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UFL/COEL-TR/111
SEDIMENT-ASSOCIATED CONSTITUENT RELEASE
AT THE MUD-WATER INTERFACE DUE TO
MONOCHROMATIC WAVES
by
Yigong Li
Dissertation
1996
SEDIMENT-ASSOCIATED CONSTITUENT RELEASE AT
THE MUD-WATER INTERFACE DUE TO MONOCHROMATIC WAVES
By
YIGONG LI
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
1996
ACKNOWLEDGMENT
First of all, I would like to express my deepest gratitude to my advisor and the
chairman of my supervisory committee Professor Ashish J. Mehta, for his constructive
direction, enthusiasm, advice and unflagging support throughout this four year study, which
has been a challenging, joyful and unforgettable experience in my life.
I wish to thank Professors Kirk Hatfield and Robert G. Dean for their continuously
valuable advice, suggestions and discussions. Thanks are also due to the other committee
members, including Professor Peter Y. Sheng and Professor Brij M. Moudgil for their advice,
comments and patience in reviewing this dissertation.
Appreciation is extended to all other faculty members in the department, as well as
those in the Department of Aerospace Engineering Mechanics and Engineering Science,
Civil Engineering, Agricultural Engineering, Soil Science and Chemical Engineering for
supplying various components of knowledge essential for the pursuit of this study through
their creative teaching efforts.
Gratitude is due to departmental staff, especially, Mr. Sydney Schofield, Mr. Jim
Joiner and Mr. George Chappell at the Coastal Engineering Laboratory for their cooperation
and help during the experimental phase of this research. Support also came from many
friends and fellow research assistants.
Special thanks go to Sandra Bivins, Becky Hudson, Lucy Hamm, Cynthia Vey, John
Davis and Helen Twedell for their kindness, which helped directly or indirectly in the
completion of this study. Deep appreciation goes to Dr. Feng Jiang, Dr. Xinjian Chen and
Dr. Say-Chong Lee, with whose help I got through the hard initial period of my stay in
Gainesville. Appreciation also goes to Mr. Paul Devine, Mr. Ahmad Tarigan, Mr. Hogo
Rodriguez, Ms. Xu Wang, Ms. Jie Zheng, Dr. Taerim Kim, Mr. Eduardo Yassuda, Mr. Mike
Krecic and Mr. Albert Browder for their assistance and useful discussions.
Funding for this work came from U.S. Army Engineer Waterways Experiment
Station at Vicksburg, Mississippi. The contaminant portion of the work was funded by
contract DACW39-95-K-0023 through the Environmental Laboratory, and the sediment part
by contract DACW39-95-K-0022 through the Hydraulics Laboratory. Assistance provided
especially by Dr. Mark Dortch and Dr. T.M. Parchure in processing the contracts, managing
them and providing valuable technical guidance is sincerely acknowledged.
My final acknowledgement is reserved for those whom I probably owe the most, my
wife, Wendy S. Tan, for her love, support, encouragement and patience throughout these
four years, and my parents, who instilled in me the work ethic and values that have allowed
me to make it this far, and have supported me all my life.
TABLE OF CONTENTS
ACKNOWLEDGMENT ................................... ......... ii
LIST OFFIGURES .................................................... viii
LIST OFTABLES .................................................... xiii
LIST OF SYMBOLS ................................................... xv
ABSTRACT ........................................................ xxx
1 INTRODUCTION ................................................... 1
1.1 Problem Statement ........................................... 1
1.2 Objective, Tasks and Scope ....................................... 5
1.3 Outline of Presentation ........................................ 7
2 BACKGROUND INFORMATION ...................................... 9
2.1 Introduction ............ .......................................9
2.2 Mud Rheology and Wave-Mud Interaction Modeling ................. 9
2.2.1 Mud Rheology .........................................10
2.2.2 Wave-Mud Interaction Modeling .................................. 19
2.3 Mud Fluidization and Fluid Mud .................................23
2.4 Processes Governing the Vertical Structure of Suspension ............. 28
2.4.1 Settling ................ ........................... 28
2.4.2 Diffusion ..... .................... .................... 31
2.4.3 Deposition Rate .................................... .35
2.4.4 Entrainment .......................................... 36
2.5 Sorption Models ............................................... 40
2.5.1 Equilibrium Sorption Isotherm ........................... 40
2.5.2 Kinetic Sorption Models ................................ 43
2.5.3 Advanced Sorption Models ............................. 46
2.6 Sediment-Associated Contaminant Transport ........................ 49
2.6.1 Contaminant Transport in Water .......................... 49
2.6.2 Contaminant Transport in Bottom Sediment ................. 52
2.6.3 Fluxes Across the Sediment-Water Interface ................ 58
2.6.4 Fluxes Across the Fluid Mud-Bed Interface .................. 62
3 MUD FLUIDIZATION BY WAVES ................................... 64
3.1 Introduction ................................................. 64
3.2 Mechanics of Fluidization ................ .................... 65
3.2.1 Vertical and Horizontal Motions of Bed under Waves ......... 65
3.2.2 Bed Failure by Horizontal Forces ......................... 66
3.2.3 Bed Fluidization by Vertical Forces ....................... 68
3.2.4 A Heuristic Criterion for Bed Fluidization Under Waves ....... 70
3.3 Voigt and Extensional-Voigt Models .............................72
3.4 Bed as a Spring-Dashpot-Mass System ............................ 74
3.4.1 Case 1: G,, and pn Independent of Depth .................... 75
3.4.2 Case 2: G,, and pn Increase with Depth ..................... 77
3.5 Dynamic Response of Bed .....................................79
3.5.1 Wave over Bed: One-Degree of Freedom ................... 79
3.5.2 Wave over Fluid Mud above Bed: Two-Degrees of Freedom .... 80
3.6 Fluidization Depth ............................................ 83
3.6.1 M odel Results ........................................ 83
3.6.2 Comparison with Data ................................ 90
3.7 Application to Field Data ..................................... 93
3.7.1 Field Conditions ......................................93
3.7.2 Model Application .................................... 96
3.8 Conclusions ................................................. 99
4 SEDIMENT TRANSPORT MODELING ............................... 102
4.1 Introduction ................................................ 102
4.2 Three-Layered System ........................................ 102
4.3 Wave-Averaged Transport Equation for Constituent Transport ......... 103
4.4 Sediment Transport Model ................................... 107
4.4.1 Vertical Fine-grained Sediment Transport Equation .......... 107
4.4.2 Boundary Conditions .................................. 110
4.5 Fluid Mud Entrainment Rate ................................... 112
4.5.1 Entrainment Rate Formulation ........................... 112
4.5.2 Laboratory Experiment ............................... 115
4.5.3 Critical Global Richardson Number for Entrainment ......... 118
4.5.4 Comparison with Experimental Data ...................... 124
4.6 Laboratory Simulations ...................................... 128
4.7 Model Application to Field Data .............................. 132
4.7.1 Coast of Louisiana ................ .................. 133
4.7.2 Southwest Coast of India ............................... 142
5 SORPTION KINETICS .............................................. 152
5.1 Introduction ............................................152
5.2 Dyes, Sediments and Fluid Characteristics ......................... 153
5.3 Sorption Test Procedure ..................................... 155
5.3.1 Sampling of Sediment-Dye-Water Mixture ................. 155
5.3.2 Photographic Method for Dye Concentration Determination
in Water ................ ......................... 156
5.3.3 Calibration Experiments ............................... 158
5.4 Results and Discussion ................ ...................... 160
5.4.1 Sorption Isotherms ................................... 160
5.4.2 Sorption Kinetics ..................................... 164
5.4.3 Discussion .......................................... 169
6 CONSTITUENT TRANSPORT MODELING ........................... 174
6.1 Introduction ................................................ 174
6.2 Constituent Transport Model .................................. 174
6.2.1 Formulation ......................................... 174
6.2.2 Governing Equations .................................. 176
6.2.3 Boundary and Interfacial Connective Conditions ............ 178
6.3 Wave-Induced Diffusivity in Fluid Mud, Ka ........................ 184
6.4 Mass Transfer Coefficient, K,2 ................................ 190
6.5 Solution Technique .......................................... 196
7 FLUME EXPERIMENTS ........................................... 198
7.1 Introduction ................................................ 198
7.2 Experimental Equipment ......................................198
7.2.1 Wave Flume ......................................... 198
7.2.2 Colorimeter ......................................... 199
7.3 Test Conditions and Procedure .................................. 204
7.3.1 Test Conditions ................ .................... 204
7.3.2 Procedure ................ ......................... 204
7.4 Data Analysis ............................................ 206
7.4.1 Suspended Sediment ................................. 206
7.4.2 Dye Release in Water ................................... 215
8 RESULTS, DISCUSSION AND CONCLUSIONS ......................... 225
8.1 Introduction ................................................225
8.2 Results of Model Simulations and Calibrations ..................... 225
8.2.1 Model Conditions ............ ........................225
8.2.2 Suspended Sediment Concentrations ...................... 230
8.2.3 Dye Concentrations ................ .................. 233
8.3 Analysis and Discussion ..................................... 238
8.3.1 Release Flux and Sub-fluxes ........................... 238
8.3.2 Effects on Release Sub-fluxes ........................... 243
8.4 Summary and Conclusions ...................................... 250
8.5 Recommendations for Future Studies ............................. 254
APPENDICES
A FLUME EXPERIMENTAL DATA ....................................257
B SIMULATED AND MEASURED SUSPENDED SEDIMENT
AND DYE CONCENTRATION PROFILES ........................ 269
B.1 Suspended Sediment Profiles .................................. 269
B.2 Dye Concentration Profiles in Water ............................. 280
BIBLIOGRAPHY .................................................... 296
BIOGRAPHICAL SKETCH ........................................... 313
LIST OF FIGURES
1.1 Total contaminant flux and sub-fluxes. .................................. 2
2.1 Common two-parameter viscoelastic models. ............................. 13
2.2 Jeffrey models. ..................................................... 14
2.3 A typical wave-mud system (after Hwang, 1989) ........................... 23
2.4 Schematic of instantaneous stress profile in a wave-mud system
(after Mehta et al., 1994). ......................................... 24
2.5 Three-layered water-fluid mud-bed system and
vertical sediment transport processes under waves........................ 25
2.6 A schematic description of settling velocity and flux variation
with suspension concentration (after Hwang, 1989)....................... 29
2.7 Sediment-associated contaminant transport processes. ....................... 49
3.1 Forces on a particle or floc........................................... 68
3.2 Bed simulated as a S-D-M system with known vertical distributions of bed density,
p, extensional elastic modulus, Gn, and extensional viscous coefficient, i,. ... 74
3.3 Bed layer represented by a one-degree of freedom system. ................... 80
3.4 Fluid mud and bed layers represented by a two-degrees of freedom system ....... 81
3.5 A schematic description of the criterion for equilibrium fluidization depth, z'c. ... 83
3.6 Fluidization depth variation with wave frequency and amplitude
with negligible cohesion and without considering initial fluid mud effects .... 84
3.7 Fluidization depth variation with wave frequency and bed elastic modulus
with negligible cohesion and without considering initial fluid mud effects. ... 85
3.8 Fluidization depth variation with wave frequency and bed viscous coefficient
with negligible cohesion and without considering initial fluid mud effects. .... 85
3.9 Fluidization depth variation with wave frequency and amplitude
with negligible cohesion, but including initial fluid mud effects. ............. 87
3.10 Fluidization depth variation with wave frequency and bed elastic modulus
with negligible cohesion, but including initial fluid mud effects. ............ 87
3.11 Fluidization depth variation with wave frequency and bed viscous coefficient
with negligible cohesion, but including initial fluid mud effects. ............. 88
3.12 Fluidization depth variation with wave frequency and cohesion parameter. ..... 89
3.13 Comparison.between calculated and flume-measured fluidization depths. ...... 92
3.14 Bathymetric map of Lake Okeechobee. Depth are relative to a datum
which is 3.81 m above msl (after Mehta and Jiang, 1990) ................. 94
3.15 Mud thickness contour map of Lake Okeechobee (after Kirby et al., 1989). ..... 95
3.16 A representative bottom mud density profile in Lake Okeechobee. ............ 97
3.17 Calculated fluidization depth as a function of water depth in Lake Okeechobee
and band of measured values. ..................................... 98
3.18 An example of fluidization depth calculation for Lake Okeechobee at a site
where water depth was 1.43 m. ..................................... 98
4.1 Three-layered system and vertical sediment transport processes considered. .... 102
4.2 Side view of wave flume used in entrainment experiments................... 115
4.3 Critical Richardson Number for entrainment by waves ...................... 121
4.4 Critical Richardson Number estimated from the experiments of Maa (1986) .... 121
4.5 Dimensionless entrainment rate as a function of Richardson Number. ......... 125
4.6 Entrainment flux as a function of wave height and mud viscosity. ............. 127
4.7 Comparison between simulated and measured data for Run 4 of Maa (1986). ... 130
4.8 Comparison between simulated and measured data for Run 5 of Maa (1986). ... 130
4.9 Comparison between simulated and measured data for Run 6 of Maa (1986). ... 131
4.10 Location map of the study area of Kemp (1986). ........................ 135
4.11 Simulated and measured data (Kemp, 1986) for the Louisiana coast
during frontal passage. ............................................ 139
4.12 Simulated and measured data (Kemp, 1986) for the Louisiana coast
during the post-frontal period. ..................................... 140
4.13 Location map of the study area of Mathew (1992) off Alleppey
in the State of Kerala, India. ..................................... 143
4.14 A conceptual model of mudbank evolution ............................. 143
4.15 Settling velocity data used for mudbank turbidity simulation ............... 150
4.16 Simulated suspended sediment concentration profiles and data of Mathew (1992)
in the vicinity of Alleppey Pier...................................... 151
5.1 Schematic diagram of the sorption procedure. ............................. 156
5.2 Schematic diagram for photography in sorption tests. ...................... 157
5.3 Procedure for determining dye concentration ............................. 157
5.4 Calibration curve for Rhodamine B solutions. ............................ 159
5.5 Calibration curve for Erioglaucine A solutions ............................ 160
5.6 Linear isotherm of Rhodamine B sorbed onto kaolinite. ..................... 161
5.7 Linear isotherm of Rhodamine B sorbed onto AK mud. ..................... 161
5.8 Linear isotherm of Erioglaucine A sorbed onto kaolinite. ................... 162
5.9 Linear isotherm of Erioglaucine A sorbed onto AK mud. ................... 163
5.10 Effect of temperature on sorption of Rhodamine B on kaolinite ............. 163
5.11 Comparison between kinetic sorption model and experimental data. ......... 167
5.12 Comparison between kinetic sorption model and experimental .............. 167
5.13 Comparison between kinetic sorption model and experimental data. ......... 168
5.14 Comparison between kinetic model and experimental results................ 168
5.15 Comparison between kinetic sorption model and experimental data. ......... 169
5.16 Chemical structures of Rhodamine B and Erioglaucine A .................. 169
5.17 Idealized clay geometries of kaolinite and attapulgite. ...................... 170
6.1 Sediment-associated contaminant transport processes considered. ............. 175
6.2 Contaminant fluxes across the water-fluid mud and fluid mud-bed interfaces. ... 179
6.3 Schematic representation of the mode of indirect dissolved and
particulate contaminant transport across the water-mud interface ........... 183
6.4 Schematic of energy transport and dissipation, velocity distribution and
momentum transport, and constituent concentration distribution and
momentum-analogous constituent mass transport ...................... 188
6.5 Schematic shown of bottom diffusive sub-layer and wave boundary layer in water,
and direct convective flux........................ .......... ......... 191
7.1 Wave flume used in experiments ........................ .............. 199
7.2 Relationship between incident wave length and transmittance for Rhodamine B.. 201
7.3 Relationship between incident wave length and transmittance for Erioglaucine A. 201
7.4 Calibration of Rhodamine B concentration with selected light wave length,
LRB= 550nm.................................................. 202
7.5 Calibration of Erioglaucine A concentration with selected light wave length,
LEA = 625 nm.................................................... 202
7.6 Suspended sediment concentration profile variations with time at B, C and D
in the duration range of 240 ~ 480 minutes ............................ 207
7.7 Suspended sediment concentration profile variations with time at A and E
in the duration range of 240 480 minutes. ......................... 208
7.8 Suspended sediment concentration profile variations with time at B, C and D
in the duration range of 480 ~ 720 minutes. ........................... 209
7.9 Suspended sediment concentration profile variations with time at A and E
in the duration range of 480 ~ 720 minutes. ........................... 210
7.10 Comparison between the calculated suspended sediment concentration
using three methods (Eqs. 7.7,7.10 and 7.11). ......................... 214
7.11 Dye concentration profile variations with time at B, C and D
in the duration range of 0 240 minutes. .......................... .215
7.12 Dye concentration profile variations with time at A and E
in the duration range of 0 ~ 240 minutes. ............................. 217
7.13 Dye concentration profile variations with time at B, C and D
in the duration range of 240 480 minutes ............................ 218
7.14 Dye concentration profile variation with time at A and E
in the duration range of 240 480 minutes ............................ 219
7.15 Dye concentration profile variations with time at B, C and D
in the duration range of 480 720 minutes. ........................... 220
7.16 Dye concentration profile variations with time at B, C and D
in the duration range of 480 720 minutes. .......................... 221
7.17 Comparison between calculated dye concentration using three methods
(Eqs. 7.12, 7.15 and 7.16).............................. ........... 223
8.1 Comparison between simulated and measured depth-averaged
suspended sediment concentrations in Test 2 ........................... 230
8.2 Comparison between simulated and measured depth-averaged
suspended sediment concentrations in Test 3 ........................... 231
8.3 Comparison between simulated and measured depth-averaged
suspended sediment concentrations in Test 4 ............
8.4 Comparison between simulated and measured depth-averaged
suspended sediment concentrations in Test 5 ............
8.5 Comparison between simulated and measured depth-averaged
suspended sediment concentrations in Test 6 ............
8.6 Comparison between simulated and measured depth-averaged
dye concentrations in Test 1. .....................
8.7 Comparison between simulated and measured depth-averaged
dye concentrations in Test2. ..........................
8.8 Comparison between simulated and measured depth-averaged
dye concentrations in Test 3. .......................
8.9 Comparison between simulated and measured depth-averaged
dye concentrations in Test4. .......................
8.10 Comparison between simulated and measured depth-averaged
dye concentrations in Test 5 ......................
8.11 Comparison between simulated and measured depth-averaged
dye concentrations in Test 6. ..... ..... ...........
8.12 Dye release flux and sub-fluxes in Tests 1-1 (0-240 min)
and 1-2(240-480 min) ..............................
.............. 231
.............. 232
.............. 232
..............235
..............235
..............236
..............236
..............237
..............237
..............238
8.13 Dye release flux and sub-fluxes in Tests 2-1(0-240 min), 2-2 (240-480 min)
and 2-3 (480-600 min). ..........................................239
8.14 Dye release flux and sub-fluxes in Tests 3-1 (0-240 min)
and 3-2(240-480 min). ..........................................239
8.15 Dye release flux and sub-fluxes in Test 4 (0-720 min). ..................... 240
8.16 Dye release flux and sub-fluxes in Tests 5-1(0-240 min), 5-2 (240-480 min)
and 5-3 (480-600 min) ........................................ 240
8.17 Dye release flux and sub-fluxes in Tests 6-1(0-240 min), 6-2 (240-480 min)
and 6-3 (480-600 min) ...........................................241
8.18 Schematic shown characteristic profiles and coefficients
related to release fluxes............................................ 244
8.19 Relationship between the direct convective flux coefficient, 12c,
and the bottom boundary layer Richardson Number, Ri ................... 246
8.20 Relationship between dimensionless diffusivity in fluid mud, D/v,,
and the dimensionless wave damping parameter, H2ki/(h2-h,) .............. 248
B. 1 Comparison between simulated and measured suspended sediment profiles
in Test 2-2. .................................................... 269
B.2 Comparison between simulated and measured suspended sediment profiles
in Test 2-3..................................................... 270
B.3 Comparison between simulated and measured suspended sediment profiles
in Test 3-2. .................................................... 271
B.4 Comparison between simulated and measured suspended sediment profiles
In Test 3-3. .................................................... 272
B.5 Comparison between simulated and measured suspended sediment profiles
in Test 4-1 during time range of 0 300 min. .......................... 273
B.6 Comparison between simulated and measured suspended sediment profiles
in Test 4-1 during time range of 300 660 min. ........................ 274
B.7 Comparison between simulated and measured suspended sediment profiles
in Test 5-2 during time range of 240 480 min ........................ 275
B.8 Comparison between simulated and measured suspended sediment profiles
in Test 5-3 during time range of 480 720 min ........................ 276
B.9 Comparison between simulated and measured suspended sediment profiles
in Test 6-1 during time range of 0 240 min .......................... 277
B.10 Comparison between simulated and measured suspended sediment profiles
in Test 6-2 during time range of 240 480 min. ........................ 278
B. 11 Comparison between simulated and measured suspended sediment profiles
in Test 6-3 during time range of 480 ~ 720 min. ........................ 279
B.12 Comparison between simulated and measured dye profiles in Tests 1-1. ...... 280
B.13 Comparison between simulated and measured dye profiles in Tests 1-2. ...... 281
B.14 Comparison between simulated and measured dye profiles in Tests 2-1 ....... 282
B.15 Comparison between simulated and measured dye profiles in Test 2-2. ....... 283
B.16 Comparison between simulated and measured dye profiles in Test 2-3. ...... 284
B.17 Comparison between simulated and measured dye profiles in Test 3-1. ....... 285
B.18 Comparison between simulated and measured dye profiles in Test 3-2. ...... 286
B.19 Comparison between simulated and measured dye profiles in Test 3-3 ....... 287
B.20 Comparison between simulated and measured dye profiles in Test 4
during time range of 0 300 min. .................................. 288
B.21 Comparison between simulated and measured dye profiles in Test 4
during time range of 300 660 min. .................................289
B.22 Comparison between simulated and measured dye profiles in Test 5-1
during time range of 0 240 min ......................................290
B.23 Comparison between simulated and measured dye profiles in Test 5-2
during time range of 240 480 min. ................................ 291
B.24 Comparison between simulated and measured dye profiles in Test 5-3
during time range of 480 720 min ................................ 292
B.25 Comparison between simulated and measured dye profiles in Test 6-1
during time range of 0 ~ 240 min ...................................293
B.26 Comparison between simulated and measured dye profiles in Test 6-2
during time range of 240 480 min. ................................. 294
B.27 Comparison between simulated and measured dye profiles in Test 6-3
during time range of 480 720 min ................................. 295
LIST OF TABLES
2.1 Diffusion coefficients for some tracers in some typical marine mud beds. ....... 57
3.1 Wave and bed conditions and calculated and measured fluidization depths. ...... 91
3.2 Summary of selected fluid mud generation experiments in flumes from Ross(1988)
and Lindenberg et al. (1989) ........................................91
3.3 Flume tests of Lindenberg et al. (1989) without fluid mud generation .......... 93
4.1 Experimental results for determination of the critical entrainment condition. .... 117
4.2 Equilibrium depth-averaged suspended sediment concentrations
for specified wave conditions. ..................................... 119
4.3 Parameters for simulating Runs 4, 5 and 6 of Maa (1986). .................. 134
4.4 Measured wave and suspended sediment data from Louisiana coast
(based on Kemp, 1986). .......................................... 136
4.5 Parameters for simulating data from the Louisiana coast. ................... 138
4.6 Parameters for simulating field data of Mathew (1992)
from the southwest coast of India. .................................. 147
5.1 Physical and chemical properties of Rhodamine B and Erioglaucine A. ........ 153
5.2 Chemical compositions of kaolinite and attapulgite (%) .................... 154
5.3 Chemical composition of tap water. ................................... 155
5.4 Equilibrium sorption distribution coefficient, Kd (1/g) ...................... 161
5.5 First order rate coefficient for sorption kinetics ............. ............ 172
7.1 Tests Conditions ..................................
8.1 Characteristic parameters for modeling of Tests 1 and 2....
8.2 Characteristic parameters for modeling of Tests 3 and 4....
8.3 Characteristic parameters for modeling for Tests 5 and 6...
A. 1 Sediment concentration (g/1) data for Test 1 ............
A.2 Sediment concentration (g/1) data for Test 2 ............
A.3 Sediment concentration (g/1) data for Test 3 ............
A.4 Sediment concentration (g/1) data for Test 4 ............
A.5 Sediment concentration (g/1) data for Test 5 ............
A.6 Sediment concentration (g/l) data for Test 6 ............
A.7 Dye concentration (mg/1) data for Test 1 ...............
A.8 Dye concentration (mg/1) data for Test 2 ...............
A.9 Dye concentration (mg/1) data for Test 3 ...............
A. 10 Dye concentration (mg/1) data for Test 4 ..............
A. 11 Dye concentration (mg/1) data for Test 5 ..............
A. 12 Dye concentration (mg/1) data for Test 6 ..............
........... 203
............. 227
............. 228
............. 229
............. 257
............. 257
............. 258
............. 259
............. 259
............. 260
.......... 261
............. 262
............. 263
............. 265
............. 266
............. 267
A. 13 Temperature measurements (oC) ..................................... 268
LIST OF SYMBOLS
Ae = Empirical coefficient
ab = Amplitude of near-bed wave-induced horizontal excursion
B = Coefficient in relationship between constituent concentration and light intensity
B', B" = Characteristic coefficients in relationship between constituent concentration and
light intensity
C, Cd = Constituent concentration in water
Cb = Constituent concentration just outside the bottom diffusive layer in water
Ceq = Constituent concentration in water at equilibrium
Cdo = Initial constituent concentration in water
Cid = Constituent concentration in water layer
C2d = Constituent concentration in pore water in fluid mud
C3d = Constituent concentration in pore water in bed
Cd. = Final or equilibrium constituent concentration in water
CF = Consolidation coefficient
Ch = Cohesion
Cr = Shear resistance
C, = Constituent concentration at mud-water interface
Ci = Concentration of component i
C = Time-averaged constituent concentration
CA = Depth-averaged dye concentration in water at location A
CB = Depth-averaged dye concentration in water at location B
Cc = Depth-averaged dye concentration in water at location C
CD = Depth-averaged dye concentration in water at location D
CE = Depth-averaged dye concentration in water at location E
Cm = Depth-average dye concentration in water over mud trench
Co = Depth-averaged dye concentration in water beyond mud trench
Ct = Average suspended sediment concentration
C = Wave oscillating part of constituent concentration
C' = Turbulent fluctuation part of constituent concentration
ce = Equivalent viscous coefficient
c2 = Equivalent viscous coefficient of fluid mud
c3 = Equivalent viscous coefficient of bed
Ca.5 = Sediment concentration at about one-half water depth at location A
Cb.5 = Sediment concentration at about one-half water depth at location B
co0.5 = Sediment concentration at about one-half water depth at location C
cd.5 = Sediment concentration at about one-half water depth at location D
ceo.5 = Sediment concentration at about one-half water depth at location E
c, = Constant coefficient
D = Diameter of particle or floc
De = Empirical coefficient
Df = Wave-induced diffusivity in fluid mud
Dm = Molecular diffusivity
Di, Di = Diffusion coefficients of component i
Dr = Sediment deposition flux
d = Bed thickness
dl, d2 = Empirical coeffcients
de = Effective grain size
Ei, = Strain tensor
E, = Sediment entrainment flux
E, = Non-dimensional entrainment rate
E'o = Amplitude of E'ij
E',j = Deviatoric components of strain
Ew = Non-dimensional entrainment rate
e = Void ratio
F = Static force
F,, F2, F3, F4 = Normal forces from adjacent particles
Fl2m = Molecular diffusive flux of constituent
F12c = Direct convective flux of constituent
F12d = Indirect convective flux of constituent
Fi2e = Indirect convective flux of constituent associated with entrainment
Fi2p = Particle-bound flux of constituent
F,2s = Indirect convective flux of constituent associated with settling
xvii
Fe = Particle-bound flux of constituent associated with entrainment
FIs2s = Particle-bound flux of constituent associated with settling
FA = Advective flux
FD = Diffusive flux
Fd = Sediment diffusive flux in water
F, = Mass flux of component i
Fp = Sediment deposition flux
Fr = Sediment flux corresponding to the fraction of entrained sediment that is returned
to fluid mud by deposition
F'r = Sediment flux corresponding to the fraction of "settled" sediment that is entrained
into water
F, = Sediment settling flux in water
f = Wave frequency (= o/2rn)
f, f2, f, f4 = Shear forces from adjacent particles
fr = Fraction of sediment exchange between water and fluid mud
fric = Bottom friction coefficient
G = Elastic modulus
GI, G2 = Elastic moduli in standard solid model
Gn = Extensional elastic modulus
G' = Bed rigidity
G' = Elastic energy storage modulus
G" = Viscous energy dissipation modulus
xviii
G* = Complex shear modulus
g = Gravitational acceleration
g' = Effective gravitational acceleration
H = Wave height
Hs = Significant wave height
Hrs = Root mean square wave height
h, = Depth of water layer
h2 = Depth of water and fluid mud
h3 = Total depth of water, fluid mud and bed layers
hk = Thickness of clay particles
I = Light intensity through a fluid medium
Io = Light intensity through clear water
IA = Hue value of Erioglaucine A
IB = Hue value of Rhodamine B
HID = Second invariant of E'j
Jo = Instantaneous compliance
Ji = Diffusive flux of component i
JMa = Mass flux in fluid mud
Jmo = Momentum flux in fluid mud
Jt = Total mass flux in fluid mud
K = Coefficient in relationship between constituent concentration and light intensity
Km = Ratio of floc volume concentration to sediment concentration
KI2c = Convective mass transfer coefficient
Kd = Constituent distribution coefficient
Kn = Neutral mass diffusivity
Kx = Diffusivity in x-direction
K, = Diffusivity in y-direction
Kz = Diffusivity in z-direction
Ki = Diffusivity in z-direction in water
KI2 = Diffusivity in z-direction in fluid mud
Kz3 = Diffusivity in z-direction in bed
K', K" = Characteristic constants in relationship between constituent concentration and
light intensity
k = Wave number
k2 = Equivalent elasticity of fluid mud
k3 = Equivalent elasticity of bed
k, = Convective mass transfer coefficient under current
ke = Equivalent elasticity
ki = Wave damping coeficient or imaginery part of complex wave number
k, = Wave number or real part of complex wave number
k, = Permeability
k, = Yield strength
k'i,(i=8) = Empirical coefficients
LA = Length range represented by flume location A
L, = Length range represented by flume location B
Lc = Length range represented by flume location C
L, = Cohesive resistance
L, = Length range represented by flume location D
LE = Length range represented by flume location E
LEA = Optimal light wave length for Rhodamine B dilution
Li = Inertial force
Lmt = Length of mud trench
L, = Optimal light wave length for Erioglaucine A dilution
Mf = Empirical coefficient
mp = Particle mass
me = Equivalent mass
m2 = Equivalent mass of fluid mud
m3 = Equivalent mass of bed
n = Porosity
ni = Porosity in water layer
n2 = Porosity in fluid mud
n3 = Porosity in bed
nm = Manning's bed resistance coefficient
P = Constituent concentration sorbed on sediment
Po = Initial constituent concentration sorbed on sediment
P, = Constituent concentration sorbed on solids in water
P2 = Constituent concentration sorbed on solids in fluid mud
P3 = Constituent concentration sorbed on solids in bed
P. = Final or equilibrium constituent concentration sorbed on sediment
P" = Maximum concentration of constituent that can be sorbed
p = Deposition probability
Po = Amplitude of wave-induced cyclic pressure
pp = Wave-induced excess pore pressure
p, = Coefficient related to viscoelastic model
p, = Wave-induced cyclic pressure
Q = non-dimensional bouyancy flux
q = Bouyancy flux
q, = Coefficient related to viscoelastic model
R = Radius of clay particle
R2 = Coefficient of determination
Reb = Bottom wave Reynolds Number over rigid bed
Re'b = Bottom wave Reynolds Number over soft bed
Re*b = Bottom wave Reynolds Number over soft bed
Rex = Reynolds Number for current
Re. = Reynolds Number of wave boundary layer
Ri = Richardson Number
Rig = Global Richardson Number
Riw = Wave Richardson Number
xxii
Ri,, = Critical wave Richardson Number for entrainment
Ri, = Critical Richardson Number for entrainment under current
Ri = Rate of reaction of component i
Rr = Relative roughness of mud surface
r = First-order rate coefficient of sorption
r, = Ratio of bottom boundary layer across coefficient, o12c, to this coefficent at static
condition, al2co
S = Sediment concentration
S, = Sediment concentration in water
S2 = Sediment concentration in fluid mud
S3 = Sediment concentration in bed
Sc = Schmidt Number
Scf = Schmidt Number in fluid mud
Sv2 = Sediment volume fraction in mud
S'i = Upper limit concentration for free settling
S*2 = Upper limit concentration for flocculation settling
S*3 = Upper limit concentration for hindered settling
S*4 = Lower limit concentration for negligible settling
S = Depth-averaged sediment concentration
SA = Depth-averaged suspended sediment concentration at location A
S = Depth-averaged suspended sediment concentration at location B
Sc = Depth-averaged suspended sediment concentration at location C
xxiii
SD = Depth-averaged suspended sediment concentration at location D
SE = Depth-averaged suspended sediment concentration at location E
Sm = Depth-average suspended sediment concentration over mud trench
So = Depth-averaged suspended sediment concentration beyond mud trench
St = Average suspended sediment concentration
s = Surface renewal factor
Sao.5 = Sediment concentration at about one-half water depth at location A
sb.5 = Sediment concentration at about one-half water depth at location B
sco.5 = Sediment concentration at about one-half water depth at location C
Sdo.5 = Sediment concentration at about one-half water depth at location D
seo.5 = Sediment concentration at about one-half water depth at location E
T = Wave period
TS = Total suspended sediment mass in flume
Tij = Stress tensor
T'o = Amplitude of T'ij
T'ij = Deviatoric components of stress
U = Mean mixed layer horizontal velocity
u = Horizontal velocity in x-direction
u, = Horizontal velocity in water layer in x-direction
u2 = Horizontal velocity in fluid mud in x-direction
u2max = Maximum horizontal velocity in fluid mud in x-direction
u3 = Horizontal velocity in bed in x-direction
xxiv
u. = Mean longitudinal velocity outside boundary layer
Ub = Maximum wave-induced horizontal velocity just outside wave boundary layer
ue = Rate of downward propagation of interface
u,, = Pore water pressure
u, = Bottom friction velocity
u2* = Characteristic horizontal momentum velocity
u2" = Characteristic horizontal energy velocity
u = Time-averaged velocity in x-direction
Q = Wave oscillating velocity in x-direction
u' = Turbulent fluctuation velocity in x-direction
V = Phase velocity of a high frequency shear wave
v = Horizontal velocity in y-direction
v, = Horizontal velocity in water layer in y-direction
v2 = Horizontal velocity in fluid mud in y-direction
v3 = Horizontal velocity in bed in y-direction
v = Time-averaged velocity in y-direction
V = Wave oscillating velocity in y-direction
v' = Turbulent fluctuation velocity in y-direction
w = Vertical velocity in z-direction
wi = Vertical velocity in water layer in z-direction
w2 = Vertical velocity in fluid mud in z-direction
w3 = Vertical velocity in bed in in z-direction
xxV
w, = Settling velocity of sediment
Wsf = Free settling velocity of fine-grained sediment
= Vertical velocity of flow relative to sediment-water interface
w = Time-averaged velocity in x-direction
S = Wave oscillating velocity in x-direction
w' = Turbulent fluctuation velocity in x-direction
z'c = Fluidization depth
a = Coefficient modulating submerged weight of particle
aI2c = Bottom boundary layer constituent transport coefficient
a12cO = Bottom boundary layer constituent transport coefficient in static condition
ai (i=18) = Empirical coefficients
af = Empirical coefficient
an = Empirical coefficient
a, = Empirical coefficient
Pi(i=1-8) = Empirical coefficients
Pf = Empirical coefficient
Pn = Empirical coefficient
Pv = Empirical coefficient
Yi = Empirical coefficient
y, = Empirical coefficient
, = Shear strain rate
E = Empirical coefficient
xxvi
Es = Specific area of particles
Ex = Current-associated momentum diffusion coefficient in x-direction
Ey = Current-associated momentum diffusion coefficient in y-direction
Ez = Current-associated momentum diffusion coefficient in z-direction
ED = Wave energy dissipation rate per unit area
Ox = Wave-induced momentum diffusion coefficient in x-direction
6, = Wave-induced momentum diffusion coefficient in y-direction
Oz = Wave-induced momentum diffusion coefficient in z-direction
A = Empirical coefficient
Au = Absolute value of velocity difference across mud-water interface
(Au), = Critical value of Au for bottom mud entrainment
Auw = Excess pore water pressure
Ab = Buoyancy jump across mud-water interface
Ap = Density jump across mud-water interface
8 = Boundary layer thickness
81 = Empirical coefficient
8c = Thickness of bottom diffusive layer
8f = Thickness of bottom boundary layer for "film theory"
8i6 = Kronecker delta
E = Empirical coefficient
C = Vertical visplacement of particle
i = Water surface dispalcement
xxvii
Col, Co = Wave amplitudes
C2 = Vertical fluid mud surface dispalcement
C02 = Amplitude of fluid mud surface dispalcement
C3 = Vertical bed surface dispalcement
Co3 = Amplitude of bed surface dispalcement
0 = Tortuosity
K = von Karman constant
p = Dynamic viscocity
pL. = Constant viscosity at the limit of high (theoretically infinite) shear rate
Pa = Apparent dynamic viscosity
Plb = Dynamic viscosity
Pn = Extensional viscosity
Pp = Apparent dynamic viscosity
P' = Dynamic viscosity
p" = Second viscosity
p* = Complex viscosity
v = Kinematic viscosity
ve = Apparent kinematic viscosity
p = Density
p, = Density of water
P2 = Density of fluid mud
p3 = Density of bed
xxviii
p,, p, = Density of water
p, = Specific gravity
o = Total normal stress
o' = Effective normal stress
1 = Shear stress
Tb = Applied bed shear stress
Tcd = Critical shear stress for deposition
Tcm = Critical maximum shear stress for deposition of non-uniform sediment
Tcn = Critical minimum shear stress for deposition of non-uniform sediment
T, = Shear strength
T y = Yield stress
Ya, Tb = Shape functions
= Density stratification correction factor
= Friction angle of sand
() 2, ()2, = Phase shifts
w = Weighting coefficient for wave diffusivity in combined diffusivity of waves and
current
4)c = Weighting coefficient of current diffusivity in combined diffusivity of waves and
current
o = Angular wave frequency
oo = Characteristic resonance frequency
6oo = Coefficient related to tortuosity
xxix
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
SEDIMENT-ASSOCIATED CONSTITUENT RELEASE AT
THE MUD-WATER INTERFACE DUE TO MONOCHROMATIC WAVES
By
YIGONG LI
May, 1996
Chairperson: Dr. Ashish J. Mehta
Major Department: Coastal and Oceanographic Engineering
Constituent release fluxes across the wave-modulated fluid mud-water interface from
bottom sediment into water have been determined through a combination of theoretical
analyses and laboratory experiments.
The dependence of the equilibrium fluid mud depth on characteristic wave and bed
theological parameters is examined for water-bed and water-fluid mud-bed systems.
Theoretical results are compared with data obtained in a wave flume, and an order of
magnitude agreement between measured and predicted fluid mud thicknesses is achieved.
A three-layered, water-fluid mud-bed system is set up for modeling vertical fine-grained
sediment transport. Model results are shown to be in good agreement with laboratory data,
and are also shown to successfully simulate field measurements from two coastal sites.
A vertical constituent transport model, which includes sorption kinetics, is developed
and is combined with the sediment transport model. The total constituent release flux from
xxx
mud under wave action comprises mainly three sub-fluxes including direct convective flux
of the dissolved constituent, indirect convective flux of dissolved constituent and indirect
particle-bound constituent flux. The behavior of the mass transfer coefficient of the direct
convective flux is examined through dimensional analysis, and a Chilton-Colburn type
relation is obtained for determining this coefficient as a function of measurable parameters.
Two clayey sediments are used to prepare bottom mud, and two conservative dyes
are used as constituent surrogates. By calibrating the constituent transport model against
wave flume data obtained for the rate of release of dyes from bottom mud, release sub-fluxes
under wave loading are calculated. The indirect convective sub-flux is found to be an order
of magnitude lower than the directive convective and particle-bound sub-fluxes. However,
the direct convective sub-flux can be amplified several times due to interfacial oscillation,
and the interfacial flux coefficient is found to be related to the Richardson Number. Particle-
bound sub-flux can dominate the constituent release process when significant resuspension
occurs. However, resettling of the sediment flocs and entrapped water tends to considerably
reduce the net release of constituent to the ambient water column. Finally, wave-induced
diffusivity in mud, which controls the rate of upward supply of the constituent to the
interfacial region, is found to be proportional to wave energy dissipation rate per unit volume
of mud, and is shown to be two or three orders of magnitude greater than molecular
diffusion.
xxxi
CHAPTER 1
INTRODUCTION
1.1 Problem Statement
Considerable effort has been made in recent years toward reducing the discharge of
contaminants into fresh water and marine systems. However, these systems, which should
be the beneficiary of an abatement of contaminants, have not always improved as expected.
One reason is that external contaminants have already accumulated in bottom sediments and,
as a result, after the external sources have been removed or reduced, bottom sediments have
become the main sources of contaminants to the water column. Bottom-contaminated
sediments affect the water quality directly via diffusion and convection of the contaminants
into water, or indirectly through resuspension of sediment particles (Fig. 1.1). From Fig. 1.1
we observe that the bottom contaminant released flux can be divided into two categories: (1)
direct flux, due to diffusion and convection and (2) indirect flux, which depends on bottom
sediment suspension. The direct flux has two components: molecular diffusion, which
always exists but is very small compared with the other three fluxes, and convection, which
is due to external wave and/or current action. The sediment suspended-related indirect flux
also has two components: particle-bound flux, in which the contaminant sorbs on sediment
and is transported together with sediment suspension, and indirect convective flux, which
is due to water flow into or out of the bed.
Molecular diffusion Direct co vective flux Indirect coi vective flufarticle ound flux
(due t( eddy and (due to water exchange) (influ nced by
wave-indu ed diffusion) sorptiol /desoprtion)
Dire t flux Indir xt flux
(of dissolve d contaminant) (due to resu pension, including
erosion/entr inment and settling)
Water
Interface Tota: flux
(connective condition between water and bottom mud)
Bottom mud
Figure 1.1 Total contaminant flux and sub-fluxes.
Some examples of bottom contaminated sediment as sources of contaminant to
aquatic systems are summarized in Table 1.1. From this table we note that many types of
contaminants have already accumulated in bottom sediments. Since these sediments
represent an in-place potential contaminant source, quantification of the direct as well as
indirect releases of the contaminant is essential for making an assessment of water quality,
and for predicting further contamination or recovery of aquatic systems (Eadie and Robbins,
1987; Basmadjian and Quan, 1988; Burton, 1991; Reible and Savant-Malhiet, 1993; and
Brownawell and Flood, 1994). For such a quantification, the release fluxes constitute the
crucial boundary or connective conditions at the sediment-water interface in water quality
and contaminant transport models, such as the hydrodynamic and water quality model
WASP5 developed by the Environmental Protection Agency (Ambrose et al., 1991), the
Green Bay integrated exposure model GBTOX for the Great Lakes (DePinto, 1994), and the
3
3-D hydrodynamic, sediment transport and nutrient dynamics models that have been applied
to Lake Okeechobee and Tampa Bay, Florida (Chen, 1994).
Table 1.1 Examples where bottom sediment is the main contaminant source.
Location Contaminants Source
Detroit River, Michigan Metals Theis et al. (1979)
Lake Tarli Karng, Targo Trace metals Hart and Davies (1981)
Reservoir and East Basin Lake,
Australia
Toronto and Hamilton Harbors, Metals Nriagu et al. (1983)
Canada
Ninigret Pond, New York Cadmium and Fluoranthene Di Toro et al. (1989)
Long Island Sound, New York Cadmium Di Toro et al. (1989)
Godavari River Basin, India Heavy metals Biksham et al. (1991)
Guan-Tin Reservoir, China Heavy metals Xian-Chen et al. (1991)
Lake Ontario, Canada PCBs and Mirex Brownawell et al. (1994)
Massachusetts Bay and Heavy Metals, organic pollutants, Butman (1994)
Cape Cod Bay, Massachusetts and sewage contaminants
Lake Okeechobee and Nutrients, e.g., Phosphorus Chen (1994)
Tampa Bay, Florida and Nitrogen
Mikawa Bay, Japan Nutrient salts, e.g., Inoue et al. (1994)
inorganic phosphorus (PO4-P)
Tampa Bay, Florida Trace metals, PAHs, and PCBs Long et al. (1994)
Boston Harbor, Massachusetts Mercury and Lead Manheim (1994)
Among sediments, fine-grained, especially cohesive, materials play a key role in
sediment-associated contaminant transport, because these sediments possess large specific
surface areas and net electrical charges on their surfaces and edges that are essential for the
sorption of contaminants. A variety of factors can influence the release of contaminants from
the bottom sediment to the water column, e.g., hydrodynamic conditions (current velocity,
wave height and frequency, water depth, turbulent mixing, etc.), bottom sediment
4
characteristics (sediment mineral composition, density, theological properties, cation
exchange capacity, organic content, sorptive properties, etc.), suspended sediment
characteristics (sediment mineral composition, flocculation and aggregation, sorptive
properties, etc.), water-sediment exchange processes (settling, deposition, erosion,
entrainment, fluidization, dewatering, etc.), contaminant characteristics (molecular
diffusivity, partitioning properties, etc.), and water properties (pH value, salinity, etc.).
Field evidence and model simulations show that, typically, the top 10-30 cm layer of
bottom mud in the marine and lacustrine environments is active in storing and releasing
contaminants is usually soft and oscillates with water waves (Mehta and Jiang, 1992).
Contaminant fluxes across the sediment-water interface are affected considerably by this
motion and by the thixotropic changes that ensue within the oscillating bed. As a case in
point, Vanderborght et al. (1977) noted that due to bottom stirring the diffusion coefficient
of dissolved silica in the pore water within the top 3.5 cm of the shallow North Sea sediment
was two orders of magnitude greater than that due to molecular diffusion.
For sorptive contaminants, e.g., hydrophobic organic compounds (DDT, PBCs,
PAHs, etc.) or heavy metals (Pb, Cs, etc.), the release flux is also controlled by the reworking
of bottom sediment surface due to sediment entrainment and deposition (Reible and Savant-
Malhiet, 1993 and Brownawell and Flood, 1994). Since the transport processes--settling,
diffusion, deposition and entrainment--are greatly affected by wave loading (Thimakorn,
1983; Maa, 1986; Ross, 1988; Hwang, 1989; Mimura, 1993 and Winterwerp, 1994), waves
can also affect the indirect release of contaminants from the bed through resuspending
bottom sediment, in addition to direct transfer at the interface.
5
Thus, it is evident that contaminant transport models must accurately incorporate the
physical and the physico-chemical mechanisms that operate in the wave-governed dynamical
environment. Typically however, wave-enhanced fluxes are conventionally accounted for
merely by calibrating transport models that are only valid for conditions specified by fixed
beds having constant (time-independent) properties (Berner, 1980). Therefore, given the
considerable importance of properly modeling bottom fluxes for particle-associated
contaminant transport in wave-dominated waters, the need for improving the state of the art
in prediction of fluxes clearly exists.
1.2 Objective. Tasks and Scope
The primary objective of this research is to investigate the role of water waves in
governing the direct and indirect contaminant fluxes across the interface between soft mud
and water layers. Specifically, through modeling and laboratory experiments, the inter-
relationships between wave parameters, contaminant kinetics and mud properties will be
examined for specific contaminant surrogates and bottom muds. This investigation should
also provide an accurate mechanistic description of the connective or boundary condition of
the contaminant mass transfer flux across the wave-modulated water-mud interface for water
quality or contaminant transport models. The specific tasks to accomplish the work were as
follows:
1. Since the mud bed may be fluidized under wave motion, and because once the bed
is fluidized the theological properties and mass transport characteristics of fluid mud can be
6
significantly different from those of the non-fluidized bed, a fluidization model was
developed. Using this model, the fluidization depth under wave motion was predicted.
2. Because aquatic contaminant transport is characteristically associated with sediment
transport, a sediment transport model was set up in a three-layered water-fluid mud-bed
system, which incorporated the effects of fluid mud.
3. Sorption kinetics between sediments and dyes, to be used as contaminant surrogates
in experiments, were determined based on sorption experiments.
4. A sediment-associated contaminant transport model was developed in the three-
layered system. This model incorporated sorption kinetics and was used in conjunction with
the sediment transport model.
5. By calibrating the contaminant transport model against flume experimental data, the
total bottom contaminant releasing flux and its sub-fluxes under wave motions were
determined for various conditions specified by sediment and contaminant compositions and
wave conditions. The relevent coefficients, which determine the magnitudes of the release
fluxes, were obtained for the selected sediments and contaminants under wave motion.
6. An effort was made to explain the results in terms of a phenomenological model
relating the effect of wave and sediment properties on the release fluxes.
To achieve the above tasks, the scope of this study was defined as follows:
1. In the development of the mud fluidization model, only normal stresses were
considered and shear stresses neglected. Thus, this model is only applicable to short period
waves. Also, from this model only the final equilibrium fluidization depth, as would result
7
from steady and sustained wave action, was obtained. The time-dependent process of
fluidization is not predicted by this model.
2. The sediment transport model in the three-layered water-fluid mud-bed system did
not consider sediment exchange between fluid mud and bed, because the fluid mud depth
was considered to be its equilibrium value.
3. Only sorption experiments were carried out for sediments and dyes. The desorption
process were assumed to be the reverse of sorption.
4. Contaminant exchange was studied experimentally by initially inoculating the bottom
sediment with dye and tracking dye release in water under wave loading.
5. The contaminant transport model was calibrated only against laboratory data. The
contaminant surrogates, i.e., were use in the laboratory experiments because their properties
are easily measurable.
1.3 Outline of Presentation
Chapter 2 reviews relevant previous topics and studies on sediment transport, i.e.,
mud rheology, wave-mud interaction, mud fluidization and resuspension of bottom sediment.
Prior investigations on generic sorption models and release of bottom contaminants are also
discussed in this chapter. The fluidization model is developed in Chapter 3. It is compared
with expeirmental data and applied to field, i.e., Lake Okeechobee, in Florida. In Chapter
4, a sediment transport model in the water-fluid mud-bed system is developed. The model
is first calibrated against laboratory data and then applied to field conditions, i.e., the
southwest coast of India and the coast of Louisiana. Chapter 5 describes sorption
8
experiments for dyes, i.e., Rhodamine B and Erioglaucine A, sorbed onto selected sediments,
i.e., a kaolinite and a mixture of kaolinite and an attapulgite. The sorptive properties
between these dyes and sediments are obtained and incorporated in the transport model
described in Chapter 6. In Chapter 6 the sediment-associated contaminant transport model
is developed, which includes sorption kinetics in conjunction with the sediment transport
model developed in Chapter 4. Chapter 7 describes the flume experimental conditions for
bottom contaminant release under wave motion and presents laboratory data for both
sediment and contaminant transport under wave action. To obtain the bottom contaminant
release flux under wave motion, the contaminant transport model developed in Chapters 4
and 6 was calibrated against the flume data, and the results presented in Chapter 7. In
Chapter 8, the release flux and sub-fluxes are compared and correlated to wave conditions
and mud theological properties, and phenomenological explanations are provided for these
corrections. Also in this Chapter, overall study conclusions are presented and
recommendations for future studies are made.
CHAPTER 2
BACKGROUND INFORMATION
2.1 Introduction
The release of conservative contaminants stored in bottom sediments depends on
external hydrodynamic factors, bottom sediment properties and contaminant sorptive
properties. Sediment transport characteristics also play an important role in contaminant
release of contaminants. Thus, theoretical and modeling studies done in the sediment
transport area, i.e., mud rheology, wave-mud interaction, mud fluidization and resuspension
of bottom sediment, are reviewed first in this chapter. Following this review, generic sorption
models and sorptive properties of the selected dyes, i.e., Rhodamine B, Erioglaucine A, used
in this study are described. Field and laboratory experimental observations and previous
studies on the release of bottom contaminants are also described in this chapter.
2.2 Mud Rheology and Wave-Mud Interaction Modeling
The interaction between waves and mud beds is an important issue in understanding
coastal processes. When waves propagate over a mud bed, wave attenuation, energy
dissipation, change in wave kinematics due to corresponding changes in mud properties
under oscillatory loading and bed fluidization, mud mass transport etc. are all due to the
interactions between the wave and bottom mud. Also, vertical sediment transport, due to
10
settling, diffusion, erosion, deposition, entrainment, etc., are influenced by wave-mud
interactions. For modeling these interactions it is necessary to construct a closed set of
equations under wave loading, which include continuity and momentum equations for water
and mud motions. These equations must be solved analytically or numerically. A key step
in formulating the equations is prescribing the momentum equation for mud motion, which
requires a constitutive equation to describe the theological properties of mud.
In contrast to a sandy bed, the mud bed is almost impermeable. Thus, mud is usually
considered as a single-phase continuum. Mud rheology supplies the constitutive equation
to relate the applied load with the mud internal response, which thus describes the macro-
scale behavior of mud under loads resulting from the micro-scale structure of the mud
matrix. Then, substituting the constitutive equation into the equations of motion, the general
dynamic equations for a continuum are obtained.
Thus, the important steps for modeling wave-mud interactions are (1) developing a
mud theological model and (2) solving the closure equations for water and mud under wave
motion with specific boundary conditions.
2.2.1 Mud Rheology
The importance of rheologic properties of cohesive sediment has been recognized in
the coastal seas and estuaries where oscillatory forcing is typically dominant (Mehta, 1991).
Natural cohesive sediments are mainly composed of minerals, colloids and organic materials.
They settle as mud with density determined by the degree of dewatering or consolidation.
The theological properties of such sediments are quite complicated and depend on the
loading shear stress and the reactive shear strain or shear strain rate (Chou,1989) and on the
11
loading frequency (Jiang, 1993), and may in addition be time-dependant (Williams and
Williams, 1992). The general constitutive equation for a continuum is (Malvern, 1969)
Tj = f(Ej) (2.1)
where Ti are the applied deviatoric stresses, Ej are the reactive deviatoric strains and
subscripts i and j denote directions in conjunction with ji, which implies a second order
tensor. Due to the complexity of mud rheology many theological models have been
proposed, as briefly described next.
Linear viscoelastic models
Linear viscoelastic models are widely used at present, and can be generally expressed
as:
M ar N As ,
E p-(Tj) qs -(E ) (2.2)
r= atr s=O at
where superscripts r and s denote orders of partial differentiation, p, and qs are coefficients
related to the selected viscoelastic model, and M and N are the specified maximum
differential orders of the chosen model. Some special and widely used cases of Eq. 2.2 are
as follows:
(1) Elastic material. With po =1, qo = 2G and all other pr and qr equals to zero, the
constitutive equation for an elastic material is (Mallard and Dalrymple, 1977; Dawson, 1978;
Foda, 1989)
Tj! = 2GEj. (2.3)
where G is the elastic modulus.
(2) Newtonian fluid. With po = 1, q, = 2p and all other Pr andq, equal to zero,
the constitutive equation for a Newtonian fluid is (Dalrymple and Liu, 1978; Shibayama et
al., 1986)
Tj = 2pE (2.4)
where p is the viscosity and E. is the deviatoric strain rate, i.e., the time derivative of Ej
with the dot denoting a time-derivative.
(3) Voigt model. With po=l, qo=2G, qt=2p and all other Pr andqs equal to zero, the
constitutive equation for the well known Voigt model is (Fig.2.1a, Carpenter et al., 1973;
Hsiao and Shemdin, 1980; McPherson, 1980; Maa, 1986; Sakakiyama and Bijker, 1989)
Tj; = 2GEj + 2p (2.5)
(4) Maxwell model. With po = 1/2p, p, = 1/2G, q, = 1 and all other p, and q,
equal to zero, the constitutive equation for the well known Maxwell model is (Fig. 2. b)
1 1 .t -,
T--Tj + -20 = E-i (2.6)
2p 2G
2p
4 1 2G 2p
2G
( a) Voigt model (b ) Maxwell model
Figure 2.1 Common two-parameter viscoelastic models.
(5) Jeffrey model. The three-parameter Jeffrey model is an extension of the Voigt
and Maxwell models. Modulating the Voigt model by an additional elastic modulus, 2G,
(Fig. 2.2a), with po = (G +G2)/p2, pP = 1, q0 = 2GG2,/P2, q1 = 2G, and all other p, and
q, equal to zero, the constitutive equation for the Standard Solid model is obtained as
(Keedwell, 1984; Kolsky, 1992; Jiang, 1993)
GI+G2 T + ./ 2GIG2, 2/ /
TI + ji + 2GIEi (2.7)
112 P-2
Constraining the Maxwell model by an elastic modulus, 2G, (Fig. 2.2b), with po = G2/p,2
pl = 1, qo = 2GIG2/p2 and q, = 2(G, +G2) and all other p, andqs equal to zero, the
resulting Jeffrey-Maxwell constitutive equation is
S + G2 / ./ 2GIG2 /
112 Tji + '2Eji + 2(G +G2)Eji (2.8)
P2 P2
2G2 2G2 2[2
L2Gi \A----
2P2 2GI
(a) Standard Solid Model (b ) Jeffrey-Maxwell Model
Figure 2.2 Jeffrey models.
When the external loading is cyclic, e.g., due to water waves, the general constitutive
equation of linear viscoelasticity can be simplified. Assuming that
Tj; = Toexp[-i()t)]
(2.9)
Ejl = Eoexp[-i(wt-8)]
where 0 is the angular frequency of loading, 8 is the phase shift, T/ and Eo/ are the
amplitudes of Tj and El, respectively, and I = (-1)2, the general relation between the
deviatoric stress and the strain is simplified as
N
E q(-iw)s
Tj = E = 2(G'-iG")E. = 2G* Ej (2.10)
M ji (2.10)
E p,(-i )r
r=0
where G* is called the complex modulus, G' is the elastic energy storage modulus and G" is
the viscous energy dissipation modulus. As an alternative to Eq. 2.10, the general relation
between the deviatoric stress and the strain rate can be obtained as
N
Sq%(-i")s
T ,-- q- ;= 2( p+ip.ll' = 2p* j/ (2.11)
Ti M (2.11)
r=O
where p* is called the complex dynamic viscosity, p' is the dynamic viscosity and p" is
called the second viscosity. Accordingly, the different theological models described can be
obtained as follows:
(1) Elastic material,
P = Pp ; i" = 0 (2.12)
(2) Newtonian fluid,
p' = o ; pH = G (2.13)
(3) Voigt model:
pt = p ; Pl G (2.14)
(4) Maxwell model,
G G )
_P = //w (2.15)
2 +( G 2+( G)
(5) Standard Solid model,
G\ G, (G1+G2)G2 +W2
G W 2
P +G 2 2 2 (2.16)
G G, + +G,
P2 P2
(6) Jeffery-Maxwell model,
G 1GIG 2
G ) i +o2 (G+G2)
G 2 ItI
/ .= 2 [ 2 2 (2.17)
G G2 2
I2 2 2
As mud rheology is complicated, none of the above models can describe mud
response under all types of loading conditions. However, for the case of mud response to
cyclic loads (Eqs. 2.10 and 2.11), a practical way to characterize mud rheology is by carrying
out a dynamical oscillatory test (Chou, 1989; Jiang, 1993). Through this test the complex
modulus, G*, or the complex dynamic viscosity, pI*, can be obtained under specific
conditions for practical use, without considering a particular model describing mud rheology.
Thus, for example, Chou (1989) related the complex modulus, G*, with strain amplitude by
using controlled strain rheometry. Then, he incorporated the resulting strain-dependant
rheologic model into his wave-mud interaction model to predict mud response to wave
motion. Jiang (1993) related the three parameters of the standard solid model to wave
loading frequency and then obtained the complex dynamic viscosity, P*, which also therefore
depends on the loading frequency, for his wave-mud interaction model. Such
17
experimentally based relations will continue to play an important role in modeling mud
rheology, before a fuller theoretical understanding of mud rheology is obtained.
Theoretically, predictive accuracy can be increased by introducing more and more
elastic/viscous elements used for modeling mud rheology. However, the more the elements
in a model, the greater the number of coefficients that must be determined. Thus, greater
complicated experimental procedures are required, which in turn limit the use of large-
element models. On the other hand, some nonlinear properties, e.g., plasticity, time-
dependant properties, etc., are often important in mud theological response to loading. As
a result, some nonlinear viscoelastic models have also been proposed for mud rheology.
Nonlinear and Time-Dependent Models
The three-dimensional structure of soft mud beds can resist small shear stresses at
rest. To account for this resistance, viscoplastic models have been used (Engelund and Wan,
1984; Suhayda, 1986; Mei and Liu, 1987; Tsuruya, 1987; Liu and Mei, 1989; Shibayama et
al., 1990). For small strains, the constitutive equation for a viscoplastic material can be
expressed as (Malvem, 1969)
T,= 2.p + k E (2.18)
where p p is the apparent viscosity, ky is the yield strength and IID is the second invariant of
E'. For the one-dimensional condition, Eq.2.18 can be simplified to (Wilkinson, 1960)
Ji"
T = Ty + Lp /
(2.19)
18
where r is the applied shear stress, Ty is the yield stress and ? = au/az. Here u is the
velocity in the same direction as T.
For the commonly found shear-thinning behavior of mud, i.e, one in which the
apparent viscosity decreases as the shear strain rate increases under increased shear stress
loading, mud can be modeled by a power-law equation (Sisko, 1958; Feng, 1992)
T = t. + c" (2.20)
where the apparent viscosity, pp, = u + cY"-', p-is the constant viscosity at the limit of
high (theoretically infinite) shear rate, c is a measure of the consistency of the material and
n < 1 is a mud-specific parameter.
James et al. (1987) conducted creep-compliance tests for illitic suspensions by using
a combination of an applied stress rheometer and a miniature vane geometry. They found
the instantaneous compliance, Jo = y/t, to be related to the applied shear stress. The
material was predominantly elastic at low shear stresses and was essentially viscous at high
shear stresses. Chou (1989) measured soft mud theological properties composed of a
kaolinite and a montomorillonite and found that the storage modulus, G', and the loss
modulus, G", changed with the shear strain amplitude.
By using shear wave rheometry, Williams and Williams (1992) measured the G' and
G" values of mud under oscillatory loads, and found that G' and G" changed with the wave
loading time, thus quantitatively demonstrating that mud has a time-dependant theological
response. Mehta et al. (1995) correlated the phase velocity of a high-frequency shear wave
in a mud bed with bed rigidity. The fall in the velocity, starting with an undisturbed bed
19
under monochromatic wave, indicated that the bed rigidity decreased as a result of bed
fluidization under wave motion. Thus, the bed had a time-dependant theological response
to wave motion.
2.2.2 Wave-Mud Interaction Modeling
Prior to the late 1950s, wave energy dissipation was mainly considered in connection
with bottom friction, percolation, wave refraction and shoaling. Brestschneider and Reid
(1954) presented graphic solutions for obtaining wave attenuation by these factors.
However, it must be pointed out that a mud bottom is usually not rigid and is almost
impermeable to water flow. The dissipation of propagating wave energy over the mud bed
is mainly due to internal friction within mud, which is characterized by mud rheology and
wave loading.
Based on Biot's (1941) 2-D, nonporous, perfect-elastic soil theory, Mallard and
Dalrymple (1977) modeled mud bed as an elastic material to study soil stresses and
displacement under wave motion. In their modeling, soil inertia was neglected and there was
no energy dissipation since mud viscosity was not included. Taking soil inertia into account,
Dawson (1978) also modeled mud as an elastic material. However, wave energy dissipation
was not considered.
An early theoretical and laboratory study on the effects of a nonrigid, impermeable
bottom on the surface wave was carried out by Gade (1958), who modeled the water layer
as an inviscid fluid overlying a viscous fluid layer of greater density. Solution for the
shallow water, single harmonic wave indicated that wave height decayed exponentially with
distance traveled, and that wave energy dissipation over the soft bottom was substantially
20
greater than that over a rigid bed. Dalrymple and Liu (1978) developed a theory for a small
amplitude linear wave propagating in a two-layered viscous fluid system. Both the governing
equations of continuity and motion and the surface, interface and bottom boundary
conditions were linearized, and an analytical approach was used for solving the equations.
The results showed that most of the energy dissipation was due to internal friction within the
soft mud as a result of high viscosity, and 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. Based on Dalrymple and Liu's work and boundary layer approximation,
Jiang and Zhao (1989) and Jiang et al. (1990) developed analytical approaches for calculating
wave attenuation for viscous mud beds under shallow water solitary and cnoidal waves.
These modeling studies included the nonlinear properties of waves in shallow water.
By considering both the viscous and the elastic properties, Hsiao and Shemdin (1980)
and McPherson (1980) independently applied the Voigt model to represent mud rheology and
used analytical approaches to solve for the assumed inviscid water wave propagating over
a linear viscoelastic bed. McPherson introduced a viscoelastic parameter, ve = */p (see Eq.
2.11), which is essential for the formulation of the problem based on the viscoelastic
assumption. For the Voigt model, ve = p/p +iG/cop (see Eq. 2.14). McPherson'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.
Hsiao and Shemdin compared their results with field measurement reported by Tubman and
Suhayda (1976) in East Bay, Louisiana, and obtained reasonable predictions. Considering
mud density increases with depth, a multi-layered Voigt model was used by Mehta and Maa
21
(1986) to model mud bed motion under a linear wave. The calculated water-mud interfacial
bed shear stresses was found to be larger than those obtained by assuming mud to be rigid.
Chou (1989) developed a three-phased viscoelastic model for mud rheology, and
under linear water wave motion a four-layered wave-mud interaction model was set up. The
depths of the second, viscous layer, and the third, viscoelastic layer, layers were determined
as part of the solution through an iterative technique. The model showed that the mud
fluidization depth and wave attenuation rate increase with wave height for a partially
consolidated mud. Using controlled-stress rheometry, Jiang (1993) modeled mud as a
Standard Solid (see Eq. 2.16). An analytical solution and laboratory tests were used to study
wave-mud interaction, wave attenuation and mud mass transport. The modeling results
compared favorably with laboratory experimental data and with field data from Lake
Okeechobee in Florida and the southwest coast of India.
Besides linear viscoelastic models, some nonlinear viscoelastic, viscoplastic and
other models have also been used in wave-mud interaction modeling and mud behavior in
general. 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) considered bottom mud to be viscoplastic. The special case of
a shallow mud layer under the action of long waves propagating in water was considered.
It was found that under certain conditions the dynamics of a plug flow layer, unsheared in
the domain with stresses below the yield stress, contributed measurably to wave attenuation
22
over a long fetch. In particular, as waves propagated mud motion could change from
continuous to intermittent.
Tsuruya et al. (1987) extended Dalrymple and Liu's (1978) viscous-fluid model to
a Bingham fluid. The equivalent viscosity, Pa which takes non-Newtonian mud rheology
into account, was defined as
au au
T a = t0 + Pb (2.21)
tz 8z
where to is the yield stress, Pb is the dynamic viscosity and the apparent dynamic viscosity
P + Pb (T > To)
Pa u (2.22)
az
Thus, the Bingham fluid was introduced into the Newtonian equation of motion in an
approximate fashion. Laboratory tests on wave attenuation and mud mass transport were
also conducted.
Using the viscoplastic Bingham model for a thin mud layer under solitary waves, Liu
and Mei (1989) analytically solved the wave-mud interaction problem through a boundary
layer approximation. They found that wave dissipation is due to the bottom mud shear layer
and the turbulent interfacial stress. Continuous to intermittent mud motion could be
predicted from this model.
Studying linear wave propagation over a soft mud, Shibayama et al. (1990) modeled
mud as a viscoelastic and/or viscoplastic material depending on the loading shear stress.
Fluid mud was assumed to be viscoelastic when the stress was less than the yield stress, and
23
was considered to be viscoplastic when the stress exceeded the yield stress. The resulting
numerical model could predict mud mass transport under wave action.
SMWL
Mobile Suspension
Entrainment Settling
SFluidization Dewatering
.^.1 A----L------------.
LutoclineFluid Mud
-- --------------
^^ -~" -- ---- -^ -----------
Deforming Bed
Stationary Bed
Figure 2.3 A typical wave-mud system (after Hwang, 1989)
2.3 Mud Fluidization and Fluid Mud
The interactions between unsteady flows and soft bottom mud are not well
understood at present, especially when a fluid mud occurs at the bottom. Fluid mud is a
concentrated sediment slurry having non-Newtonian theological properties. It may be
generated by rapid deposition of suspended sediment or by fluidization of the mud bed by
waves (Winterwerp, 1994). In Fig. 2.3, a qualitative description of the mud-water system
is given, in which u is the horizontal wave orbital velocity. The bulk density of the water-
mud mixture, p, varies from the water surface to the bed bottom. A sharp density gradient,
or lutocline (Parker and Kirby, 1982), separates the upper column suspension from fluidized
24
mud below. The bed just below fluid mud tends to undergo deformation. Below the level
at which the depth of penetration of the wave orbit ends, the bed remains uninfluenced by
wave motion.
Mean Water Surface
Mobile Suspension
S\ Fluid Mud Surface (Lutocline)
S' = 0 Fluid Mud
Bed Surface
Wh "% % w Bed
STRESS
Figure 2.4 Schematic of instantaneous stress profile in a wave-mud system
(after Mehta et al., 1994).
An instantaneous view of the stress profiles in the mud-water system is shown
schematically in Fig. 2.4, in which oh is the hydrostatic pressure, Au, is the wave-averaged
excess pore pressure, o' is the wave-averaged effective normal stress, uw is the pore pressure
and the total stress, a = uw + o = oh + Auw + Considering the time-variation of the
normal stress in the bed at a given point, as the excess wave-mean pore water pressure builds
up gradually under wave motion, the effective stress reduces accordingly. When the pore
water pressure equals the total pressure at some elevation, i.e. when the effective normal
stress is zero, the bed at and above this elevation is considered to be fluidized (Ross, 1988;
25
Feng, 1992). Thus, fluid mud is actually supported by water and has fluid-like properties,
while the non-fluidized bed retains a structured sedimentary matrix.
Because the theological properties and the mass transport characteristics of fluid mud
can be significantly different from those of the non-fluidized bed, fluid mud must be modeled
separately from the bed. Therefore, to investigate water-mud bed interactions, once the bed
is fluidized, a three-layered water-fluid mud-bed system must be set up (see Fig. 2.5).
Water layer DifBia.t s-tl 8
Wata-fluid Tedterfutcs E et f Depotsiion
Fluid mud layer
Fluid mud-bed Interfa FdizitionI Dewatetin
Bed layer
Bed bottom
Figure 2.5 Three-layered water-fluid mud-bed system and vertical sediment transport
processes under waves.
To investigate vertical mass transport in the three-layered system, several mass flux-
associated processes must be determined quantitatively. In the water layer, the sediment
settling velocity and the buoyancy-stabilized diffusivity must be specified. The entrainment
rate of fluid mud and the deposition rate of sediment onto the surface of fluid mud must be
modeled. To determine the fluid mud layer thickness and the mass fluxes of sediment across
the fluid mud-bed interface, fluidization characteristics of the bed and dewatering of fluid
mud by consolidation must be understood.
26
Fluid mud thickness can affect both the velocity distribution in the water column
through wave-fluid mud interaction (Jiang, 1993), and mass exchange between water and
bed. However, a fully operational model to predict fluidization under waves has not yet been
developed. Roberts (1993) used the three-layered system to develop a two-dimensional
horizontal model to predict fluid mud movement under storm waves in the coastal zone. The
thickness of fluid mud was assumed to be equal to the wave boundary layer height by way
of the argument that the water-fluid mud interface coincides with the top of the wave
boundary layer. However, this assumption is not always appropriate, since in general the
interface can rise above the height of the boundary layer or occur below the latter. By
measuring the pore and total pressures within the bed under wave motion in a laboratory
flume, Feng (1992) developed an empirical formula for determining fluid mud thickness.
Combining the experimental data of Feng (1992) and the theological properties of mud
measured by a miniature shearometer (Williams and Williams, 1992), Mehta et al. (1994)
also developed an empirical formula to predict time-varying fluid mud thickness during
fluidization. These two empirical relations are limited to the types of sediment and wave
conditions used in the experiments for which they were derived. Therefore, to determine the
necessary coefficients for other sediments or wave conditions, additional experiments must
be carried out.
For modeling the depth of fluidization, Foda et al. (1993) applied a wave-bed
interaction model incorporating the assumption that the viscoelastic properties of mud are
dependent on the wave-induced strains within the bed. This model can only predict the final
equilibrium fluidization depth, which is actually a time-dependent process starting with a bed
27
over which wave motion commences. By using a miniature rheometer to measure the phase
velocity, V, of a high-frequency shear wave in a mud bed and then correlating V with bed
rigidity, G', Mehta et al. (1995) obtained the empirical relation
G r(t) [V(t)]2 2(2.23)
G = -t ) exp(-at) (2.23)
G (O) LV(0)
where a, and P, are empirical coefficients dependent on bed theological properties and the
wave characteristics. It was found that the trends of increasing measured fluid mud thickness
with time, d(t), were quantitatively in agreement with the trend of decreasing V(t) and were
empirically expressed as
dr(t) = M[1 -exp(-at ')] (2.24)
whereM, a, and P, are empirical coefficients. However, a theoretical explanation for Eq.
2.24 was not provided. In fact, since the fluidization process is not well understood, in the
present development the fluid mud layer will be considered to be of the final equilibrium
thickness.
Dewatering is the process by which fluid mud consolidates and the bed structure is
rebuilt. Experimental measurements of Feng (1992) and de Wit (1994a, 1994b) indicate that,
as long as wave motion continues, fluid mud can be maintained and may not dewater. Since
in the present case only the equilibrium thickness of fluid mud is to be considered, the
dewatering process will not be examined. Thus, no portion of fluid mud will be considered
to dewater to become part of the bed layer. For the same reason, no bed sediment will be
28
fluidized. Therefore, sediment mass transport across the interface between the fluid mud and
the bed will be neglected and the fluid mud-bed interface will be considered to be fixed in
elevation.
2.4 Processes Governing the Vertical Structure of Suspension
Referring to Fig. 2.5, in the water layer settling and upward diffusion of the
suspended sediment determine the vertical distribution of sediment concentration. The
entrainment rate of fluid mud or bed and the deposition rate of suspended sediment onto the
surface of fluid mud or the bed govern sediment exchange between the water and fluid mud
or bed.
2.4.1 Settling
The settling velocity of cohesive sediments strongly varies with the concentration in
suspension (Mehta, 1988a). A descriptive plot is shown in Fig. 2.6 for the relationship that
is typically found between the settling velocity, ws, and the suspended sediment
concentration, S. Also shown is the variation of the corresponding settling flux, F, = w,S.
The settling velocity regime can be divided into three sub-ranges depending on the
concentration and the mode of settling, which can be free settling, flocculation settling or
hindered settling.
Free settling occurs in the very low suspension concentration range, when S < S,
(see Fig. 2.6). In this range the particles or flocs settle independently and the settling velocity
is not a function of suspension concentration. For cohesive sediments, the upper
29
concentration limit, S 1, is considered to be in the range of 0.1 to 0.3 g/1 (Mehta, 1988). The
settling velocity in the free settling range is usually assumed as a constant, i.e.:
w = ws = a S = const
S f a I
(S < S *)
where at and P, are coefficients which will be explained next.
i4 Settling Flux
O
o
0 n
SWo ......----- ----
w, = Const. Settling Velocityl
I I
W Wf F o
to Free Flocculation Settling I Hindered
o4 settlmgi ` I Settlipg
(2.25)
Log(CONCENTRATION)
Figure 2.6 A schematic description of settling velocity and flux
variation with suspension concentration (after Hwang, 1989).
Flocculation settling occurs in the middle suspension concentration range specified
by S < S < S In this range increasing concentration leads to increasing interparticle
collision and enhanced particle aggregation. Thus, the settling velocity increases with
concentration due to the formation of stronger, denser and larger flocs or aggregates. The
typical relationship between the settling velocity and suspended sediment concentration is
Ws = a1S P S; < S < S (2.26)
where pi is 4/3 theoretically (Krone, 1962), although the actual value typically varies
between about 0.8 and 2 (Mehta, 1988a). The proportionality coefficient, bn, can vary by
an order of magnitude depending on sediment composition and the flow environment.
Hindered settling occurs in the high concentration range, when S > S ;. In this range,
the occurrence of an aggregate network hinders the upward transport of interstitial water.
Thus, w, decreases with increasing S ( Kynch, 1952). A widely accepted relationship for the
settling velocity as a function of concentration was developed by Richardson and Zaki (1954)
w, = ws( -la/ S)P2 (S > S) (2.27)
where w0s is the initial or reference settling velocity, a/' is a coefficient depending on the
sediment composition and 32 = 5. Above the concentration S4*, the settling flux is
negligible.
Hwang (1989) measured the settling velocity of sediments collected from Lake
Okeechobee in Florida, and parameterized the effects of flocculation and hindered settling
on the settling velocity by way of a nonlinear formula dependent on the sediment
concentration, and expressed as
a S1
w (2.28)
(S 2 + Y)2
31
where a,, PI yi and 8, are empirical constants which must be experimentally determined for
different sediments.
2.4.2 Diffusion
The rate of upward diffusion of suspended fine-grained sediment is governed by the
mass diffusivity determined by hydrodynamic forcing and also by density stratification,
which tends to damp diffusion. The diffusivity, KI, is usually expressed as the product of the
neutral mass diffusivity, Kn, and the density stratification correction factor, D (Rossby and
Montgomery, 1935; Munk and Anderson, 1948; Ross, 1988), i.e.
Kz = K, D (2.29)
The most commonly applied expression of the vertical variation of the neutral mass
diffusivity for turbulent unidirectional flow was developed by Rouse (Vanoni, 1975).
Following von Karman's assumption of a linear shear stress distribution with depth and a
logarithmic velocity profile, the neutral diffusivity is expressed as
Kn = Ku*z 1- (2.30)
where K is von Karman's constant, u, is the friction velocity, h is the water depth and the
coordinate z is upward originating from bed surface. However, under wave motion, the
diffusivity becomes complicated and has not been fully parameterized thus far. By analogy
with the mixing length theory in unidirectional flows, Homma et al. (1962) gave a plausible
expression for the wave diffusivity. To overcome the shortcomings found in the derived
32
expression, Homma et al. (1965) modified the expression by introducing a mixing length
following the von Karman hypothesis. The resulting neutral wave diffusivity under the linear
waves was expressed as
K H sinh3k(h+z) (2.31)
Sksinhkh cosh2k(h+z)
where P2 is a constant, H is the wave height, o is the wave frequency and k is the wave
number.
Hwang and Wang (1982) summarized the models available for the turbulent diffusion
coefficient outside the wave boundary layer and indicated that Prandtl's mixing length theory
may not be applicable due to the large scale of wave motion. They assumed that the wave
diffusivity is proportional to the vertical velocity component and the orbital radius of the
water particle in the vertical direction. Using the linear wave theory, the expression for wave
diffusivity was obtained as
K = 2H 2 sinh2kz
Kn = a2H' o- (2.32)
2 sinh2kh
where a2 is a constant. From the concept based on energy in a turbulent field, Thimakorn
(1984) expressed the eddy diffusivity as
Kn = 0.885 H -2 n (2.33)
sinh3kh
33
which is similar to Hwang and Wang's expression, with a2 = 1.77/sinhkh.
Stratification due to density variation in suspension can measurably alter the vertical
momentum and mass mixing characteristics. In the case of stable density stratification in
which the density decreases upward, vertical diffusion is impeded by the stabilizing
gravitational force of sediment suspension acting against the destabilizing force due to fluid
shear. The local gradient Richardson number, Ri, is usually used as a measure of stability,
and is defined as
ap
Ri -g az
p ( (2.34)
iaz
where g is the gravitational acceleration, p is the density and z is positive upward. Positive
values of Ri indicate stable stratification, negative values indicate unstable stratification and
Ri = 0 implies a neutral condition. The greater of the positive value of Ri, the greater the
suppression of vertical diffusion.
A number of models for the density stratification correction factor, D, in Eq. 2.27,
and therefore for stratified diffusivity, have been proposed. One of the most frequently used
expression was developed by Munk and Anderson (1948), which is
K = Kn(1 +33Ri)al (2.35)
where a3 and 33 are empirical coefficients and, typically, a3 varies from -0.5 to -2.0 and P3
varies from 0.3 to 30 (Rossby and Montgomery, 1934; Holzman, 1943; Munk and Anderson,
34
1948; Nelson, 1972; French, 1979; French and McCutcheon, 1983) depending on sediment
composition and the flow environment. Thus, the diffusive flux, Fd, can be expressed as
Fd = -Kz -Kn(1 + Ri)a3 a- (2.36)
z z z
Without the effect of buoyancy stabilization, Ri = 0 and Kz = Kn, so that the diffusive flux
is linearly proportional to the vertical concentration gradient, aS/az. However, since the
local gradient Richardson number, Ri, is proportional to ap/az a 9S/az, the stratified
diffusivity, KI, is inversely proportional to the vertical concentration gradient with a3 < 0.
Thus, nonlinearity between the diffusive flux, Fd, and aS/az develops when gravitational
stability is considered. Also, from the theoretical results of Eq. 2.34, the diffusivity flux
initially increases with low values of aS/az, reaches a maximum and then slowly decreases
with high values of aS/az. The formation of the lutocline is strongly related to the nonlinear
dependence of Fd on aS/az (Eq. 2.34) (Ross, 1988; Hwang, 1989; Costa, 1989; Scarlatos and
Mehta, 1990).
Ross (1988) and Hwang (1989) used Eqs. 2.30 and 2.34 in their vertical fine-grained
sediment transport modeling. Model results were found to be in agreement with laboratory
data, and with field data from the Severn Estuary and tributaries, United Kingdom (Kirby,
1986; Kirby and Parker, 1977) and from Lake Okeechobee in Florida (Hwang, 1989). Also,
lutoclines could be simulated with the models.
2.4.3 Deposition Rate
The deposition rate of suspended sediment onto fluid mud surface can be expressed
as the product of the settling rate of sediment near the mud surface and the probability of
deposition (Krone, 1962; Mehta, 1988b). Thus, deposition rate, Fp, is expressed as
Fp = -wSPI mud surface (2.37)
where p is the deposition probability. The concept of this probability implies that deposition
occurs through the sorting of suspended sediment aggregates, which in turn occurs because
of the high rates of fluid shear near the bottom. When the aggregates are strong enough to
withstand the near-bed shear stress, they stick to the bottom mud; if not, they are disrupted
and resuspended.
Krone (1962) first related the deposition probability to the relative shear stress as
0 (Tb Tcd)
p = b < (2.38)
1- (b
where Tb is the bed shear stress and "cd is the critical shear stress above which no deposition
occurs. This stress typically varies from 0.04 0.15 Pa.
Mehta (1986) and Mehta and Lott (1987) studied nonuniform sediment deposition
and defined the critical shear stress for deposition of such sediment. With -rm and Tc defined
as the maximum and minimum values, respectively, of the critical shear stress for deposition,
tci, for tb > cm no initially suspended sediment can deposit, while for Tb < cn the entire of
36
suspended material will eventually deposit. For on < Tb < cm a fraction of the initially
suspended sediment for which li < Tb will not deposit, while the remainder will finally
deposit at steady state. Using this approach the instantaneous concentration variation with
time can be predicted depending on the initial suspension concentration, settling velocity,
sediment size distribution and the flow condition.
Sanford and Halka (1993) investigated sediment resuspension in Chesapeake Bay and
found that the resuspension model with the assumption of no critical stress for deposition
described the field data better than one including the critical stress. They suggested that
deposition always occurs, and is proportional to the near bottom suspension concentration.
Thus, even at steady state when the suspended sediment concentration is constant, sediment
exchange between the water column and the bed exists. The deposition rate in this case is
expressed as
Fp = -wsS mud surface (2.39)
which is a special case of Eq. 2.37 with the deposition probability equal to one.
2.4.4 Entrainment
The problem of erosion of cohesive sediments has been extensively investigated
(Grissinger, 1966; Kelly and Gularte, 1981; Parchure and Mehta, 1985; Sheng, 1986; Maa
and Mehta, 1987; Shaikh et al, 1988), with different types of deposited or remolded clay beds
under current or wave motion, in fresh or saline water. Erosion of mud bed is due to: (1)
frictional, gravitational and cohesive bonding forces of bed being exceeded by the
hydrodynamic erosive force, which causes bed surface erosion, and (2) the bed bulk shear
37
stress being exceeded by the flow-induced shear stress,.which causes mass erosion (Mehta,
1986).
In the above context it should be noted that the mechanism by which fluid mud is
entrained by shear flow differs from that governing the rate of erosion of the settled mud
having a structured matrix (Mehta and Srinivas, 1994 and Kranenburg, 1994). Entrainment
is due to the breakup of the fluid mud-water interface, which is characterized by the ratio of
the destabilizing factor, the mechanical mixing energy, and the stabilizing factor, the
potential energy stored in density stratification (Scarlatos and Mehta, 1993 and Mehta and
Srinivas, 1993).
Kato and Phillips (1969) applied a constant shear stress, T = p u at the surface of
an initially linearly stratified fluid in an annular flume. This resulted in the development of
an upper homogeneous layer and a lower stratified fluid with an interfacial buoyancy jump
Ab. They arrived at the expression
u.
E, =_e=K1Ri,-7 (2.40)
where E, is the non-dimensional entrainment rate, ue is the rate of downward propagation of
the interface, u* is the friction velocity and the Richardson Number, Ri, = hAb/u2, and h
is the depth of the upper homogeneous layer. Moore and Long (1971) ran two-layered
experiments in a race-track shaped flume with density and velocity profiles being uniform
in the longitudinal direction. Over the range of the global/overall Richardson Number, Ri ,
38
studied, they found that the non-dimensional buoyancy flux, Q = q/Ab (2AU), was related
to the overall Richardson Number according to
Q ~ Rig-' (2.41)
where q is the buoyancy flux, 2AU is the velocity difference between the two layers and
Rig = lIAb/(2AU), in which H is the total depth.
In the field, Bedford et al. (1987) measured sediment entrainment in Long Island
Sound, and examined its dependence on the bed shear stress, the Reynolds stress, and the
turbulent and wave kinetic energies. Only the time-patterns of turbulent and wave kinetic
energies correlated with the entrainment time pattern.
Scarlatos and Mehta (1993) approximated the mud-water system by a two-layered,
slightly viscous, horizontal system with water flowing over fluid mud. By introducing a
slight disturbance along the interface, the vortex sheet initially developed a wave-like pattern,
typical of Kelvin-Helmholz instability. Then the vortex sheet stretched, folded and
eventually caused a thickening and vigorous mixing of the interface. It was demonstrated
that the dynamic behavior of the vortex sheet is controlled by the velocity and density
gradients across the interface.
Mehta and Srinivas (1993) examined the entrainment behavior of fluid mud subjected
to turbulent shear flows based on theoretically analyzing the turbulent kinetic energy
equation and using experimental data. A non-dimensional formula for the entrainment rate
dependent on the global Richardson Number was obtained as
39
E = AeRig--DeRig (2.42)
where E = ue/U, Rig = hAb/U2, Ab is the buoyancy jump across the water-fluid mud
interface, U is the mean mixed layer velocity, h is the mixed layer depth, and Ae and De are
experimentally determined coefficients. The first term on the right side represents the
interaction between mechanical mixing energy and potential energy stored in density
stratification. The second term arises from influences of particle settling, cohesion, viscosity
difference between mud and water etc. It should be noted that according to Eq. 2.40, at low
values of Rig the upward flux of sediment mass is proportional to the cube of the mean
mixed layer velocity. In contrast, the rate of erosion of cohesive bed depends on U2 in
turbulent flows (Partheniades, 1965; Mehta, 1989).
Winterwerp (1994) carried out experiments on the entrainment of soft mud layers in
an annular flume and also found the entrainment rate of fluid mud to be governed by the
global Richardson number. The non-dimensional entrainment rate, E. = uc/u,, was
correlated with Ri.-1/2 for small Ri. and with Ri,-' for large Ri, where Ri, = hAb/u2.
Kranenburg (1994) derived an entrainment model for fluid mud by integrating the
equation for turbulence kinetic energy over the mixed layer and introducing some modeling
assumptions. Two basic conditions for entrainment were considered. In case 1, the water
layer was selected to be the turbulent mixing layer and fluid mud entrained into water layer.
In case 2, turbulence was considered to be produced primarily in the fluid mud layer due to
the shear stress at the bottom of the fluid mud layer. Water entrained downward into fluid
40
mud layer increased the fluid mud depth with time. Such behavior has also been reported
in the field by Wolanski et al. (1988) and Le Hir (1994).
2.5 Sorption Models
Sorption processes include adsorption, chemisorption, absorption and ion exchange
(Fetter, 1992). Adsorption is the process by which a solute (sorbate) clings the solid
sorbentt), i.e., sediment, surface. Chemisorption occurs when the solute is incorporated on
a solid surface by a chemical reaction. Absorption describes the process that the solute goes
beyond the solid surface and diffuses into the porous interior compartment of the solid. Ion
exchange is the process that charged solutes exchanges with ion on solids surface. From a
practical point of view the important aspects are the removal of the solute from solution to
sorb onto or into solid, or the release of solute from solid into solution, irrespective of the
details of process mechanics. If the rate of sorption is much faster than advective-diffusive
transport rate of solute, the process can be described by an equilibrium sorption isotherm.
If the rate of sorption is slow or of the same order of magnitude as advective-diffusive
transport rate of solute, the process must be described by a kinetic sorption model. Some
sorption processes may be more complicated and must be described by more complicated
models which may include multi-step reaction and/or diffusion steps (Wu and Gschwend,
1986).
2.5.1 Equilibrium Sorption Isotherm
Linear Sorption Isotherm
41
The simplest and most widely used equilibrium sorption isotherm is the linear
sorption isotherm which is given by a linear relationship between the solute concentration
in solution, C, and sorbed on solid, P
P = KdC (2.43)
where Kd is the distribution coefficient, which is a measure of the retention of solute by the
solid.
The linear isotherm model has been used frequently to describe the sorption of
radioactive materials by soils (Burkhoider, 1976; Van De Pol et al., 1977). Davison et al.
(1968), Scott and Phillips (1972) and Sleim et al. (1977) have employed Eq. 2.43 in studies
of the movement of herbicides in the soil column. Lai and Jurinak (1972) and Lai et al.
(1978) have used the linear isotherm for the sorption of metals, i.e. Na and Mg2', on soils
in their studies of cation sorption during convective-diffusive flow through soil. Overman
et al. (1976) have used this isotherm in their study of phosphorous transport. Since the linear
isotherm can be conveniently incorporated into solute transport models and is easily
determined, it has also been used in a number of studies to predict the rate of movement of
solute fronts (Anderson, 1979; Faust and Mercer, 1980 and Srinivasan and Mercer, 1988).
However, a limitation of this isotherm is that it does not include any upper limit to the
amount of solute that can be sorbed onto the solid.
Freundlich Sorption Isotherm
A more general equilibrium sorption isotherm is the Freundlich isotherm, which is
defined by a nonlinear relationship
P = k'lCN (2.44)
where k', and N are constants. When k', = Kd and N = 1, Eq. 2.44 reduces to the linear
sorption isotherm (Eq. 2.43).
The Freundlich isotherm has been widely applied to model the sorption of various
metals such as copper, zinc and cadmium by Sidle et al. (1977), molybdenum by Jarrell and
Sabey (1977) and cadmium by Garcia-Miragaya and Page (1976) and Street et al. (1977).
Bowman and Sans (1977), Van Bladel and Moreale (1977) and Yaron (1978) proposed the
use of Eq. 2.44 to describe the sorption of herbicides and pesticides on soils. Fitter and
Sutton (1975), Harter and Foster (1976) and White and Taylor (1977) fitted data on
phosphorous sorption on soil to Freundlich isotherm. Nathwani and Phillips (1977) reported
the equilibrium distribution of the hydrocarbons benzene, toluene, o-xylene and n-
hexadecane between the liquid and sorbed phase in soil following the Freundlich isotherm.
Note however that this sorption isotherm suffers from the same fundamental problem as the
linear isotherm, that is, there is no upper limit to the amount of the solute that can be sorbed,
so that the solid is never saturated.
Langmuir Sorption Isotherm
The Langmuir isotherm was developed by Langmuir (1918) based on the concept that
a solid surface processes a finite number of sorption sites. When all the sorption sites are
filled, the surface will no longer sorb the solute. The standard form of this isotherm is
p pm
C I + C (2.45)
k'2
where pm is the maximum amount of solute that can be sorbed and k'2 is a sorption constant
related to the binding energy.
The Langmuir model has been used extensively to describe the sorption of solutes,
i.e., metals, phosphorous etc., by soil. Thus, for example copper, lead, cadmium and zinc
sorbed to soil were shown to follow the Langmuir isotherm by Singh and Sekhon (1977a and
1977b), Harter (1979) and Shukla and Mittal (1979) in their studies on metals distribution
and transport by flow in soil. This isotherm was also used by Enfield and Bledsoe (1975)
and Novak et al. (1975) to model the movement of phosphorous in soils resulting from the
renovation of waste water by a land application treatment system.
All the equilibrium models assume that the rate of change of concentration due to
sorption is much greater than that due to the advective and diffusive transport. If this is not
the case, a kinetic model is needed.
2.5.2 Kinetic Sorption Models
Reversible Linear Kinetic Sorption Model
The most frequently used kinetic sorption model is the kinetic form of the linear
sorption isotherm, which is called a reversible linear kinetic sorption model,
P k'3C k',P (2.46)
dt
44
where k'3 and k'4 are forward and backward sorption rate coefficients. Equation 2.46 can
also be written in a different form (Nielsen et al. 1986)
k'4(KdC P) (2.47)
dt
where k'4 is the first-order rate coefficient. If sufficient time is available for the system to
reach equilibrium, dP/dt 0, so that P = KdC, which is the linear sorption isotherm.
One of the most frequent applications of Eqs. 2.46 or 2.47 is in the description of
sorption kinetics of phosphorous in soil. Among those who have used this model to describe
the movement of phosphorous through soil are Enfield and Bledsoe (1975), Enfield et al.
(1976), Shah et al. (1975) and Novak and Petschauer (1979). Leistra and Dekkers (1977)
applied this model to describe simultaneously fast and slow sorption phenomena that occur
with the addition of pesticides in solution to soil. Cho (1971) also used Eq. 2.47 for
calculations the convective transport of various oxides of Nitrogen in soil.
Reversible Nonlinear Kinetic Sorption Model
Corresponding to Freundlich sorption isotherm, the reversible nonlinear kinetic
sorption model is expressed as
dP = k'5C N k'6P (2.48)
dt
where k's and k'6 are the forward and backward sorption rate coefficients, respectively, and
N is a constant. When N = 1, Eq. 2.48 reduces to the reversible linear kinetic sorption
model. If the system reaches equilibrium, dP/dt -0, we obtain P = (k'5/k'6)C N, which
is the Freundlich sorption isotherm.
45
Homsby and Davidson (1973) used this model to describe the transport of the organic
pesticide fluometuron in soils. The distributions of the sorbed and solution phases of the
pesticide were well described at high flow rates. At low flow rates, where equilibrium
conditions existed, the kinetics of the sorption was not too important, and the process was
described equally well by the Freundlich isotherm. Enfield et al. (1977) and Fiskell et al.
(1979) found that phosphorous movement in the soil could be described with the reversible
nonlinear kinetic model with the value of N less than unity.
Bilinear Kinetic Sorption Model
The kinetic version of the Langmuir isotherm is given by the bilinear kinetic sorption
model:
d k' C (P m P) k'8P (2.49)
dt
where k'7 and k'8 are the constant. The theoretical foundation for Eq. 2.49 is the same as that
for the Langmuir sorption isotherm. If the time is sufficient, dP/dt 0, and Eq. 2.49 is
reduced to the Langmuir isotherm.
Gupta and Greenkorn (1973) and Kuo and Loste (1974) have applied the model to
the study of sorption of phosphorous on clay minerals. Despite its strong theoretical
foundation, Eq. 2.49 has not received widespread application. One reason is that the
constituent transport equation coupled with Eq. 2.49 is difficult to solve for.
Mass Transfer Model
If the rate of solute sorption is controlled by the mass transport process from solution
to solid surface, the mass transfer model is applied
dP k'8(C C*) (2.50)
dt
where k's is a sorption rate parameter that accounts for the diffusive transport of the solute
through a liquid layer surrounding the solid, and C* is the liquid phase concentration of the
solute in immediate contact with the solid surface. The mass transfer model has been used
by Navok and Pelschauer (1979) to describe orthophosphate sorption and used by Vilker
(1980) to describe the capacity of activated sludge to sorb pathogenic viruses.
All the isotherms and kinetic models discussed above are called one-box or one-step
models, in which sorption is a function of the solute concentration sorbed on the sorbent
(viewed as a completely mixed box) and concentration in solution (Lapidus and Amundson,
1952 and Oddson et al., 1970). However, some sorption processes are complicated, e.g.,
when a rapid initial intake is followed by a slower approach to equilibrium (Karickhoff,
1980), and for these cases the one-step model is not accurate enough to describe processes
kinetics. As a results, advanced model, which typically include more than one step, have
been applied.
2.5.3 Advanced Sorption Models
Diffusion-Sorption Model
To describe the sorption of hydrophobic organic chemicals, i.e., chlorobenzene
congeners, to and from suspended sediment and soil particles, Wu and Gschwend (1986)
combined diffusive penetration and the linear isotherm to develop their two-step sorption
model. The model is based on the following assumptions
47
(1) The ambient fluid is sufficiently turbulent that an exterior boundary layer does not limit
sorptive exchange.
(2) The porous sorbent, i.e., suspended sediment and soil particles, is spherical and
homogeneous. The rate at which the sorbate molecules diffuse through the pore fluids held
in the interstices of the porous sorbent is expressed as
9[(1-n)p,P'(r) + nC'(r)] = Dn 2C/(r) 2 C(r) (2.51)
at ar2 r ar
where C' is the solute concentration in the pore fluid, P' is the solid bound solute
concentration, n is the porosity of the sorbent, r is radial distance, p, is the specific gravity
of the sorbent, Dm is the molecular diffusion coefficient and (1-n) pP'(r) + nC'(r) is the local
total volumetric concentration in the sorbent.
(3) The solute concentrations in pore fluid and that bounded by solid are locally in
equilibrium and are described by linear isotherm
P' = KdC' (2.52)
By incorporating Eqs. 2.51 and 2.52 into the convective-diffusive transport equation, Wu and
Gschwend (1988) simulated the sorption of nonpolar organic chemicals transported in an
aqueous system containing a spectrum of particle sizes.
ME-SORB Model
For assessments of the impact of bottom sediment resuspension events on an aquatic
ecosystem, DePinto et al. (1994) developed the ME-SORB model. This model is intended
48
to be used in formulating a mechanistic description of metal ion sorption onto and/or
desorption from sediments. The following five steps are examined and included in the
model.
(1) Bulk transport--transfer of material from liquid bulk solution to the liquid surface film
surrounding the particle. In a well mixed suspension, this step is normally rapid and not rate
limiting.
(2) Film transport--transfer of the material through the surface film, which usually is a thin
layer of quiescent liquid lying between the bulk solution and the sorbent particle surface.
(3) Surface sorption--sorption of the material on to the surface of sorbent.
(4) Pore diffusion--transport of the sorbate radially between the particle surface and the
center of the particle through the intra-particle pore space.
(5) Pore sorption--sorption of diffused material on to the walls of the pores.
Each of these steps can occur in either the forward or reverse direction depending on
the sorbate gradient at a particular point. One or more steps may be insignificant compared
to others at a particular condition. The model was calibrated by resuspension experiments
carried in the Trenton Channel (Theis, 1988), and was applied to predict bulk solution
concentration versus time relations for lead, nickel and cobalt desorption from sediment
during a resuspension event.
Since the advanced sorption models include more transport and sorption processes,
they are usually more accurate in simulating the real process. However, the calibration
procedure for these models becomes more difficult and complicated than for simpler but less
accurate models. That is the reason why simpler sorption models are still widely used.
2.6 Sediment-Associated Contaminant Transport
The transport of contaminant in the three-layered water fluid mud bed system
shown in Fig. 2.7 must include dissolved contaminant transport, particle-bound contaminant
transport (related to sediment transport) and sediment-contaminant interaction, i.e.,
sorption/desorption. In this system, contaminant transport exhibits different characteristics
within each of the three layers. Contaminant fluxes across the water-fluid mud interface and
the fluid mud-bed interface represent the connective conditions for each of the two adjacent
layers and control the rates of contaminant exchange between them. The fluxes across the
water surface and bed bottom represent the boundary conditions.
2.6.1 Contaminant Transport in Water
In the water layer under wave motion, dissolved contaminant transport is governed
by the diffusion process, which includes molecular diffusion and wave-induced diffusion,
and also by sorption/desorption kinetics. Since particle-bound contaminant transport occurs
together with the sediment to which the contaminant is sorbed, the transport of particle-
bound contaminant is determined by sediment transport and the kinetics of sorption/
desorption. Sorption/desorption of contaminants onto/from the sediment is dependent on
contaminant (sorbate) characteristics, sediment sorbentt) characteristics and on water quality
(pH value, salinity, temperature, etc.).
Figure 2.7 Sediment-associated contaminant transport processes.
Considering the water layer as a well mixed layer and ignoring the vertical transport
and distribution in water, Onishi (1982), O'Connor (1988a, 1988b and 1988c), Basmadjian
and Quan (1988) and Wang et al. (1991) used the depth-averaged contaminant concentration
of water layer in their transport models. Onishi (1982) formulated a horizontal two-
dimensional sediment and contaminant transport model named FETRA, by coupled a
sediment transport sub-model, a dissolved contaminant transport sub-model and a particulate
contaminant transport sub-model. The dissolved contaminant transport sub-model includes
(1) advection and diffusion/dispersion of dissolved contaminants; (2) sorption (uptake) of
dissolved contaminants by both moving and stationary sediment or desorption from sediment
into water; and (3) chemical and biological degradation, or radio nuclide decay of
contaminants. The particulate contaminant transport sub-model includes (1) advection and
51
diffusion/dispersion of particulate contaminants; (2) sorption (uptake) of dissolved
contaminants by sediments or desorption from sediments into water; (3) chemical and
biological degradation, or radio nuclide decay of contaminants; and (4) deposition of
particulate contaminants onto the river bed or erosion from the river bed. The sorption of
contaminant between the water and solid phases is described by the linear sorption isotherm.
The model was applied to the James River estuary to simulate the transport of sediment and
the pesticide Ketone.
O'Connor (1988a) has developed the equations that define the steady state
distribution of solids and sorptive chemicals in fresh water systems. The dissolved and
particulate phases of chemicals, as well as the sediment, are considered and yield a series of
six simultaneous equations for depth-averaged values in the water and bed layers considered
to be totally mixed. Instantaneous equilibrium is applied to the chemical sorption between
the dissolved and particulate states. The solution of the equations is applied to steady state
and time-variable conditions in (1) lakes and reservoirs (O'Connor, 1988b) and (2) streams
and rivers (O'Connor, 1988c) under different flow and wastewater discharge conditions.
Basmadjian and Quan (1988) developed the differential equations that describe the
fate of chemicals in rivers as they transfer, react and volatilize, both during contamination
and recovery. The sorption reaction in the water layer is given by linear isotherm. The water
column is assumed to be well-mixed and vertical and lateral concentration changes in the
water layer are neglected. Diffusive transport in the direction of flow is neglected compared
to advective transport in this direction. An explicit analytical solution is obtained for the 1-D
unsteady distribution of chemicals in both the aqueous and sediment phases.
52
Considering the vertical transport and distribution of contaminants in the water layer,
Hayter and Pakala (1989), Huber and Dickinson (1992), Sheng (1993) and Chen (1994) used
3-D models to simulate the contaminant transport in water column. The diffusion
coefficients of dissolved contaminants in water are same as the eddy diffusivity. The
particulate part of contaminants transports together with the sorbent, i.e. sediment, which can
be obtained by the coupled 3-D hydrodynamic and sediment transport models. The first-
order reversible kinetic model is used in these models to describe the sorption process.
Hayter and Pakala (1989) applied their CONTAM-3D model to a partially stratified tidal
river, Sampit River, South Carolina, to predict the lead transport and distribution in this river.
Huber and Dickinson (1988) used their water quality model for simulating various
phosphorus component distributions in Lake Okeechobee, Florida. Sheng (1993) and Chen
(1994) applied his model to Lake Okeechobee and Tampa Bay for prediction of phosphorus
and nitrogen transport transformation and distributions in both water and bottom sediments.
2.6.2 Contaminant Transport in Bottom Sediment
Compared to the movement in water, the movement of interstitial water in the bottom
layer tends to be considered slower. Thus, Onishi (1982) neglected contaminant transport
and sorption within the bottom layer. In his FETRA model only sorption/desorption between
water and bottom sediment surface is considered. However, since bottom sediment may be
an important sink and/or source of contaminants, contaminant transport and transformation
within the bottom layer must be taken into account. Contaminant transport in bottom
sediment includes dissolved contaminants transport, particulate contaminant transport and
sorption.
53
For modeling the vertical movements of pore water and dissolved constituents in the
bottom sediment layer, Berner (1980) gave the general diagenetic equation, which is
expressed as
ai z) a( Ci) + (2.53)
at az az
where Ci is the concentration of component i in terms of mass per unit volume of total
sediment (solids plus water), Di is the diffusion coefficient of component i, is the
velocity of flow relative to sediment-water interface, Ri is the rate of reaction and z is the
coordinate with origin at the sediment-water interface and directing vertical downward. The
first term on the right side of Eq. 2.53 is the diffusion term, the second term is the advection
term and the third one is the term for chemical, biological and radiogenic reactions. The
relative flow velocity and diffusion coefficient depend on bottom sediment characteristics,
overlying water motion and contaminant properties, etc. and are site-specific. To solve Eq.
2.53, \i, )i, as well as ai, must be provided.
For a mud bed, the self-weight consolidation is an important way by which advective
transport of water and its dissolved constituents is produced. A series of studies for one-
dimensional consolidation of saturated clays was carried out by Gibson et al. (1967, 1981),
Znidarcic et al. (1984) and Schiffman et al. (1984). Base on these studies, the general
consolidation equation in term of the void ratio, e, is expressed as:
P '1 d kw (e) ae a k,(e) do' ee e
1 [ 1+e] ae+ L a e) da/]e + ae = 0 (2.54)
p" de 1+e -z 9z pf(1 +e) de az at
where k, is the permeability which is a function of void ratio e, a' is the effective normal
stress and pf is the pore water density.
To solve Eq. 2.54 requires the effective stress/void ratio, o/(e), and the
permeability/void ratio, k,(e), relationships. By introducing the assumption that the
effective stress/void ratio and permeability/void ratio are relationships linear and that the
consolidation coefficient, CF = [-k(e)/pf(l +e)] do'/de, is a constant, Lee and Sills (1981)
solved Eq. 2.54 analytically. It should be noted, however, that the o/(e) andk(e)
relationships are in general nonlinear functions for real soils and are also site-specific.
Furthermore, the consolidation coefficient, CF, is usually not a constant. Thus, numerical
modeling is generally needed for solving Eq. 2.54. In that connection, Gibson et al. (1967
and 1981) have developed the theory and solutions for the finite nonlinear consolidation of
both thin and thick homogeneous layers. Macay et al. (1986) applied the theory to quiescent
consolidation of phosphatic waste clays. Since the o'(e) and k(e) relationships are site-
specific, experiments are required to supply these relationships for numerical modeling.
Znidarcic et al. (1984) has summarized existed testing procedures for laboratory
consolidation experiments and found that the current laboratory methods are restricted in
their applicability and only give approximate soil properties for theoretical calculation and
numerical modeling.
55
In the vicinity of the sediment-water interface, advective transport of pore water and
dissolved constituents is affected by the overlying water motion and thus is enhanced.
Savant et al. (1987) carried out laboratory experiments and numerical simulations to
investigate the transport of chemically inert, non-sorbed contaminants in a river sediment and
showed that interstitial fluid advection controlled contaminant transport within the stable
sediment bed with an in-bed Peclet number on the order of 100-1,000. These high Peclet
numbers indicated the negligible influence of molecular diffusion under the conditions
examined. Through an analytical solution of the two-dimensional Navier-Stokes equation
for oscillatory flow over a rippled bed, it was examined the transport of reactive solutes
(governed by a first-order kinetic model below the rippled bed) under wave motion and the
enhanced exchange of non-reactive solutes across the water-sediment interface. Shum
concluded that the wave-induced pore water circulation below the rippled bed and the total
wave-induced exchange across the water-sediment interface increased with the wave height
and decreased with the wave period.
For the mud bed, the permeability is much lower than that of sand bed, so that the bed
is typically considered to be impermeable over comparatively short time-scales, e.g., of the
order of wave motion. However, the mud bed may be fluidized under wave motion and form
a fluid mud layer. Thus, although advective transport may be negligible in the fluid mud
layer, the mixing process will be obviously enhanced because of its slurry-like properties and
oscillation under wave motion.
Contaminant diffusion in the interstitial water takes place in accordance with Fick's
laws of diffusion (Crank, 1975), which are
(1) first law:
Ji = -D-i (2.55)
(2) second law:
a. i a(Di z) (2.56)
at az az
where Ji is the diffusion flux of component i.
Within the undisturbed sediment that is relatively far below the sediment-water
interface and is not affected by the overlying water motion, diffusion can be only due to
molecular diffusion. Due to the presence of particles, it is not easy for dissolved
contaminants to diffuse in any direction within the sediment as they do in the overlying water
layer. Instead, diffusion is hindered by the solid particles and follows tortuous paths of fluid
flowing between and around particles. Mathematically, tortuosity, 0, is defined as (Fetter,
1981)
dl
0 (2.57)
dz
where dl is the length of the actual sinuous path over a depth interval dz. The diffusion
coefficient of component i in terms of tortuosity, is expressed as
Dm
Di = 0D (2.58)
02
57
where o9 is a coefficient that is related to the tortuosity (Bear, 1972). Perkins and Johnson
(1963) found that wo was equal to 0.7 in sand columns studies using a uniform sand. In
laboratory studies for geologic materials, Freeze and Cherry (1979) found that ao ranged
from 0.5 to 0.01. Also, according to Li and Gregory (1974) and McDuff and Ellis (1979),
an average 0 value for deep sea clays is 1.8. The molecular diffusion coefficients of some
ionic constituents in typical muds with porosity, n, are listed in table 2.1.
The typical molecular diffusion coefficient is seen to be in the range of 10"10 ~ 10'9
m2/s, while the kinematic viscosity, v, of water is around 10 Ih /s. Thus, the Schmidt
Number, Sc, which is the ratio of mass transfer rate to momentum transfer rate, i.e.,
Sc = v/D in the nonturbulent environment is about 1/1,000 1/10,000, an extremely low
range.
Table 2.1 Diffusion coefficients for some tracers in some typical marine mud beds.
Component Porosity, n Dm (10'1 m2/s)
Na 0.71 7.4
Ca+ 0.71 4.4
Cl- 0.71 10.2
SO4- 0.71 5.0
S04- 0.772 5.0
SO4- 0.64 4.0
NH4+ 0.72 9.8
HPO4- 0.72 3.6
Near the sediment-water interface, due to the effect of overlying water motion,
diffusion is not solely due to molecular diffusion. Vanderborght et al. (1977) analyzed the
58
vertical transport of dissolved constituents in a large muddy zone along the Belgian North
Sea coast. They selected a diffusion coefficient that was 100 times larger than that for
molecular diffusion down to 3.5 cm depth from the sediment-water interface. In the lower
compact layer diffusion was only due to molecular movement. This two-layered model of
bottom sediments was applied to simulate the vertical transport and profile of dissolved
silica, oxygen, sulfate, nitrate and ammonium.
DePinto (1994) in his Green Bay study classified the range of 0 ~ 4 cm depth below
bottom sediment surface as surface-mixed sediment layer, and the lower layer as subsurface-
mixed sediment layer. The fate and transport properties of contaminants in these two layers
were recognized and.modeled separately in his GBTOX model. Berger and Heath (1968)
used an "box-model" approach to represent the active surfacial sediment layer in which a fast
rate of mixing occurs over a certain depth over which sediment properties are uniform.
In the three-layered system shown in Fig. 2.6, the fluid mud oscillates under wave
motion. In this layer, contaminant diffusion will include the molecular diffusion and wave-
induced diffusion as a result of the effect from overlying water motion. Therefore, the
production, decay and transport of fluid mud can be expected greatly affect contaminant
transport between water and bottom sediment.
2.6.3 Fluxes Across the Sediment-Water Interface
Contaminant flux across the water-fluid or bed interface controls the rate of
contaminant mass transfer between water and bottom sediment. The accumulation of
contaminants in bottom sediments and the release of bottom contaminant are both controlled
by this flux. Generally, this flux is composed of dissolved and particulate contaminant
59
subfluxes. The dissolved part is transported by diffusion and advection, whereas the
particulate part is transported together with the sorbent, i.e., sediment. Thus, resuspension
and deposition of sediment can strongly influence the particulate contaminant flux across the
interface.
The transport of materials across a solid boundary surface and a moving fluid or a
interface between two relatively immiscible moving fluids is defined as convection (Welty,
et al., 1969). The flux of convective mass transfer is expressed in the form
Fi = k ACi (2.59)
where F, is the mass flux of component i across the interface, ACi is the concentration
difference of constituent i across the interface and kc is the convective mass-transfer
coefficient.
The so called film theory for mass transfer is based on the assumption of the presence
of a fictitious film at the interface that offer the same resistance to mass transfer as actually
exists in the entire field. Accordingly, the convective mass transfer coefficient is modeled
as
D.
kc =- (2.60)
8f
where Di is the molecular diffusion coefficient of component i and 86 is the film thickness.
Thus, the mass transfer coefficient, ke, is a function of molecular diffusion coefficient Dm and
the chemical and hydrodynamic parameters that determine the film thickness, namely, water
60
viscosity, v, bottom friction velocity, u,, and the Schmidt number, Sc (Boudreau and
Guinasso, 1982). Expressions for kc are generally expressed as
kc = l2c u,Sc P 12 (2.61)
where coefficient a12c is in the range of 0.04 1.0 and P12c is in the range of 2/3 1.
The so-called penetration theory was first proposed by Higbie (1935) based on the
idea that the diffusive component only penetrates a short distance into the phase of interest.
This theory has been subsequently applied to turbulent flow by Danckwerts (1951). The
convective mass-transfer coefficient based on this theory is expressed as
kc = (2.62)
where s is the surface-renewal factor. Toor and Marchello (1958) have pointed out that the
penetration concept is valid only when surface renewal is relatively rapid, in which case
kc ~ For relatively slow surface renewal, the film theory is valid in which kc ~ Di.
In reality the convective mass-transfer coefficient is proportional to a power of the molecular
mass diffusivity, Di, between 0.5 and 1.0.
In studying diffusive fluxes across the sediment interface, Morse (1974) considered
that there exists a stagnant benthic boundary layer just over the sediment-water interface.
The transport of dissolved constituents across this layer is purely by molecular diffusion,
while the transport in the main water column overlying the boundary layer is by turbulent
mixing. The maximum possible thickness of the stagnant layer is on the order of 1 cm,
which is within the limit of the calculations of Wimbush (1970) for the South Pacific.
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Santschi et al. (1983) investigated the resistance of chemical transport within the deep-sea
boundary layer and concluded that the equivalent stagnant boundary film thickness is about
0.47 mm. The result is based on the directly measurements of CaCO3 dissolution and Mn"
released from bottom sediment at a site in the eastern Pacific.
Under wave motion the exchange of water and dissolved constituents between water
and bottom sediments is evidently enhanced due to wave-induced diffusion and advection
in the near interface layer. Savant et al. (1987) and Shum (1992,1993) examined
contaminant release from sandy beds under wave motion. They found that wave-induced
advection, which causes a flux across the interface, is so great that molecular diffusion can
be neglected.
In contaminant transport modeling, DePinto (1994) used a diffusion process which
included the effects of diffusion and advection induced by external hydrodynamics to
describe contaminant transport across the sediment-water interface. Hayter and Pakala
(1989) considered contaminant sorption/desorption onto/from the bottom sediment surface
and advection due to bottom sediment consolidation in their CONTAM-3D model. Chen
(1994) only included advective transport due to the hydrodynamic effect on the bottom but
neglected the effect of diffusion in his three-dimensional contaminant transport model.
Particle-bound contaminant flux has been shown to be related to the sediment
entrainment and deposition rates and to contaminant sorptive properties in water and fluid
mud or bed (Onishi, 1981; O'Connor, 1988; Hayter and Pakala, 1989, DePinto, 1994 and
Chen, 1994). However, one issue that has not always been fully accounted for by most
investigators is that together with the entrainment and deposition of sediment, water is also
62
entrained and deposited. Thus, sediment mass transfer between the water and fluid mud or
bed layers can not only induce particle-bound contaminant transfer between the two layers,
but also enhance dissolved contaminant transfer. This is the dissolved contaminant mass
transfer through water exchange induced by sediment entrainment and deposition, which
must be considered when entrainment and/or deposition take place.
2.6.4 Fluxes Across the Fluid Mud-Bed Interface
Contaminant transfer across the fluid mud -bed interface can also be divided into two
parts: dissolved part and particulate part. When the bed is fluidized or fluid mud dewaters
to become part of bed, the particle-bound contaminants will be effectively transported across
the interface. However, under those situations in which only an equilibrium fluidization
depth occurs, as in the present case, there will be no sediment mass transfer is considered
across the interface and also no particulate contaminant transport across the interface.
For dissolved contaminants, since around the fluid mud-bed interface there is
typically no disturbance from the overlying water motion, molecular diffusion may be the
only diffusion process across the interface. In the GBTOX model (DePinto, 1994), the
transfer of unbounded PCB between the surface-mixed and the subsurface-mixed sediment
layers is only by molecular diffusion. Vanderborght et al. (1977) also considered only the
molecular diffusion to describe dissolved silica flux across the interface between the top 3.5
cm bed layer and the more compact lower layer of the bed.
Consolidation may be the main advective process for dissolved contaminants across
the fluid mud-bed interface. However, typically consolidation is a slow process, in
63
comparison with wave-induced transport. Thus, for example for an episodic event,
consolidation is usually neglected.
CHAPTER 3
MUD FLUIDIZATION BY WAVES
3.1 Introduction
The theological and mass transport characteristics of fluid mud can be significantly
different from those of the non-fluidized bed. Thus, once fluid mud develops, it must be
considered separately from the non-fluidized bed, as noted in Section 2.3.
The main step for setting up the three-layered water-fluid mud-bed system is the
determination of the thickness of the fluid mud layer. Fluidization is a time-dependent
process starting with a bed over which wave motion commences initially in clear water.
Thus, it is a function of the wave characteristics and bottom mud properties. In this chapter,
by assuming the mud bed to be a single-phase continuum, fluidization is considered in terms
of forcing due to cyclic normal pressure loading and the horizontal pressure gradient. First,
the mechanism of fluidization of mud beds by waves is examined. The bed is considered to
fluidize when the effective submerged weight of the particle or floc is exceeded by the
upward inertia force due to bed vibration. Then, by characterizing the bed as a uniaxial
extensional-Voigt solid, it is represented in terms of an equivalent spring-dashpot-mass
system. The dependence of the fluidization depth on characteristic wave and bed theological
parameters is examined for a water-bed and a water-fluid mud-bed system represented,
respectively, as one-degree and two-degrees of freedom analogs. Theoretical results obtained
65
are compared wave flume data obtained by Ross (1988), Lindenberg et al. (1989) and Feng
(1992) on fluid mud thickness. Finally the model is applied to lake Okeechobee in Florida,
where fluid mud thicknesses were measured previously.
3.2 Mechanics of Fluidization
3.2.1 Vertical and Horizontal Motions of Bed under Waves
For a mud bed under waves, two aspects of wave motion can be distinguished (Ross,
1988): "pumping" due to cyclic normal stress on bed surface and "shaking"due to cyclic
shear stress on bed surface. Pumping is responsible for bed motion in the vertical direction,
while shaking may cause bed motion in the horizontal direction.
The gradients of pressure and shear stress on the bed provide the forcing for the
horizontal motion of the bed. Dalrymple and Liu (1978) compared results on water wave
attenuation from their wave-soft bed interaction model with results from Gade's (1957)
inviscid, shallow water model, and found that despite Gade's neglect of viscosity in the upper
fluid, and therefore absence of shear stress at the interface, both solutions agreed reasonably
well with the laboratory data collected by Gade. A conclusion can therefore be drawn that
the principal mode of energy transfer to the mud is the normal stress rather than the shear
stress, due to wave-induced work on the lower medium. Isobe et al. (1992) carried out
experiments to check the predominance of forces for mud bed horizontal motion. In one
experiment, the bed was covered with a nylon sheet which transferred the pressure and but
not the shear stress. A second experiment was done without cover. The results showed that
wave height decay with and without the cover were almost same over a wide range of
66
incident wave heights. This also implies that the pressure gradient due to wave motion is the
dominant external force for the mud horizontal motion and wave energy dissipation.
Under wave motion mud beds are known to fluidize, and a fluid mud layer of
equilibrium thickness is observed to occur after a certain elapsed time depending on the
initial conditions with respect to the mud-water system and the forcing wave characteristics.
Thus, a mechanistic framework for explaining mud bed fluidization must be based on a
consideration of forces required to cause particle-particle separation, hence a loss of effective
stress. These forces can be considered in three ways: (1) forces in the vertical direction, (2)
forces in the horizontal direction and (3) combination of vertical and horizontal forces.
3.2.2 Bed Failure by Horizontal Forces
Madsen (1974) hypothesized that a large horizontal pressure gradient can cause bed
failure. The criterion for failure was expressed in a non-dimensional form:
SP tanpw (3.1)
Pwg ax crit Pw
where p, is the wave-induced pore pressure in excess of the hydrostatic pressure, p is the
water density, g is the gravitational acceleration, p3 is the bed density and ( is the friction
angle. This shear failure criterion is applicable only to non-cohesive sediment, because the
effects of cohesion and rheology are not included. Modifying Eq. 3.1 by introducing
cohesion, Ch, and the multiplying factor pgD for both sides of the equation, we obtain a
more general, stress-form, criterion
( D= pwg'D tan +Ch (3.2)
where the effective gravitational acceleration, g = g(p3 Pw)/Pw and D is the diameter of
the particle or floc. By using the definition of the effective stress a' = pg'D and referring
to the well-known Coulomb equation for shear strength, Ts = o/tan) + Ch, Eq. 3.2 can be
expressed as
D = pg /D tan + Ch
ax crit (3.3)
= a/tanO + C =
Thus, when the horizontal pressure difference equals the bed shear strength, the mud bed
fails. The inter-connections between the particles or flocs are cut off at that level where
Eq.3.3 is satisfied and the bed above that level is fluidized at that instant.
Although Madsen (1974) successfully used the criterion, -(ap /x)crit/pwg = 0.5,
for loose sand liquefaction, the use of Eq. 3.3 for the mud bed may lead to an
underestimation of fluidization. The reason is that Eqs. 3.1, 3.2 and 3.3 are obtained by
considering forces in the horizontal direction only. For sandy beds this may be acceptable
because there may be practically no vertical motion of the bed under waves. However, for
the mud bed, oscillations under wave action are typically significant and cannot be neglected.
In the following development, we show that a vertical oscillation of the mud bed
produces an upward inertia stress, oi, that can greatly reduce the effective stress. Also, the
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buildup of the excessive pore pressure, Au,, will reduce the effective stress. Thus, Eq. 3.2
can be applied for the mud bed under long wave action, in which motion in the vertical
direction is negligible and the excessive pore pressure buildup is minor.
FF
SFluidized Layer
F jf2g
f4 mpg' F2 p
Bed In vertical direction
(a) (b) (c)
Figure 3.1 Forces on a particle or floc. F,, F2, F3 and F4 = normal forces from adjacent
particles; f,, f2, f3 and f 4 = shear forces from adjacent particles; Li = inertia force; L
cohesive resistance; mp = particle mass and g' = effective gravity: a) Bed and fluidized mud
with tagged particle forming a part of the lower boundary of fluid mud; b) forces on tagged
particle; c) forces in the vertical direction.
3.2.3 Bed Fluidization by Vertical Forces
Consider a particle (or a cohesive floe treated as an integral particle) mass mp in the
mud bed under wave motion as shown in Fig. 3.1. Four types of forces acting on this particle
must be recognized: (1) effective gravity or buoyancy in water, (2) inter-particle cohesive
forces, (3) normal and shear forces due to contacts between adjacent particles, and (4) inertia
force due to vibration, L,(t), under wave loading. Considering the forces in the vertical
direction, the normal and shear forces (F1, F2, F3, F4, fl, f2, f3, and f4) due to contacts
between adjacent, almost randomly positioned, particles can be represented together as a
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