curves; mea-
sured data by asterisk (hi = 16 cm, h2 = 18 cm) and plus (hi =
16cm h2 = 12cm ) ........................

Acceleration profile for AK mud: a = 1.0 cm, f =
16cm, h2 = 18cm; solid curve: theoretical result;
measured data.....................

Acceleration profile for AK mud: a = 1.0 cm, f =
16 cm, h2 = 18 cm; solid curve: theoretical result;
measured data.....................

Acceleration profile for AK mud: a = 0.5 cm, f =
16 cm, h2 = 18 cm; solid curve: theoretical results;
measured data.....................

Acceleration profile for AK mud: a = 1.0 cm, f =
16 cm, h2 = 18 cm; solid curve: theoretical results;
measured data.....................

Acceleration profile for AK mud: a = 1.5 cm, f =
16 cm, h2 = 18 cm; solid curve: theoretical results;
measured data.....................

0.5 Hz, hi =
asterisk sign:

0.9 Hz, hi =
asterisk sign:

1.0 Hz, hi =
asterisk sign:

1.0 Hz, hi =
asterisk sign:

1.0 Hz, hi =
asterisk sign:
. . .

7.14

7.15

7.16

7.17

7.18

7.19

7.20

7.21

7.22

7.23

7.24

7.25

8.1

7.12

7.13

Vertical distribution of mud mass transport velocity for AK mud:
a = 1.25 cm, f = 0.5 Hz, hi = 14 cm, h2 = 17 cm. . . .

Vertical distribution of mud mass transport velocity for AK mud:
a = 2.0 cm, f = 0.5 Hz, hi = 14 cm, h2 = 17 cm . . .

Vertical distribution of mud mass transport velocity for AK mud:
a = 2.0 cm, f = 1.0 Hz, hi = 14 cm, h2 = 17 cm . . .

Vertical distribution of mud mass transport velocity for AK mud:
a = 3.0cm, f = 1.0 Hz, hi = 14cm, h2 = 17cm . . .

Vertical distribution of mud mass transport velocity for AK mud:
a = 1.25 cm, f = 0.5 Hz, hi = 14cm, h2 = 17cm; dashed line:
p = 1.22 kg/m3; solid line: p = 1.18 kg/m3.........

Measured time-variation of wave-mud interface slope along flume
for AK mud: a = 1.5 cm, f = 1.0 Hz, hi = 16 cm, h2 = 17.3 cm..

Equilibrium water-mud interfacial slope along flume for AK mud:
theoretical simulation represented by the curve; measured data by
dots . . . . . . . ..

Bathymetric map of Lake Okeechobee, Florida. . . .

8.2 Wave spectral density (cm2.s) as a function of frequency at 1 hour. 165

8.3 Wave spectral density (cm2.s) as a function of frequency at 5 hours.166

8.4 Dynamic viscosity relative to water, //u,, as a function of density
for Okeechobee mud.......................... 167

8.5 Dispersion relationship from the shallow water model. . 170

8.6 Wave attenuation relationship from the shallow water model. 171

8.7 Predicted profiles of velocity amplitude, Un, by the shallow water
model. ..................... ........... 171

8.8 Mean rate of wave energy dissipation, Ed (kg/s3), obtained by the
shallow water model ...................... 172

8.9 Predicted profiles of the phase (radian) of velocity amplitude, Un,
using the shallow water model. . . . . .... 172

8.10 Water velocity spectral density (cm2/s) as a function of frequency
at 1 hour; solid curve: measurement; dashed curve: simulation
using the shallow water model. ................... 175

8.11 Mud acceleration spectral density (cm2/s3) as a function of fre-
quency at 1 hour; solid curve: measurement; dashed curve: simu-
lation using the shallow water model. . . . ... 176

8.12 Water velocity spectral density (cm2/s) as a function of frequency
at 5 hours; solid curve: measurement; dashed curve: simulation
using the shallow water model. . . . . ... 177

8.13 Mud acceleration spectral (cm2/s3) as a function of frequency at 5
hours; solid curve: measurement; dashed curve: simulation using
the shallow water model .................... 177

8.14 Water velocity spectral density (cm2.s) as a function of frequency
at 1 hour; solid curve: measurement; dashed curve: simulation
using the second order model. . . . ...... 178

8.15 Mud acceleration spectral density (cm2/s3) as a function of fre-
quency at 1 hour; solid curve: measurement; dashed curve: simu-
lation using the second order model. . . . ... 179

8.16 Water velocity spectral density (cm2/s) as a function of frequency
at 5 hours; solid curve: measurement; dashed curve: simulation
using the second order model. . . . ...... 179

8.17 Mud acceleration spectral density (cm2/s3) as a function of fre-
quency at 5 hours; solid curve: measurement; dashed curve: sim-
ulation using the second order model . . . .... 180

8.18 Mud acceleration spectral density (cm2/s3) as a function of fre-
quency at 1 hours; solid curve: measurement; long-dashed curve:
simulation using the shallow water model; short-dashed curve:
simulation using the second order model. . . .... 181

8.19 Bottom topography off Alleppey, Kerala, India. . ... 184

8.20 Schematic profile of a mudbank region (after Nair, 1988). Note
the 'exponential' shape of the mudbank profile . .... 184

8.21 Conceptual model of mudbank dynamics in response to variable
coastal wave action (after Mathew, 1992). . . .... 185

8.22 Typical non-mudbank monsoonal wave spectra (after Mathew,
1992) . . . . . . . ... 186

8.23 Comparison between synchronous nearshore and offshore wave
spectra without a mudbank (after Mathew, 1992). . ... 187

8.24 Mudbank wave spectra (after Mathew, 1992). . . .... 188

8.25 Comparison between synchronous offshore and mudbank wave spec-
tra (after Mathew,1992). ...................... 189

8.26 G1, G2 and p as functions of frequency for Alleppey mud. . 190

8.27 Computed attenuation coefficient against wave frequency for the
Alleppey field condition ........................ .. 191

8.28 Comparison between model calculated and measured nearshore
wave spectra (m2/s) for 7/27/1987 in the absence of the mudbank. 192

8.29 Comparison between model calculated and measured nearshore
wave spectra (m2/s) for 7/1/1989 (1200 hours) in the presence of
the mudbank .......................... 192

8.30 Comparison between model calculated and measured nearshore
wave spectra (m2/s) for 7/2/1989 (1200 hours) in the presence of
the mudbank .......................... 193

8.31 Comparison between model calculated and measured nearshore
wave spectra (m2/s) for 7/3/1989 (1200 hours) in the presence of
the mudbank .......................... 193

8.32 Hypothesized progression of mudbank profile toward the shoreline. 196

LIST OF TABLES

Chemical Compositions of Attapulgite and Kaolinite

Chemical Composition of Water . .

Mud Composition in the Flume Tests .....

Chemical Composition of Bentonite . .

Wave Attenuation Tests for Bed Material AK .

Wave Attenuation Tests for Bed Material AK .

Wave Attenuation Tests for Bed Material ABK

Mud Acceleration Tests for Bed Material AK .

Mud Mass Transport Tests ...........

Wave Attenuation Coefficient Data . .

Mud Acceleration Data .............

Mud Acceleration Data (Cont'd) . .

Mud Mass Transport Data ...........

Interfacial Elevation Change with Time .

6.13 Interfacial Elevation Change with Time (Cont'd

8.1 Muddy coast physical parameters for Louisiana
nam and Korea (after Wells, 1983; Mathew, 199

. . . 58

. . . 110

. . . 111

. . . 112

. . . 113

. . . 114

. . . 115

. . . 115

. . . 123

. . . 124

. . . 125

. . . 125

. . . 126

) . . 126

, SW
'2) .

India, Suri-

4.2

6.1

6.2

6.3

6.4

6.5

6.6

6.7

6.8

6.9

6.10

6.11

6.12

LIST OF SYMBOLS

Ao = Wave amplitude at z = 0

Ac = Acceleration

As = Output of the accelerometer in my

ao = Wave amplitude at x = 0

a, = Wave amplitude at any distance x

b = Interfacial amplitude at x = 0

Cf = Proportionality constant

Cg = Group velocity

Co = Amplitude of the vertical acceleration of the water surface

d = Distance between two plates in a pulse shearometer

E = Young's modulus

Eo = Initial wave energy

Em = Energy term in Equation 7.25

E, = Energy term in Equation 7.26

f = Wave frequency (Hz)

Fr = Froude number

G = Shear modulus of elasticity

g = Gravitational acceleration

Go = Instantaneous shear modulus

G' = Storage modulus

G" = Loss modulus

G* = Complex shear modulus

h = Water depth

hi = Depth in water layer

h2 = Mud thickness

h; = Elevation of the water-mud interface

Jo = Instantaneous compliance

Ip = Intrinsic permeability coefficient

k = Complex wave number

ki = Wave attenuation coefficient

kr = Wave number

Mf = Momentum flux

n = Volumetric porosity

P = Dynamic pressure

Pb = Maximum dynamic pressure at the interface
ph = Hydrostatic pressure

PJ = Total pressure

P'') = First order dynamic pressure
P,2) = Second order dynamic pressure

PJ(1 = First order total pressure

i( = Second order total pressure
Vp = Gradient of the pore pressure

' = Discharge velocity

r = Normalized density jump

Re = Reynolds number

R,, = Wave Reynolds number

S = Slope

Su,,1 = Water velocity spectrum

Suw, = Mud acceleration spectrum

Sn = Surface wave energy spectrum
Te = Elapsed time (hour)

ul = Wave-induced velocity in water column

u2 = Wave-induced velocity in mud layer

Ub = Maximum particle velocity

u; = Velocity in the x; direction

uj = Velocity in the xj direction

um = Wave-induced velocity amplitude

iP) = Horizontal component of the first order orbital velocity

iL(2) = Horizontal component of the second order orbital velocity

UL = Mass transport velocity

v = Shear wave phase velocity

0(1) = Vertical component of the first order orbital velocity

b(2) = Vertical components of the second order orbital velocity

x = Horizontal axis located at the mean water level or the rigid bottom

z = Vertical axis located at the mean water level or the rigid bottom

ai = Material parameter

#2 = Positive constant

#i = Material parameter

I = Complex wave propagation constant

7 = Strain

7 = Strain amplitude

7 = Strain rate

- = Rate of strain rate

6 = Phase angle between shear stress and shear strain

Ed = Mean rate of energy dissipation

(b = Maximum horizontal excursion of the water particle

7/0 = Wave amplitude

7r1 = Surface variation about its mean value

772 = Interface variation about its mean value

It = Dynamic viscosity

Pa = Apparent viscosity

I/* = Complex viscosity

/ej = Equivalent dynamic viscosity
v = Kinematic viscosity; Poisson's ratio

Vj = Equivalent kinematic viscosity

vt = Eddy viscosity

Pi = Water density

P2 = Mud density

p, = Density of elastic bed

0r = Angular frequency

7 = Shear stress

T = Stress amplitude

T0 = Yield stress

ri = Inertia

Tij = Stress in the xj direction in the plane perpendicular to xi

7y = Apparent yield stress

X = Normalized mud layer thickness

xvii

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

BOTTOM MUD TRANSPORT DUE TO WATER WAVES

By

Feng Jiang

May, 1993

Chairman: Dr. A. J. Mehta
Major Department: Coastal and Oceanographic Engineering

The interactions between surface water waves and a soft mud bottom have been

investigated in this study through mathematical modeling, laboratory experiments

and model applications to field observations.

Based on the viscoelastic theory and laboratory tests, a constitutive model has

been developed to describe mud rheology. The model simulates observed strain creep

and stress relaxation responses, and it is shown that the viscoelastic properties of

mud depend strongly on wave frequency. The developed model is reduced to the

commonly used but more restrictive Voigt model at comparatively high frequencies.

Such frequencies however are of uncommon interest in relevant coastal environments.

A second order hydrodynamic model has been developed to examine important

wave-mud interactions. The water column is considered to be viscid and mud to

be viscoelastic. Due to the second order consideration, the usual restriction of a

small wave amplitude is removed in the model. In conjunction with the developed

constitutive model, the hydrodynamic model predicts surface wave attenuation, mud

acceleration and mass transport. These predictions have been compared with labora-

tory experiments, and agreements found to be satisfactory. To further demonstrate

the applicability of the model, it has been applied to predict wave attenuation over

xviii

coastal mudbanks. Comparison with measured wave spectra shows reasonable agree-

ment. Wave-related data from Lake Okeechobee, Florida have also been simulated

with a reasonable degree of success.

Based on a force balance, a theoretical expression for the water-mud interface set-

up has been developed. The predicted equilibrium interface profile generally agrees

with that measured in the laboratory. Further, a hypothesis regarding the time-

evolution of the interface has been examined as a mechanism to explain the observed

cross-shore motion of the above mentioned coastal mudbanks.

CHAPTER 1
INTRODUCTION

1.1 Problem Statement

Problems associated with the movement of cohesive sediment or mud in coastal

seas, estuaries, reservoirs and shallow lakes are of increasing interest in recent years

due to various concerns of the ecological environment, coastal flooding, shoreline

protection, port sedimentation, structural failure and so on. With respect to the

latter, for instance, two of Shell oil company's platforms in the South Pass off the

Mississippi River Delta, where the bottom was composed of mud, were toppled during

Hurricane Camille (Sterling and Strohbeck,1973). It was suggested that the problem

may have been caused by slope failure or bottom fluidization initiated by hurricane-

generated waves. Since that event, interest in understanding the interactions between

water waves and soft mud bottoms has apparently increased significantly.

At many shallow marine sites where mudbanks occur, such as the northeastern

coast of South America and the southwestern coast of India, wave attenuation is found

to be tremendous when waves propagate toward the shoreline in the presence of a

compliant, deformable bottom (Mathew, 1992). One of the positive consequences of

the drastic wave energy dissipation is that the corresponding beach is often signifi-

cantly protected from wave erosion even during storm events, while nearby beaches

with sandy or rigid bottoms suffer from severe wave activity (Nair, 1988). On the

other hand, soil mud having the potential to absorb contaminants can be transported

under the action of waves and currents to undesirable locations such as recreational

beaches, navigation channels and harbors, thus causing serious environmental prob-

lems. It is usually quite expensive to remove the deposited, contaminated mud from

2
channels and harbors by dredging or other engineering operations. Therefore, there

is a strong technical need to have the capability to predict and examine in advance

the response of the wave-mud systems once any changes to the system, such as storm

impacts or coastal oil spills, occur in a region where the bottom is composed of a cohe-

sive soil, thus providing efficient ways to control both the physical and the ecological

impacts to the region.

While mud deposition and erosion by currents have been studied extensively by

many investigators in the past, only limited knowledge of mud dynamics under the

action of waves, which often constitute the dominant forcing in the coastal seas and

estuaries, has been gained. Generally, in the process of wave-mud interaction, surface

water waves can cause the bottom mud to oscillate and be advected, and in turn the

bottom mud can affect the wave climate by absorbing and dissipating wave energy.

Interfacial phenomena including diffusion and entrainment of sediment as well occur

in the meantime. However, the rates of upward mass diffusion and particulate entrain-

ment above the wave boundary layer, which is usually thin, are generally small and

typically result in comparatively very low concentrations of suspended sediment over

most of the water column. It has in fact been found, based on laboratory experiments

and field observations, that wave loading required to generate measurable bottom mo-

tion can be quite moderate compared with that required to cause significant upward

diffusion or entrainment (Suhayda, 1986; Ross, 1988). Hence, the wave-bottom mud

interaction problem can be conveniently treated to be practically unaffected by the

resuspension of bottom sediment (van Rijn, 1985; Maa and Mehta, 1987). Thus, for

example, the velocity field in the wave-mud system can be first determined without

the consideration of interfacial diffusion and entrainment, and such a velocity field

can be used later as a given forcing for the resuspension process.

It has further been realized that the rheological properties of mud play a critical

role in governing the wave attenuation rate, hence mud mass transport and so on

3
(Mehta, 1991). However, the theological behavior of mud is fairly complicated and

not well understood at present, although it is generally agreed that mud has both

viscous and elastic behaviors (Maa, 1986; Chou, 1989). The Voigt model and the

Maxwell model have been commonly used to represent the constitutive viscoelastic

behavior. Unfortunately, however, it has been shown that neither model is suitable for

representing mud behavior accurately (Chou, 1989). Therefore a better description

of mud theological behavior is naturally one of topics in any wave-mud interaction

study, such as this one.

When waves enter an intermediate or shallow water zone, the bottom starts to

respond dynamically to surface forcing. In the case of a mud bottom or mudbank,

as mentioned above, mud mass transport, in general strongly dependent upon wave

height and mud properties, occurs as a result of wave-mud bottom interaction. With

regard to mudbank motion, field observations have been made, for example, in the

southwest coast of India in recent years (Mathew, 1992). While there is a tendency for

longshore mud transport, which is believed to be a result of wave refraction, mudbank

activity occurs primarily in the cross-shore direction, and it is believed that the time-

evolution of the mudbank surface is determined by the mud mass transport in the

cross-shore direction, which is therefore the focus of attention in this study.

In the traditional linear wave theories, wave amplitude is restricted to be a small

value, which is typically violated in reality. Since the mass transport rate is a non-

linear quantity with respect to the wave height, the actual amount of mass transport

produced by surface waves with finite amplitudes may be significantly different from

the prediction based on a linear theory. Therefore, the ultimate objective of this work

is the physical understanding and accurate prediction of wave attenuation, mud mass

transport and other high order quantities such as setups of the wave-mud interface

and the mean water level. This objective is pursued here through the development of

a second order wave-mud interaction model, coupled with an improved mud rheologi-

4
cal model, so that the usual limitations of the small amplitude wave can be removed,

and a proper description of mud properties taken into account. Following the theo-

retical development, it is naturally necessary to verify the model through laboratory

experiments. As one of the major purposes, the verified model is then shown to be

applicable to field situations, through comparisons between model results and field

data. Based on a theoretical analysis and laboratory observations, the mechanism of

mudbank cross-shore motion is finally examined as the last goal of this study.

1.2 Tasks and Scope

The specific tasks can be described as follows:

1. As an initial exercise, a simple shallow water wave-mud interaction model is

developed, in which the water column is assumed to be inviscid and bottom

mud to be viscous. As shown, such a model can be conveniently applied, given

the shallow water assumption, to water areas such as margins of large lakes or

coastal seas with relatively small water depths.

2. In order to eliminate one of limitations of above simple model, namely that mud

is a purely viscous fluid, an improved constitutive model is developed for better

describing the theological properties of mud under oscillatory action.

3. A second order wave model, in which restrictions of small water depth and small

amplitude wave inherent in the shallow water wave model are removed, is de-

veloped in conjunction with the improved mud theological model, to accurately

simulate the mud response especially for the nonlinear quantities such as mass

transport.

4. The theoretical model results are examined and verified through comparisons

with laboratory flume measurements.

5
5. Selected field observations are utilized to investigate the applicability of the

developed model.

6. A mechanism for cross-shore motion of mudbank is proposed through theoretical

analysis and laboratory observations.

To achieve the above goals, the scope of this study is therefore defined as follows:

1. In the development of a second order wave model, the water column has been

considered to be viscid and mud layer to be viscoelastic. The model includes

second order effects of kinematic and dynamic boundary conditions which allows

a finite wave amplitude to be considered.

2. The developed mud theological model is based on considerations of linear vis-

coelastic properties and laboratory observations, combined with mechanical sim-

ulations.

3. Two different types of mud including a mixture of commercial attapulgite and

kaolinite clays, and soil samples collected from the bottom of Lake Okeechobee,

Florida, are used in theological tests which were performed with a controlled

stress rheometer at the U.S. Army Engineer Waterways Experiment Station

Hydraulics Laboratory in Vicksburg, Mississippi.

4. In order to verify the second order wave model, laboratory wave flume experi-

ments were carried out, in which the clay mixture, monochromatic nonbreaking

waves and tap water as fluid are used. Wave attenuation, mud acceleration,

mud mass transport and the variation of the water-mud interface along the

wave propagation direction were recorded and examined.

5. The application of the theoretical model has been examined through previously

obtained field observations. Two selected field areas for the investigation of

wave-mud problem are Lake Okeechobee, and the southwest coast in India.

6
While Lake Okeechobee is a shallow water, wave-dominated environment with a

soft mud bottom, the southwest Indian coast is an open water area of mudbanks

which are active especially during the monsoon season.

1.3 Outline of Presentation

Chapter 2 reviews previous field and laboratory observations and dynamic sim-

ulation models based on five basic theological constitutive relationships related to

wave-mud interaction. Prior investigations on mass transport, an important result of

wave-bottom interaction, is also discussed in this chapter. The bases of mud rheology

are presented in Chapter 3, which introduces linear viscoelastic models, static and

dynamic tests and past studies. In Chapter 4, a new theological model is developed

and relevant parameters are determined through laboratory experiments. The shal-

low water model and the second order model are developed in Chapter 5. Chapter 6

describes laboratory experiments for the verification of the second order model and

presents corresponding data on wave attenuation, mud acceleration and mud mass

transport. Results and analysis of laboratory experiments, including the evolution of

water-mud interface, are presented in Chapter 7. Applications of the theoretical mod-

els to some field data from Lake Okeechobee and the coastal area of southwest India

are presented and, further, a hypothesis regarding mudbank motion in the cross-shore

direction is proposed on the basis of theoretical analysis and laboratory observations

in Chapter 8. The last chapter includes a summary, conclusions and recommendations

for future studies.

CHAPTER 2
BACKGROUND

2.1 Introduction

Surface wave disturbed water column together with a deformable bottom consti-

tutes a coupled system in which wave-bottom interaction can occur, and in it the

constitutive behavior for the bottom material plays an important role in governing

the response of the system. Laboratory and field observations, and theoretical work

done in the past in this area are reviewed in this chapter. Five basic constitutive rheo-

logical models for bottom material, namely elastic, poro-elastic, viscous, viscoplastic,

and viscoelastic models are described. For muds the viscoelastic model is believed

to be a relatively better choice, in which effects of both elasticity and viscosity are

included in keeping with the rheology of typical cohesive sediments. As a result of

wave action and the response of mud bottom, mass transport in the bottom layer,

which has been one of the important research topics of interest in recent years, is also

discussed.

2.2 Laboratory and Field Observations

Ewing and Press (1949) reported that a type of wave motion in soft bottom

sediments is likely to occur under intermediate and shallow water conditions, and

that it would affect the characteristics of the overlying surface waves. Lhermitte

(1958) later, based on laboratory observations, indicated that cohesive bottom silt

may loose its shear resistance due to the penetration of surface wave action into

the bottom, and therefore sediment transport may occur by wave-induced residual

velocity, even without a tidal current (see Figure 2.1).

Figure 2.1: Sediment mass transport under wave action (after Lhermitte, 1958).

9
Migniot (1968) demonstrated that in laboratory flume tests, wave-induced orbital

motion does occur in the mud layer and that soft layers can exhibit properties similar

to those found in fluids. He also observed that muds have an 'initial rigidity', which he

defined as the yield strength. Mitchell et al. (1972) carried out laboratory experiments

to evaluate the stability of slopes of submarine bentonite clay under wave forcing. It

was shown that wave action could initiate slope failure in bottom sediments. The

investigators suggested that the mechanism of failure may involve strength reduction

due to cyclic shearing stresses induced by wave action. As an efficient way to examine

the response of a silt bed to wave loading, laboratory measurements of pore water

pressure were made by Clukey et al. (1983). It was observed that under the action

of waves, the pore water pressure rose continuously and eventually resulted in the

liquefaction of the silt. The investigators suggested that wave conditions required for

liquefaction of normally consolidated silty marine sediments at low overburden stresses

can be substantially lower than that required for sand liquefaction, and therefore,

utilization of data for sands could lead to an unrealistic design practice in offshore

regions composed of silty materials.

Maa and Mehta (1987) showed through laboratory observations that the wave

orbits could penetrate the mud bed, thereby leading to deformations of the upper part

of the otherwise stationary bed. Likewise, laboratory results by Doyle (1973) showed

that bottom sediment movement varies considerably with depth below the water-

mud interface, and that wave-induced bottom pressures were considerably altered

from those expected over a rigid bottom. Ross and Mehta (1990) indicated, based on

laboratory experiments, that wave conditions required to generate measurable bottom

motion can be quite moderate compared with conditions required to cause significant

particulate erosion, and therefore the contribution of the latter to resuspension is

often negligible.

10
Tzang et al. (1992) extended the work of Clukey et al. in greater detail. It was

found that a large increase in the mean value of the pore pressure above its hydrostatic

value was accompanied with a substantial jump in the energy level of pressure for a

certain wave frequency, which indicated liquefaction of the soil bed. It was suggested

that the significant amplification in the pore pressure amplitude for the certain wave

frequency is related to the cavity structure of soil, and when the wave frequency

approaches one of the natural frequencies of the cavity, internal resonance occurs,

which in turn leads to liquefaction.

Feng (1992) measured both total pressure and pore water pressure in the labo-

ratory to examine the fluidization of mud layer by water waves. It was shown that

the rate of bed fluidization was dependent on the rate of input of wave energy, hence

on the wave height. It was also found that the time of bed consolidation prior to

initiation of wave action was an important controlling factor in fluidization process.

Jiang and Zhao (1989) studied solitary waves travelling over fluid-mud seabeds.

The experimental data showed that solitary waves attenuated much more quickly

than the result of the theory for rigid smooth bed by Keulegan (1948). Sakakiyama

and Bijker (1989) further examined the relationship between wave attenuation and

wave height based on their experimental data. It was reported that wave attenuation

decreased as the surface wave height increased.

While evidences of wave-mud bottom interaction have been studied extensively

in the laboratory, useful field observations have also been carried out by many inves-

tigators. For instance, Bea et al. (1973) and Bea (1983) reported that storm waves

can cause massive mud flows and that hurricane-generated wave heights dramati-

cally decreased from 20-23m in deep water to 3-4m in water depths of 12-21m in the

Mississippi River Delta area.

Tubman and Suhayda (1976) and Suhayda et al. (1976) conducted field experi-

ments in East Bay, Louisiana, in order to observe the calming effect on the surface

11
waves due to energy absorption by bottom which was composed of fine-grained sedi-

ments. Wave characteristics and the resulting bottom mud oscillations were measured

simultaneously. The investigators found that bottom oscillations on the order of 2-

3cm in amplitude occurred for seas having a significant wave height of 0.9m and a

period of 5 seconds. The results further showed that a relatively much greater amount

of wave energy was dissipated on a muddy coast at intermediate water depths than on

a sandy coast, based on the analysis of field data. Suhayda (1977) further indicated

that measured bottom pressures were up to 35% larger than predicted by the linear

wave theory, and out of phase with bottom mud motion by about 200 degrees.

Forristall and Reece et al. (1980) and Forristall and Reece (1985) obtained mea-

surements of wave attenuation and bottom motion as the waves travelled from deep

water to relatively shallow water, at platform VV off the Mississippi Delta. Very high

rates of wave attenuation were observed. Spectral comparisons showed that theoreti-

cally calculated refraction and shoaling could explain the change in the spectra when

the wave height was low, but as the wave height increased, a non-linear attenuation

mechanism became strong in the sense that attenuation was a strong function of the

deep-water wave height. The investigators claimed that wave frequency was also an

important variable, even though it seemed that the attenuation was a weak function of

frequency in that case. A phase lag between bottom motion and the surface wave was

found. The results also showed that the magnitude of the transfer function between

surface wave and bottom motion was lower than the value predicted by linear wave

theory, and that this variation increased as the frequency became lower. A phase

difference of 180 degrees between the pressure near the bottom and bottom vertical

motion indicated that the bottom moved down as the pressure increased.

Wells and Coleman (1981) reported simultaneous measurements of waves, tide

elevation, suspended-sediment concentration and variations in mud density off the

coast of Surinam, South America. The results showed that soft intertidal and subtidal

12
muds were suspended at both tidal and wave frequencies. The strong attenuation

of shallow-water waves by mud provided conditions that were favorable for further

sedimentation. Wells and Kemp (1986), based on data from Louisiana and Surinam,

further indicated that interactions between surface waves and cohesive sediments

might govern the processes in areas of recent and future erosion and accretion in

open coast environments, and could lead to extraordinary high rates of sediment

transport, even under relatively weak currents.

2.3 Simulation Models

Prior to the late 1950's, wave energy attenuation was primarily considered in

connection with bottom friction, percolation, wave refraction and shoaling. It was

commonly accepted that the friction coefficient is 0.01 for a sandy bottom, and greater

than 0.01 for a mud bottom. Water wave attenuation due to percolation of water

through porous beds has been treated by Putnam (1949) and Reid and Kajiura (1957),

and later by Liu (1973). In those studies, the bed is assumed to be a rigid porous

material so that the dispersion relationship of water wave remains the same as that

for a smooth rigid bottom.

Bretschneider and Reid (1954) presented solutions in graphical form for obtaining

wave attenuation by bottom friction, percolation, refraction and shoaling by consid-

ering the value of friction coefficient to be greater than 0.01 for mud bottom. It must

be pointed out however, that mud bottom is generally not rigid but deforms under

wave loading, and there is dissipation of wave energy due to internal friction. Conse-

quently, instead of bottom friction, the theological properties of bottom mud play a

critical role in governing the rate of wave energy dissipation.

Based on different theoretical considerations or assumptions, basically five rhe-

ological models (noted next), describing the deformation properties of soil bottom

material exist. In the following contents, the word 'model' is used to mean a certain

constitutive relationship between shear stress and strain for the bottom material.

2.3.1 Elastic Model

The equation relating stress and strain in a linearly-elastic isotropic material, i.e.,

Hooke's law, can be written as

c, = [(1 + v)ao,- v (oax + ayy + azz)]/E

yy = [(1 + v)ayy v (o.. + oyy + z)]/E

,zz = [(1 + V)a,, v(Ox. + oy,, + cz )]/E

-xy = 7y / G, 7Y- = -Z / G, 7~y = Ty/ G (2.1)

where the Young's modulus, E, Poisson's ratio, v, and the shear modulus, G, are

related by
E
G = (2.2)
2(1 + v)

and E~, ey, y y,, 7xz, and y7, are the six components of the strain tensor, while

axx, Oyy7, azz y, -z, and yz, are the six components of the corresponding stress
tensor.

Based on the consolidation theory by Biot (1941) for a 2-D, non-porous, per-

fect elastic soil, Mallard and Dalrymple (1977), presented a linear analytic solution

for a periodic water wave passing over a deformable bottom in which stress equilib-

rium within the soil was assumed. Soil stresses and displacements along with related

water wave kinematics were obtained. The analytic solutions demonstrated that a

deformable bottom has a significant effect on water kinematics and dynamics, e.g.,

the wave number increases as the shear modulus of mud decreases. Dawson (1978)

added the effects of soil inertia to the above problem in which the soil beneath wa-

ter was regarded as an elastic solid. He claimed that such effects cannot generally

be ignored. However, wave energy dissipation was not taken into account in either

solution.

Chen (1983) theoretically examined the response of an elastic bed to water waves

on the basis of small-amplitude wave theory. It was shown that bed response to waves

14
was strongly dependent on the dimensionless shear modulus

G -G
G =(2.3)
Ps 92

of the elastic bed, where a is the angular frequency, p, the density of elastic bed, and

g the gravitational acceleration.

Foda (1989) considered a compliant bed to be elastic having inhomogeneous prop-

erties with a vertically stratified profile. He extended the theory of sideband instability

of gravity waves over a rigid bed to the case of wave propagation over a soft bed. It

was shown that the carrier gravity wave would lose energy to sideband oscillations

which could be contaminated with small amplitude but very short, elastic waves, and

that these shear waves would interact with the viscous boundary layer at the bed-

water interface, thus significantly enhancing the viscous attenuation of wave energy

in the boundary layer.

2.3.2 Poro-elastic Model

Poro-elastic material has both elastic and porous properties. Generally the flow

in the porous bed is assumed to be governed by Darcy's law, and the soil skeleton to

behave as an elastic material (via Hooke's law). Darcy's law for an unsteady state

can be expressed as

Vp n t (2.4)

in which Vp is the gradient of the pore pressure, q the discharge velocity, pL the

dynamic viscosity, Kp the intrinsic permeability coefficient representing the charac-

teristics of the porous media, p the density of fluid, and n the volumetric porosity of

the medium.

Blot (1956) theoretically studied the linear response of fluid-filled porous media

under dynamic loading. He found that dissipative poro-elastic waves exist in addition

to the usual elastic waves. The presence of such dissipative waves was later confirmed

experimentally by Plona (1980) and Mayes et al. (1986).

15
Yamamoto etal. (1978) and Madsen (1978) assumed the soil bed to be poro-

elastic. The former developed a solution based on the 3-D consolidation theory of

Biot (1941). Madsen took into account the compressibility of the soil skeleton and the

anisotropic medium, and further indicated that the effect of hydraulic anisotropy was

appreciable only for soils coarser than silt. In common, their solutions showed that

bed response was dependent both on the permeability and the stiffness. However,

they ignored the interaction between waves and sediment.

Dalrymple and Liu (1982) improved the poro-elastic model in which they included

both soil and water wave dynamics. In their work, the effect of inertia was also

considered. It was concluded that wave energy attenuation was largely due to energy

losses in the porous medium rather than from boundary layer losses.

Yamamoto (1981, 1982, 1985) suggested that the internal loss of energy is due to

Coulomb attenuation or solid-to-solid friction between the grains, and is nearly inde-

pendent of the loading frequency. The seabed was assumed to be a homogeneous and

isotropic half space of a Coulomb-damped poro-elastic medium. The linear harmonic

problem in time and space was considered. Linearized Coulomb attenuation was com-

bined with the linear consolidation theory of Biot (1941). Yamamoto (1981) applied

the model to wave spectral transform for the case of a one -dimensional Bretschneider

spectrum. However, the occurrence of a bed shear stress was ignored due to the as-

sumption of potential flow for water. Yamamoto further considered one-dimensional

ocean wave spectrum transformation by an arbitrary slope of an inhomogeneous poro-

elastic medium with non-linear elastic moduli and non-linear Coulomb attenuation.

It was shown that the attenuation of water wave was small in sandy beds, and large

and sensitive to wave height in a normally consolidated clay. It was further concluded

that larger waves damp out rapidly while small ones damp slowly. However in very

soft mud (which behaves like a fluid) wave attenuation was found to become nearly

independent of the wave height.

16
Foda (1987) analyzed dissipative waves inside poro-elastic media and showed that

under certain conditions the developed free waves could lead to conditions of local-

ized liquefaction in an infinite homogeneous medium, and that significant fluid-solid

relative flow inside the porous body could occur, so that significant attenuation of

wave energy can result.

2.3.3 Viscous Model

A linear relation between stress and strain rate represents the property of a viscous

material. The constitutive equation for a viscous material can be expressed as

9ui 0uj
S= ( + a (2.5)

where ui is the velocity in the x; direction, uj the velocity in the xj direction, Tj

the stress in the xj direction in the plane perpendicular to xi, and ft the dynamic

viscosity.

An early theoretical and laboratory study on the effects of a nonrigid, imperme-

able bottom on surface waves in shallow water was made by Gade (1958). Gade's

formulation included a layer of inviscid fluid overlying a viscous fluid of greater density.

Solutions for a single harmonic wave indicated that wave height decayed exponentially

with traveled distance, and the rate of decay had a maximum value when the dimen-

sionless parameter, X = h2oa2v, had a value of 1.2, h2 being the thickness of the

lower viscid layer, ac the angular frequency, and v the kinematic viscosity of the lower

fluid. The results showed that energy dissipation due to soft mud was much greater

than that due to a rigid sandy bottom.

Dalrymple and Liu (1978) developed a theory which was for a small amplitude,

linear wave propagating in a two-layered viscous fluid system. The problem was

characterized as one of laminar flow for both the upper and the lower layers. The

governing equations are as follows:

-" + = 0 (2.6)
9x 5Z '

uj 1 oPj o2uj 92uj
a-- + Vj (--- + ) (2.7)
9vy 1 9P, ,29U 92U .
= + v (2v + aj) (2.8)
at pj 9z x2 9z2
where the subscripts j = 1, 2 represent the upper and the lower layers, respectively,

vj is the kinematic viscosity, x, z are the horizontal and the vertical axes located at

the mean water level, pj is the density and Pj is the dynamic pressure defined as

Pj = P + pjgz + PO (2.9)

where g is the gravitational gravity acceleration, Pj the total pressure and

P 0, for j = 1 (2.10)
S(p2-p1)gh, forj = 2

where h is the depth of the water layer. Both the governing equations and the surface,

interface and bottom boundary conditions were assumed to be linear. The results

showed that extremely high wave attenuation rates are possible when the thickness

of the lower layer is of the same order as the internal boundary layer thickness, and

when the lower layer is thick.

Jiang and Zhao (1989) introduced a three-boundary-layer viscous model to in-

vestigate solitary wave propagation over fluid-mud bottom. They pointed out that

the interaction between nonlinear waves and fluid-mud bottom can be remarkable.

However, only the first order approximation was taken into account for convenience,

in the solutions for the irrotational core and the boundary layers.

2.3.4 Viscoplastic Model

In the case of a simple shear flow, the viscoplastic or Bingham material is repre-

sented by the nonlinear stress-strain relationship as

Ou 0, iflr- < 70
9z r ro sgn(Ou/Oz), if r > To )

where T0 is the yield stress and p the dynamic viscosity.

18
Engelund and Wan (1984) used the viscoplastic model to study the instability

of hyperconcentrated flows. They claimed that the Bingham model was valid for

the description of the instability of the fluid surface elevation. Mei and Liu (1987)

also considered bottom mud to be viscoplastic. The special case of a shallow mud

layer under the action of long gravity waves propagating in the water layer above was

considered. It was found that under certain conditions the dynamics of a plug flow

layer, which was unsheared within the yield stress, contributed measurably to wave

attenuation over a long fetch. In particular, as waves propagated mud motion could

change from continuous to intermittent.

Sakakiyama and Bijker (1989) treated mud to be a Bingham material for small

strain rates. Based on the analogy of the viscosity of a Newtonian fluid, they intro-

duced an apparent viscosity pa defined as

Ou du
T = /a z = TO + Jlb (2.12)

where
Bu
Pa = Pb + 70/ (2.13)
az
By the selection of the apparent viscosity, they actually assumed a viscous mud bot-

tom. However, they ignored the fact that To and Ou/az should maintain same numer-

ical sign under the wave action, because a Bingham material always has a tendency,

represented by the yield stress, to resist any forcing, no matter in which direction the

force acts.

2.3.5 Viscoelastic Model

Both viscous and elastic properties exist in a viscoelstic material. Basically there

are two linear viscoelastic elements: the Voigt element and the Maxwell element. The

corresponding constitutive equations, in a simple shear flow, are

(2.14)

1

for the Voigt element, and

7 + ( ) = (2.15)

for the Maxwell element, where 7 is the shear stress, 7 the strain, and the dot indicates

derivative with respect to time.

Carpenter et al. (1973) applied the concept of viscoelasticity to marine sediments,

thus considering that a material is viscoelastic if the total deformation is calculated

from the sum of the elastic and the viscous deformations. They further considered

that marine sediments can be characterized as viscoelastic, having both the properties

of a solid and a liquid. An empirical nonlinear viscoelstic formula was used for data

analysis, which led to the result that the shear resistance of marine sediment taken

in the Gulf of Mexico was the sum of an exponential function of shear deformation

and a function of the one-third power of the rate of shear deformation.

Stevenson (1973) presented a practical method for determining the modulus of a

linear viscoelastic model for submarine sediments. He showed that the Voigt element

was applicable for his soil samples.

Schapery and Dunlap (1978) applied the concept of 'equivalent linearization' to

relate the linear viscosity properties to the inherent nonlinear properties of the sedi-

ment. With this approach, they calculated soil displacement due to water waves and

predicted the associated drag forces exerted on the pile legs of offshore structures.

Hsiao and Shemdin (1980) and MacPherson (1980) independently investigated the

same problem i.e., that of water waves propagating over a linear viscoelastic bed.

The problem was analyzed on the basis of the small-amplitude wave theory. In the

lower layer mud was considered to respond in both an elastic and a viscous manner.

The overlying layer was regarded as inviscid. Linear governing equations together

with linearized boundary conditions were solved. MacPherson (1980) introduced a

viscoelastic parameter, Ve, for the Voigt element as follows:
G
Ve = v + i (2.16)
0ap

20
where G denotes the shear modulus of elasticity, and i = vr-T. The introduction of

Ve is significant for the formulation of the problem based on the viscoelastic assump-

tion. MacPherson's results showed that depending on the elasticity and the viscosity

of the sea bed, wave attenuation can be of the same or a larger order of magnitude

than that due to bottom friction or percolation in a permeable bed. It was indicated

that for the viscosity dominant case, there are no elastic forces to restore the bed to its

undisturbed position, and as a result extremely rapid rates of attenuation can occur

whereby waves are almost completely damped within several wavelengths. Hsiao and

Shemdin (1980) compared their results with field measurements reported by Tubman

and Suhayda (1976) in East Bay, Louisiana. They found that the observed wave

energy dissipation was predicted reasonably by the viscoelastic mud model.

Suhayda (1986) presented a simplified technique based on an empirical nonlinear

viscoelastic model, and predicted surface wave attenuation, soil shear stress and shear

strain profiles. The shear modulus and viscosity were computed from a formula

as a function of shear strain in the soil. The results showed that the variation of

wave height with water depth could be quite large. Horizontal soil movements under

hurricane waves were predicted to be up to 0.5m in magnitude, depending on the

shear strength profile used. Motion was predicted to occur to a water depth of over

30m. It was shown that the bottom pressure was shifted from being in phase with

the surface wave as it would be over a rigid bottom. Horizontal and vertical mud

motions also showed large phase shifts relative to the surface wave.

Maa and Mehta (1988) experimentally investigated the dynamic properties of

soft muds. They demonstrated that the Voigt and the Maxwell elements are possible

choices. It was however suggested that the Voigt element is better for modeling

soft mud bed responses compared with the Maxwell model. Based on the linear

viscoelastic Voigt model, Maa and Mehta (1986, 1990) applied a linearized, multi-

layered hydrodynamic model for simulating depth-varying bed properties, the shear

21
stress at the mud-water interface and wave energy dissipation. The calculated water-

mud interfacial bed shear stresses were found to be larger than those obtained by

assuming mud to be rigid, due to out-of-phase motion between water and mud. The

results showed a resonance phenomenon with respect to the motion of the wave-mud

interface due to the elastic characteristics of the Voigt element. However, as the

investigators noted, the boundary conditions applied in the model were obtained by

Taylor series expansion, but only the harmonic term was considered; thus, nonlinear

features of the water-mud system were ignored.

Chou (1989) developed a four-layered viscoelastic model. The depths of the sec-

ond viscous layer and the third viscoelastic layer were determined as part of the so-

lution through an iterative technique. The model showed that sediment fluidization

depth and wave attenuation rate increased with wave height for a partially consoli-

dated sediment. It should be noted, however, that the iterative method that was used

makes the solution dependent on the initial theological state of bottom mud, which

is unknown a prior. In other words, for a different assumed initial condition of mud,

the result could be significantly different.
2.4 Mass Transport

Mud mass transport occurs as a result of a nonlinear effect under the action of

waves. In the case of a rigid bottom, the water mass transport velocity is predicted

to be proportional to the square of wave height, as a first order approximation based

on the small amplitude (linear) wave theory (see Dean and Dalrymple, 1991). For

the soft mud bed, the prediction of mud mass transport can be expected to be much

more complicated than that for the rigid bottom, due to water-mud interaction. Gen-

erally, the total mass transport of sediment consists of two parts, including suspended

sediment transport in the water column and sediment transport within the bottom

layer. Under wave loading, horizontal or near-horizontal mass transport that occurs

within the mud layer is normally much more significant compared with the transport

22
of the suspension resulting from mass exchange across the water-mud interface due

to particulate erosion and deposition.

Yamamoto (1984) presented experimental results on wave-induced mass transport

in clay beds. It was reported that wave dispersion was uniquely governed by the

mudline motion, and that the rate of mass transport in the bed was proportional to

the rate of wave energy dissipation in the bed.

Shibayama et al. (1987) calculated wave-induced sediment mass transport in the

water layer by solving the 1-D diffusion equation for sediment concentration, and

the velocity of sediment transport in the mud layer by using an analytical solution

for the mud velocity field due to wave action obtained from the model of Dalrymple

and Liu (1978). The investigator found that for kaolinite as the mud material, mass

transport in the mud layer occurred over a wide range of water content ratio, while

for a bentonite, mass transport in the mud layer occurred at relatively high water

contents.

Nakano et al. (1987) investigated the transport of a mud layer into a trench due to

waves and a current in same direction as that of wave propagation. The investigators

found, as expected, that the magnitude of mass transport was measurably greater

under the simultaneous action of waves and current than that due to waves alone,

and that a maximum value of mass transport occurred in the case of a favorable

current combined with the wave action.

Sakakiyama and Bijker (1989) applied a two-layered viscous model to evaluate

mud mass transport by using a pertinent apparent viscosity noted in the previous

section. They found that the measured profile of mass transport in the mud layer was

more uniform than that obtained by the theoretical model; in particular, the difference

between the two was significant near the rigid bottom, and that the measured mud

mass transport was proportional to a higher power of the surface wave height than

the square of that wave height predicted by the linear wave theory. This discrepancy

23
was explained by the nonlinearity of the relationship between the shear stress and the

strain rate in the mud.

Shibayama et al. (1989, 1990) applied a viscoelastic and the so-called visco-elastic-

plastic model to calculate mud transport rate under waves and wave-current action,

respectively. An oscillating type viscoelastic meter was used to obtain the values of

the elasticity and the viscosity. The investigators concluded that transport in the

mud layer was measurably greater than suspended mud transport in water layer, and

that the wave effect on the total transport rate was greater than that of current.

Model results were compared with the results of their experiments. However, as in

the case of Sakakiyama and Bijker (1989) as noted above, it was not recognized that

the numerical signs of the mud yield stress and the gradient of horizontal velocity

must be the same, an important requirement for the viscoplastic or Bingham model,

especially in the case of wave action.

2.5 Discussion

Summarizing the knowledge of deformable bottom response to water waves, it is

noted that in spite of considerable theoretical and experiment research, there are still

some important issues that remain unsolved and require further examination. Thus

for instance, it has generally been recognized that soil or mud bed can be fluidized

or liquified, as represented by a jump in the value in the mean pore water pressure

under wave loading. The fluidized mud will interact with the surface wave, and

this interaction may eventually lead to a final, steady depth of the fluidizied mud.

During the fluidization process, the water-mud interaction is time-dependent relative

to a time scale greater than the wave period. However, after a steady fluidized mud

depth is reached, the characteristics of the water-mud system can be regarded as

time-independent in the sense that, for example, wave decay and mud theological

properties are no longer functions of time in any significant way, as was selected to

be the case in the present study.

24
In the process of the mud response to the water waves, surface wave energy is

dissipated significantly due to the viscous motion in the bed, and mass exchange and

diffusion across the water-bed interface result in sediment suspension in the water

column. However, in comparison with the interfacial entrainment of sediment particles

into the upper water column, a measurable motion of the fluidized mud bed can

be generated by comparatively moderate wave condition. Therefore, the problem

of water-mud interaction may be conveniently separated from interfacial phenomena

such as entrainment of sediment. Consequently in this study, bottom mud motion has

been treated without invoking the effects of particulate entrainment at the interface.

There is no doubt that the constitutive relationship between stress and strain

is very important in governing the response of soft bottoms to surface waves. In

reality, the theological properties of natural compliant bottom materials are rather

complicated. For instance, when well-consolidated clayey soils are subjected to cyclic

shear loading, the relation between shear stress and strain is naturally described by a

hysteresis loop (Hardin and Drnevich, 1972). Fortuitously, in large water bodies such

as a bay or an estuary the bottom material is relatively weakly-consolidated, with

typically high water content and low shear strength. In that case, the description of

deformational behavior of the bottom material can be relatively simplified compared

with that for a well-consolidated bed. As noted earlier, five basic theological models

have been developed and often applied to characterize the constitutive properties of

the bottom material.

In the elastic model, no energy dissipation is predicted in the bottom layer due

to the pure elasticity assumption, which is obviously not appropriate for the present

case. As noted earlier, wave attenuation has been extensively observed both in the

field and in laboratory experiments.

Elasticity and permeability are the properties of interest in the poro-elastic model.

Yamamoto (1982) proposed a mechanism of energy loss by solid-solid friction within

25
the bottom layer, which is different from the mechanism of viscous friction. However,

mud beds have typically very low permeabilities and hence can be reasonably consid-

ered to be a continuum. Therefore, the theological behavior of mud may not be well

described by the poro-elastic model.

In the viscous model, mud is simply considered to be a pure viscid fluid, by

neglecting elasticity. It is evident from the evidence presented earlier that mud has

elastic properties due to the solid phase, even in the case of fluid mud which is totally

supported by the water or the fluid. Therefore, while the viscous model has the ability

to predict the wave energy dissipation, the effect of elasticity for example on the wave

attenuation is ignored, which can lead to a significant error in prediction.

By using the Bingham model, mud motion has been considered to be not con-

tinuous, but intermittent (Mei and Liu, 1987). However, field and laboratory data

show that bottom motions seem to have a continuous form similar to a wave record

(Suhayda et al., 1976; Maa, 1986). The Bingham model is really a simplification of

the pseudo-plastic model in which the stress-strain rate is continuous.

It is in fact reasonable to consider that soft bottom materials have both an elastic

property representing a restoring force (which has a direct influence on the wave num-

ber) due to the solid phase, and a viscous property of the fluid phase representing

dissipative force, which is directly related to wave energy attenuation. The elastic

model and the viscous model are special cases of the viscoelastic model. Two the-

oretical linear viscoelastic models, which are the Voigt and Maxwell models, have

often been applied. Unfortunately, those two models have their own limitations in

terms of deviations of the theoretical response curves based on these models from the

corresponding theological experimental data, and so on. Further discussion on this

matter will be given in Chapter 3.

It has been recognized that nonlinear water waves play an important role in many

coastal and ocean engineering problems. Based on a higher order approximation of

26
the boundary conditions, a linear surface wave with small amplitude is modified to be

one of finite amplitude, such as Stokes' wave in the case of a rigid bottom. For a de-

formable bottom, the surface wave is significantly modulated by the nonlinear effects

as well. Therefore, the velocity field, the pressure distribution, and the associated

mass transport should be considered along with the finite wave amplitude.

With regard to mass transport, as indicated in Section 2.4, the mass transport

velocity is a function of the square of surface wave height for a linear small amplitude

wave traveling over a rigid bottom. Therefore, the effect of the second order bound-

ary conditions on the mass transport velocity can be even more remarkable, due to

the nonlinear dependence on wave height, compared with other quantities which are

linear functions of the wave height, such as pressure. Unfortunately, in most previ-

ous modeling related to wave-soft bottom interaction, only linear waves with small

amplitudes have been considered. That is the reason why the accurate prediction of

soft bottom response under the wave action through higher order modeling is one of

the major issues examined in this study.

CHAPTER 3
MUD RHEOLOGY

3.1 Introduction

The importance of rheologic properties of cohesive sediment has been recognized

in problems associated with wave attenuation and mud mass transport in the coastal

seas and estuaries where oscillatory forcing is often dominant (Mehta, 1991). Natural

cohesive sediments are mainly composed of colloids, minerals and organic materials.

They appear as flocculated suspensions with relatively low densities, and as settled

soft beds with relatively high densities that are related to the degree of consolida-

tion. The rheologic properties of such sediments are rather complicated and not well

understood, even though there exist several simple theological constitutive models

(see Figure 3.1) that are commonly used to represent mud rheologic behavior. In
those models, only the shear stress and the shear strain rate are related to each other,

which means that elasticity is not taken into account explicitly. However, all real

materials can be generally assumed to be viscoelastic. In other words, both viscous

and elastic properties exist in all materials. In fact, viscoelastic models have been

widely applied to study the rheologic properties of polymers and soils in chemical and

soil engineering (Ferry, 1980; Hearle, 1982).

Based on both a general viscoelastic relationship and mechanical simulations, four

typical theological models, namely the Voigt model, the Maxwell model, the Jeffreys

model and the Burgers model are described in this Chapter. Static tests including

the creep test and the relaxation test, dynamic tests including the oscillatory test and

the wave propagation test, and the corresponding theoretical responses are presented.

Finally, the theological behavior of muds and the validity of the Voigt model and the

28
Maxwell model mechanically representing muds are discussed in conjunction with

previous studies.

3.2 Viscoelastic Models

The general differential constitutive equation for viscoelasticity can be expressed

as

r + an + a2_r + #'" = 1o + /31i + #27 + "'" (3.1)

where 7 is the shear stress; 7 is the strain; a, and /; are material parameters; and the

dot indicates derivative with respect to time. In the above equation, if the coefficients

have no dependence on variables such as strain, strain rate or shear stress and so on,

then the material is said to possess linear viscoelasticity; otherwise, it has nonlinear

viscoelastic properties. For the linear viscoelastic material the constitutive equation is

linear, and the corresponding mathematical theory can be based on the 'superposition

principle', which means that the deformation (strain) at any time is proportional to

the value of the initiating forcing. In other words, doubling the stress will double

the strain. In what follows, the material is assumed to be linear viscoelastic for

convenience of description.

It can be easily seen from Equation 3.1 that, if #0 were the only non-zero param-

eter, one would have the equation of Hookean elasticity, i.e., linear solid behavior,

with /o as the rigidity modulus. If #1 were the only non-zero parameter, the equation

could indicate Newtonian viscous response; /1 being the dynamic viscosity.

Equation 3.1 can be represented by a general mechanical model which is made

up of combinations of springs and dashpots. The force on a spring is proportional to

strain (7) and the force on a dashpot is proportional to strain rate (4). Consequently

the springs and the dashpots in a model represent the elastic and viscous properties

of the material, respectively. In the following, four special cases of Equation 3.1 with

corresponding mechanical models are discussed.

A Bingham Model
B Shear Thinning with a Yield Stress
C Shear Thinning Model
D Newtonian Model

UC

RATE OF STRAIN

Figure 3.1: Typical relationships between shear stress and strain rate in common
constitutive theological models for muds.

3.2.1 Voigt Model and Maxwell Model

If /o and /3 are both non-zero, whilst the other parameters are zero, we obtain one

of the simplest models of viscoelasticity, i.e. the Voigt model or the Kelvin model.

Another very simple model is the so called Maxwell model which can be derived

from the general equation by allowing ac and /1 to be the only non-zero material

parameters. Figure 3.2 is the mechanical presentation of both models.

The Voigt model results from a parallel combination of a spring and a dashpot.

In this case, both the spring element and the dashpot element have the same defor-

mation (strain) and the total stress is equal to the sum of the stress on each element.

Therefore, the following constitutive equation for the Voigt model, in simple shear

flow, is obtained:

7 = G~ + pj (3.2)

In the Maxwell model, a spring is connected in series with a dashpot. Both the

spring element and the dashpot element experience the same stress and the total

strain rate is the sum of strain rates of the two elements. Thus, the constitutive

equation for the Maxwell model, in simple shear, can be represented as:

+(+( )+ = /- (3.3)

3.2.2 Jeffreys Model and Burgers Model

The Jeffreys model and the Burgers model are extensions of the Voigt model and

the Maxwell model, respectively, obtained by adding an extra dashpot or a spring in

series or in parallel. From the general equation of viscoelasticity, if a1, /1 and /2 are

the only non-zero parameters, we have the Jeffreys model as

7 + a I = 01 + #2 7 (3.4)

In this case we can construct two alternative spring-dashpot models, A and B,

which correspond to the same mechanical behavior as represented by Equation 3.4.

G Il~

Figure 3.2: Voigt model (left) and Maxwell model (right).

GI A B

G1

G2 1-2

Figure 3.3: Jeffreys model (A and B) and Burgers model (C and D).

33
An equation of form of 3.4 was derived theoretically by Oldroyd (1953), when he

investigated the elastic and the viscous properties of emulsions and suspensions of

one Newtonian fluid in another. The model was applied successfully by Toms and

Strawbridge (1953) to describe the behavior of certain polymer solutions.

Similarly, the so called Burgers model can be obtained from the general equation

of viscoelasticity. Two alternative, mechanically-equivalent, forms (C and D) are

shown in Figure 3.3. In this case, the relationship between the shear stress and the

shear strain has the form

r + al + a02 = #1P + 32 7 (3.5)

Compared with a two-parameter model or a three-parameter model, the Burgers

model is more complex with four parameters, which can potentially cause difficulty

in determining all the parameters from experimental data.

3.3 Static and Dynamic Tests

Two different types of rheometric methods, the static test and the dynamic test,

are often applied to determine the viscoelasticity of a material or material parameters

in a viscoelastic model. The theoretical relationships corresponding to the static

test and the dynamic test can be obtained mathematically according to the different

mechanical models. The following results are based on the Voigt model and the

Maxwell model. The theoretical curves for the Jeffreys model and the Burgers model

can be derived similarly.

3.3.1 Static Test

The static test in general includes a material creep test at a constant stress and

a material relaxation test at a constant strain. Theoretically, the input stress or

strain, whether an increase or a decrease, is assumed to be applied instantaneously.

Obviously, this is impossible in practice because of inertia effects in the loading and

the measuring systems and the delay in transmitting the signal across the test sample,

C,,
L. _^_ ~----- -------------------.--- .--------

0 - - - - - -

Time
Figure 3.4: Responses of the Voigt model and the Maxwell model in creep test.

as determined by the speed of sound. In general, however, the time required for the
input signal to reach its steady value must be short compared to the time over which
the ultimate varying output is to be recorded. The inertia effect can be estimated by
adding weights and checking the derived viscoelastic functions.
Creep Test
In the creep test, there are two stages of input stress. The loading stage (0 < t <
t') is often called the creep curve. The unloading stage (t > t,) is referred to as recoil
or creep recovery. The input stress is presented as

o70, for0 0, fort, < t (3.6)
For the Voigt model, it can be shown that the strain response is given by
TO G
7= [1 -exp(--t)] (0 G I
If t~ is large enough, the strain approaches a steady value, and
r0 G
S= exp[--(t tc) (t,

35

Solid Line. Voigt Model
Dashed Line: Maxwell Model

vU

S .-" / ---------------

Time

Figure 3.5: Responses of the Voigt model and the Maxwell model in relaxation test.

where I/G is called the retardation time representing the characteristic time scale

for strain creep. It is easy to show that in the first stage the strain can reach the

final value, ro/G, exponentially and that the viscosity retards the increasing strain to

reach this final value; in the second stage the strain can return to the initial (zero)

value completely.

For the Maxwell model, the strain response is linear with time:

7 = t (0 L t< t) (3.9)

O = -t (t, < t) (3.10)

In this case, there is no equilibrium strain value due to the viscosity-induced continu-

ous deformation. In the first stage, the strain increases linearly with a slope of ro//.

In the second stage, the strain remains constant.

Relaxation Test

The input strain rate for the relaxation test is

o, forO 0, for t, < t

For the Voigt model, the responses corresponding to the loading strain rate and

the unloading strain rate are

T = Got + Po (0
T = Gotr (t > tr) (3.13)

The stress in the loading stage increases with a slope of G'ro, then suddenly decreases

by an amount yo at t = tr when the strain rate is suddenly removed, and remains

constant thereafter. Thus there exists a residual stress after the unloading of the

imposed strain rate.

For the Maxwell model, the response curves read

T = ito[1 exp(--t)] (0 < ttr) (3.14)

If t, is large enough, the stress approaches a steady value, and

G
= tfo exp[-- ( t,)] (t, < t) (3.15)

where j//G is called the relaxation time which is the characteristic time scale for stress

relaxation. In the loading stage, the stress increases exponentially and there occurs an

equilibrium value of stress when the time t becomes large enough. After the removal

of the strain rate, the shear stress reduces to the initial value (zero) which is different

from what occurs in the Voigt model.

3.3.2 Dynamic Test

The dynamic test includes two different sub-tests: the oscillatory test and the

shear wave propagation test. A rheometer and a shearometer are used for conducting

the oscillatory test and the wave propagation test, respectively.

Oscillatory Test

There are two types of oscillatory tests available depending on the input oscilla-

tion. One is the controlled-strain oscillatory test in which the input is the oscillating

strain. The other is the controlled-stress oscillatory test in which the input is the os-

cillating stress. However, in both tests the stress always exhibits phase shift, 6, ahead

of the strain. 6 has values between 0 and rr/2 radians. When 6 is 0, the response

of the material is purely elastic. 6 = 7r/2 represents the pure viscous material. In

between, 0 < 6 < 7r/2, the material is in the viscoelastic range.

In a controlled-strain test, the input strain and output stress can be expressed as

Y = 0o exp(-iot) (3.16)

7 = rT exp[-i(at + 6)] (3.17)

The input stress and output strain in a controlled-stress test are

r = rexp(-iat) (3.18)

7 = 70 exp[-i(at 6)] (3.19)

where i = /1T, r7 is the stress amplitude, -y is the strain amplitude, and a is the

oscillation frequency.

We may write

7 = G*7 (3.20)

where G* is called the 'complex shear modulus'. G* can be written as

G* = G' iG" (3.21)

where G' and G" are referred to as 'storage modulus' and 'loss modulus', respectively.

We further let

T = (3.22)

where g* is called 'equivalent viscosity' or 'complex viscosity'. 1p* can be expressed as

P* = it' + if" (3.23)

where i' is usually called 'dynamic viscosity', and y" has no special name.

For the Voigt model, it can be shown that, for both types of tests,

G'= G (3.24)

G" = /a (3.25)

I' = ft (3.26)

t" G (3.27)
0"

and

T0 sin
S= To sin6 (3.28)
'o o"

G = 70 cos 6 (3.29)
7o

For the Maxwell model, it can be similarly shown that, for both types of tests,

u2o2G
G' = 2 (3.30)
G 2 2 + G2

G" = I G2 (3.31)
YG2 2 + G2

S t=G2 (3.32)
G2 +p2C02

39

= G (3.33)
G2 + 22(333

and

To 1
o = (3.34)
7o sin 6

To 1
G (3.35)
7o cos S

For both the Voigt and the Maxwell models, it can be further shown that, for both

types of tests,

G' = o cos 6 (3.36)
7o

G" = sin 6 (3.37)
70

As a matter of fact, it can be proven that Equations 3.36 and 3.37 are always true

for any viscoelastic model on the basis of Equations 3.16, and 3.17, or Equations 3.18,

3.19 and 3.21.

It can be shown that for the Voigt model and the Maxwell model the two param-

eters (/ and G) can be determined by the oscillatory test alone from Equation 3.28

and 3.29, or from Equation 3.34 and 3.35. However, for the Jeffreys model with three

parameters and the Burgers model with four parameters, additional conditions are

needed which can be provided by the static test.

Wave Propagation Test

The wave propagation test can be performed by using a pulse shearometer which

is designed to measure the shear modulus of a viscoelastic material by the propagation

of a small amplitude, high frequency shear wave through a small gap filled with the

material (Goodwin et al. 1976). Basically, the material is placed between two parallel

metal plates, each connected to a piezoelectric crystal. Using a pulse generator,

a shear wave is generated at the left plate at time zero (see Figure 3.6). This wave

Figure 3.6: Gap between two parallel plates in a pulse shearometer.

propagates through the material and is detected at the right plate at time t. Therefore,

the shear wave speed, v, is

v = d (3.38)
t
where d is the distance between the two plates. If assuming that the material follows

the Voigt model, the modulus, G, is obtained from

G = pv2 (3.39)

where p is the density of the material.

Equation 3.39 is derived as follows:

Starting with the momentum equation for this case
a2 x 1 r
=2 X 1 (3.40)
dt2 p dz

Let r = G*Y, where the shear strain, 7, can be expressed as
Ox
7 = (3.41)
8z

By substitutions, we have a wave equation as
a2x G* 02x
-(3.42)
9t2 p 9z2

This equation is solved as

x = Ao exp(- z) exp(-i t) (3.43)

where Ao is the wave amplitude at z = 0, and F is the complex wave propagation

constant, the square of which is expressed as

2 = 2 (3.44)
G*

It is convenient to let

r = ki + i kr (3.45)

where obviously kr is the wave number, k, is the wave attenuation coefficient. Then
the solution becomes

x = Ao exp(-ki z) exp[-i (at + k, z)] (3.46)

From Equation 3.44, we can have

G' = p k (3.47)
r( k+ k?)2
2 k, k-
G" = p a2 k (3.48)
( k + v ()

Next assume

1 (3.49)
k,
This assumption is based on the knowledge that for typical wave flume experiments

ki is on the order of 10-1 (see Chapter 6) and kr on the order of 103 (Feng et al.

1992). Then ki/kr is on the order 10-4. G' and G" are thus simplified as

G' = p v2 (3.50)

k
G" = 2 pv2 (3.51)
k,

42
where v is the shear wave phase velocity and equal to cr/kr. It is known from Equa-

tions 3.36 and 3.37 that G"/G' = tan 6, therefore, for a given 6, the wave attenuation

coefficient can be determined from

ar k
k= tan6 = tan 6 (3.52)
2v 2

Wave attenuation is primarily related to the density of the material and the wave

frequency. In this case, since the pulse shear wave is of high frequency, the wave

attenuation coefficient, ki is expected to be quite small.

Based on Equations 3.50 and 3.51, we are able to determine the elastic modulus

and the viscosity for a given theological model. If the Voigt model is applied, we have

G = G' = pv2 (3.53)

tan 6 2 (
S= p v2 (3.54)

For the Maxwell model, it can be easily shown that

G = (1 + tan2 )pv2 (3.55)

1 + tan2 6
= tan p v2 (3.56)
a tan 6

It is readily seen that if the shear wave phase velocity, v, and the phase angle

between shear stress and shear strain, 6, can be measured or calculated experimentally,

then we are able to obtain G and p for a given theological model or G' and G".

3.4 Previous Studies

In this section, rheometric techniques used to investigate the properties of a vis-

coelastic material, applications of the Voigt model and the Maxwell model, and the

corresponding experimental results are briefly discussed.

Christensen and Wu (1964) and Abdel-Hady and Herrin (1966) applied a rheo-

logical model which was a combination of the Maxwell model and the Voigt model

43
to investigate creep behavior of clays. In this theory, instead of a Newtonian viscous

response, the rate process theory (see e.g. Glasstone et al., 1941) was applied for

viscous deformation, in which the shear stress is related to the strain rate nonlinearly.

However, since the shear stress is related to the shear strain rate alone, the elasticity

of the material is ignored.

Applied stress rheometers have been used in creep tests to investigate the vis-

coelastic properties of cohesive materials under a range of gradually increasing levels

of applied stresses (Davis etal., 1968; James etal., 1987; Williams and Williams,

1989). Typical test curves are shown in Figure 3.7, with reference to which the creep

compliance J(t) is defined as

J(t) = ) (3.57)
7-
and y is defined as the apparent yield stress related to the strength of the space-filling

soil network structure. Jo represents the region of instantaneous compliance due to

pure elastic response. As time increases, the compliance increases and is retarded

due to the viscous effect. When 7 > 7y, thixotropic breakdown of material structure

occurs, i.e., the network structure collapses with the corresponding disappearance of

measurable elastic response. It should be noted that the apparent yield stress is not

the same as the Bingham yield stress below which no viscous flow occurs.

Buscall etal. (1987) studied the rheology of strongly-flocculated suspensions.

Creep tests using an applied stress rheometer showed that the material exhibited

solid-like viscoelastic behavior at sufficiently small stresses. On increasing the stress

there was a change from solid-like to fluid-like behavior over a very narrow range of

the applied stress. The instantaneous modulus, Go, which is the inverse of instanta-

neous compliance, Jo, was measured by two independent methods, wave propagation

and creep. The results from those two methods were compared at small strains and

good agreement was obtained, which may imply that the instantaneous modulus is

independent of the forcing frequency. However, the analysis of the wave propagation

I Il!

Jo

Stress tress TIME
Applied Removed

Figure 3.7: Changing creep-compliance with increasing applied stress (after James
et al., 1987; Williams and Williams, 1989).

45
test data was made with the assumptions that attenuation was negligible and G' was

equal to Go.

James etal. (1987) used a combination of an applied stress rheometer and a

miniature vane geometry to measure the static yield properties of illitic suspensions.

The advantage of a vane geometry is that it avoids the wall-slip problem which occurs

in rotational measuring devices such as a bob. From the applied stress tests, the

instantaneous compliance, Jo, was plotted against the applied stress. It was found

that under small applied stresses the behavior of the suspension was predominantly

elastic, becoming essentially viscous at higher stresses. Below a certain stress Jo was

found to be almost independent of stress, which enabled the identification of the range

of linearity of response to stressing.

Williams et al. (1989) used the complementary techniques of constant applied

stress rheometry and shear wave propagation to investigate the elastic and yield

behaviors of K-illite suspensions. The elastic modulus, G, was determined by using

Equation 3.39 from shearometry. The creep test was shown to be characterized by

an instantaneous strain followed by a period of retarded elasticity. It is worth noting

that Equation 3.39, which they used for the shear wave test, is only valid for the

Voigt model, and that the wave rigidity modulus, G, was assumed to be equal to the

instantaneous modulus, Go. It appears that Go can approximate G because both of

them represent a very high frequency response, namely a pulse response. It should be

noted that a basic assumption made for the shear wave rheometry is that the material

properties are independent of the forcing frequency. With respect to the applicability

of the Voigt model to the high frequency forcing, further discussion will be given in

Chapter 4.

Maa (1986) used a Brookfield viscometer and miniature vanes to conduct shear

stress relaxation tests in which the shear stress and the angular displacement of mud

sample were measured. It was found that the mud exhibited a finite residual stresses.

46
Based on this observation, the investigator claimed that the selected mud did not

behave as a Maxwell element since the mud resisted an applied shear stress. Based

on the assumption of the Voigt element, values of the shear modulus, G, and the

viscosity, /, from an averaged shear rate were estimated. The results showed that the

shear modulus was well correlated with mud density, and that the viscosity generally

decreased as the density increased, which was consistent with a decrease in the rate

of energy dissipation with increasing density.

Ramsay (1986) investigated the theological properties of Laponite dispersions by

conducting controlled-strain oscillatory tests with a Weissenberg rheogoniometer. The

investigator examined the effects of dispersion concentration, and input strain ampli-

tude, and showed that the elastic or storage modulus, G', increased and eventually

reached an equilibrium value as the concentration increased. It was also found that

at very small strain amplitudes the response of the clay was totally elastic, and that

at higher strain amplitudes a nonlinear response, typical of plastic behavior, occurred

due to extensive break-down of the soil structure.

Schreuder et al. (1986) observed the deformation during oscillatory shearing of

concentrated suspensions. It was indicated that structures in concentrated suspen-

sions formed by shear can support elastic stresses. The investigators found that at

small deformations the material followed a linear viscoelastic behavior which was

characterized by the dynamic viscosity, ti' = G"/cr, which at low frequencies was

inversely proportional to the frequency, a, and at high frequencies approached a con-

stant value. The result was explained through the suggestion that at low frequencies

sheet formation occurred during shear, and that at high frequencies sheet formation

was less developed.

Chou (1989) conducted measurements of the theological properties of soft muds

composed of kaolinite and montmorillonite, under oscillatory shearing. He found that

the tested muds had rheologic responses which depended on the amplitude of the

1031-

102

10-2 10-1 11
STRAIN AMPLITUDE

Figure 3.8: Storage modulus, G', and loss modulus, G" of Laponite dispersions at
different strain amplitudes. Solid symbols denote G", open symbols G' (after Ramsay,
1985).

101-

10'3

48
wave-induced strains and the mud density. Under intermediate amplitudes of strains,

muds responded as viscoelastic materials. He showed that the storage modulus, G',

and the loss modulus, G" were not sensitive to the forcing frequency. The shear stress

in experiments with a constant loading period (0 < t < t,) was a steady value instead

of monotonically increasing as predicted by the Voigt model, and a residual stress

was found. Therefore, both the Voigt model and the Maxwell model were rejected for

the selected muds. However, no new constitutive model was proposed and the Voigt

model was applied in the governing equation of motion for the water-mud system

under wave action that was investigated.

3.5 Discussion

It can be seen that the strain creep test, stress relaxation test, oscillatory test

and shear wave propagation test are suitable rheometrical techniques to study the

viscoelastic properties of muds. Those techniques could be complementary; in other

words, one type of test can be combined with another to gain an insight into mud

rheology (Williams et al., 1989).

Muds behave as elastic solids under relative small strain amplitudes, and as vis-

cous fluids under large strain amplitudes. In general, mud is a viscoelastic material,

and its elasticity becomes more important as density increases. Linear viscoelastic

properties can be expected under small deformations. As forcing becomes large, non-

linearity becomes pronounced. It appears that mud behavior is also related to the

forcing frequency since energy dissipation, which affects the mud properties, is a func-

tion of frequency. The elastic modulus, G, in the pulse shear wave propagation test

and the instantaneous modulus, Go, in the creep test, are indeed representatives of

high frequency responses of mud.

The experimental results of exponential type of strain decay in the creep test and

exponential type of stress decay in the relaxation test cannot be explained by the

Voigt model or the Maxwell model alone. The independence of the storage modulus,

49
G', on frequency (Chou, 1989) implies that the elasticity, G, is irrelevant to frequency

if the Voigt model is applied. Therefore neither the Voigt model nor the Maxwell

model is suitable for describing the rheological behavior of mud. It is apparent that,

a more general model, such as the Burgers model, can be a possible better choice

for the mechanical simulation of mud because greater response flexibility would be

provided in such a model. For the purpose of the present study, this issue is examined

in the next chapter by first examining the response of selected sediments in tests using

an applied stress rheometer.

CHAPTER 4
A CONSTITUTIVE MODEL FOR MUD

4.1 Mechanical Simulation

It was noted in the previous chapter that neither the Voigt model nor the Maxwell

model is appropriate for describing the theological properties of soft mud. Compared

with these two-parameter models, the four-parameter Burgers model, for which the

constitutive equation 3.5 and the mechanical description are given in Chapter 3, can

be examined as a means to better describe the behavior of viscoelastic materials,

especially mud. Two typical cases of the Burgers model will be considered here. In

the first, if the value of G1 is infinite, Equation 3.5 can be reduced to the Jeffreys

equation with the form:

T + -P= 1 +7 + 7 (4.1)
G G

where the parameters I1, P2 and G are introduced before. It is seen that, for strain

creep test in which + is zero, the strain rate, j, and the rate of strain rate, 7, are

correlated linearly. However, from the experimental data to be described later (see

Figure 4.1), it is observed that 7 was almost independent of 7, which implies that

either /2 should be zero or G has to approach infinity. Therefore, the above Jeffreys

model becomes either the Maxwell model (t2 = 0) or the viscous model (G -+ oo) in

this case, which obviously is not the correct answer sought here.

As a second approach, in the case of a very large value of /j, the Burgers model

can be changed to yield a new constitutive equation:

7 + ai + = #o 7 + 01 (4.2)

0.004 0.008 0.012 0.016

STRAIN RATE

IVI I I I I I I

0.10(

0.060

0.02C

-0.020

-0.06C

-0.100C
0.

0.060

0.020

-0.020

-0.060

-0.100

I I I II I
I I
Sample: Okeechobee Mud
Applied Stress: 1.00 Pa
Density: 1.15 g/cm3
I I I

I I II

I I I

0.004 0.006 0.008 0.010

STRAIN RATE

0.012

Figure 4.1: Typical relationship between 7 (1/min2) and 7 (1/min) for mud.

0.020

I- -

) -
iI ii) Iii lt ii iK il I
-Sample: Okeechobee Mud
Applied Stress: 2.00 Pa
-Density: 1.12 g/cm3
1 -- l -- I i I ; I ,

000

n 1nn

I
I I

0.014

0.014

i I ii o

I I I i

wI

52
In comparison with the general viscoelastic constitutive Equation 3.1, it is seen that

terms of second and higher order time derivatives of shear strain and shear stress are

not included in the above expression. Based on strain creep test data, as for example

shown in Figure 4.1, it is noted that 7 is almost a constant close to zero, which

indicates that higher order derivatives of shear strain can be ignored. Therefore, it

is generally acceptable to apply the above constitutive equation to describe the mud

viscoelastic behavior in the present case.

The mechanical model corresponding to Equation 4.2 is shown in Figure 4.2.

Based on the force balance and mechanical principle, the coefficients ac, 3o and P/

can be expressed as

a, = (4.3)
G1 + G2

o = (4.4)
G1 + G2

1 = G2 (4.5)
G1 + G2
where the material parameters i, G1 and G2 are shown in Figure 4.2.

It can be shown that this model predicts an exponential type of strain decay in

the strain creep test, and an exponential type of stress decay with a residual stress

in the stress relaxation test, as observed in previous laboratory studies (Chou, 1989).

It is seen that in the model the strain rate, 7, is a linear function of strain, 7, with

a negative slope, -S, in the strain creep test under a constant shear stress (" = 0),

which agrees with the results obtained in laboratory experiments reasonably well.

Details are given in Section 4.3. The slope, S, is thus defined as

S = o (4.6)

From this definition, it is clear that S represents the ratio of elasticity to viscosity.

From Equation 4.2, the following analytical relationships can be obtained for the

controlled-stress oscillatory test:
cos sin = o (4.7)
cos 6 a, sin 6 = #o (4.7)
TO

sin 6 + a a cos 6 = r 31 -O (4.8)

where 6 is the phase shift, a is the oscillatory frequency, 70 is the stress amplitude

and -0 is the strain amplitude. Based on the above three equations, G1, G2 and p

can be calculated as follows:

TO a
G 1= (4.9)
o a cos 6 S sin (4.9)

To S
G2 = (4.10)
-t sin 6 ( 02 + S2 )

S 70 sin 6 (a2 + S2 (4.11)

The equivalent viscosity, /* = r/7, for this model is:

(/3i /3oai)0, + i(00o + alo) 1 (4.
20 =2 (4.12)
= (I + a~l 2

4.2 Laboratory Experiments

4.2.1 Test Equipment

Both creep tests and oscillatory tests were conducted in a Carri-Med CSL con-

trolled stress rheometer shown in Figure 4.3, at the US Army Engineer Waterways

Experiment Station in Vicksburg, Mississippi. Full details regarding the rheometer

can be obtained from the original complete manual. A brief description is given below.

The rheometer, which requires a computer with a corresponding software and a

water bath for successful operation, is capable of testing materials under either uni-

directional shear mode with a strain rate range of 1 x 10-6 to 5 x 10-3 s-1, or the

oscillatory mode in the frequency range of 1 x 10-3 to 10 Hz and a minimum strain

amplitude of 1 x 10-4. Under both modes, the shear stress can be varied from 0.08

to 2.54 x 104 Pa.

The rheometer is almost completely controlled by the software, resulting in no

external controls except an on or off switch. Clean, regulated, compressed air is

Figure 4.2: A Mechanical Rheological Model.

55
required in order for the air-bearing to function properly, and water to or from the

water bath and the Peltier system to provide thermo-regulation of the measuring

plate.

The oscillatory tests are performed by feeding into the rheometer a digital, computer-

generated sinusoidal wave signal based on the torque and the frequency values selected

by the user. The rheometer applies the corresponding stress wave to the sample. The

movement of the measuring geometry is scanned by an optical encoder and the values

of the sinusoidal displacement wave form are stored in memory for later evaluation.

Normally the computer takes the rheometer through five cycles of oscillation and the

last cycle is stored and used for analysis. The stress and strain data can be obtained

from the information on the amplitude of the measured displacement wave and the

phase difference between it and the input torque wave. Creep tests can be carried

out on the same principle, but under a uni-directional mode.

4.2.2 Test Samples

Two different kinds of samples were used in the theological test, including a natu-

ral mud and a clay mixture (AK mud). The former was collected from the bottom of

Lake Okeechobee, Florida, where a field investigations on the mud bottom response

to surface water wave were conducted as discussed in Chapter 8. This mud (OK

mud), having the granular density, ps, equal to 2.14 g/cm3, mainly consists of miner-

als including kaolinite, sepiolite and montmorillonite, and organic material (Hwang,

1989). Among clay minerals, kaolinite is the predominant constituent. The fraction

of organic matter, which was determined by loss on ignition, is fairly high, ranging

from 36% to 41%, which results in the relatively low granular density compared with

the typical dry soil density of 2.65 g/cm3. The dispersed median diameter, d50, was

in the range of 3.4 to 14.4 ,m.

The second type of mud sample (AK) consisted of (by weight) 50% attapulgite

(palygorskite) of greenish-white color, and 50% kaolinite (pulverized kaolin). The

56

9 \ 10 11 12 13 II
V L

Figure 4.3: Carri-Med CSL Rheometer: 1) rigid cast metal stand/base; 2) air bearing
support pillar; 3) motor drive spindle; 4) air bearing housing; 5) drive motor stator;
6) air bearing; 7) optical displacement encoder; 8) draw rod; 9) shorting strap; 10)
inlet/outlet for Peltier water supply; 11) bottom plate of cone and plate assembly; 12)
automatic sample presentation system; 13) gap-setting micrometer wheel; 14) height
adjusting micrometer scale; 15) liquid crystal display; 16) adjustable levelling feet.

Table 4.1: Chemical Compositions of Attapulgite and Kaolinite

Ions Attapulgite Kaolinite
A1203 9.67% 37.62%
CaO 1.65% 0.25%
FeO 0.19%
Fe203 2.32% 0.51%
H20 10.03%
K20 0.10% 0.40%
MgO 8.92% 0.16%
Na20 0.10% 0.02%
NH20- 9.48%
P205 0.19%
SiO2 55.2% 46.5%
SO3 0.21%
TiO2 0.36%
V20s 0.001%

cation exchange capacity (CEC) of the attapulgite was 28 milliequivalents per 100

grams. The kaolinite was a light beige-colored powder of CEC in the range of 5.2-

6.5 milliequivalents per 100 grams. The chemical compositions of attapulgite and

kaolinite are shown in Table 4.1. The median particle diameter of attepulgite and

kaolinite were 0.86/im, 1.10pm, respectively (Feng, 1992).
The essential properties of cohesive sediments can be described in terms of grain

size, mineralogical composition, organic content, and the cation exchange capacity

(Mehta, 1986). It is seen that while both types of samples (natural mud and AK) are

typical cohesive sediments, differences between them arise from grain size, mineral
composition, CEC and especially organic content. Sample AK is a pure clay mixture

without any organic matter. It was anticipated that the occurrence of organic material

in the natural mud would have measurable effects on not only the granular density

but also on the rheological properties, since organic matter is very different from

clay minerals both chemically and physically. Therefore, those two samples together

Table 4.2: Chemical Composition of Water

Ions Content (ppm)
Al 1.2
Ca 24.4
Fe 0.2
Mg 16.2
Na 9.6
Si 11.4
Total salts 278

represent a comparatively wide range of properties of cohesive sediments.

Tap water was used to prepare mud samples. The pH and the conductivity

of the tap water were 8 and 0.284 milihos, respectively (Feng, 1992). Its chemical

composition is given in Table 4.2

4.2.3 Test Procedures and Conditions

The power supply to the water bath was turned on first in order to provide

water for the Peltier system. Clean air with a proper pressure was then supplied to

the rheometer, followed by turning on the power to the rheometer and the computer.

Having switched on the above three supplies and ensured that all functioned correctly,

the air-bearing was freed. After switching on the power to the CSL (see Figure 4.3),

the liquid-crystal display (15) showed the number of standard tests being carried out

internally. After the mud sample was loaded with a selected measuring system, both

the creep test and the oscillatory test were easily carried out by following the menu

shown on the computer screen. The outcome of each test was copied onto disks for

later data analysis.

Two concentric cylinders with the stator inner radius 23 mm and rotor outer

radius 25 mm were used as the measuring geometry, and the cylinder immersed height

was 30 mm. The lower cone with uniformly loaded sample was first fitted into the

59
bottom plate (11), then the upper (inner) cylinder was lowered down to the designated

position and contained the sample within the gap between the two cylinders.

In order to examine the effect of density on the theological properties of cohesive

sediment, a range of densities, 1.05-1.15 g/cm3 for Okeechobee mud and 1.02-1.22

g/cm3 for AK, were selected. For each sample of a given density, input stress am-

plitudes in the oscillatory test and input stresses in the creep test were varied over

different ranges depending on density, to examine the effect of shear stress or strain.

In order to cover frequencies corresponding to representative coastal and ocean waves,

a forcing frequency range of 0.02 to 2.0 Hz in the oscillatory test was chosen. The

temperature was controlled at 20c during the performance of the tests.

4.3 Results and Analysis

Data (plots) are presented in this section to demonstrate typical experimental

results. Figures 4.4(A) and 4.5(A) show two typical examples of creep curves of

strain against time consisting of stage I with an instantaneous strain and stage II.

Stage I corresponds to tc > t > 0; stage II to t > tc (see Chapter 3). It can be

seen that the strain shows an exponential type increase with time in stage I while the

forcing stress is constant. At the end of stage I, the stress is suddenly released and

stage II begins. An exponential type decay of the strain is found after stress release

such that the decrease of the strain slows down with increasing time. From the creep

curve, the strain rate, -, and the rate of strain rate, 7, can be calculated.

In Figures 4.4(B), 4.5(B), and 4.8 it is observed that there is a reasonably linear

relationship between strain, 7, and strain rate, j, except in the low strain portion of

stage I and in the high strain portion of stage II. It is noted that those two nonlinear

portions correspond to the initial periods of stages I and II, respectively. An attempt

is made to explain these features in the creep curve as follows:

When a mass starts to move from the still state or stop moving, the inertia

force can be significant during the initial period according to Newton's second law of

I I I I I I I I I I I I I I

0.050

0.040

0.030

0.020

0.010

0.000

' Sample: O0
Applied Str
Density: 1.

' i l l s a a

200.

400.

600.

800.

Time (s)

0.0200

0.0144

" 0.0088
-I0 0

0.0032

"-0.0024
Vn

-0.0080 1-
0.0000

. I I I I I I I I I I I i i
+

Plus for stage I
Star for stage II -

+ +

+4.

(B)

0.0090 0.0180 0.0270 0.0360 0.0450
0.0090 0.0180 O.0270 0.0360 0.0450

Strain

Figure 4.4: Strain, 7, as a function of time, t, and strain rate, 7, as a function of
strain, 7, for Lake Okeechobee mud.

CEECHOBEE MUD
ress: 0.10 (Pa)
08 (g/cm-3) (A)
I I I I I I f I I

1000.

I

I I t I I f I I

100. 200. 300. 400.

Time (s)

0.2500

0.1900

-'0.1300

0 0.0700

S0.0100
CO

-0.0500
0.00

o00

0.0600

0.1200 0.1800 0.2400 0.3000

Strain

Figure 4.5: Strain, 7,
strain, 7, for AK mud.

as a function of time, t, and strain rate, 7, as a function of

0.250

0.200

0.150

0.100

0.050

0.000

500.

I I I I I I I I

Plus for stage I
Star for stage II -

+
-
-+

-:

(B)
SI i I I i i I I t I t I' I i-

- -

62
motion. In the present case, the inertia, ri, can be expressed as

T; = /2 2 (4.13)

where /2 is a positive constant. From Equation 4.2, for 7 = ro, the strain rate is

given by

0 =_ ^o +ro (4.14)

As far as the inertia effect is concerned, the total stress can be represented by

ro = f0o + 31 + /2 (4.15)

Based on Equation 4.15, let us consider two simple cases to examine how the rela-

tionship between the strain rate and the strain can be influenced by inertia. In case

1, if the rate of strain rate, 4, is equal to a constant, 70, then the strain rate, 4, can

be restated as

go 7 P 0(
= o + (4.16)

By comparing Equation 4.14 with Equation 4.16, it is seen that both straight lines

have different intercepts, but the same slopes (-S = -/3o/13). In case 2, if the rate

of strain rate is a linear function of strain rate i.e., 7 = a + by, where a and b are

constants, the strain rate becomes

S o + #2 b ro #2a
= + (4.17)
#101

In Equations 4.14 and 4.17, we can see that neither the slope nor the intercept is the

same in those two straight lines.
In the plots of rate of strain rate against strain (Figures 4.6 and 4.7), it is noted

that during the initial periods of stages I and II, j is a linear function of 7 which

corresponds to case 2, and afterwards 4 becomes a constant which means that case 1

applies. Therefore, the slope, S, during initial period of both stages is distorted due to

0.0150

0.0120

0.0090

0.0060

0.0030

n_ nnn

w

z-
I-

ucJ

Cr

zm

0-
,,)

0

I-

0.0160 0.0220 0.0280 0.0340
STRAIN

U.U3UU

0.0180

0.0060-

0.0060 -

0.0180-
n_-nn

0.0100

0.0160 0.0220 0.0280 0.0340

STRAIN

0.0

0.c

400

)400

Figure 4.6: Strain rate, j, and rate of strain rate, 7, in stage I as functions of strain
for Lake Okeechobee mud.

I I I I i
4. Sample: Okeechobee Mud -
Applied Stress: 0.10 Pa
Density: 1.08 g/cm3

+ +
m-

+H4.^
__________ ----I----- -----I----I----\----I-----I---

0.0100

64
the inertia effect and cannot be used for solving for the parameters of the theological

model. However, S, after this initial period can be regarded as independent of the

inertia effect even though the intercept is altered due to inertia.

It is interesting to note that in the creep tests the slopes of stages I and II are very

close to each other except in the initial period (see Figures 4.4), which indicates that

the stress has little effect on the slope, S. In Figure 4.9, it is also seen that S is very

weakly dependent on stress. This weak dependence may imply that elasticity and

viscosity respond in similar ways when the stress is varied. In other words, when the

stress increases, both elasticity and viscosity may decrease at the same rate, or remain

unchanged. However, S is a strong function of mud density. As the density increases,

S also becomes large. In the case of a pure elastic material with a high density, S

approaches infinity, since fl is zero. When the density decreases to 1.00 g/cm3, which

would mean that the material is pure (viscous) water, Po is zero, therefore S is also

zero. The empirical relationship between slope, S, and density, p, for Okeechobee

mud can be expressed approximately as

S = c (p 1)C2 (4.18)

in which the coefficients, cl = 1.621 and c2 = 0.650, are obtained through the least

squares method. The strong correlation between S and p indicates that elasticity

and viscosity react in the opposite sense when the density changes. For example, the

elasticity decreases, but the viscosity increases, when the density becomes smaller.

Based on the instantaneous strain representing instantaneous elastic response

in the creep test, the variation of the compliance, Jo, which is the inverse of the

instantaneous modulus, Go, with the applied stress was obtained and the results are

shown in Figures 4.10 and 4.11. As would be expected, Jo values are strongly related

to mud density. For the Okeechobee mud of density 1.08 g/cm3, the yield stress is

observed to be possibly about 0.3 Pa, above which a relatively more rapid increase

in Jo is suggested. The yield stresses of other samples of Okeechobee mud cannot be

65
determined due to limited data. For the AK mud, it is clearly shown in Figure 4.11

that the yield stress is a function of density. When the stress is less than the yield

stress, generally Jo (or Go) is observed to be weakly related to the stress, which

reflects the linear range of theological properties of the mud. Comparing Figure 4.10

with Figure 4.11, it is also seen that for a given density Okeechobee mud is more

compliant than AK mud.

From the oscillatory test, the storage modulus G' and the loss modulus G" are

plotted against frequency in Figure 4.12. It is seen that G' and G" are almost indepen-

dent of the frequency, which means that in the Voigt model, the elastic modulus, G,

equal to G', has no dependence on the frequency, although the viscosity, /, strongly

depends on the frequency. In fact, as we shall see next the Voigt model may be

suitable for simulating the rheological response under high frequency forcing.

Based on the results of the creep tests and the oscillatory tests, G1, G2 and P can

be computed from Equations 4.9, 4.10 and 4.11. From Figures 4.13 and 4.14, it is

seen that the coefficients of the viscoelstic model depend on the frequency such that

G2 and p decrease, but G1 increases with increasing frequency. For high frequency os-

cillations, very high values of G1 and relatively low values of G2 are obtained, which

implies that the proposed model is reduced to the Voigt model for high frequency

motion. In fact, when G1 is greater than 3000 Pa, which corresponds to the forcing

frequency greater than 10 Hz for clay mixture AK (see Figure 4.14), the wave attenu-

ation coefficient is no longer affected by G1 under wave flume conditions (see Section

7.2.4).

In general, the rate of energy dissipation per unit volumn, e, for a viscoelastic

material is given by E = (72/8)tp (Schreuder et al., 1986). The frequency depen-

dence of the material parameters implies that the energy dissipation is a function

of frequency (Krizek, 1971). At low or high frequencies, the energy dissipations are

low due to essentially an isothermal or adiabatic process, respectively. In the in-

0.0010

-0.0002

-0.0014

-0.0026

-0.0038

-0.0050
0.0250

0.0200

0.0120-

0.0040

-0.0040-

-0.0120-

-0.0200
0.0250

SI I i I I

++. +
++.
+ +

1 I I

4-+

Sample: Okeechobee Mud -
Applied Stress: 0.10 Pa
Density: 1.08 g/cm3

+
4-

0.0270 0.0290 0.0310 0.0330

STRAIN

0.0270 0.0290 0.0310 0.0330

STRAIN

0.0350

0.0350

Figure 4.7: Strain rate, 7, and rate of strain rate, 7, in stage II as functions of strain
for Lake Okeechobee mud.

I I I I I I I

IE

I 7 II

19"11:r

-

I I _T I_ J_ _^ __ I _

0.1500

0.1000

S0.0500

= 0.0000

-0.0500

-0.1000
0.0850

0.0100

0.0073

S0.0046

. 0.0019
-0

-0.0008

-0 0035

0.1180 0.1510 0.1840 0.2170 0.2500

Strain

0.0000 0.0060 0.0120 0.0180 0.0240 0.0300

Strain

Figure 4.8: Strain rate, 7, in stage I as functions of strain for AK mud.

I i i i I I
+

-4--

4-
-+ + +++++++

Sample: AK MUD
Applied Stress: 0.80 (Pa)
Density: 1.10 (g/cmr3)

- i 1 I I I i I i I l i I l I I

- -
+ +
S++ + +

+ ++ ++++++++ ++
+ 4

Sample: AK MUD
Applied Stress: 7.50 (Pa)
Density: 1.20 (g/cm r)
I I I i I t I i I I I I i

- ~---

.

68
termediate frequency range, the energy dissipation is appreciable as a result of a

neither-isothermal-nor-adiabatic process.

It is seen that in Figures 4.13, curve 6 is rather different in terms of the curve

shape and much smaller value of G1, G2 and / compared with other curves, and is

considered to be beyond the linear range of theological behavior. However, based

on the logarithmic scale, straight lines may be considered in both Figures 4.13 and

4.14, except for curve 6. In other words, there is a power relationship between the

moduli (G1, G2), viscosity (p), and the forcing frequency. In the case of Okeechobee
mud of 1.12 g/cm3 density, the average G1(Pa), G2(Pa) and it(Pa.s) as functions of

frequency can be expressed as

G, = exp(5.548) x fo.127 (4.19)

G2 = exp(0.318) x f-0.687 (4.20)

S= exp(5.290)x f-0.687 (4.21)

For AK mud of 1.2 g/cm3 density, a similar relationship, which will be used in the

hydrodynamic model simulation presented in Chapter 7, is found as

G1 = exp(8.049) x fo.14 (4.22)

G2 = exp(2.604) x f-0.490 (4.23)

S= exp(8.222) x f-0.490 (4.24)

where f is frequency (Hz).

It is observed, for example, for Okeechobee mud of 1.12 g/cm3 density, that

the measured value of instantaneous modulus, Go, equal to 1/Jo, is close to the

corresponding, G1, calculated at high frequency (about 10 Hz in this case) based on

the developed theological model and experimental data, which is not surprising since
the instantaneous modulus really represents the high frequency response. Figures 4.15

and 4.16 show G1, G2 and y as functions of stress amplitude. It is observed from

0.20
Stress (Pa)

' 10.00
Stress (Pa)

1.10
Density (g/cm3)

Figure 4.9: Slope, S, as a function of density and stress for Lake Okeechobee mud.

--------------------------

- -

Plus for density-1.05 (g/cm^3)
Star for density-1.08 (g/cm-3)

0.00

0.10

0.500

0.250

0.000

0.750

0.500

0.250

0.000

0.750

0.500

0.250

0.000

Plus for density-1.12 (g/cmr3)
Star for density=1.15 (g/cmr3)
I ___i _

0.00

5.00

0.30

15.00

1.15

0.40

20.00

1.20

En

1.00

1.05

70
Figure 4.15 that there exists a turning point at which G1, G2 and P start to decrease

dramatically with increasing frequency. However, when the stress is less than a certain

critical stress, T7, corresponding to the turning point, G1, G2 and p are weak functions

of stress. This critical stress may reveal the breakdown threshold of the material

structure. For example, when the density is 1.08 g/cm3, the critical stress is around

0.3 Pa, which agrees with the result from the instantaneous modulus point of view

(see Figure 4.10).
From Figures 4.13 and 4.14, as would be expected, while the OK mud and the

AK mud exhibit similar behaviors, differences between them are apparent as well. In

comparison with AK, OK mud is relatively fluid-like with a lower modulus, G2, which

is believed to be due to the presence of a high percentage of organic material in OK.

4.4 Summary of Observations

The selected theological model predicts an exponential decay of strain in the

strain creep test, and an exponential decay of the stress in the stress relaxation

test, both of which agree with the experimental results by Chou (1989).

Both the slope, S, representing the ratio of elasticity to viscosity, and the in-

tercept in the creep test, representing the instantaneous elastic response, are

distorted due to the inertia effect during the initial period of stages I and II

shown in Figure 4.4(A).

The slope, S, is almost independent of the applied stress, but a strong function

of mud density, which suggests that elasticity and viscosity respond in a way

such that elasticity increases while viscosity decreases when the mud density

becomes larger.

The Voigt model is only suitable for describing the theological response of mud

under the action of high frequency waves, for example, greater than 10 Hz for

71
AK mud in this case. In fact, the proposed model is reduced to the Voigt model

when the forcing frequency is reasonably large.

* The instantaneous modulus, Go, can be predicted at high frequency by the

model. In other words, the elastic modulus, G, in the Voigt model can be

approximated by the instantaneous modulus, since both moduli correspond to

high frequency oscillations.

* Linear viscoelastic properties can be expected when the forcing stress is less

than a critical stress, Tr, since in that case the parameters G1, G2 and iu are

weak functions of stress.

* G1, G2 and [ are functions of the forcing frequency and mud density. Owing to

the organic content, Okeechobee mud is less elastic, thus more fluid-like, than

AK mud. It is worth noting that the strong dependence on frequency suggests

that in the wave-mud system the wave frequency plays an even more important

role in the process of wave-mud interaction than anticipated previously.

1.0000

0.1000

0.0100

0.0010

0.0001
0.0

1

0.10

1.00

10.00

100.00

Stress (Pa)

Figure 4.10: Instantaneous compliance, Jo, as a function of stress for Lake Okeechobee
mud.

1 I 1111111 I I 1T 111ilI I I I I I I III l
I

1 for Density 1.05 (g/cnms)
2 for Density 1.08 (g/cnms)
1 3 for Density 1.12 (g/cmn) -
4 for Density 1.15 (g/cmns)

t I I itll I I I3-t1-- I- fil

1.00000 = i

1 for Density 1.05 (g/cnma)
0. 10000 2 for Density 1.10 (g/crrr3)
3 for Density 1.20 (g/cnra)
4 for Density 1.31 (g/crm3)
a
S0.01000 -

2

0.00100
0 3

= ~4--+-4--4^
0.00001

0.00001 1i111 l i 1 fllll
0.01 0.10 1.00 10.00 100.00

Stress (Pa)

Figure 4.11: Instantaneous compliance, Jo, as a function of stress for AK mud.

Sample: AK Mud
Density 1.20 g/cm3
Applied Stress: 5.00 Pa
16 Solid Line: Storage Modulus G"
Dashed Line: Loss Modulus G'

12

8

4

0 I I I I I I I
0.00 0.40 0.80 1.20 1.60 2.00
FREQUENCY (Hz)

gn

0 1

z1

0
2

0 1

- 1

0
O
2^

U i I i i i I i
Sample: AK Mud
Density 1.20 g/cm3
Applied Stress: 0.60 Pa
6 Solid Une: Storage Modulus G"
Dashed ine: Loss Modulus G'

2-

8
.-.------------

4

. I I I I I ;I I

0.00 0.40 0.80 1.20 1.60 2.00
FREQUENCY (Hz)

20

o 16

12

S12
CI 8

04
O

Sample: Okeechobee Mud
Density 1.15 g/cm3
Applied Stress: 7.00 Pa
6 Solid Une: Storage Modulus G"
Dashed Line: Loss Modulus G'

2-

8

4

n I I I I I I I I I

0.00 0.40 0.80 1.20 1.60 2.00
FREQUENCY (Hz)

1~

I I I I I I I i I
Sample: Okeechobee Mud
SDensity 1.08 g/cm3
Applied Stress: 0.30 Pa
SSolid ine: Storage Modulus G" -
Dashed Line: Loss Modulus G'

I I I I I I I I I

0.00 0.40 0.80 1.20 1.60 2.00
FREQUENCY (Hz)

Figure 4.12: Storage modulus, G', and loss modulus, G", as functions of frequency
for different densities and stress amplitudes for Okeeechobee mud.

10000.

1000.

100.

10.
0.010

1000.0 e

100.0

10.0

1.0

0.1 L
0.010

100000. e-

10000.

1000.
0
o
100.

10.
0.010

I I __ I I_ Ii III li

I for Strem 0.20 (Pa)
2 for Stres 0.40 (Pa)
3 for Stres 0.50 (Pa)

4 for Strss
5 for Stres

I 1 111111 I I I 11111 I II I

0.100

1.000

0.100 1.000

0.100 1.000

Frequency (Hz)

Figure 4.13: G1, G2 and, as functions of frequency for Lake Okeechobee mud.

I I i I I 1
1.00 (P.)
2.00 (Pa)

10.000

10.000

10.000

I I I I

SI I I I I 1 1I

76

100000.
for Stram 0.30 (Pa)
+ for Stress 0.60 (Pa)
> for Stress 5.00 (Pa)
S10000.

1000.

0
0 100.

10.
0.010 0.100 1.000 10.000
1000.0 i i

C 100.0 >

10.0

0
S1.0

0.010 0.100 1.000 10.000

1000000. -- I i I
Sample: AK MUD
Densityr 1.20 (g/ens)
,100000. -

0 =
>

1000.

100.

10.
0.010 0.100 1.000 10.000

Frequency (Hz)

Figure 4.14: G1, G2 and y as functions of frequency for AK mud.

77

1000. i 1 1

1
S 100.

I for Frequency 0.05 (Hz)
0 2 for Frequency 0.88 (Hz)
S3 for rrequency 2.00 (Hz)

10. I I i i -11
0.01 0.10 1.00 10.00
100.000

- 0.1000 1

1.000

0.100
o
0
S0.010

0.001 I I 1 ii I I il I II I
0.01 0.10 1.00 10.00
10000.0 i i i

S1000.0 1

100.0 3

0 10.0

> 1.0 Sample: OKEECHOBEE MUD
Density? I.1 (g/cmre)

0.1 1 1 I I I II I III I I I 1 1111
0.01 0.10 1.00 10.00

Stress (Pa)

Figure 4.15: G1, G2 and y as functions of applied stress for Lake Okeechobee mud.

100000. i i

S10000.

1000.
Z
I for Frequency 0.01 (Hz)
S 100. 2 for rrequency 2.15 (Hz)
3 for rrequency 10.0 (Hz)

10. I i i i ll
0.01 0.10 1.00 10.00

100.0

100.0

0 2
10.0
o 1.0 r3
0o o 3

0.1
0.01 0.10 1.00 10.00
1000000. i i i i

,100000. r

10000.

o 1000.

S100. Sample: AK MUD
Densityr 1.20 (g/cam)

10 1 1 1 I I l I I I I I I I I I I I
0.01 0.10 1.00 10.00

Stress (Pa)

Figure 4.16: G1, G2 and i as functions of applied stress for AK mud.

CHAPTER 5
WAVE-MUD INTERACTION MODELING

5.1 Introduction

In this chapter, a shallow water wave-mud inetraction model is introduced first

as an initial attempt to simulate the response of a soft mud bottom under progressive

shallow water waves with small amplitudes, in which an inviscid water column and a

viscous fluid mud are assumed. The evident advantage of this model is its simplicity

of application, since the solutions are analytic. With the considerations of finite

amplitude wave resulting from second order boundary conditions, a viscid water layer

and viscoelastic mud, a second order wave-mud interaction model is then developed,

so that the limitations in the shallow water model are removed. This development

allows the wave-mud modeling to be applicable over a much wider range of natural

conditions.

In order to have a better general description of the wave-mud interaction problem,

it is helpful to consider Figure 5.1, in which the density (p) profile of the sediment-

water mixture and the profile of the horizontal component of the wave-induced veloc-

ity amplitude (ur) in the water column and bottom mud are depicted. The density

profile is idealized to have three zones with boundaries corresponding to two signifi-

cant density gradients. In the upper layer, in which the pressure gradient and inertia

forces are generally more important than the shear stress in governing motion, the

suspension with a density close to that of water is practically a Newtonian fluid due

to typically quite low density, very close to 1.0 g/cm3. The boundary between the

water layer and the fluidized mud layer is characterized by a rather significant gradi-

ent in density, or lutoline, below which the density of mud is considerably higher, up

80
to about 1.20 g/cm3. Below the fluidized mud there occurs the cohesive bed in which

the density is even higher. In response to continuous wave loading, the thickness of

the fluidized layer increases in reality, due to an elastic deformation of the bed coupled

with a buildup of the excess pore pressure. The mud is then a mainly fluid-supported,

non-Newtonian slurry.

Noting the typically low concentrations of sediment over the entire water column

except near the water-mud interface, the problem of wave-mud interaction modeling

can be conveniently considered to be practically independent of entrainment of bottom

sediment particles into the upper water column (Mehta, 1991). Consequently, a water

wave-mud system forced by a progressive, non-breaking surface wave of periodicity

specified by frequency, cr, is considered in Figure 5.1. The upper layer is clear water,

while the low layer consists of mud with a sharp interface. The datum below the

lower mud layer is assumed to be rigid. It was argued by Foda (1989) that, based

on a physical scale analysis for the self-selected representative parameters, viscous

dissipation in the bed layer may occur over a relatively thin boundary layer just

below the interface. In the present case, the entire mud layer is considered to be

moveable; therefore wave energy is considered to be dissipated over the entire mud

bottom. In fact, due to the high viscosity of mud, the thickness of the wave boundary

layer in the mud layer is commonly quite large. In other words, the entire lower layer

can be the boundary layer, depending upon the wave frequency and the mud layer

thickness.

5.2 A Shallow Water Model

In the shallow water model, the problem is restricted to shallow water waves. In

the case of a rigid bottom, the shallow water condition is often defined as kh < 7r/10.

Combined with the shallow water dispersion relationship a/k = V/gh, the condition

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