UFL/COELTR/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 DACW3990K0010.
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 Poroelastic 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 WAVEMUD 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 Setups . . . .... 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 WaterMud 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 WAVEMUD 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 creepcompliance 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 CarriMed 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) gapsetting 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 waterfluid 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 (VibroMeter, 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 timeseries and corre
sponding acceleration signal timeseries; 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 (R117R124; 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 twolayered, waveforced 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 timevariation of wavemud interface slope along flume
for AK mud: a = 1.5 cm, f = 1.0 Hz, hi = 16 cm, h2 = 17.3 cm..
Equilibrium watermud 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; longdashed curve:
simulation using the shallow water model; shortdashed 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 nonmudbank 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 watermud 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 = Waveinduced velocity in water column
u2 = Waveinduced velocity in mud layer
Ub = Maximum particle velocity
u; = Velocity in the x; direction
uj = Velocity in the xj direction
um = Waveinduced 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
wavemud 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. Waverelated 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 watermud 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
crossshore 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 wavemud 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 wavemud 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 wavebottom 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 wavemud 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 wavemud 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 wavemud 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 crossshore direction, and it is believed that the time
evolution of the mudbank surface is determined by the mud mass transport in the
crossshore 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 wavemud interface
and the mean water level. This objective is pursued here through the development of
a second order wavemud 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 crossshore 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 wavemud 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 crossshore 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 watermud 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
wavemud problem are Lake Okeechobee, and the southwest coast in India.
6
While Lake Okeechobee is a shallow water, wavedominated 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
wavemud interaction. Prior investigations on mass transport, an important result of
wavebottom 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
watermud 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 crossshore
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 wavebottom 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, poroelastic, 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 waveinduced 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, waveinduced 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 waveinduced 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 fluidmud 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 wavemud 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 hurricanegenerated wave heights dramati
cally decreased from 2023m in deep water to 34m in water depths of 1221m 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 finegrained 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 nonlinear attenuation
mechanism became strong in the sense that attenuation was a strong function of the
deepwater 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, suspendedsediment 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 shallowwater 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 linearlyelastic 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 2D, nonporous, 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 smallamplitude 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 Poroelastic Model
Poroelastic 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 fluidfilled porous media
under dynamic loading. He found that dissipative poroelastic 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 3D 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 poroelastic 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 solidtosolid 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 Coulombdamped poroelastic 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 onedimensional
ocean wave spectrum transformation by an arbitrary slope of an inhomogeneous poro
elastic medium with nonlinear elastic moduli and nonlinear 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 poroelastic 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 fluidsolid
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 twolayered 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(p2p1)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 threeboundarylayer viscous model to in
vestigate solitary wave propagation over fluidmud bottom. They pointed out that
the interaction between nonlinear waves and fluidmud 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 stressstrain 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 onethird 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 smallamplitude 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 = vrT. 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 depthvarying bed properties, the shear
21
stress at the mudwater 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 outofphase motion between water and mud. The
results showed a resonance phenomenon with respect to the motion of the wavemud
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 watermud system were ignored.
Chou (1989) developed a fourlayered 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 watermud 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 nearhorizontal 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 watermud interface due
to particulate erosion and deposition.
Yamamoto (1984) presented experimental results on waveinduced 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 waveinduced sediment mass transport in the
water layer by solving the 1D 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 twolayered 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 socalled viscoelastic
plastic model to calculate mud transport rate under waves and wavecurrent 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 watermud interaction is timedependent relative
to a time scale greater than the wave period. However, after a steady fluidized mud
depth is reached, the characteristics of the watermud system can be regarded as
timeindependent 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 waterbed 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 watermud 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 wellconsolidated 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 weaklyconsolidated, 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 wellconsolidated 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 poroelastic model.
Yamamoto (1982) proposed a mechanism of energy loss by solidsolid 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 poroelastic 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 pseudoplastic model in which the stressstrain 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 wavesoft 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 nonzero 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 nonzero 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 nonzero, 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 nonzero 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 nonzero parameters, we have the Jeffreys model as
7 + a I = 01 + #2 7 (3.4)
In this case we can construct two alternative springdashpot 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 12
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, mechanicallyequivalent, 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 twoparameter model or a threeparameter 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 viscosityinduced 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 subtests: 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 controlledstrain oscillatory test in which the input is the oscillating
strain. The other is the controlledstress 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 controlledstrain 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 controlledstress 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 101 (see Chapter 6) and kr on the order of 103 (Feng et al.
1992). Then ki/kr is on the order 104. 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 AbdelHady 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 spacefilling
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 stronglyflocculated suspensions.
Creep tests using an applied stress rheometer showed that the material exhibited
solidlike viscoelastic behavior at sufficiently small stresses. On increasing the stress
there was a change from solidlike to fluidlike 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 creepcompliance 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 wallslip 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 Killite 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 controlledstrain 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 breakdown 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
102 101 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
waveinduced 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 watermud 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 twoparameter models, the fourparameter 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
controlledstress 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 CarriMed 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 106 to 5 x 103 s1, or the
oscillatory mode in the frequency range of 1 x 103 to 10 Hz and a minimum strain
amplitude of 1 x 104. 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 airbearing to function properly, and water to or from the
water bath and the Peltier system to provide thermoregulation 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 unidirectional 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 greenishwhite color, and 50% kaolinite (pulverized kaolin). The
56
9 \ 10 11 12 13 II
V L
Figure 4.3: CarriMed 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) gapsetting 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 beigecolored 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 airbearing was freed. After switching on the power to the CSL (see Figure 4.3),
the liquidcrystal 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, 1.051.15 g/cm3 for Okeechobee mud and 1.021.22
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/cm3) (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 II\II
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
neitherisothermalnoradiabatic 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 f0.687 (4.20)
S= exp(5.290)x f0.687 (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 f0.490 (4.23)
S= exp(8.222) x f0.490 (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 density1.05 (g/cm^3)
Star for density1.08 (g/cm3)
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 density1.12 (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 fluidlike 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 fluidlike, than
AK mud. It is worth noting that the strong dependence on frequency suggests
that in the wavemud system the wave frequency plays an even more important
role in the process of wavemud 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 I3t1 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+44^
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
WAVEMUD INTERACTION MODELING
5.1 Introduction
In this chapter, a shallow water wavemud 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 wavemud interaction model is then developed,
so that the limitations in the shallow water model are removed. This development
allows the wavemud modeling to be applicable over a much wider range of natural
conditions.
In order to have a better general description of the wavemud 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 waveinduced 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 fluidsupported,
nonNewtonian slurry.
Noting the typically low concentrations of sediment over the entire water column
except near the watermud interface, the problem of wavemud 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
wavemud system forced by a progressive, nonbreaking 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 selfselected 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
